Function gradients via reverse-mode automatic differentiation. More...

Classes
struct	fvar

class	partials_vari

struct	OperandsAndPartials
	A variable implementation that stores operands and derivatives with respect to the variable. More...

class	chainable
	Abstract base class for variable implementations that handles memory management and applying the chain rule. More...

class	op_ddv_vari

class	op_dv_vari

class	op_dvd_vari

class	op_dvv_vari

class	op_matrix_vari

class	precomp_v_vari

class	precomputed_gradients_vari
	This is a var implementation class that takes precomputed gradient values. More...

class	op_v_vari

class	op_vd_vari

class	op_vdd_vari

class	op_vdv_vari

class	op_vector_vari

class	op_vv_vari

class	op_vvd_vari

class	op_vvv_vari

class	LDLT_alloc
	This object stores the actual (double typed) LDLT factorization of an Eigen::Matrix<var> along with pointers to its vari's which allow the *_ldlt functions to save memory. More...

class	stored_gradient_vari
	A var implementation that stores the daughter variable implementation pointers and the partial derivative with respect to the result explicitly in arrays constructed on the auto-diff memory stack. More...

class	var
	Independent (input) and dependent (output) variables for gradients. More...

class	chainable_alloc
	A chainable_alloc is an object which is constructed and destructed normally but the memory lifespan is managed along with the arena allocator for the gradient calculation. More...

class	vari
	The variable implementation base class. More...

Typedefs
typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > ::Index	size_type

typedef Eigen::Matrix< fvar < double >, Eigen::Dynamic, Eigen::Dynamic >	matrix_fd

typedef Eigen::Matrix< fvar < var >, Eigen::Dynamic, Eigen::Dynamic >	matrix_fv

typedef Eigen::Matrix< fvar < fvar< double > >, Eigen::Dynamic, Eigen::Dynamic >	matrix_ffd

typedef Eigen::Matrix< fvar < fvar< var > >, Eigen::Dynamic, Eigen::Dynamic >	matrix_ffv

typedef Eigen::Matrix< fvar < double >, Eigen::Dynamic, 1 >	vector_fd

typedef Eigen::Matrix< fvar < var >, Eigen::Dynamic, 1 >	vector_fv

typedef Eigen::Matrix< fvar < fvar< double > >, Eigen::Dynamic, 1 >	vector_ffd

typedef Eigen::Matrix< fvar < fvar< var > >, Eigen::Dynamic, 1 >	vector_ffv

typedef Eigen::Matrix< fvar < double >, 1, Eigen::Dynamic >	row_vector_fd

typedef Eigen::Matrix< fvar < var >, 1, Eigen::Dynamic >	row_vector_fv

typedef Eigen::Matrix< fvar < fvar< double > >, 1, Eigen::Dynamic >	row_vector_ffd

typedef Eigen::Matrix< fvar < fvar< var > >, 1, Eigen::Dynamic >	row_vector_ffv

typedef Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic >	matrix_v
	The type of a matrix holding `stan::agrad::var` values. More...

typedef Eigen::Matrix< var, Eigen::Dynamic, 1 >	vector_v
	The type of a (column) vector holding `stan::agrad::var` values. More...

typedef Eigen::Matrix< var, 1, Eigen::Dynamic >	row_vector_v
	The type of a row vector holding `stan::agrad::var` values. More...

Functions
template<typename T , typename F >
void	derivative (const F &f, const T &x, T &fx, T &dfx_dx)
	Return the derivative of the specified univariate function at the specified argument. More...

template<typename T , typename F >
void	partial_derivative (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int n, T &fx, T &dfx_dxn)
	Return the partial derivative of the specified multiivariate function at the specified argument. More...

template<typename F >
void	gradient (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_fx)
	Calculate the value and the gradient of the specified function at the specified argument. More...

template<typename T , typename F >
void	gradient (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad_fx)

template<typename F >
void	jacobian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, Eigen::Matrix< double, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &J)

template<typename T , typename F >
void	jacobian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, Eigen::Matrix< T, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &J)

template<typename F >
void	hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &H)

template<typename T , typename F >
void	hessian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &H)

template<typename T1 , typename T2 , typename F >
void	gradient_dot_vector (const F &f, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &v, T1 &fx, T1 &grad_fx_dot_v)

template<typename F >
void	hessian_times_vector (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &v, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &Hv)

template<typename T , typename F >
void	hessian_times_vector (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &Hv)

template<typename F >
void	grad_tr_mat_times_hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &M, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_tr_MH)

template<typename T >
fvar< T >	abs (const fvar< T > &x)

template<typename T >
fvar< T >	acos (const fvar< T > &x)

template<typename T >
fvar< T >	acosh (const fvar< T > &x)

template<typename T >
fvar< T >	asin (const fvar< T > &x)

template<typename T >
fvar< T >	asinh (const fvar< T > &x)

template<typename T >
fvar< T >	atan (const fvar< T > &x)

template<typename T >
fvar< T >	atan2 (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	atan2 (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	atan2 (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	atanh (const fvar< T > &x)

template<typename T >
fvar< T >	bessel_first_kind (int v, const fvar< T > &z)

template<typename T >
fvar< T >	bessel_second_kind (int v, const fvar< T > &z)

template<typename T >
fvar< T >	binary_log_loss (const int y, const fvar< T > &y_hat)

template<typename T >
fvar< T >	binomial_coefficient_log (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	binomial_coefficient_log (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	binomial_coefficient_log (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	cbrt (const fvar< T > &x)

template<typename T >
fvar< T >	ceil (const fvar< T > &x)

template<typename T >
fvar< T >	cos (const fvar< T > &x)

template<typename T >
fvar< T >	cosh (const fvar< T > &x)

template<typename T >
fvar< T >	digamma (const fvar< T > &x)

template<typename T >
fvar< T >	erf (const fvar< T > &x)

template<typename T >
fvar< T >	erfc (const fvar< T > &x)

template<typename T >
fvar< T >	exp (const fvar< T > &x)

template<typename T >
fvar< T >	exp2 (const fvar< T > &x)

template<typename T >
fvar< T >	expm1 (const fvar< T > &x)

template<typename T >
fvar< T >	fabs (const fvar< T > &x)

template<typename T >
fvar< T >	falling_factorial (const fvar< T > &x, const fvar< T > &n)

template<typename T >
fvar< T >	falling_factorial (const fvar< T > &x, const double n)

template<typename T >
fvar< T >	falling_factorial (const double x, const fvar< T > &n)

template<typename T >
fvar< T >	fdim (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	fdim (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	fdim (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	floor (const fvar< T > &x)

template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 > ::type >	fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
	The fused multiply-add operation (C99). More...

template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 > ::type >	fma (const T1 &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...

template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 > ::type >	fma (const fvar< T1 > &x1, const T2 &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...

template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 > ::type >	fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...

template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 > ::type >	fma (const T1 &x1, const T2 &x2, const fvar< T3 > &x3)
	See all-var input signature for details on the function and derivatives. More...

template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 > ::type >	fma (const fvar< T1 > &x1, const T2 &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...

template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 > ::type >	fma (const T1 &x1, const fvar< T2 > &x2, const T3 &x3)
	See all-var input signature for details on the function and derivatives. More...

template<typename T >
fvar< T >	fmax (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	fmax (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	fmax (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	fmin (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	fmin (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	fmin (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	fmod (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	fmod (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	fmod (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	gamma_p (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	gamma_p (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	gamma_p (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	gamma_q (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	gamma_q (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	gamma_q (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	hypot (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	hypot (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	hypot (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	inv (const fvar< T > &x)

template<typename T >
fvar< T >	inv_cloglog (const fvar< T > &x)

template<typename T >
fvar< T >	inv_logit (const fvar< T > &x)

template<typename T >
fvar< T >	inv_sqrt (const fvar< T > &x)

template<typename T >
fvar< T >	inv_square (const fvar< T > &x)

template<typename T >
int	is_inf (const fvar< T > &x)
	Returns 1 if the input's value is infinite and 0 otherwise. More...

template<typename T >
int	is_nan (const fvar< T > &x)
	Returns 1 if the input's value is NaN and 0 otherwise. More...

template<typename T >
fvar< T >	lbeta (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	lbeta (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	lbeta (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	lgamma (const fvar< T > &x)

template<typename T >
fvar< typename stan::return_type< T, int > ::type >	lmgamma (int x1, const fvar< T > &x2)

template<typename T >
fvar< T >	log (const fvar< T > &x)

template<typename T >
fvar< T >	log10 (const fvar< T > &x)

template<typename T >
fvar< T >	log1m (const fvar< T > &x)

template<typename T >
fvar< T >	log1m_exp (const fvar< T > &x)

template<typename T >
fvar< T >	log1m_inv_logit (const fvar< T > &x)

template<typename T >
fvar< T >	log1p (const fvar< T > &x)

template<typename T >
fvar< T >	log1p_exp (const fvar< T > &x)

template<typename T >
fvar< T >	log2 (const fvar< T > &x)

template<typename T >
fvar< T >	log_diff_exp (const fvar< T > &x1, const fvar< T > &x2)

template<typename T1 , typename T2 >
fvar< T2 >	log_diff_exp (const T1 &x1, const fvar< T2 > &x2)

template<typename T1 , typename T2 >
fvar< T1 >	log_diff_exp (const fvar< T1 > &x1, const T2 &x2)

template<typename T >
fvar< T >	log_falling_factorial (const fvar< T > &x, const fvar< T > &n)

template<typename T >
fvar< T >	log_falling_factorial (const double x, const fvar< T > &n)

template<typename T >
fvar< T >	log_falling_factorial (const fvar< T > &x, const double n)

template<typename T >
fvar< T >	log_inv_logit (const fvar< T > &x)

template<typename T >
fvar< T >	log_rising_factorial (const fvar< T > &x, const fvar< T > &n)

template<typename T >
fvar< T >	log_rising_factorial (const fvar< T > &x, const double n)

template<typename T >
fvar< T >	log_rising_factorial (const double x, const fvar< T > &n)

template<typename T >
fvar< T >	log_sum_exp (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	log_sum_exp (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	log_sum_exp (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	log_sum_exp (const std::vector< fvar< T > > &v)

template<typename T >
fvar< T >	logit (const fvar< T > &x)

template<typename T >
fvar< T >	modified_bessel_first_kind (int v, const fvar< T > &z)

template<typename T >
fvar< T >	modified_bessel_second_kind (int v, const fvar< T > &z)

template<typename T >
fvar< T >	multiply_log (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	multiply_log (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	multiply_log (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	owens_t (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	owens_t (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	owens_t (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	Phi (const fvar< T > &x)

template<typename T >
fvar< T >	pow (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	pow (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	pow (const fvar< T > &x1, const double x2)

template<typename T >
double	primitive_value (const fvar< T > &v)
	Return the primitive value of the specified forward-mode autodiff variable. More...

template<typename T >
fvar< T >	rising_factorial (const fvar< T > &x, const fvar< T > &n)

template<typename T >
fvar< T >	rising_factorial (const fvar< T > &x, const double n)

template<typename T >
fvar< T >	rising_factorial (const double x, const fvar< T > &n)

template<typename T >
fvar< T >	round (const fvar< T > &x)

template<typename T >
fvar< T >	sin (const fvar< T > &x)

template<typename T >
fvar< T >	sinh (const fvar< T > &x)

template<typename T >
fvar< T >	sqrt (const fvar< T > &x)

template<typename T >
fvar< T >	square (const fvar< T > &x)

template<typename T >
fvar< T >	tan (const fvar< T > &x)

template<typename T >
fvar< T >	tanh (const fvar< T > &x)

template<typename T >
fvar< T >	tgamma (const fvar< T > &x)

template<typename T >
fvar< T >	trunc (const fvar< T > &x)

template<typename T >
T	value_of (const fvar< T > &v)
	Return the value of the specified variable. More...

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 >	columns_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 >	columns_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 >	columns_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, 1, C >	columns_dot_self (const Eigen::Matrix< fvar< T >, R, C > &x)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, C, C >	crossprod (const Eigen::Matrix< fvar< T >, R, C > &m)

template<typename T , int R, int C>
fvar< T >	determinant (const Eigen::Matrix< fvar< T >, R, C > &m)

template<typename T1 , typename T2 >
stan::return_type< T1, T2 >::type	divide (const T1 &v, const T2 &c)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	divide (const Eigen::Matrix< fvar< T >, R, C > &v, const fvar< T > &c)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	divide (const Eigen::Matrix< fvar< T >, R, C > &v, const double c)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	divide (const Eigen::Matrix< double, R, C > &v, const fvar< T > &c)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	operator/ (const Eigen::Matrix< fvar< T >, R, C > &v, const fvar< T > &c)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	operator/ (const Eigen::Matrix< fvar< T >, R, C > &v, const double c)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	operator/ (const Eigen::Matrix< double, R, C > &v, const fvar< T > &c)

template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)

template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)

template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)

template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2, size_type &length)

template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2, size_type &length)

template<typename T , int R1, int C1, int R2, int C2>
fvar< T >	dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2, size_type &length)

template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< fvar< T > > &v2)

template<typename T >
fvar< T >	dot_product (const std::vector< double > &v1, const std::vector< fvar< T > > &v2)

template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< double > &v2)

template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< fvar< T > > &v2, size_type &length)

template<typename T >
fvar< T >	dot_product (const std::vector< double > &v1, const std::vector< fvar< T > > &v2, size_type &length)

template<typename T >
fvar< T >	dot_product (const std::vector< fvar< T > > &v1, const std::vector< double > &v2, size_type &length)

template<typename T , int R, int C>
fvar< T >	dot_self (const Eigen::Matrix< fvar< T >, R, C > &v)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	inverse (const Eigen::Matrix< fvar< T >, R, C > &m)

template<typename T , int R, int C>
fvar< T >	log_determinant (const Eigen::Matrix< fvar< T >, R, C > &m)

template<typename T >
Eigen::Matrix< fvar< T > , Eigen::Dynamic, 1 >	log_softmax (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &alpha)

template<typename T , int R, int C>
fvar< T >	log_sum_exp (const Eigen::Matrix< fvar< T >, R, C > &v)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_left (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_left (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_left (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)

template<int R1, int C1, int R2, int C2, typename T2 >
Eigen::Matrix< fvar< T2 >, R1, C2 >	mdivide_left_ldlt (const stan::math::LDLT_factor< double, R1, C1 > &A, const Eigen::Matrix< fvar< T2 >, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_left_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_left_tri_low (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_left_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 >	mdivide_right_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	mdivide_right_tri_low (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)

template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m, const fvar< T > &c)

template<typename T , int R2, int C2>
Eigen::Matrix< fvar< T >, R2, C2 >	multiply (const Eigen::Matrix< fvar< T >, R2, C2 > &m, const double c)

template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const Eigen::Matrix< double, R1, C1 > &m, const fvar< T > &c)

template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const fvar< T > &c, const Eigen::Matrix< fvar< T >, R1, C1 > &m)

template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const double c, const Eigen::Matrix< fvar< T >, R1, C1 > &m)

template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 >	multiply (const fvar< T > &c, const Eigen::Matrix< double, R1, C1 > &m)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m1, const Eigen::Matrix< fvar< T >, R2, C2 > &m2)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m1, const Eigen::Matrix< double, R2, C2 > &m2)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 >	multiply (const Eigen::Matrix< double, R1, C1 > &m1, const Eigen::Matrix< fvar< T >, R2, C2 > &m2)

template<typename T , int C1, int R2>
fvar< T >	multiply (const Eigen::Matrix< fvar< T >, 1, C1 > &rv, const Eigen::Matrix< fvar< T >, R2, 1 > &v)

template<typename T , int C1, int R2>
fvar< T >	multiply (const Eigen::Matrix< fvar< T >, 1, C1 > &rv, const Eigen::Matrix< double, R2, 1 > &v)

template<typename T , int C1, int R2>
fvar< T >	multiply (const Eigen::Matrix< double, 1, C1 > &rv, const Eigen::Matrix< fvar< T >, R2, 1 > &v)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, R >	multiply_lower_tri_self_transpose (const Eigen::Matrix< fvar< T >, R, C > &m)

template<typename T >
Eigen::Matrix< fvar< T > , Eigen::Dynamic, Eigen::Dynamic >	qr_Q (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &m)

template<typename T >
Eigen::Matrix< fvar< T > , Eigen::Dynamic, Eigen::Dynamic >	qr_R (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &m)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 >	rows_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 >	rows_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 >	rows_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, 1 >	rows_dot_self (const Eigen::Matrix< fvar< T >, R, C > &x)

template<typename T >
Eigen::Matrix< fvar< T > , Eigen::Dynamic, 1 >	softmax (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &alpha)

template<typename T >
std::vector< fvar< T > >	sort_asc (std::vector< fvar< T > > xs)

template<typename T >
std::vector< fvar< T > >	sort_desc (std::vector< fvar< T > > xs)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	sort_asc (Eigen::Matrix< fvar< T >, R, C > xs)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	sort_desc (Eigen::Matrix< fvar< T >, R, C > xs)

template<typename T , int R, int C>
fvar< T >	sum (const Eigen::Matrix< fvar< T >, R, C > &m)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, R >	tcrossprod (const Eigen::Matrix< fvar< T >, R, C > &m)

template<typename T >
fvar< T >	to_fvar (const T &x)

template<typename T >
fvar< T >	to_fvar (const fvar< T > &x)

matrix_fd	to_fvar (const stan::math::matrix_d &m)

matrix_fd	to_fvar (const matrix_fd &m)

matrix_fv	to_fvar (const matrix_fv &m)

matrix_ffd	to_fvar (const matrix_ffd &m)

matrix_ffv	to_fvar (const matrix_ffv &m)

vector_fd	to_fvar (const stan::math::vector_d &v)

vector_fd	to_fvar (const vector_fd &v)

vector_fv	to_fvar (const vector_fv &v)

vector_ffd	to_fvar (const vector_ffd &v)

vector_ffv	to_fvar (const vector_ffv &v)

row_vector_fd	to_fvar (const stan::math::row_vector_d &rv)

row_vector_fd	to_fvar (const row_vector_fd &rv)

row_vector_fv	to_fvar (const row_vector_fv &rv)

row_vector_ffd	to_fvar (const row_vector_ffd &rv)

row_vector_ffv	to_fvar (const row_vector_ffv &rv)

template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C >	to_fvar (const Eigen::Matrix< T, R, C > &val, const Eigen::Matrix< T, R, C > &deriv)

template<int RD, int CD, int RA, int CA, int RB, int CB, typename T >
fvar< T >	trace_gen_quad_form (const Eigen::Matrix< fvar< T >, RD, CD > &D, const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)

template<int RA, int CA, int RB, int CB, typename T >
stan::agrad::fvar< T >	trace_quad_form (const Eigen::Matrix< stan::agrad::fvar< T >, RA, CA > &A, const Eigen::Matrix< stan::agrad::fvar< T >, RB, CB > &B)

template<typename T >
fvar< T >	operator+ (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	operator+ (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	operator+ (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	operator/ (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	operator/ (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	operator/ (const double x1, const fvar< T > &x2)

template<typename T >
bool	operator== (const fvar< T > &x, const fvar< T > &y)

template<typename T >
bool	operator== (const fvar< T > &x, double y)

template<typename T >
bool	operator== (double x, const fvar< T > &y)

template<typename T >
bool	operator> (const fvar< T > &x, const fvar< T > &y)

template<typename T >
bool	operator> (const fvar< T > &x, double y)

template<typename T >
bool	operator> (double x, const fvar< T > &y)

template<typename T >
bool	operator>= (const fvar< T > &x, const fvar< T > &y)

template<typename T >
bool	operator>= (const fvar< T > &x, double y)

template<typename T >
bool	operator>= (double x, const fvar< T > &y)

template<typename T >
bool	operator< (const fvar< T > &x, double y)

template<typename T >
bool	operator< (double x, const fvar< T > &y)

template<typename T >
bool	operator< (const fvar< T > &x, const fvar< T > &y)

template<typename T >
bool	operator<= (const fvar< T > &x, const fvar< T > &y)

template<typename T >
bool	operator<= (const fvar< T > &x, double y)

template<typename T >
bool	operator<= (double x, const fvar< T > &y)

template<typename T >
fvar< T >	operator* (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	operator* (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	operator* (const fvar< T > &x1, const double x2)

template<typename T >
bool	operator!= (const fvar< T > &x, const fvar< T > &y)

template<typename T >
bool	operator!= (const fvar< T > &x, double y)

template<typename T >
bool	operator!= (double x, const fvar< T > &y)

template<typename T >
fvar< T >	operator- (const fvar< T > &x1, const fvar< T > &x2)

template<typename T >
fvar< T >	operator- (const double x1, const fvar< T > &x2)

template<typename T >
fvar< T >	operator- (const fvar< T > &x1, const double x2)

template<typename T >
fvar< T >	operator- (const fvar< T > &x)

double	calculate_chain (const double &x, const double &val)

static void	set_zero_all_adjoints ()
	Reset all adjoint values in the stack to zero. More...

static void	grad (chainable *vi)
	Compute the gradient for all variables starting from the specified root variable implementation. More...

template<typename T_result , class Policy >
bool	check_pos_definite (const char function, const Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic > &y, const char name, T_result *result, const Policy &)

var	abs (const var &a)
	Return the absolute value of the variable (std). More...

var	acos (const var &a)
	Return the principal value of the arc cosine of a variable, in radians (cmath). More...

var	acosh (const stan::agrad::var &a)
	The inverse hyperbolic cosine function for variables (C99). More...

int	as_bool (const agrad::var &v)
	Return 1 if the argument is unequal to zero and 0 otherwise. More...

var	asin (const var &a)
	Return the principal value of the arc sine, in radians, of the specified variable (cmath). More...

var	asinh (const stan::agrad::var &a)
	The inverse hyperbolic sine function for variables (C99). More...

var	atan (const var &a)
	Return the principal value of the arc tangent, in radians, of the specified variable (cmath). More...

var	atan2 (const var &a, const var &b)
	Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath). More...

var	atan2 (const var &a, const double b)
	Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath). More...

var	atan2 (const double a, const var &b)
	Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath). More...

var	atanh (const stan::agrad::var &a)
	The inverse hyperbolic tangent function for variables (C99). More...

var	bessel_first_kind (const int &v, const var &a)

var	bessel_second_kind (const int &v, const var &a)

var	binary_log_loss (const int y, const stan::agrad::var &y_hat)
	The log loss function for variables (stan). More...

var	cbrt (const stan::agrad::var &a)
	Returns the cube root of the specified variable (C99). More...

var	ceil (const var &a)
	Return the ceiling of the specified variable (cmath). More...

var	cos (const var &a)
	Return the cosine of a radian-scaled variable (cmath). More...

var	cosh (const var &a)
	Return the hyperbolic cosine of the specified variable (cmath). More...

var	digamma (const stan::agrad::var &a)

var	erf (const stan::agrad::var &a)
	The error function for variables (C99). More...

var	erfc (const stan::agrad::var &a)
	The complementary error function for variables (C99). More...

var	exp (const var &a)
	Return the exponentiation of the specified variable (cmath). More...

var	exp2 (const stan::agrad::var &a)
	Exponentiation base 2 function for variables (C99). More...

var	expm1 (const stan::agrad::var &a)
	The exponentiation of the specified variable minus 1 (C99). More...

var	fabs (const var &a)
	Return the absolute value of the variable (cmath). More...

var	falling_factorial (const var &a, const double &b)

var	falling_factorial (const var &a, const var &b)

var	falling_factorial (const double &a, const var &b)

var	fdim (const stan::agrad::var &a, const stan::agrad::var &b)
	Return the positive difference between the first variable's the value and the second's (C99). More...

var	fdim (const double &a, const stan::agrad::var &b)
	Return the positive difference between the first value and the value of the second variable (C99). More...

var	fdim (const stan::agrad::var &a, const double &b)
	Return the positive difference between the first variable's value and the second value (C99). More...

var	floor (const var &a)
	Return the floor of the specified variable (cmath). More...

var	fma (const stan::agrad::var &a, const stan::agrad::var &b, const stan::agrad::var &c)
	The fused multiply-add function for three variables (C99). More...

var	fma (const stan::agrad::var &a, const stan::agrad::var &b, const double &c)
	The fused multiply-add function for two variables and a value (C99). More...

var	fma (const stan::agrad::var &a, const double &b, const stan::agrad::var &c)
	The fused multiply-add function for a variable, value, and variable (C99). More...

var	fma (const stan::agrad::var &a, const double &b, const double &c)
	The fused multiply-add function for a variable and two values (C99). More...

var	fma (const double &a, const stan::agrad::var &b, const double &c)
	The fused multiply-add function for a value, variable, and value (C99). More...

var	fma (const double &a, const double &b, const stan::agrad::var &c)
	The fused multiply-add function for two values and a variable, and value (C99). More...

var	fma (const double &a, const stan::agrad::var &b, const stan::agrad::var &c)
	The fused multiply-add function for a value and two variables (C99). More...

var	fmax (const stan::agrad::var &a, const stan::agrad::var &b)
	Returns the maximum of the two variable arguments (C99). More...

var	fmax (const stan::agrad::var &a, const double &b)
	Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...

var	fmax (const double &a, const stan::agrad::var &b)
	Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...

var	fmin (const stan::agrad::var &a, const stan::agrad::var &b)
	Returns the minimum of the two variable arguments (C99). More...

var	fmin (const stan::agrad::var &a, const double &b)
	Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...

var	fmin (const double &a, const stan::agrad::var &b)
	Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...

var	fmod (const var &a, const var &b)
	Return the floating point remainder after dividing the first variable by the second (cmath). More...

var	fmod (const var &a, const double b)
	Return the floating point remainder after dividing the the first variable by the second scalar (cmath). More...

var	fmod (const double a, const var &b)
	Return the floating point remainder after dividing the first scalar by the second variable (cmath). More...

var	gamma_p (const stan::agrad::var &a, const stan::agrad::var &b)

var	gamma_p (const stan::agrad::var &a, const double &b)

var	gamma_p (const double &a, const stan::agrad::var &b)

var	gamma_q (const stan::agrad::var &a, const stan::agrad::var &b)

var	gamma_q (const stan::agrad::var &a, const double &b)

var	gamma_q (const double &a, const stan::agrad::var &b)

var	hypot (const stan::agrad::var &a, const stan::agrad::var &b)
	Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...

var	hypot (const stan::agrad::var &a, const double &b)
	Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...

var	hypot (const double &a, const stan::agrad::var &b)
	Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...

var	ibeta (const var &a, const var &b, const var &x)
	The normalized incomplete beta function of a, b, and x. More...

var	if_else (bool c, const var &y_true, const var &y_false)
	If the specified condition is true, return the first variable, otherwise return the second variable. More...

var	if_else (bool c, double y_true, const var &y_false)
	If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable. More...

var	if_else (bool c, const var &y_true, const double y_false)
	If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar. More...

var	inv (const var &a)
	$\mbox{inv}(x) = \begin{cases} \frac{1}{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$ More...

var	inv_cloglog (const stan::agrad::var &a)
	Return the inverse complementary log-log function applied specified variable (stan). More...

var	inv_logit (const stan::agrad::var &a)
	The inverse logit function for variables (stan). More...

var	inv_sqrt (const var &a)
	$\mbox{inv\_sqrt}(x) = \begin{cases} \frac{1}{\sqrt{x}} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$ More...

var	inv_square (const var &a)
	$\mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$ More...

int	is_inf (const var &v)
	Returns 1 if the input's value is infinite and 0 otherwise. More...

int	is_nan (const var &v)
	Returns 1 if the input's value is NaN and 0 otherwise. More...

bool	is_uninitialized (var x)
	Returns `true` if the specified variable is uninitialized. More...

var	lgamma (const stan::agrad::var &a)
	The log gamma function for variables (C99). More...

var	lmgamma (int a, const stan::agrad::var &b)

var	log (const var &a)
	Return the natural log of the specified variable (cmath). More...

var	log10 (const var &a)
	Return the base 10 log of the specified variable (cmath). More...

var	log1m (const stan::agrad::var &a)
	The log (1 - x) function for variables. More...

var	log1m_exp (const stan::agrad::var &a)
	Return the log of 1 minus the exponential of the specified variable. More...

var	log1p (const stan::agrad::var &a)
	The log (1 + x) function for variables (C99). More...

var	log1p_exp (const stan::agrad::var &a)
	Return the log of 1 plus the exponential of the specified variable. More...

var	log2 (const stan::agrad::var &a)
	Returns the base 2 logarithm of the specified variable (C99). More...

var	log_diff_exp (const stan::agrad::var &a, const stan::agrad::var &b)
	Returns the log sum of exponentials. More...

var	log_diff_exp (const stan::agrad::var &a, const double &b)
	Returns the log sum of exponentials. More...

var	log_diff_exp (const double &a, const stan::agrad::var &b)
	Returns the log sum of exponentials. More...

var	log_falling_factorial (const var &a, const double &b)

var	log_falling_factorial (const var &a, const var &b)

var	log_falling_factorial (const double &a, const var &b)

var	log_rising_factorial (const var &a, const double &b)

var	log_rising_factorial (const var &a, const var &b)

var	log_rising_factorial (const double &a, const var &b)

var	log_sum_exp (const stan::agrad::var &a, const stan::agrad::var &b)
	Returns the log sum of exponentials. More...

var	log_sum_exp (const stan::agrad::var &a, const double &b)
	Returns the log sum of exponentials. More...

var	log_sum_exp (const double &a, const stan::agrad::var &b)
	Returns the log sum of exponentials. More...

var	log_sum_exp (const std::vector< var > &x)
	Returns the log sum of exponentials. More...

var	modified_bessel_first_kind (const int &v, const var &a)

var	modified_bessel_second_kind (const int &v, const var &a)

var	multiply_log (const var &a, const var &b)
	Return the value of a*log(b). More...

var	multiply_log (const var &a, const double b)
	Return the value of a*log(b). More...

var	multiply_log (const double a, const var &b)
	Return the value of a*log(b). More...

var	owens_t (const var &h, const var &a)
	The Owen's T function of h and a. More...

var	owens_t (const var &h, const double &a)
	The Owen's T function of h and a. More...

var	owens_t (const double &h, const var &a)
	The Owen's T function of h and a. More...

var	Phi (const stan::agrad::var &a)
	The unit normal cumulative density function for variables (stan). More...

var	Phi_approx (const stan::agrad::var &a)
	Approximation of the unit normal CDF for variables (stan). More...

var	pow (const var &base, const var &exponent)
	Return the base raised to the power of the exponent (cmath). More...

var	pow (const var &base, const double exponent)
	Return the base variable raised to the power of the exponent scalar (cmath). More...

var	pow (const double base, const var &exponent)
	Return the base scalar raised to the power of the exponent variable (cmath). More...

double	primitive_value (const agrad::var &v)
	Return the primitive double value for the specified auto-diff variable. More...

var	rising_factorial (const var &a, const double &b)

var	rising_factorial (const var &a, const var &b)

var	rising_factorial (const double &a, const var &b)

var	round (const stan::agrad::var &a)
	Returns the rounded form of the specified variable (C99). More...

var	sin (const var &a)
	Return the sine of a radian-scaled variable (cmath). More...

var	sinh (const var &a)
	Return the hyperbolic sine of the specified variable (cmath). More...

var	sqrt (const var &a)
	Return the square root of the specified variable (cmath). More...

var	square (const var &x)
	Return the square of the input variable. More...

var	step (const stan::agrad::var &a)
	Return the step, or heaviside, function applied to the specified variable (stan). More...

var	tan (const var &a)
	Return the tangent of a radian-scaled variable (cmath). More...

var	tanh (const var &a)
	Return the hyperbolic tangent of the specified variable (cmath). More...

var	tgamma (const stan::agrad::var &a)
	Return the Gamma function applied to the specified variable (C99). More...

var	trunc (const stan::agrad::var &a)
	Returns the truncatation of the specified variable (C99). More...

double	value_of (const agrad::var &v)
	Return the value of the specified variable. More...

var	precomputed_gradients (const double value, const std::vector< var > &vars, const std::vector< double > &gradients)
	This function is provided for Stan users that want to compute gradients without using Stan's auto-diff. More...

matrix_v	crossprod (const matrix_v &M)
	Returns the result of pre-multiplying a matrix by its own transpose. More...

template<int R, int C>
var	determinant (const Eigen::Matrix< var, R, C > &m)

double	divide (double x, double y)
	Return the division of the first scalar by the second scalar. More...

template<typename T1 , typename T2 >
var	divide (const T1 &v, const T2 &c)

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< var, R, C >	divide (const Eigen::Matrix< T1, R, C > &v, const T2 &c)
	Return the division of the specified column vector by the specified scalar. More...

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value, var >::type	dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
	Returns the dot product. More...

template<typename T1 , typename T2 >
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value, var >::type	dot_product (const T1 v1, const T2 v2, size_t length)
	Returns the dot product. More...

template<typename T1 , typename T2 >
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value, var >::type	dot_product (const std::vector< T1 > &v1, const std::vector< T2 > &v2)
	Returns the dot product. More...

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value, Eigen::Matrix < var, 1, C1 > >::type	columns_dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value, Eigen::Matrix < var, R1, 1 > >::type	rows_dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)

template<int R, int C>
var	dot_self (const Eigen::Matrix< var, R, C > &v)
	Returns the dot product of a vector with itself. More...

template<int R, int C>
Eigen::Matrix< var, 1, C >	columns_dot_self (const Eigen::Matrix< var, R, C > &x)
	Returns the dot product of each column of a matrix with itself. More...

void	grad (var &v, Eigen::Matrix< var, Eigen::Dynamic, 1 > &x, Eigen::VectorXd &g)
	Propagate chain rule to calculate gradients starting from the specified variable. More...

void	initialize_variable (var &variable, const var &value)
	Initialize variable to value. More...

template<int R, int C>
void	initialize_variable (Eigen::Matrix< var, R, C > &matrix, const var &value)
	Initialize every cell in the matrix to the specified value. More...

template<typename T >
void	initialize_variable (std::vector< T > &variables, const var &value)
	Initialize the variables in the standard vector recursively. More...

template<int R, int C>
var	log_determinant (const Eigen::Matrix< var, R, C > &m)

template<int R, int C>
var	log_determinant_ldlt (stan::math::LDLT_factor< var, R, C > &A)

template<int R, int C>
var	log_determinant_spd (const Eigen::Matrix< var, R, C > &m)

Eigen::Matrix< var, Eigen::Dynamic, 1 >	log_softmax (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &alpha)
	Return the softmax of the specified Eigen vector. More...

template<int R, int C>
var	log_sum_exp (const Eigen::Matrix< var, R, C > &x)
	Returns the log sum of exponentials. More...

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_ldlt (const stan::math::LDLT_factor< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_ldlt (const stan::math::LDLT_factor< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_ldlt (const stan::math::LDLT_factor< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
	Returns the solution of the system Ax=b given an LDLT_factor of A. More...

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_spd (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_spd (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)

template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_spd (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)

template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)

template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)

template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 >	mdivide_left_tri (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)

template<typename T1 , typename T2 >
boost::enable_if_c < (boost::is_scalar< T1 > ::value\|\|boost::is_same< T1, var >::value)&&(boost::is_scalar < T2 >::value\|\|boost::is_same < T2, var >::value), typename boost::math::tools::promote_args < T1, T2 >::type >::type	multiply (const T1 &v, const T2 &c)
	Return the product of two scalars. More...

template<typename T1 , typename T2 , int R2, int C2>
Eigen::Matrix< var, R2, C2 >	multiply (const T1 &c, const Eigen::Matrix< T2, R2, C2 > &m)
	Return the product of scalar and matrix. More...

template<typename T1 , int R1, int C1, typename T2 >
Eigen::Matrix< var, R1, C1 >	multiply (const Eigen::Matrix< T1, R1, C1 > &m, const T2 &c)
	Return the product of scalar and matrix. More...

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value, Eigen::Matrix < var, R1, C2 > >::type	multiply (const Eigen::Matrix< T1, R1, C1 > &m1, const Eigen::Matrix< T2, R2, C2 > &m2)
	Return the product of the specified matrices. More...

template<typename T1 , int C1, typename T2 , int R2>
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value, var >::type	multiply (const Eigen::Matrix< T1, 1, C1 > &rv, const Eigen::Matrix< T2, R2, 1 > &v)
	Return the scalar product of the specified row vector and specified column vector. More...

matrix_v	multiply_lower_tri_self_transpose (const matrix_v &L)

template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c < boost::is_same< TA, var > ::value\|\|boost::is_same< TB, var >::value, Eigen::Matrix < var, CB, CB > >::type	quad_form (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)

template<typename TA , int RA, int CA, typename TB , int RB>
boost::enable_if_c < boost::is_same< TA, var > ::value\|\|boost::is_same< TB, var >::value, var >::type	quad_form (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, 1 > &B)

template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c < boost::is_same< TA, var > ::value\|\|boost::is_same< TB, var >::value, Eigen::Matrix < var, CB, CB > >::type	quad_form_sym (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)

template<typename TA , int RA, int CA, typename TB , int RB>
boost::enable_if_c < boost::is_same< TA, var > ::value\|\|boost::is_same< TB, var >::value, var >::type	quad_form_sym (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, 1 > &B)

var	sd (const std::vector< var > &v)
	Return the sample standard deviation of the specified standard vector. More...

template<int R, int C>
var	sd (const Eigen::Matrix< var, R, C > &m)

Eigen::Matrix< var, Eigen::Dynamic, 1 >	softmax (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &alpha)
	Return the softmax of the specified Eigen vector. More...

std::vector< var >	sort_asc (std::vector< var > xs)
	Return the specified standard vector in ascending order with gradients kept. More...

std::vector< var >	sort_desc (std::vector< var > xs)
	Return the specified standard vector in descending order with gradients kept. More...

template<int R, int C>
Eigen::Matrix< var, R, C >	sort_asc (Eigen::Matrix< var, R, C > xs)
	Return the specified eigen vector in ascending order with gradients kept. More...

template<int R, int C>
Eigen::Matrix< var, R, C >	sort_desc (Eigen::Matrix< var, R, C > xs)
	Return the specified eigen vector in descending order with gradients kept. More...

template<int R1, int C1, int R2, int C2>
var	squared_distance (const Eigen::Matrix< var, R1, C1 > &v1, const Eigen::Matrix< var, R2, C2 > &v2)

template<int R1, int C1, int R2, int C2>
var	squared_distance (const Eigen::Matrix< var, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)

template<int R1, int C1, int R2, int C2>
var	squared_distance (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< var, R2, C2 > &v2)

void	stan_print (std::ostream *o, const var &x)

template<int R, int C>
var	sum (const Eigen::Matrix< var, R, C > &m)
	Returns the sum of the coefficients of the specified matrix, column vector or row vector. More...

matrix_v	tcrossprod (const matrix_v &M)
	Returns the result of post-multiplying a matrix by its own transpose. More...

var	to_var (const double &x)
	Converts argument to an automatic differentiation variable. More...

var	to_var (const var &x)
	Converts argument to an automatic differentiation variable. More...

matrix_v	to_var (const stan::math::matrix_d &m)
	Converts argument to an automatic differentiation variable. More...

matrix_v	to_var (const matrix_v &m)
	Converts argument to an automatic differentiation variable. More...

vector_v	to_var (const stan::math::vector_d &v)
	Converts argument to an automatic differentiation variable. More...

vector_v	to_var (const vector_v &v)
	Converts argument to an automatic differentiation variable. More...

row_vector_v	to_var (const stan::math::row_vector_d &rv)
	Converts argument to an automatic differentiation variable. More...

row_vector_v	to_var (const row_vector_v &rv)
	Converts argument to an automatic differentiation variable. More...

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename T3 , int R3, int C3>
boost::enable_if_c < boost::is_same< T1, var > ::value\|\|boost::is_same< T2, var >::value\|\|boost::is_same < T3, var >::value, var > ::type	trace_gen_inv_quad_form_ldlt (const Eigen::Matrix< T1, R1, C1 > &D, const stan::math::LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
	Compute the trace of an inverse quadratic form. More...

template<typename TD , int RD, int CD, typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c < boost::is_same< TD, var > ::value\|\|boost::is_same< TA, var >::value\|\|boost::is_same < TB, var >::value, var > ::type	trace_gen_quad_form (const Eigen::Matrix< TD, RD, CD > &D, const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)

template<typename T2 , int R2, int C2, typename T3 , int R3, int C3>
boost::enable_if_c < boost::is_same< T2, var > ::value\|\|boost::is_same< T3, var >::value, var >::type	trace_inv_quad_form_ldlt (const stan::math::LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
	Compute the trace of an inverse quadratic form. More...

template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c < boost::is_same< TA, var > ::value\|\|boost::is_same< TB, var >::value, var >::type	trace_quad_form (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)

template<int R, int C>
const Eigen::Matrix< double, R, C > &	value_of (const Eigen::Matrix< double, R, C > &M)
	Convert a matrix to a matrix of doubles. More...

template<int R, int C>
Eigen::Matrix< double, R, C >	value_of (const Eigen::Matrix< var, R, C > &M)
	Convert a matrix to a matrix of doubles. More...

var	variance (const std::vector< var > &v)
	Return the sample variance of the specified standard vector. More...

template<int R, int C>
var	variance (const Eigen::Matrix< var, R, C > &m)

var	operator+ (const var &a, const var &b)
	Addition operator for variables (C++). More...

var	operator+ (const var &a, const double b)
	Addition operator for variable and scalar (C++). More...

var	operator+ (const double a, const var &b)
	Addition operator for scalar and variable (C++). More...

var	operator/ (const var &a, const var &b)
	Division operator for two variables (C++). More...

var	operator/ (const var &a, const double b)
	Division operator for dividing a variable by a scalar (C++). More...

var	operator/ (const double a, const var &b)
	Division operator for dividing a scalar by a variable (C++). More...

bool	operator== (const var &a, const var &b)
	Equality operator comparing two variables' values (C++). More...

bool	operator== (const var &a, const double b)
	Equality operator comparing a variable's value and a double (C++). More...

bool	operator== (const double a, const var &b)
	Equality operator comparing a scalar and a variable's value (C++). More...

bool	operator> (const var &a, const var &b)
	Greater than operator comparing variables' values (C++). More...

bool	operator> (const var &a, const double b)
	Greater than operator comparing variable's value and double (C++). More...

bool	operator> (const double a, const var &b)
	Greater than operator comparing a double and a variable's value (C++). More...

bool	operator>= (const var &a, const var &b)
	Greater than or equal operator comparing two variables' values (C++). More...

bool	operator>= (const var &a, const double b)
	Greater than or equal operator comparing variable's value and double (C++). More...

bool	operator>= (const double a, const var &b)
	Greater than or equal operator comparing double and variable's value (C++). More...

bool	operator< (const var &a, const var &b)
	Less than operator comparing variables' values (C++). More...

bool	operator< (const var &a, const double b)
	Less than operator comparing variable's value and a double (C++). More...

bool	operator< (const double a, const var &b)
	Less than operator comparing a double and variable's value (C++). More...

bool	operator<= (const var &a, const var &b)
	Less than or equal operator comparing two variables' values (C++). More...

bool	operator<= (const var &a, const double b)
	Less than or equal operator comparing a variable's value and a scalar (C++). More...

bool	operator<= (const double a, const var &b)
	Less than or equal operator comparing a double and variable's value (C++). More...

var	operator* (const var &a, const var &b)
	Multiplication operator for two variables (C++). More...

var	operator* (const var &a, const double b)
	Multiplication operator for a variable and a scalar (C++). More...

var	operator* (const double a, const var &b)
	Multiplication operator for a scalar and a variable (C++). More...

bool	operator!= (const var &a, const var &b)
	Inequality operator comparing two variables' values (C++). More...

bool	operator!= (const var &a, const double b)
	Inequality operator comparing a variable's value and a double (C++). More...

bool	operator!= (const double a, const var &b)
	Inequality operator comparing a double and a variable's value (C++). More...

var	operator- (const var &a, const var &b)
	Subtraction operator for variables (C++). More...

var	operator- (const var &a, const double b)
	Subtraction operator for variable and scalar (C++). More...

var	operator- (const double a, const var &b)
	Subtraction operator for scalar and variable (C++). More...

var &	operator-- (var &a)
	Prefix decrement operator for variables (C++). More...

var	operator-- (var &a, int)
	Postfix decrement operator for variables (C++). More...

var &	operator++ (var &a)
	Prefix increment operator for variables (C++). More...

var	operator++ (var &a, int)
	Postfix increment operator for variables (C++). More...

var	operator- (const var &a)
	Unary negation operator for variables (C++). More...

bool	operator! (const var &a)
	Prefix logical negation for the value of variables (C++). More...

var	operator+ (const var &a)
	Unary plus operator for variables (C++). More...

void	print_stack (std::ostream &o)
	Prints the auto-dif variable stack. More...

static void	grad (chainable *vi)

static bool	empty_nested ()
	Return true if there is no nested autodiff being executed. More...

static void	recover_memory ()
	Recover memory used for all variables for reuse. More...

static void	recover_memory_nested ()
	Recover only the memory used for the top nested call. More...

static void	start_nested ()
	Record the current position so that `recover_memory_nested()` can find it. More...

static size_t	nested_size ()

Variables
std::vector< chainable * >	var_stack_

std::vector< chainable * >	var_nochain_stack_

std::vector< chainable_alloc * >	var_alloc_stack_

memory::stack_alloc	memalloc_

std::vector< size_t >	nested_var_stack_sizes_

std::vector< size_t >	nested_var_nochain_stack_sizes_

std::vector< size_t >	nested_var_alloc_stack_starts_

Detailed Description

Function gradients via reverse-mode automatic differentiation.

Typedef Documentation

typedef Eigen::Matrix<fvar<double>,Eigen::Dynamic,Eigen::Dynamic> stan::agrad::matrix_fd

Definition at line 18 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<double> >,Eigen::Dynamic,Eigen::Dynamic> stan::agrad::matrix_ffd

Definition at line 26 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<var> >,Eigen::Dynamic,Eigen::Dynamic> stan::agrad::matrix_ffv

Definition at line 30 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<var>,Eigen::Dynamic,Eigen::Dynamic> stan::agrad::matrix_fv

Definition at line 22 of file typedefs.hpp.

typedef Eigen::Matrix<var,Eigen::Dynamic,Eigen::Dynamic> stan::agrad::matrix_v

The type of a matrix holding stan::agrad::var values.

Definition at line 21 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<double>,1,Eigen::Dynamic> stan::agrad::row_vector_fd

Definition at line 50 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<double> >,1,Eigen::Dynamic> stan::agrad::row_vector_ffd

Definition at line 58 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<var> >,1,Eigen::Dynamic> stan::agrad::row_vector_ffv

Definition at line 62 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<var>,1,Eigen::Dynamic> stan::agrad::row_vector_fv

Definition at line 54 of file typedefs.hpp.

typedef Eigen::Matrix<var,1,Eigen::Dynamic> stan::agrad::row_vector_v

The type of a row vector holding stan::agrad::var values.

Definition at line 37 of file typedefs.hpp.

typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >::Index stan::agrad::size_type

Definition at line 14 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<double>,Eigen::Dynamic,1> stan::agrad::vector_fd

Definition at line 34 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<double> >,Eigen::Dynamic,1> stan::agrad::vector_ffd

Definition at line 42 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<var> >,Eigen::Dynamic,1> stan::agrad::vector_ffv

Definition at line 46 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<var>,Eigen::Dynamic,1> stan::agrad::vector_fv

Definition at line 38 of file typedefs.hpp.

typedef Eigen::Matrix<var,Eigen::Dynamic,1> stan::agrad::vector_v

The type of a (column) vector holding stan::agrad::var values.

Definition at line 29 of file typedefs.hpp.

Function Documentation

template<typename T >

fvar<T> stan::agrad::abs ( const fvar< T > & x )

inline

Definition at line 19 of file abs.hpp.

var stan::agrad::abs ( const var & a )

inline

Return the absolute value of the variable (std).

Delegates to fabs() (see for doc).

$\mbox{abs}(x) = \begin{cases} |x| & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{abs}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } x < 0 \\ 0 & \mbox{if } x = 0 \\ 1 & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable input.

Returns: Absolute value of variable.

Definition at line 35 of file abs.hpp.

template<typename T >

fvar<T> stan::agrad::acos ( const fvar< T > & x )

inline

Definition at line 16 of file acos.hpp.

var stan::agrad::acos ( const var & a )

inline

Return the principal value of the arc cosine of a variable, in radians (cmath).

The derivative is defined by

$\frac{d}{dx} \arccos x = \frac{-1}{\sqrt{1 - x^2}}$ .

$\mbox{acos}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arccos(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{acos}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\,\arccos(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial \, \arccos(x)}{\partial x} = -\frac{1}{\sqrt{1-x^2}}$

Parameters

a	Variable in range [-1,1].

Returns: Arc cosine of variable, in radians.

Definition at line 60 of file acos.hpp.

template<typename T >

fvar<T> stan::agrad::acosh ( const fvar< T > & x )

inline

Definition at line 17 of file acosh.hpp.

var stan::agrad::acosh ( const stan::agrad::var & a )

inline

The inverse hyperbolic cosine function for variables (C99).

For non-variable function, see boost::math::acosh().

The derivative is defined by

$\frac{d}{dx} \mbox{acosh}(x) = \frac{x}{x^2 - 1}$ .

$\mbox{acosh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \cosh^{-1}(x) & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{acosh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \frac{\partial\, \cosh^{-1}(x)}{\partial x} & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right)$

$\frac{\partial \, \cosh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2-1}}$

Parameters

a	The variable.

Returns: Inverse hyperbolic cosine of the variable.

Definition at line 67 of file acosh.hpp.

int stan::agrad::as_bool ( const agrad::var & v )

inline

Return 1 if the argument is unequal to zero and 0 otherwise.

Parameters

v Value.

Returns: 1 if argument is equal to zero (or NaN) and 0 otherwise.

Definition at line 15 of file as_bool.hpp.

template<typename T >

fvar<T> stan::agrad::asin ( const fvar< T > & x )

inline

Definition at line 16 of file asin.hpp.

var stan::agrad::asin ( const var & a )

inline

Return the principal value of the arc sine, in radians, of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$ .

$\mbox{asin}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arcsin(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{asin}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\,\arcsin(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial \, \arcsin(x)}{\partial x} = \frac{1}{\sqrt{1-x^2}}$

Parameters

a	Variable in range [-1,1].

Returns: Arc sine of variable, in radians.

Definition at line 60 of file asin.hpp.

template<typename T >

fvar<T> stan::agrad::asinh ( const fvar< T > & x )

inline

Definition at line 16 of file asinh.hpp.

var stan::agrad::asinh ( const stan::agrad::var & a )

inline

The inverse hyperbolic sine function for variables (C99).

For non-variable function, see boost::math::asinh().

The derivative is defined by

$\frac{d}{dx} \mbox{asinh}(x) = \frac{x}{x^2 + 1}$ .

$\mbox{asinh}(x) = \begin{cases} \sinh^{-1}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{asinh}(x)}{\partial x} = \begin{cases} \frac{\partial\, \sinh^{-1}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right)$

$\frac{\partial \, \sinh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2+1}}$

Parameters

a	The variable.

Returns: Inverse hyperbolic sine of the variable.

Definition at line 65 of file asinh.hpp.

template<typename T >

fvar<T> stan::agrad::atan ( const fvar< T > & x )

inline

Definition at line 16 of file atan.hpp.

var stan::agrad::atan ( const var & a )

inline

Return the principal value of the arc tangent, in radians, of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}$ .

$\mbox{atan}(x) = \begin{cases} \arctan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{atan}(x)}{\partial x} = \begin{cases} \frac{\partial\, \arctan(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial \, \arctan(x)}{\partial x} = \frac{1}{x^2+1}$

Parameters

a	Variable in range [-1,1].

Returns: Arc tangent of variable, in radians.

Definition at line 56 of file atan.hpp.

template<typename T >

fvar<T> stan::agrad::atan2	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 16 of file atan2.hpp.

template<typename T >

fvar<T> stan::agrad::atan2	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 27 of file atan2.hpp.

template<typename T >

fvar<T> stan::agrad::atan2	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 37 of file atan2.hpp.

var stan::agrad::atan2	(	const var &	a,
		const var &	b
	)

inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath).

The partial derivatives are defined by

$\frac{\partial}{\partial x} \arctan \frac{x}{y} = \frac{y}{x^2 + y^2}$ , and

$\frac{\partial}{\partial y} \arctan \frac{x}{y} = \frac{-x}{x^2 + y^2}$ .

Parameters

a	Numerator variable.
b	Denominator variable.

Returns: The arc tangent of the fraction, in radians.

Definition at line 64 of file atan2.hpp.

var stan::agrad::atan2	(	const var &	a,
		const double	b
	)

inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath).

The derivative with respect to the variable is

$\frac{d}{d x} \arctan \frac{x}{c} = \frac{c}{x^2 + c^2}$ .

Parameters

a	Numerator variable.
b	Denominator scalar.

Returns: The arc tangent of the fraction, in radians.

Definition at line 80 of file atan2.hpp.

var stan::agrad::atan2	(	const double	a,
		const var &	b
	)

inline

Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath).

The derivative with respect to the variable is

$\frac{\partial}{\partial y} \arctan \frac{c}{y} = \frac{-c}{c^2 + y^2}$ .

$\mbox{atan2}(x,y) = \begin{cases} \arctan\left(\frac{x}{y}\right) & \mbox{if } -\infty\leq x \leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{atan2}(x,y)}{\partial x} = \begin{cases} \frac{y}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{atan2}(x,y)}{\partial y} = \begin{cases} -\frac{x}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	Numerator scalar.
b	Denominator variable.

Returns: The arc tangent of the fraction, in radians.

Definition at line 121 of file atan2.hpp.

template<typename T >

fvar<T> stan::agrad::atanh ( const fvar< T > & x )

inline

Definition at line 16 of file atanh.hpp.

var stan::agrad::atanh ( const stan::agrad::var & a )

inline

The inverse hyperbolic tangent function for variables (C99).

For non-variable function, see boost::math::atanh().

The derivative is defined by

$\frac{d}{dx} \mbox{atanh}(x) = \frac{1}{1 - x^2}$ .

$\mbox{atanh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \tanh^{-1}(x) & \mbox{if } -1\leq x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{atanh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \tanh^{-1}(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\tanh^{-1}(x)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$

$\frac{\partial \, \tanh^{-1}(x)}{\partial x} = \frac{1}{1-x^2}$

Parameters

a	The variable.

Returns: Inverse hyperbolic tangent of the variable.

Definition at line 65 of file atanh.hpp.

template<typename T >

fvar<T> stan::agrad::bessel_first_kind	(	int	v,
		const fvar< T > &	z
	)

inline

Definition at line 15 of file bessel_first_kind.hpp.

var stan::agrad::bessel_first_kind	(	const int &	v,
		const var &	a
	)

inline

Definition at line 27 of file bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::agrad::bessel_second_kind	(	int	v,
		const fvar< T > &	z
	)

inline

Definition at line 15 of file bessel_second_kind.hpp.

var stan::agrad::bessel_second_kind	(	const int &	v,
		const var &	a
	)

inline

Definition at line 27 of file bessel_second_kind.hpp.

template<typename T >

fvar<T> stan::agrad::binary_log_loss	(	const int	y,
		const fvar< T > &	y_hat
	)

inline

Definition at line 15 of file binary_log_loss.hpp.

var stan::agrad::binary_log_loss	(	const int	y,
		const stan::agrad::var &	y_hat
	)

inline

The log loss function for variables (stan).

See stan::math::binary_log_loss() for the double-based version.

The derivative with respect to the variable $\hat{y}$ is

$\frac{d}{d\hat{y}} \mbox{logloss}(1,\hat{y}) = - \frac{1}{\hat{y}}$ , and

$\frac{d}{d\hat{y}} \mbox{logloss}(0,\hat{y}) = \frac{1}{1 - \hat{y}}$ .

$\mbox{binary\_log\_loss}(y,\hat{y}) = \begin{cases} y \log \hat{y} + (1 - y) \log (1 - \hat{y}) & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0,1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{binary\_log\_loss}(y,\hat{y})}{\partial \hat{y}} = \begin{cases} \frac{y}{\hat{y}}-\frac{1-y}{1-\hat{y}} & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0,1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases}$

Parameters

y	Reference value.
y_hat	Response variable.

Returns: Log loss of response versus reference value.

Definition at line 69 of file binary_log_loss.hpp.

template<typename T >

fvar<T> stan::agrad::binomial_coefficient_log	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 16 of file binomial_coefficient_log.hpp.

template<typename T >

fvar<T> stan::agrad::binomial_coefficient_log	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 45 of file binomial_coefficient_log.hpp.

template<typename T >

fvar<T> stan::agrad::binomial_coefficient_log	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 69 of file binomial_coefficient_log.hpp.

double stan::agrad::calculate_chain	(	const double &	x,
		const double &	val
	)

inline

Definition at line 8 of file calculate_chain.hpp.

template<typename T >

fvar<T> stan::agrad::cbrt ( const fvar< T > & x )

inline

Definition at line 16 of file cbrt.hpp.

var stan::agrad::cbrt ( const stan::agrad::var & a )

inline

Returns the cube root of the specified variable (C99).

See boost::math::cbrt() for the double-based version.

The derivative is

$\frac{d}{dx} x^{1/3} = \frac{1}{3 x^{2/3}}$ .

$\mbox{cbrt}(x) = \begin{cases} \sqrt[3]{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{cbrt}(x)}{\partial x} = \begin{cases} \frac{1}{3x^{2/3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Specified variable.

Returns: Cube root of the variable.

Definition at line 51 of file cbrt.hpp.

template<typename T >

fvar<T> stan::agrad::ceil ( const fvar< T > & x )

inline

Definition at line 15 of file ceil.hpp.

var stan::agrad::ceil ( const var & a )

inline

Return the ceiling of the specified variable (cmath).

The derivative of the ceiling function is defined and zero everywhere but at integers, and we set them to zero for convenience,

$\frac{d}{dx} {\lceil x \rceil} = 0$ .

The ceiling function rounds up. For double values, this is the smallest integral value that is not less than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

$\mbox{ceil}(x) = \begin{cases} \lceil x\rceil & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{ceil}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Input variable.

Returns: Ceiling of the variable.

Definition at line 60 of file ceil.hpp.

template<typename T_result , class Policy >

bool stan::agrad::check_pos_definite	(	const char *	function,
		const Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic > &	y,
		const char *	name,
		T_result *	result,
		const Policy &
	)

inline

Definition at line 15 of file check_pos_definite.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, 1, C1> stan::agrad::columns_dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2
	)

inline

Definition at line 18 of file columns_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, 1, C1> stan::agrad::columns_dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< double, R2, C2 > &	v2
	)

inline

Definition at line 34 of file columns_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, 1, C1> stan::agrad::columns_dot_product	(	const Eigen::Matrix< double, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2
	)

inline

Definition at line 50 of file columns_dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>

boost::enable_if_c<boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value, Eigen::Matrix<var, 1, C1> >::type stan::agrad::columns_dot_product	(	const Eigen::Matrix< T1, R1, C1 > &	v1,
		const Eigen::Matrix< T2, R2, C2 > &	v2
	)

inline

Definition at line 261 of file dot_product.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,1,C> stan::agrad::columns_dot_self ( const Eigen::Matrix< fvar< T >, R, C > & x )

inline

Definition at line 15 of file columns_dot_self.hpp.

template<int R, int C>

Eigen::Matrix<var,1,C> stan::agrad::columns_dot_self ( const Eigen::Matrix< var, R, C > & x )

inline

Returns the dot product of each column of a matrix with itself.

Parameters

x Matrix.

Template Parameters

T	scalar type

Definition at line 89 of file dot_self.hpp.

template<typename T >

fvar<T> stan::agrad::cos ( const fvar< T > & x )

inline

Definition at line 15 of file cos.hpp.

var stan::agrad::cos ( const var & a )

inline

Return the cosine of a radian-scaled variable (cmath).

The derivative is defined by

$\frac{d}{dx} \cos x = - \sin x$ .

$\mbox{cos}(x) = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{cos}(x)}{\partial x} = \begin{cases} -\sin(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable for radians of angle.

Returns: Cosine of variable.

Definition at line 50 of file cos.hpp.

template<typename T >

fvar<T> stan::agrad::cosh ( const fvar< T > & x )

inline

Definition at line 15 of file cosh.hpp.

var stan::agrad::cosh ( const var & a )

inline

Return the hyperbolic cosine of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \cosh x = \sinh x$ .

$\mbox{cosh}(x) = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{cosh}(x)}{\partial x} = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a Variable.

Returns: Hyperbolic cosine of variable.

Definition at line 51 of file cosh.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,C,C> stan::agrad::crossprod ( const Eigen::Matrix< fvar< T >, R, C > & m )

inline

Definition at line 17 of file crossprod.hpp.

matrix_v stan::agrad::crossprod ( const matrix_v & M )

inline

Returns the result of pre-multiplying a matrix by its own transpose.

Parameters

M	Matrix to multiply.

Returns: Transpose of M times M

Definition at line 17 of file crossprod.hpp.

template<typename T , typename F >

void stan::agrad::derivative	(	const F &	f,
		const T &	x,
		T &	fx,
		T &	dfx_dx
	)

Return the derivative of the specified univariate function at the specified argument.

Template Parameters

T	Argument type
F	Function type

Parameters

[in]	f	Function
[in]	x	Argument
[out]	fx	Value of function applied to argument
[out]	dfx_dx	Value of derivative

Definition at line 25 of file autodiff.hpp.

template<typename T , int R, int C>

fvar<T> stan::agrad::determinant ( const Eigen::Matrix< fvar< T >, R, C > & m )

inline

Definition at line 21 of file determinant.hpp.

template<int R, int C>

var stan::agrad::determinant ( const Eigen::Matrix< var, R, C > & m )

inline

Definition at line 67 of file determinant.hpp.

template<typename T >

fvar<T> stan::agrad::digamma ( const fvar< T > & x )

inline

Definition at line 16 of file digamma.hpp.

var stan::agrad::digamma ( const stan::agrad::var & a )

inline

Definition at line 25 of file digamma.hpp.

template<typename T1 , typename T2 >

stan::return_type<T1,T2>::type stan::agrad::divide	(	const T1 &	v,
		const T2 &	c
	)

inline

Definition at line 17 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::divide	(	const Eigen::Matrix< fvar< T >, R, C > &	v,
		const fvar< T > &	c
	)

inline

Definition at line 23 of file divide.hpp.

double stan::agrad::divide	(	double	x,
		double	y
	)

inline

Return the division of the first scalar by the second scalar.

Parameters

[in]	x	Specified vector.
[in]	y	Specified scalar.

Returns: Vector divided by the scalar.

Definition at line 23 of file divide.hpp.

template<typename T1 , typename T2 >

var stan::agrad::divide	(	const T1 &	v,
		const T2 &	c
	)

inline

Definition at line 28 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::divide	(	const Eigen::Matrix< fvar< T >, R, C > &	v,
		const double	c
	)

inline

Definition at line 34 of file divide.hpp.

template<typename T1 , typename T2 , int R, int C>

Eigen::Matrix<var,R,C> stan::agrad::divide	(	const Eigen::Matrix< T1, R, C > &	v,
		const T2 &	c
	)

inline

Return the division of the specified column vector by the specified scalar.

Parameters

[in]	v	Specified vector.
[in]	c	Specified scalar.

Returns: Vector divided by the scalar.

Definition at line 40 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::divide	(	const Eigen::Matrix< double, R, C > &	v,
		const fvar< T > &	c
	)

inline

Definition at line 46 of file divide.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::agrad::dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2
	)

inline

Definition at line 20 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::agrad::dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< double, R2, C2 > &	v2
	)

inline

Definition at line 36 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::agrad::dot_product	(	const Eigen::Matrix< double, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2
	)

inline

Definition at line 52 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::agrad::dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2,
		size_type &	length
	)

inline

Definition at line 68 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::agrad::dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< double, R2, C2 > &	v2,
		size_type &	length
	)

inline

Definition at line 83 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

fvar<T> stan::agrad::dot_product	(	const Eigen::Matrix< double, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2,
		size_type &	length
	)

inline

Definition at line 98 of file dot_product.hpp.

template<typename T >

fvar<T> stan::agrad::dot_product	(	const std::vector< fvar< T > > &	v1,
		const std::vector< fvar< T > > &	v2
	)

inline

Definition at line 113 of file dot_product.hpp.

template<typename T >

fvar<T> stan::agrad::dot_product	(	const std::vector< double > &	v1,
		const std::vector< fvar< T > > &	v2
	)

inline

Definition at line 126 of file dot_product.hpp.

template<typename T >

fvar<T> stan::agrad::dot_product	(	const std::vector< fvar< T > > &	v1,
		const std::vector< double > &	v2
	)

inline

Definition at line 139 of file dot_product.hpp.

template<typename T >

fvar<T> stan::agrad::dot_product	(	const std::vector< fvar< T > > &	v1,
		const std::vector< fvar< T > > &	v2,
		size_type &	length
	)

inline

Definition at line 152 of file dot_product.hpp.

template<typename T >

fvar<T> stan::agrad::dot_product	(	const std::vector< double > &	v1,
		const std::vector< fvar< T > > &	v2,
		size_type &	length
	)

inline

Definition at line 164 of file dot_product.hpp.

template<typename T >

fvar<T> stan::agrad::dot_product	(	const std::vector< fvar< T > > &	v1,
		const std::vector< double > &	v2,
		size_type &	length
	)

inline

Definition at line 176 of file dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>

boost::enable_if_c<boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value, var>::type stan::agrad::dot_product	(	const Eigen::Matrix< T1, R1, C1 > &	v1,
		const Eigen::Matrix< T2, R2, C2 > &	v2
	)

inline

Returns the dot product.

Parameters

[in]	v1	First column vector.
[in]	v2	Second column vector.

Returns: Dot product of the vectors.

Exceptions

std::domain_error if length of v1 is not equal to length of v2.

Definition at line 213 of file dot_product.hpp.

template<typename T1 , typename T2 >

boost::enable_if_c<boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value, var>::type stan::agrad::dot_product	(	const T1 *	v1,
		const T2 *	v2,
		size_t	length
	)

inline

Returns the dot product.

Parameters

[in]	v1	First array.
[in]	v2	Second array.
[in]	length	Length of both arrays.

Returns: Dot product of the arrays.

Definition at line 233 of file dot_product.hpp.

template<typename T1 , typename T2 >

boost::enable_if_c<boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value, var>::type stan::agrad::dot_product	(	const std::vector< T1 > &	v1,
		const std::vector< T2 > &	v2
	)

inline

Returns the dot product.

Parameters

[in]	v1	First vector.
[in]	v2	Second vector.

Returns: Dot product of the vectors.

Exceptions

std::domain_error if sizes of v1 and v2 do not match.

Definition at line 249 of file dot_product.hpp.

template<typename T , int R, int C>

fvar<T> stan::agrad::dot_self ( const Eigen::Matrix< fvar< T >, R, C > & v )

inline

Definition at line 16 of file dot_self.hpp.

template<int R, int C>

var stan::agrad::dot_self ( const Eigen::Matrix< var, R, C > & v )

inline

Returns the dot product of a vector with itself.

Parameters

[in] v Vector.

Returns: Dot product of the vector with itself.

Template Parameters

R	number of rows or `Eigen::Dynamic` for dynamic; one of R or C must be 1
C	number of rows or `Eigen::Dyanmic` for dynamic; one of R or C must be 1

Definition at line 77 of file dot_self.hpp.

static bool stan::agrad::empty_nested ( )

inlinestatic

Return true if there is no nested autodiff being executed.

Definition at line 42 of file var_stack.hpp.

template<typename T >

fvar<T> stan::agrad::erf ( const fvar< T > & x )

inline

Definition at line 17 of file erf.hpp.

var stan::agrad::erf ( const stan::agrad::var & a )

inline

The error function for variables (C99).

For non-variable function, see erf() from math.h

The derivative is

$\frac{d}{dx} \mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \exp(-x^2)$ .

$\mbox{erf}(x) = \begin{cases} \operatorname{erf}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{erf}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erf}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$

$\frac{\partial \, \operatorname{erf}(x)}{\partial x} = \frac{2}{\sqrt{\pi}} e^{-x^2}$

Parameters

a	The variable.

Returns: Error function applied to the variable.

Definition at line 63 of file erf.hpp.

template<typename T >

fvar<T> stan::agrad::erfc ( const fvar< T > & x )

inline

Definition at line 17 of file erfc.hpp.

var stan::agrad::erfc ( const stan::agrad::var & a )

inline

The complementary error function for variables (C99).

For non-variable function, see erfc() from math.h.

The derivative is

$\frac{d}{dx} \mbox{erfc}(x) = - \frac{2}{\sqrt{\pi}} \exp(-x^2)$ .

$\mbox{erfc}(x) = \begin{cases} \operatorname{erfc}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{erfc}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erfc}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2}dt$

$\frac{\partial \, \operatorname{erfc}(x)}{\partial x} = -\frac{2}{\sqrt{\pi}} e^{-x^2}$

Parameters

a	The variable.

Returns: Complementary error function applied to the variable.

Definition at line 63 of file erfc.hpp.

template<typename T >

fvar<T> stan::agrad::exp ( const fvar< T > & x )

inline

Definition at line 16 of file exp.hpp.

var stan::agrad::exp ( const var & a )

inline

Return the exponentiation of the specified variable (cmath).

$\mbox{exp}(x) = \begin{cases} e^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{exp}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable to exponentiate.

Returns: Exponentiated variable.

Definition at line 45 of file exp.hpp.

template<typename T >

fvar<T> stan::agrad::exp2 ( const fvar< T > & x )

inline

Definition at line 16 of file exp2.hpp.

var stan::agrad::exp2 ( const stan::agrad::var & a )

inline

Exponentiation base 2 function for variables (C99).

For non-variable function, see boost::math::exp2().

The derivatie is

$\frac{d}{dx} 2^x = (\log 2) 2^x$ .

$\mbox{exp2}(x) = \begin{cases} 2^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{exp2}(x)}{\partial x} = \begin{cases} 2^x\ln2 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	The variable.

Returns: Two to the power of the specified variable.

Definition at line 53 of file exp2.hpp.

template<typename T >

fvar<T> stan::agrad::expm1 ( const fvar< T > & x )

inline

Definition at line 15 of file expm1.hpp.

var stan::agrad::expm1 ( const stan::agrad::var & a )

inline

The exponentiation of the specified variable minus 1 (C99).

For non-variable function, see boost::math::expm1().

The derivative is given by

$\frac{d}{dx} \exp(a) - 1 = \exp(a)$ .

$\mbox{expm1}(x) = \begin{cases} e^x-1 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{expm1}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	The variable.

Returns: Two to the power of the specified variable.

Definition at line 54 of file expm1.hpp.

template<typename T >

fvar<T> stan::agrad::fabs ( const fvar< T > & x )

inline

Definition at line 18 of file fabs.hpp.

var stan::agrad::fabs ( const var & a )

inline

Return the absolute value of the variable (cmath).

Choosing an arbitrary value at the non-differentiable point 0,

$\frac{d}{dx}|x| = \mbox{sgn}(x)$ .

where $\mbox{sgn}(x)$ is the signum function, taking values -1 if $x < 0$ , 0 if $x == 0$ , and 1 if $x == 1$ .

The function abs() provides the same behavior, with abs() defined in stdlib.h and fabs() defined in cmath. The derivative is 0 if the input is 0.

Returns std::numeric_limits<double>::quiet_NaN() for NaN inputs.

[ {fabs}(x) = {cases} |x| & {if } - x \[6pt] {NaN} & {if } x = {NaN} {cases} ]

[ {\,{fabs}(x)}{ x} = {cases} -1 & {if } x < 0 \ 0 & {if } x = 0 \ 1 & {if } x > 0 \[6pt] {NaN} & {if } x = {NaN} {cases} ]

Parameters

a	Input variable.

Returns: Absolute value of variable.

Definition at line 53 of file fabs.hpp.

template<typename T >

fvar<T> stan::agrad::falling_factorial	(	const fvar< T > &	x,
		const fvar< T > &	n
	)

inline

Definition at line 15 of file falling_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::falling_factorial	(	const fvar< T > &	x,
		const double	n
	)

inline

Definition at line 26 of file falling_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::falling_factorial	(	const double	x,
		const fvar< T > &	n
	)

inline

Definition at line 37 of file falling_factorial.hpp.

var stan::agrad::falling_factorial	(	const var &	a,
		const double &	b
	)

inline

Definition at line 48 of file falling_factorial.hpp.

var stan::agrad::falling_factorial	(	const var &	a,
		const var &	b
	)

inline

Definition at line 53 of file falling_factorial.hpp.

var stan::agrad::falling_factorial	(	const double &	a,
		const var &	b
	)

inline

Definition at line 58 of file falling_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::fdim	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 15 of file fdim.hpp.

template<typename T >

fvar<T> stan::agrad::fdim	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 28 of file fdim.hpp.

template<typename T >

fvar<T> stan::agrad::fdim	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 40 of file fdim.hpp.

var stan::agrad::fdim	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Return the positive difference between the first variable's the value and the second's (C99).

See stan::math::fdim() for the double-based version.

The partial derivative with respect to the first argument is

$\frac{\partial}{\partial x} \mbox{fdim}(x,y) = 0.0$ if $x < y$ , and

$\frac{\partial}{\partial x} \mbox{fdim}(x,y) = 1.0$ if $x \geq y$ .

With respect to the second argument, the partial is

$\frac{\partial}{\partial y} \mbox{fdim}(x,y) = 0.0$ if $x < y$ , and

$\frac{\partial}{\partial y} \mbox{fdim}(x,y) = -\lfloor\frac{x}{y}\rfloor$ if $x \geq y$ .

$\mbox{fdim}(x,y) = \begin{cases} 0 & \mbox{if } x < y\\ x-y & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fdim}(x,y)}{\partial x} = \begin{cases} 0 & \mbox{if } x < y \\ 1 & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fdim}(x,y)}{\partial y} = \begin{cases} 0 & \mbox{if } x < y \\ -\lfloor\frac{x}{y}\rfloor & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: The positive difference between the first and second variable.

Definition at line 112 of file fdim.hpp.

var stan::agrad::fdim	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Return the positive difference between the first value and the value of the second variable (C99).

See fdim(var,var) for definitions of values and derivatives.

The derivative with respect to the variable is

$\frac{d}{d y} \mbox{fdim}(c,y) = 0.0$ if $c < y$ , and

$\frac{d}{d y} \mbox{fdim}(c,y) = -\lfloor\frac{c}{y}\rfloor$ if $c \geq y$ .

Parameters

a	First value.
b	Second variable.

Returns: The positive difference between the first and second arguments.

Definition at line 137 of file fdim.hpp.

var stan::agrad::fdim	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Return the positive difference between the first variable's value and the second value (C99).

See fdim(var,var) for definitions of values and derivatives.

The derivative with respect to the variable is

$\frac{d}{d x} \mbox{fdim}(x,c) = 0.0$ if $x < c$ , and

$\frac{d}{d x} \mbox{fdim}(x,c) = 1.0$ if $x \geq yc$ .

Parameters

a	First value.
b	Second variable.

Returns: The positive difference between the first and second arguments.

Definition at line 160 of file fdim.hpp.

template<typename T >

fvar<T> stan::agrad::floor ( const fvar< T > & x )

inline

Definition at line 15 of file floor.hpp.

var stan::agrad::floor ( const var & a )

inline

Return the floor of the specified variable (cmath).

The derivative of the floor function is defined and zero everywhere but at integers, so we set these derivatives to zero for convenience,

$\frac{d}{dx} {\lfloor x \rfloor} = 0$ .

The floor function rounds down. For double values, this is the largest integral value that is not greater than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

$\mbox{floor}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{floor}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Input variable.

Returns: Floor of the variable.

Definition at line 60 of file floor.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1,T2,T3>::type> stan::agrad::fma	(	const fvar< T1 > &	x1,
		const fvar< T2 > &	x2,
		const fvar< T3 > &	x3
	)

inline

The fused multiply-add operation (C99).

This double-based operation delegates to fma.

The function is defined by

fma(a,b,c) = (a * b) + c.

$\mbox{fma}(x,y,z) = \begin{cases} x\cdot y+z & \mbox{if } -\infty\leq x,y,z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fma}(x,y,z)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x,y,z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fma}(x,y,z)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x,y,z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fma}(x,y,z)}{\partial z} = \begin{cases} 1 & \mbox{if } -\infty\leq x,y,z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

x1	First value.
x2	Second value.
x3	Third value.

Returns: Product of the first two values plus the third.

Definition at line 62 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1,T2,T3>::type> stan::agrad::fma	(	const T1 &	x1,
		const fvar< T2 > &	x2,
		const fvar< T3 > &	x3
	)

inline

See all-var input signature for details on the function and derivatives.

Definition at line 74 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1,T2,T3>::type> stan::agrad::fma	(	const fvar< T1 > &	x1,
		const T2 &	x2,
		const fvar< T3 > &	x3
	)

inline

See all-var input signature for details on the function and derivatives.

Definition at line 86 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1,T2,T3>::type> stan::agrad::fma	(	const fvar< T1 > &	x1,
		const fvar< T2 > &	x2,
		const T3 &	x3
	)

inline

See all-var input signature for details on the function and derivatives.

Definition at line 98 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1,T2,T3>::type> stan::agrad::fma	(	const T1 &	x1,
		const T2 &	x2,
		const fvar< T3 > &	x3
	)

inline

See all-var input signature for details on the function and derivatives.

Definition at line 110 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1,T2,T3>::type> stan::agrad::fma	(	const fvar< T1 > &	x1,
		const T2 &	x2,
		const T3 &	x3
	)

inline

See all-var input signature for details on the function and derivatives.

Definition at line 122 of file fma.hpp.

var stan::agrad::fma	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b,
		const stan::agrad::var &	c
	)

inline

The fused multiply-add function for three variables (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double,double,double) is defined in <cmath>.

The partial derivatives are

$\frac{\partial}{\partial x} (x * y) + z = y$ , and

$\frac{\partial}{\partial y} (x * y) + z = x$ , and

$\frac{\partial}{\partial z} (x * y) + z = 1$ .

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns: Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 134 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >

fvar<typename stan::return_type<T1,T2,T3>::type> stan::agrad::fma	(	const T1 &	x1,
		const fvar< T2 > &	x2,
		const T3 &	x3
	)

inline

See all-var input signature for details on the function and derivatives.

Definition at line 134 of file fma.hpp.

var stan::agrad::fma	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b,
		const double &	c
	)

inline

The fused multiply-add function for two variables and a value (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double,double,double) is defined in <cmath>.

The partial derivatives are

$\frac{\partial}{\partial x} (x * y) + c = y$ , and

$\frac{\partial}{\partial y} (x * y) + c = x$ .

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns: Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 159 of file fma.hpp.

var stan::agrad::fma	(	const stan::agrad::var &	a,
		const double &	b,
		const stan::agrad::var &	c
	)

inline

The fused multiply-add function for a variable, value, and variable (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double,double,double) is defined in <cmath>.

The partial derivatives are

$\frac{\partial}{\partial x} (x * c) + z = c$ , and

$\frac{\partial}{\partial z} (x * c) + z = 1$ .

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns: Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 184 of file fma.hpp.

var stan::agrad::fma	(	const stan::agrad::var &	a,
		const double &	b,
		const double &	c
	)

inline

The fused multiply-add function for a variable and two values (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double,double,double) is defined in <cmath>.

The derivative is

$\frac{d}{d x} (x * c) + d = c$ .

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns: Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 207 of file fma.hpp.

var stan::agrad::fma	(	const double &	a,
		const stan::agrad::var &	b,
		const double &	c
	)

inline

The fused multiply-add function for a value, variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double,double,double) is defined in <cmath>.

The derivative is

$\frac{d}{d y} (c * y) + d = c$ , and

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns: Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 230 of file fma.hpp.

var stan::agrad::fma	(	const double &	a,
		const double &	b,
		const stan::agrad::var &	c
	)

inline

The fused multiply-add function for two values and a variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double,double,double) is defined in <cmath>.

The derivative is

$\frac{\partial}{\partial z} (c * d) + z = 1$ .

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns: Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 253 of file fma.hpp.

var stan::agrad::fma	(	const double &	a,
		const stan::agrad::var &	b,
		const stan::agrad::var &	c
	)

inline

The fused multiply-add function for a value and two variables (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double,double,double) is defined in <cmath>.

The partial derivaties are

$\frac{\partial}{\partial y} (c * y) + z = c$ , and

$\frac{\partial}{\partial z} (c * y) + z = 1$ .

Parameters

a	First multiplicand.
b	Second multiplicand.
c	Summand.

Returns: Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 278 of file fma.hpp.

template<typename T >

fvar<T> stan::agrad::fmax	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 17 of file fmax.hpp.

template<typename T >

fvar<T> stan::agrad::fmax	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 38 of file fmax.hpp.

template<typename T >

fvar<T> stan::agrad::fmax	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 59 of file fmax.hpp.

var stan::agrad::fmax	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Returns the maximum of the two variable arguments (C99).

See boost::math::fmax() for the double-based version.

No new variable implementations are created, with this function defined as if by

fmax(a,b) = a if a's value is greater than b's, and .

fmax(a,b) = b if b's value is greater than or equal to a's.

$\mbox{fmax}(x,y) = \begin{cases} x & \mbox{if } x \geq y \\ y & \mbox{if } x < y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x,y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fmax}(x,y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \geq y \\ 0 & \mbox{if } x < y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x,y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fmax}(x,y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x,y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: If the first variable's value is larger than the second's, the first variable, otherwise the second variable.

Definition at line 64 of file fmax.hpp.

var stan::agrad::fmax	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

See boost::math::fmax() for the double-based version.

For fmax(a,b), if a's value is greater than b, then a is returned, otherwise a fesh variable implementation wrapping the value b is returned.

Parameters

a	First variable.
b	Second value

Returns: If the first variable's value is larger than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 101 of file fmax.hpp.

var stan::agrad::fmax	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

See boost::math::fmax() for the double-based version.

For fmax(a,b), if a is greater than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters

a	First value.
b	Second variable.

Returns: If the first value is larger than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 132 of file fmax.hpp.

template<typename T >

fvar<T> stan::agrad::fmin	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 17 of file fmin.hpp.

template<typename T >

fvar<T> stan::agrad::fmin	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 38 of file fmin.hpp.

template<typename T >

fvar<T> stan::agrad::fmin	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 59 of file fmin.hpp.

var stan::agrad::fmin	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Returns the minimum of the two variable arguments (C99).

See boost::math::fmin() for the double-based version.

For fmin(a,b), if a's value is less than b's, then a is returned, otherwise b is returned.

$\mbox{fmin}(x,y) = \begin{cases} x & \mbox{if } x \leq y \\ y & \mbox{if } x > y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x,y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fmin}(x,y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \leq y \\ 0 & \mbox{if } x > y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x,y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fmin}(x,y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \leq y \\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x,y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: If the first variable's value is smaller than the second's, the first variable, otherwise the second variable.

Definition at line 60 of file fmin.hpp.

var stan::agrad::fmin	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

See boost::math::fmin() for the double-based version.

For fmin(a,b), if a's value is less than b, then a is returned, otherwise a fresh variable wrapping b is returned.

Parameters

a	First variable.
b	Second value

Returns: If the first variable's value is less than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 95 of file fmin.hpp.

var stan::agrad::fmin	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

See boost::math::fmin() for the double-based version.

For fmin(a,b), if a is less than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters

a	First value.
b	Second variable.

Returns: If the first value is smaller than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 126 of file fmin.hpp.

template<typename T >

fvar<T> stan::agrad::fmod	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 16 of file fmod.hpp.

template<typename T >

fvar<T> stan::agrad::fmod	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 26 of file fmod.hpp.

template<typename T >

fvar<T> stan::agrad::fmod	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 39 of file fmod.hpp.

var stan::agrad::fmod	(	const var &	a,
		const var &	b
	)

inline

Return the floating point remainder after dividing the first variable by the second (cmath).

The partial derivatives with respect to the variables are defined everywhere but where $x = y$ , but we set these to match other values, with

$\frac{\partial}{\partial x} \mbox{fmod}(x,y) = 1$ , and

$\frac{\partial}{\partial y} \mbox{fmod}(x,y) = -\lfloor \frac{x}{y} \rfloor$ .

$\mbox{fmod}(x,y) = \begin{cases} x - \lfloor \frac{x}{y}\rfloor y & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fmod}(x,y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x,y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{fmod}(x,y)}{\partial y} = \begin{cases} -\lfloor \frac{x}{y}\rfloor & \mbox{if } -\infty\leq x,y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: Floating pointer remainder of dividing the first variable by the second.

Definition at line 105 of file fmod.hpp.

var stan::agrad::fmod	(	const var &	a,
		const double	b
	)

inline

Return the floating point remainder after dividing the the first variable by the second scalar (cmath).

The derivative with respect to the variable is

$\frac{d}{d x} \mbox{fmod}(x,c) = \frac{1}{c}$ .

Parameters

a	First variable.
b	Second scalar.

Returns: Floating pointer remainder of dividing the first variable by the second scalar.

Definition at line 122 of file fmod.hpp.

var stan::agrad::fmod	(	const double	a,
		const var &	b
	)

inline

Return the floating point remainder after dividing the first scalar by the second variable (cmath).

The derivative with respect to the variable is

$\frac{d}{d y} \mbox{fmod}(c,y) = -\lfloor \frac{c}{y} \rfloor$ .

Parameters

a	First scalar.
b	Second variable.

Returns: Floating pointer remainder of dividing first scalar by the second variable.

Definition at line 139 of file fmod.hpp.

template<typename T >

fvar<T> stan::agrad::gamma_p	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 16 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::agrad::gamma_p	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 52 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::agrad::gamma_p	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 87 of file gamma_p.hpp.

var stan::agrad::gamma_p	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Definition at line 91 of file gamma_p.hpp.

var stan::agrad::gamma_p	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Definition at line 96 of file gamma_p.hpp.

var stan::agrad::gamma_p	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Definition at line 101 of file gamma_p.hpp.

template<typename T >

fvar<T> stan::agrad::gamma_q	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 15 of file gamma_q.hpp.

template<typename T >

fvar<T> stan::agrad::gamma_q	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 51 of file gamma_q.hpp.

var stan::agrad::gamma_q	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Definition at line 61 of file gamma_q.hpp.

var stan::agrad::gamma_q	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Definition at line 66 of file gamma_q.hpp.

var stan::agrad::gamma_q	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Definition at line 71 of file gamma_q.hpp.

template<typename T >

fvar<T> stan::agrad::gamma_q	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 86 of file gamma_q.hpp.

static void stan::agrad::grad ( chainable * vi )

static

void stan::agrad::grad	(	var &	v,
		Eigen::Matrix< var, Eigen::Dynamic, 1 > &	x,
		Eigen::VectorXd &	g
	)

Propagate chain rule to calculate gradients starting from the specified variable.

Resizes the input vector to be the correct size.

The grad() function does not itself recover any memory. use agrad::recover_memory() or agrad::recover_memory_nested(), defined in , defined in agrad/rev/var_stack.hpp, to recover memory.

Parameters

[in]	v	Value of function being differentiated
[in]	x	Variables being differentiated with respect to
[out]	g	Gradient, d/dx v, evaluated at x.

Definition at line 27 of file grad.hpp.

static void stan::agrad::grad ( chainable * vi )

static

Compute the gradient for all variables starting from the specified root variable implementation.

Does not recover memory. This chainable variable's adjoint is initialized using the method init_dependent() and then the chain rule is applied working down the stack from this chainable and calling each chainable's chain() method in turn.

This function computes a nested gradient only going back as far as the last nesting.

This function does not recover any memory from the computation.

Parameters

vi	Variable implementation for root of partial derivative propagation.

Definition at line 110 of file chainable.hpp.

template<typename F >

void stan::agrad::grad_tr_mat_times_hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	M,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad_tr_MH
	)

Definition at line 321 of file autodiff.hpp.

template<typename F >

void stan::agrad::gradient	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad_fx
	)

Calculate the value and the gradient of the specified function at the specified argument.

The functor must implement

stan::agrad::var operator()(const Eigen::Matrix<stan::agrad::var,Eigen::Dynamic,1>&)

using only operations that are defined for stan::agrad::var. This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Time and memory usage is on the order of the size of the fully unfolded expression for the function applied to the argument, independently of dimension.

Template Parameters

F	Type of function

Parameters

[in]	f	Function
[in]	x	Argument to function
[out]	fx	Function applied to argument
[out]	grad_fx	Gradient of function at argument

Definition at line 93 of file autodiff.hpp.

template<typename T , typename F >

void stan::agrad::gradient	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	grad_fx
	)

Definition at line 117 of file autodiff.hpp.

template<typename T1 , typename T2 , typename F >

void stan::agrad::gradient_dot_vector	(	const F &	f,
		const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &	v,
		T1 &	fx,
		T1 &	grad_fx_dot_v
	)

Definition at line 252 of file autodiff.hpp.

template<typename F >

void stan::agrad::hessian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	grad,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	H
	)

Definition at line 194 of file autodiff.hpp.

template<typename T , typename F >

void stan::agrad::hessian	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	grad,
		Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	H
	)

Definition at line 223 of file autodiff.hpp.

template<typename F >

void stan::agrad::hessian_times_vector	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	v,
		double &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	Hv
	)

Definition at line 274 of file autodiff.hpp.

template<typename T , typename F >

void stan::agrad::hessian_times_vector	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	v,
		T &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	Hv
	)

Definition at line 304 of file autodiff.hpp.

template<typename T >

fvar<T> stan::agrad::hypot	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 16 of file hypot.hpp.

template<typename T >

fvar<T> stan::agrad::hypot	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 27 of file hypot.hpp.

template<typename T >

fvar<T> stan::agrad::hypot	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 38 of file hypot.hpp.

var stan::agrad::hypot	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

See boost::math::hypot() for double-based function.

The partial derivatives are given by

$\frac{\partial}{\partial x} \sqrt{x^2 + y^2} = \frac{x}{\sqrt{x^2 + y^2}}$ , and

$\frac{\partial}{\partial y} \sqrt{x^2 + y^2} = \frac{y}{\sqrt{x^2 + y^2}}$ .

Parameters

a	Length of first side.
b	Length of second side.

Returns: Length of hypoteneuse.

Definition at line 54 of file hypot.hpp.

var stan::agrad::hypot	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

See boost::math::hypot() for double-based function.

The derivative is

$\frac{d}{d x} \sqrt{x^2 + c^2} = \frac{x}{\sqrt{x^2 + c^2}}$ .

Parameters

a	Length of first side.
b	Length of second side.

Returns: Length of hypoteneuse.

Definition at line 73 of file hypot.hpp.

var stan::agrad::hypot	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

See boost::math::hypot() for double-based function.

The derivative is

$\frac{d}{d y} \sqrt{c^2 + y^2} = \frac{y}{\sqrt{c^2 + y^2}}$ .

$\mbox{hypot}(x,y) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \sqrt{x^2+y^2} & \mbox{if } x,y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{hypot}(x,y)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{x}{\sqrt{x^2+y^2}} & \mbox{if } x,y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{hypot}(x,y)}{\partial y} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{y}{\sqrt{x^2+y^2}} & \mbox{if } x,y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	Length of first side.
b	Length of second side.

Returns: Length of hypoteneuse.

Definition at line 119 of file hypot.hpp.

var stan::agrad::ibeta	(	const var &	a,
		const var &	b,
		const var &	x
	)

inline

The normalized incomplete beta function of a, b, and x.

Used to compute the cumulative density function for the beta distribution.

Partial derivatives are those specified by wolfram alpha. The values were checked using both finite differences and by independent code for calculating the derivatives found in JSS (paper by Boik and Robison-Cox).

Parameters

a	Shape parameter.
b	Shape parameter.
x	Random variate.

Returns: The normalized incomplete beta function.

Exceptions

if	any argument is NaN.

Definition at line 223 of file ibeta.hpp.

var stan::agrad::if_else	(	bool	c,
		const var &	y_true,
		const var &	y_false
	)

inline

If the specified condition is true, return the first variable, otherwise return the second variable.

Parameters

c	Boolean condition.
y_true	Variable to return if condition is true.
y_false	Variable to return if condition is false.

Definition at line 17 of file if_else.hpp.

var stan::agrad::if_else	(	bool	c,
		double	y_true,
		const var &	y_false
	)

inline

If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable.

Parameters

c	Boolean condition.
y_true	Value to promote to variable and return if condition is true.
y_false	Variable to return if condition is false.

Definition at line 29 of file if_else.hpp.

var stan::agrad::if_else	(	bool	c,
		const var &	y_true,
		const double	y_false
	)

inline

If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar.

Parameters

c	Boolean condition.
y_true	Variable to return if condition is true.
y_false	Value to promote to variable and return if condition is false.

Definition at line 44 of file if_else.hpp.

void stan::agrad::initialize_variable	(	var &	variable,
		const var &	value
	)

inline

Initialize variable to value.

(Function may look pointless, but its needed to bottom out recursion.)

Definition at line 15 of file initialize_variable.hpp.

template<int R, int C>

void stan::agrad::initialize_variable	(	Eigen::Matrix< var, R, C > &	matrix,
		const var &	value
	)

inline

Initialize every cell in the matrix to the specified value.

Definition at line 24 of file initialize_variable.hpp.

template<typename T >

void stan::agrad::initialize_variable	(	std::vector< T > &	variables,
		const var &	value
	)

inline

Initialize the variables in the standard vector recursively.

Definition at line 33 of file initialize_variable.hpp.

template<typename T >

fvar<T> stan::agrad::inv ( const fvar< T > & x )

inline

Definition at line 15 of file inv.hpp.

var stan::agrad::inv ( const var & a )

inline

$\mbox{inv}(x) = \begin{cases} \frac{1}{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{inv}(x)}{\partial x} = \begin{cases} -\frac{1}{x^2} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Definition at line 43 of file inv.hpp.

template<typename T >

fvar<T> stan::agrad::inv_cloglog ( const fvar< T > & x )

inline

Definition at line 15 of file inv_cloglog.hpp.

var stan::agrad::inv_cloglog ( const stan::agrad::var & a )

inline

Return the inverse complementary log-log function applied specified variable (stan).

See stan::math::inv_cloglog() for the double-based version.

The derivative is given by

$\frac{d}{dx} \mbox{cloglog}^{-1}(x) = \exp (x - \exp (x))$ .

Parameters

a	Variable argument.

Returns: The inverse complementary log-log of the specified argument.

Definition at line 37 of file inv_cloglog.hpp.

template<typename T >

fvar<T> stan::agrad::inv_logit ( const fvar< T > & x )

inline

Definition at line 15 of file inv_logit.hpp.

var stan::agrad::inv_logit ( const stan::agrad::var & a )

inline

The inverse logit function for variables (stan).

See stan::math::inv_logit() for the double-based version.

The derivative of inverse logit is

$\frac{d}{dx} \mbox{logit}^{-1}(x) = \mbox{logit}^{-1}(x) (1 - \mbox{logit}^{-1}(x))$ .

Parameters

a	Argument variable.

Returns: Inverse logit of argument.

Definition at line 35 of file inv_logit.hpp.

template<typename T >

fvar<T> stan::agrad::inv_sqrt ( const fvar< T > & x )

inline

Definition at line 15 of file inv_sqrt.hpp.

var stan::agrad::inv_sqrt ( const var & a )

inline

$\mbox{inv\_sqrt}(x) = \begin{cases} \frac{1}{\sqrt{x}} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{inv\_sqrt}(x)}{\partial x} = \begin{cases} -\frac{1}{2\sqrt{x^3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Definition at line 43 of file inv_sqrt.hpp.

template<typename T >

fvar<T> stan::agrad::inv_square ( const fvar< T > & x )

inline

Definition at line 15 of file inv_square.hpp.

var stan::agrad::inv_square ( const var & a )

inline

$\mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{inv\_square}(x)}{\partial x} = \begin{cases} -\frac{2}{x^3} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Definition at line 43 of file inv_square.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::inverse ( const Eigen::Matrix< fvar< T >, R, C > & m )

inline

Definition at line 21 of file inverse.hpp.

template<typename T >

int stan::agrad::is_inf ( const fvar< T > & x )

inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to stan::math::is_inf.

Parameters

x	Value to test.

Returns: 1 if the value is infinite and 0 otherwise.

Definition at line 22 of file is_inf.hpp.

int stan::agrad::is_inf ( const var & v )

inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to stan::math::is_inf.

Parameters

x	Value to test.

Returns: 1 if the value is infinite and 0 otherwise.

Definition at line 23 of file is_inf.hpp.

template<typename T >

int stan::agrad::is_nan ( const fvar< T > & x )

inline

Returns 1 if the input's value is NaN and 0 otherwise.

Delegates to stan::math::is_nan.

Parameters

x	Value to test.

Returns: 1 if the value is NaN and 0 otherwise.

Definition at line 22 of file is_nan.hpp.

int stan::agrad::is_nan ( const var & v )

inline

Returns 1 if the input's value is NaN and 0 otherwise.

Delegates to stan::math::is_nan.

Parameters

x	Value to test.

Returns: 1 if the value is NaN and 0 otherwise.

Definition at line 23 of file is_nan.hpp.

bool stan::agrad::is_uninitialized ( var x )

inline

Returns true if the specified variable is uninitialized.

This overload of the stan::math::is_uninitialized() function delegates the return to the is_uninitialized() method on the specified variable.

Parameters

x	Object to test.

Returns: true if the specified object is uninitialized.

Definition at line 25 of file is_uninitialized.hpp.

template<typename F >

void stan::agrad::jacobian	(	const F &	f,
		const Eigen::Matrix< double, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< double, Eigen::Dynamic, 1 > &	fx,
		Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &	J
	)

Definition at line 135 of file autodiff.hpp.

template<typename T , typename F >

void stan::agrad::jacobian	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		Eigen::Matrix< T, Eigen::Dynamic, 1 > &	fx,
		Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &	J
	)

Definition at line 166 of file autodiff.hpp.

template<typename T >

fvar<T> stan::agrad::lbeta	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 16 of file lbeta.hpp.

template<typename T >

fvar<T> stan::agrad::lbeta	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 28 of file lbeta.hpp.

template<typename T >

fvar<T> stan::agrad::lbeta	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 38 of file lbeta.hpp.

template<typename T >

fvar<T> stan::agrad::lgamma ( const fvar< T > & x )

inline

Definition at line 15 of file lgamma.hpp.

var stan::agrad::lgamma ( const stan::agrad::var & a )

inline

The log gamma function for variables (C99).

The derivatie is the digamma function,

$\frac{d}{dx} \Gamma(x) = \psi^{(0)}(x)$ .

Parameters

a	The variable.

Returns: Log gamma of the variable.

Definition at line 36 of file lgamma.hpp.

template<typename T >

fvar<typename stan::return_type<T,int>::type> stan::agrad::lmgamma	(	int	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 16 of file lmgamma.hpp.

var stan::agrad::lmgamma	(	int	a,
		const stan::agrad::var &	b
	)

inline

Definition at line 29 of file lmgamma.hpp.

template<typename T >

fvar<T> stan::agrad::log ( const fvar< T > & x )

inline

Definition at line 15 of file log.hpp.

var stan::agrad::log ( const var & a )

inline

Return the natural log of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \log x = \frac{1}{x}$ .

$\mbox{log}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \ln(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{log}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable whose log is taken.

Returns: Natural log of variable.

Definition at line 51 of file log.hpp.

template<typename T >

fvar<T> stan::agrad::log10 ( const fvar< T > & x )

inline

Definition at line 15 of file log10.hpp.

var stan::agrad::log10 ( const var & a )

inline

Return the base 10 log of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \log_{10} x = \frac{1}{x \log 10}$ .

$\mbox{log10}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \log_{10}(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{log10}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x \ln10} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable whose log is taken.

Returns: Base 10 log of variable.

Definition at line 55 of file log10.hpp.

template<typename T >

fvar<T> stan::agrad::log1m ( const fvar< T > & x )

inline

Definition at line 16 of file log1m.hpp.

var stan::agrad::log1m ( const stan::agrad::var & a )

inline

The log (1 - x) function for variables.

The derivative is given by

$\frac{d}{dx} \log (1 - x) = -\frac{1}{1 - x}$ .

Parameters

a	The variable.

Returns: The variable representing log of 1 minus the variable.

Definition at line 33 of file log1m.hpp.

template<typename T >

fvar<T> stan::agrad::log1m_exp ( const fvar< T > & x )

inline

Definition at line 17 of file log1m_exp.hpp.

var stan::agrad::log1m_exp ( const stan::agrad::var & a )

inline

Return the log of 1 minus the exponential of the specified variable.

Definition at line 31 of file log1m_exp.hpp.

template<typename T >

fvar<T> stan::agrad::log1m_inv_logit ( const fvar< T > & x )

inline

Definition at line 15 of file log1m_inv_logit.hpp.

template<typename T >

fvar<T> stan::agrad::log1p ( const fvar< T > & x )

inline

Definition at line 16 of file log1p.hpp.

var stan::agrad::log1p ( const stan::agrad::var & a )

inline

The log (1 + x) function for variables (C99).

The derivative is given by

$\frac{d}{dx} \log (1 + x) = \frac{1}{1 + x}$ .

Parameters

a	The variable.

Returns: The log of 1 plus the variable.

Definition at line 35 of file log1p.hpp.

template<typename T >

fvar<T> stan::agrad::log1p_exp ( const fvar< T > & x )

inline

Definition at line 15 of file log1p_exp.hpp.

var stan::agrad::log1p_exp ( const stan::agrad::var & a )

inline

Return the log of 1 plus the exponential of the specified variable.

Definition at line 29 of file log1p_exp.hpp.

template<typename T >

fvar<T> stan::agrad::log2 ( const fvar< T > & x )

inline

Definition at line 17 of file log2.hpp.

var stan::agrad::log2 ( const stan::agrad::var & a )

inline

Returns the base 2 logarithm of the specified variable (C99).

See stan::math::log2() for the double-based version.

The derivative is

$\frac{d}{dx} \log_2 x = \frac{1}{x \log 2}$ .

$\mbox{log2}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \log_2(x) & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{log2}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \frac{1}{x\ln2} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Specified variable.

Returns: Base 2 logarithm of the variable.

Definition at line 54 of file log2.hpp.

template<typename T , int R, int C>

fvar<T> stan::agrad::log_determinant ( const Eigen::Matrix< fvar< T >, R, C > & m )

inline

Definition at line 20 of file log_determinant.hpp.

template<int R, int C>

var stan::agrad::log_determinant ( const Eigen::Matrix< var, R, C > & m )

inline

Definition at line 69 of file log_determinant.hpp.

template<int R, int C>

var stan::agrad::log_determinant_ldlt ( stan::math::LDLT_factor< var, R, C > & A )

Definition at line 48 of file log_determinant_ldlt.hpp.

template<int R, int C>

var stan::agrad::log_determinant_spd ( const Eigen::Matrix< var, R, C > & m )

inline

Definition at line 116 of file log_determinant_spd.hpp.

template<typename T >

fvar<T> stan::agrad::log_diff_exp	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 14 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >

fvar<T2> stan::agrad::log_diff_exp	(	const T1 &	x1,
		const fvar< T2 > &	x2
	)

inline

Definition at line 25 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >

fvar<T1> stan::agrad::log_diff_exp	(	const fvar< T1 > &	x1,
		const T2 &	x2
	)

inline

Definition at line 36 of file log_diff_exp.hpp.

var stan::agrad::log_diff_exp	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Returns the log sum of exponentials.

Definition at line 55 of file log_diff_exp.hpp.

var stan::agrad::log_diff_exp	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Returns the log sum of exponentials.

Definition at line 62 of file log_diff_exp.hpp.

var stan::agrad::log_diff_exp	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Returns the log sum of exponentials.

Definition at line 69 of file log_diff_exp.hpp.

template<typename T >

fvar<T> stan::agrad::log_falling_factorial	(	const fvar< T > &	x,
		const fvar< T > &	n
	)

inline

Definition at line 15 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::log_falling_factorial	(	const double	x,
		const fvar< T > &	n
	)

inline

Definition at line 25 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::log_falling_factorial	(	const fvar< T > &	x,
		const double	n
	)

inline

Definition at line 35 of file log_falling_factorial.hpp.

var stan::agrad::log_falling_factorial	(	const var &	a,
		const double &	b
	)

inline

Definition at line 63 of file log_falling_factorial.hpp.

var stan::agrad::log_falling_factorial	(	const var &	a,
		const var &	b
	)

inline

Definition at line 68 of file log_falling_factorial.hpp.

var stan::agrad::log_falling_factorial	(	const double &	a,
		const var &	b
	)

inline

Definition at line 73 of file log_falling_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::log_inv_logit ( const fvar< T > & x )

inline

Definition at line 15 of file log_inv_logit.hpp.

template<typename T >

fvar<T> stan::agrad::log_rising_factorial	(	const fvar< T > &	x,
		const fvar< T > &	n
	)

inline

Definition at line 16 of file log_rising_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::log_rising_factorial	(	const fvar< T > &	x,
		const double	n
	)

inline

Definition at line 27 of file log_rising_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::log_rising_factorial	(	const double	x,
		const fvar< T > &	n
	)

inline

Definition at line 39 of file log_rising_factorial.hpp.

var stan::agrad::log_rising_factorial	(	const var &	a,
		const double &	b
	)

inline

Definition at line 49 of file log_rising_factorial.hpp.

var stan::agrad::log_rising_factorial	(	const var &	a,
		const var &	b
	)

inline

Definition at line 54 of file log_rising_factorial.hpp.

var stan::agrad::log_rising_factorial	(	const double &	a,
		const var &	b
	)

inline

Definition at line 59 of file log_rising_factorial.hpp.

template<typename T >

Eigen::Matrix<fvar<T>,Eigen::Dynamic,1> stan::agrad::log_softmax ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > & alpha )

inline

Definition at line 16 of file log_softmax.hpp.

Eigen::Matrix<var,Eigen::Dynamic,1> stan::agrad::log_softmax ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > & alpha )

inline

Return the softmax of the specified Eigen vector.

Softmax is guaranteed to return a simplex.

The gradient calculations are unfolded.

Parameters

alpha Unconstrained input vector.

Returns: Softmax of the input.

Exceptions

std::domain_error If the input vector is size 0.

Definition at line 60 of file log_softmax.hpp.

template<typename T >

fvar<T> stan::agrad::log_sum_exp	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 15 of file log_sum_exp.hpp.

template<typename T , int R, int C>

fvar<T> stan::agrad::log_sum_exp ( const Eigen::Matrix< fvar< T >, R, C > & v )

Definition at line 19 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::agrad::log_sum_exp	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 26 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::agrad::log_sum_exp	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 36 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::agrad::log_sum_exp ( const std::vector< fvar< T > > & v )

Definition at line 45 of file log_sum_exp.hpp.

template<int R, int C>

var stan::agrad::log_sum_exp ( const Eigen::Matrix< var, R, C > & x )

inline

Returns the log sum of exponentials.

Parameters

x matrix

Definition at line 59 of file log_sum_exp.hpp.

var stan::agrad::log_sum_exp	(	const stan::agrad::var &	a,
		const stan::agrad::var &	b
	)

inline

Returns the log sum of exponentials.

Definition at line 81 of file log_sum_exp.hpp.

var stan::agrad::log_sum_exp	(	const stan::agrad::var &	a,
		const double &	b
	)

inline

Returns the log sum of exponentials.

Definition at line 88 of file log_sum_exp.hpp.

var stan::agrad::log_sum_exp	(	const double &	a,
		const stan::agrad::var &	b
	)

inline

Returns the log sum of exponentials.

Definition at line 95 of file log_sum_exp.hpp.

var stan::agrad::log_sum_exp ( const std::vector< var > & x )

inline

Returns the log sum of exponentials.

Definition at line 102 of file log_sum_exp.hpp.

template<typename T >

fvar<T> stan::agrad::logit ( const fvar< T > & x )

inline

Definition at line 17 of file logit.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_left	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 25 of file mdivide_left.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_left	(	const Eigen::Matrix< double, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 69 of file mdivide_left.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_left	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Definition at line 95 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left	(	const Eigen::Matrix< var, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Definition at line 264 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left	(	const Eigen::Matrix< var, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Definition at line 288 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left	(	const Eigen::Matrix< double, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Definition at line 312 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2, typename T2 >

Eigen::Matrix<fvar<T2>,R1,C2> stan::agrad::mdivide_left_ldlt	(	const stan::math::LDLT_factor< double, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T2 >, R2, C2 > &	b
	)

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns: x = b A^-1, solution of the linear system.

Exceptions

std::domain_error if rows of b don't match the size of A.

Definition at line 25 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_ldlt	(	const stan::math::LDLT_factor< var, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns: x = b A^-1, solution of the linear system.

Exceptions

std::domain_error if rows of b don't match the size of A.

Definition at line 243 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_ldlt	(	const stan::math::LDLT_factor< var, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns: x = b A^-1, solution of the linear system.

Exceptions

std::domain_error if rows of b don't match the size of A.

Definition at line 269 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_ldlt	(	const stan::math::LDLT_factor< double, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters

A	LDLT_factor
b	Right hand side matrix or vector.

Returns: x = b A^-1, solution of the linear system.

Exceptions

std::domain_error if rows of b don't match the size of A.

Definition at line 295 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_spd	(	const Eigen::Matrix< var, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Definition at line 244 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_spd	(	const Eigen::Matrix< var, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Definition at line 268 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_spd	(	const Eigen::Matrix< double, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Definition at line 292 of file mdivide_left_spd.hpp.

template<int TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_tri	(	const Eigen::Matrix< var, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Definition at line 296 of file mdivide_left_tri.hpp.

template<int TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_tri	(	const Eigen::Matrix< double, R1, C1 > &	A,
		const Eigen::Matrix< var, R2, C2 > &	b
	)

inline

Definition at line 319 of file mdivide_left_tri.hpp.

template<int TriView, int R1, int C1, int R2, int C2>

Eigen::Matrix<var,R1,C2> stan::agrad::mdivide_left_tri	(	const Eigen::Matrix< var, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Definition at line 342 of file mdivide_left_tri.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::mdivide_left_tri_low	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 22 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::mdivide_left_tri_low	(	const Eigen::Matrix< double, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 67 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::mdivide_left_tri_low	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Definition at line 107 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_right	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 26 of file mdivide_right.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_right	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Definition at line 70 of file mdivide_right.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_right	(	const Eigen::Matrix< double, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 97 of file mdivide_right.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::mdivide_right_tri_low	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 22 of file mdivide_right_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_right_tri_low	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	A,
		const Eigen::Matrix< double, R2, C2 > &	b
	)

inline

Definition at line 67 of file mdivide_right_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::mdivide_right_tri_low	(	const Eigen::Matrix< double, R1, C1 > &	A,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	b
	)

inline

Definition at line 102 of file mdivide_right_tri_low.hpp.

template<typename T >

fvar<T> stan::agrad::modified_bessel_first_kind	(	int	v,
		const fvar< T > &	z
	)

inline

Definition at line 15 of file modified_bessel_first_kind.hpp.

var stan::agrad::modified_bessel_first_kind	(	const int &	v,
		const var &	a
	)

inline

Definition at line 24 of file modified_bessel_first_kind.hpp.

template<typename T >

fvar<T> stan::agrad::modified_bessel_second_kind	(	int	v,
		const fvar< T > &	z
	)

inline

Definition at line 15 of file modified_bessel_second_kind.hpp.

var stan::agrad::modified_bessel_second_kind	(	const int &	v,
		const var &	a
	)

inline

Definition at line 24 of file modified_bessel_second_kind.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::multiply	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	m,
		const fvar< T > &	c
	)

inline

Definition at line 21 of file multiply.hpp.

template<typename T1 , typename T2 >

boost::enable_if_c< (boost::is_scalar<T1>::value \|\| boost::is_same<T1,var>::value) && (boost::is_scalar<T2>::value \|\| boost::is_same<T2,var>::value), typename boost::math::tools::promote_args<T1,T2>::type>::type stan::agrad::multiply	(	const T1 &	v,
		const T2 &	c
	)

inline

Return the product of two scalars.

Parameters

[in]	v	First scalar.
[in]	c	Specified scalar.

Returns: Product of scalars.

Definition at line 31 of file multiply.hpp.

template<typename T , int R2, int C2>

Eigen::Matrix<fvar<T>, R2, C2> stan::agrad::multiply	(	const Eigen::Matrix< fvar< T >, R2, C2 > &	m,
		const double	c
	)

inline

Definition at line 33 of file multiply.hpp.

template<typename T1 , typename T2 , int R2, int C2>

Eigen::Matrix<var,R2,C2> stan::agrad::multiply	(	const T1 &	c,
		const Eigen::Matrix< T2, R2, C2 > &	m
	)

inline

Return the product of scalar and matrix.

Parameters

[in]	c	Specified scalar.
[in]	m	Matrix.

Returns: Product of scalar and matrix.

Definition at line 42 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::multiply	(	const Eigen::Matrix< double, R1, C1 > &	m,
		const fvar< T > &	c
	)

inline

Definition at line 45 of file multiply.hpp.

template<typename T1 , int R1, int C1, typename T2 >

Eigen::Matrix<var,R1,C1> stan::agrad::multiply	(	const Eigen::Matrix< T1, R1, C1 > &	m,
		const T2 &	c
	)

inline

Return the product of scalar and matrix.

Parameters

[in]	m	Matrix.
[in]	c	Specified scalar.

Returns: Product of scalar and matrix.

Definition at line 56 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::multiply	(	const fvar< T > &	c,
		const Eigen::Matrix< fvar< T >, R1, C1 > &	m
	)

inline

Definition at line 57 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::multiply	(	const double	c,
		const Eigen::Matrix< fvar< T >, R1, C1 > &	m
	)

inline

Definition at line 64 of file multiply.hpp.

template<typename T , int R1, int C1>

Eigen::Matrix<fvar<T>,R1,C1> stan::agrad::multiply	(	const fvar< T > &	c,
		const Eigen::Matrix< double, R1, C1 > &	m
	)

inline

Definition at line 71 of file multiply.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>

boost::enable_if_c< boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value, Eigen::Matrix<var,R1,C2> >::type stan::agrad::multiply	(	const Eigen::Matrix< T1, R1, C1 > &	m1,
		const Eigen::Matrix< T2, R2, C2 > &	m2
	)

inline

Return the product of the specified matrices.

The number of columns in the first matrix must be the same as the number of rows in the second matrix.

Parameters

[in]	m1	First matrix.
[in]	m2	Second matrix.

Returns: The product of the first and second matrices.

Exceptions

std::domain_error if the number of columns of m1 does not match the number of rows of m2.

Definition at line 76 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::multiply	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	m1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	m2
	)

inline

Definition at line 78 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::multiply	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	m1,
		const Eigen::Matrix< double, R2, C2 > &	m2
	)

inline

Definition at line 96 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>,R1,C2> stan::agrad::multiply	(	const Eigen::Matrix< double, R1, C1 > &	m1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	m2
	)

inline

Definition at line 114 of file multiply.hpp.

template<typename T1 , int C1, typename T2 , int R2>

boost::enable_if_c< boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value, var >::type stan::agrad::multiply	(	const Eigen::Matrix< T1, 1, C1 > &	rv,
		const Eigen::Matrix< T2, R2, 1 > &	v
	)

inline

Return the scalar product of the specified row vector and specified column vector.

The return is the same as the dot product. The two vectors must be the same size.

Parameters

[in]	rv	Row vector.
[in]	v	Column vector.

Returns: Scalar result of multiplying row vector by column vector.

Exceptions

std::domain_error if rv and v are not the same size

Definition at line 123 of file multiply.hpp.

template<typename T , int C1, int R2>

fvar<T> stan::agrad::multiply	(	const Eigen::Matrix< fvar< T >, 1, C1 > &	rv,
		const Eigen::Matrix< fvar< T >, R2, 1 > &	v
	)

inline

Definition at line 132 of file multiply.hpp.

template<typename T , int C1, int R2>

fvar<T> stan::agrad::multiply	(	const Eigen::Matrix< fvar< T >, 1, C1 > &	rv,
		const Eigen::Matrix< double, R2, 1 > &	v
	)

inline

Definition at line 142 of file multiply.hpp.

template<typename T , int C1, int R2>

fvar<T> stan::agrad::multiply	(	const Eigen::Matrix< double, 1, C1 > &	rv,
		const Eigen::Matrix< fvar< T >, R2, 1 > &	v
	)

inline

Definition at line 152 of file multiply.hpp.

template<typename T >

fvar<T> stan::agrad::multiply_log	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 15 of file multiply_log.hpp.

template<typename T >

fvar<T> stan::agrad::multiply_log	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 25 of file multiply_log.hpp.

template<typename T >

fvar<T> stan::agrad::multiply_log	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 35 of file multiply_log.hpp.

var stan::agrad::multiply_log	(	const var &	a,
		const var &	b
	)

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to a is log(b). The partial deriviative with respect to b is a/b. When a and b are both 0, this is set to Inf.

Parameters

a	First variable.
b	Second variable.

Returns: Value of a*log(b)

Definition at line 77 of file multiply_log.hpp.

var stan::agrad::multiply_log	(	const var &	a,
		const double	b
	)

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to a is log(b).

Parameters

a	First variable.
b	Second scalar.

Returns: Value of a*log(b)

Definition at line 90 of file multiply_log.hpp.

var stan::agrad::multiply_log	(	const double	a,
		const var &	b
	)

inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to b is a/b. When a and b are both 0, this is set to Inf.

Parameters

a	First scalar.
b	Second variable.

Returns: Value of a*log(b)

Definition at line 104 of file multiply_log.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,R> stan::agrad::multiply_lower_tri_self_transpose ( const Eigen::Matrix< fvar< T >, R, C > & m )

inline

Definition at line 17 of file multiply_lower_tri_self_transpose.hpp.

matrix_v stan::agrad::multiply_lower_tri_self_transpose ( const matrix_v & L )

inline

Definition at line 19 of file multiply_lower_tri_self_transpose.hpp.

static size_t stan::agrad::nested_size ( )

inlinestatic

Definition at line 103 of file var_stack.hpp.

bool stan::agrad::operator! ( const var & a )

inline

Prefix logical negation for the value of variables (C++).

The expression (!a) is equivalent to negating the scalar value of the variable a.

Note that this is the only logical operator defined for variables. Overridden logical conjunction (&&) and disjunction (||) operators do not apply the same "short circuit" rules as the built-in logical operators.

$\mbox{operator!}(x) = \begin{cases} 0 & \mbox{if } x \neq 0 \\ 1 & \mbox{if } x = 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable to negate.

Returns: True if variable is non-zero.

Definition at line 31 of file operator_unary_not.hpp.

template<typename T >

bool stan::agrad::operator!=	(	const fvar< T > &	x,
		const fvar< T > &	y
	)

inline

Definition at line 14 of file operator_not_equal.hpp.

template<typename T >

bool stan::agrad::operator!=	(	const fvar< T > &	x,
		double	y
	)

inline

Definition at line 21 of file operator_not_equal.hpp.

bool stan::agrad::operator!=	(	const var &	a,
		const var &	b
	)

inline

Inequality operator comparing two variables' values (C++).

$\mbox{operator!=}(x,y) = \begin{cases} 0 & \mbox{if } x = y\\ 1 & \mbox{if } x \neq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: True if the first variable's value is not the same as the second's.

Definition at line 26 of file operator_not_equal.hpp.

template<typename T >

bool stan::agrad::operator!=	(	double	x,
		const fvar< T > &	y
	)

inline

Definition at line 28 of file operator_not_equal.hpp.

bool stan::agrad::operator!=	(	const var &	a,
		const double	b
	)

inline

Inequality operator comparing a variable's value and a double (C++).

Parameters

a	First variable.
b	Second value.

Returns: True if the first variable's value is not the same as the second value.

Definition at line 39 of file operator_not_equal.hpp.

bool stan::agrad::operator!=	(	const double	a,
		const var &	b
	)

inline

Inequality operator comparing a double and a variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns: True if the first value is not the same as the second variable's value.

Definition at line 52 of file operator_not_equal.hpp.

template<typename T >

fvar<T> stan::agrad::operator*	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 14 of file operator_multiplication.hpp.

template<typename T >

fvar<T> stan::agrad::operator*	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 22 of file operator_multiplication.hpp.

template<typename T >

fvar<T> stan::agrad::operator*	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 29 of file operator_multiplication.hpp.

var stan::agrad::operator*	(	const var &	a,
		const var &	b
	)

inline

Multiplication operator for two variables (C++).

The partial derivatives are

$\frac{\partial}{\partial x} (x * y) = y$ , and

$\frac{\partial}{\partial y} (x * y) = x$ .

$\mbox{operator*}(x,y) = \begin{cases} xy & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator*}(x,y)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator*}(x,y)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable operand.
b	Second variable operand.

Returns: Variable result of multiplying operands.

Definition at line 82 of file operator_multiplication.hpp.

var stan::agrad::operator*	(	const var &	a,
		const double	b
	)

inline

Multiplication operator for a variable and a scalar (C++).

The partial derivative for the variable is

$\frac{\partial}{\partial x} (x * c) = c$ , and

Parameters

a	Variable operand.
b	Scalar operand.

Returns: Variable result of multiplying operands.

Definition at line 97 of file operator_multiplication.hpp.

var stan::agrad::operator*	(	const double	a,
		const var &	b
	)

inline

Multiplication operator for a scalar and a variable (C++).

The partial derivative for the variable is

$\frac{\partial}{\partial y} (c * y) = c$ .

Parameters

a	Scalar operand.
b	Variable operand.

Returns: Variable result of multiplying the operands.

Definition at line 114 of file operator_multiplication.hpp.

template<typename T >

fvar<T> stan::agrad::operator+	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 14 of file operator_addition.hpp.

template<typename T >

fvar<T> stan::agrad::operator+	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 21 of file operator_addition.hpp.

template<typename T >

fvar<T> stan::agrad::operator+	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 28 of file operator_addition.hpp.

var stan::agrad::operator+ ( const var & a )

inline

Unary plus operator for variables (C++).

The function simply returns its input, because

$\frac{d}{dx} +x = \frac{d}{dx} x = 1$ .

The effect of unary plus on a built-in C++ scalar type is integer promotion. Because variables are all double-precision floating point already, promotion is not necessary.

$\mbox{operator+}(x) = \begin{cases} x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator+}(x)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Argument variable.

Returns: The input reference.

Definition at line 43 of file operator_unary_plus.hpp.

var stan::agrad::operator+	(	const var &	a,
		const var &	b
	)

inline

Addition operator for variables (C++).

The partial derivatives are defined by

$\frac{\partial}{\partial x} (x+y) = 1$ , and

$\frac{\partial}{\partial y} (x+y) = 1$ .

$\mbox{operator+}(x,y) = \begin{cases} x+y & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator+}(x,y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator+}(x,y)}{\partial y} = \begin{cases} 1 & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable operand.
b	Second variable operand.

Returns: Variable result of adding two variables.

Definition at line 83 of file operator_addition.hpp.

var stan::agrad::operator+	(	const var &	a,
		const double	b
	)

inline

Addition operator for variable and scalar (C++).

The derivative with respect to the variable is

$\frac{d}{dx} (x + c) = 1$ .

Parameters

a	First variable operand.
b	Second scalar operand.

Returns: Result of adding variable and scalar.

Definition at line 98 of file operator_addition.hpp.

var stan::agrad::operator+	(	const double	a,
		const var &	b
	)

inline

Addition operator for scalar and variable (C++).

The derivative with respect to the variable is

$\frac{d}{dy} (c + y) = 1$ .

Parameters

a	First scalar operand.
b	Second variable operand.

Returns: Result of adding variable and scalar.

Definition at line 115 of file operator_addition.hpp.

var& stan::agrad::operator++ ( var & a )

inline

Prefix increment operator for variables (C++).

Following C++, (++a) is defined to behave exactly as (a = a + 1.0) does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters

a	Variable to increment.

Returns: Reference the result of incrementing this input variable.

Definition at line 35 of file operator_unary_increment.hpp.

var stan::agrad::operator++	(	var &	a,
		int
	)

inline

Postfix increment operator for variables (C++).

Following C++, the expression (a++) is defined to behave like the sequence of operations

var temp = a; a = a + 1.0; return temp;

Parameters

a	Variable to increment.

Returns: Input variable.

Definition at line 51 of file operator_unary_increment.hpp.

template<typename T >

fvar<T> stan::agrad::operator-	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 14 of file operator_subtraction.hpp.

template<typename T >

fvar<T> stan::agrad::operator- ( const fvar< T > & x )

inline

Definition at line 14 of file operator_unary_minus.hpp.

template<typename T >

fvar<T> stan::agrad::operator-	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 21 of file operator_subtraction.hpp.

template<typename T >

fvar<T> stan::agrad::operator-	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 28 of file operator_subtraction.hpp.

var stan::agrad::operator- ( const var & a )

inline

Unary negation operator for variables (C++).

$\frac{d}{dx} -x = -1$ .

$\mbox{operator-}(x) = \begin{cases} -x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator-}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Argument variable.

Returns: Negation of variable.

Definition at line 50 of file operator_unary_negative.hpp.

var stan::agrad::operator-	(	const var &	a,
		const var &	b
	)

inline

Subtraction operator for variables (C++).

The partial derivatives are defined by

$\frac{\partial}{\partial x} (x-y) = 1$ , and

$\frac{\partial}{\partial y} (x-y) = -1$ .

$\mbox{operator-}(x,y) = \begin{cases} x-y & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator-}(x,y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator-}(x,y)}{\partial y} = \begin{cases} -1 & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable operand.
b	Second variable operand.

Returns: Variable result of subtracting the second variable from the first.

Definition at line 98 of file operator_subtraction.hpp.

var stan::agrad::operator-	(	const var &	a,
		const double	b
	)

inline

Subtraction operator for variable and scalar (C++).

The derivative for the variable is

$\frac{\partial}{\partial x} (x-c) = 1$ , and

Parameters

a	First variable operand.
b	Second scalar operand.

Returns: Result of subtracting the scalar from the variable.

Definition at line 113 of file operator_subtraction.hpp.

var stan::agrad::operator-	(	const double	a,
		const var &	b
	)

inline

Subtraction operator for scalar and variable (C++).

The derivative for the variable is

$\frac{\partial}{\partial y} (c-y) = -1$ , and

Parameters

a	First scalar operand.
b	Second variable operand.

Returns: Result of sutracting a variable from a scalar.

Definition at line 130 of file operator_subtraction.hpp.

var& stan::agrad::operator-- ( var & a )

inline

Prefix decrement operator for variables (C++).

Following C++, (–a) is defined to behave exactly as

a = a - 1.0)

does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters

a	Variable to decrement.

Returns: Reference the result of decrementing this input variable.

Definition at line 39 of file operator_unary_decrement.hpp.

var stan::agrad::operator--	(	var &	a,
		int
	)

inline

Postfix decrement operator for variables (C++).

Following C++, the expression (a–) is defined to behave like the sequence of operations

var temp = a; a = a - 1.0; return temp;

Parameters

a	Variable to decrement.

Returns: Input variable.

Definition at line 55 of file operator_unary_decrement.hpp.

template<typename T >

fvar<T> stan::agrad::operator/	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 14 of file operator_division.hpp.

template<typename T >

fvar<T> stan::agrad::operator/	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 22 of file operator_division.hpp.

template<typename T >

fvar<T> stan::agrad::operator/	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 30 of file operator_division.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::operator/	(	const Eigen::Matrix< fvar< T >, R, C > &	v,
		const fvar< T > &	c
	)

inline

Definition at line 58 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::operator/	(	const Eigen::Matrix< fvar< T >, R, C > &	v,
		const double	c
	)

inline

Definition at line 64 of file divide.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::operator/	(	const Eigen::Matrix< double, R, C > &	v,
		const fvar< T > &	c
	)

inline

Definition at line 70 of file divide.hpp.

var stan::agrad::operator/	(	const var &	a,
		const var &	b
	)

inline

Division operator for two variables (C++).

The partial derivatives for the variables are

$\frac{\partial}{\partial x} (x/y) = 1/y$ , and

$\frac{\partial}{\partial y} (x/y) = -x / y^2$ .

$\mbox{operator/}(x,y) = \begin{cases} \frac{x}{y} & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator/}(x,y)}{\partial x} = \begin{cases} \frac{1}{y} & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{operator/}(x,y)}{\partial y} = \begin{cases} -\frac{x}{y^2} & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable operand.
b	Second variable operand.

Returns: Variable result of dividing the first variable by the second.

Definition at line 95 of file operator_division.hpp.

var stan::agrad::operator/	(	const var &	a,
		const double	b
	)

inline

Division operator for dividing a variable by a scalar (C++).

The derivative with respect to the variable is

$\frac{\partial}{\partial x} (x/c) = 1/c$ .

Parameters

a	Variable operand.
b	Scalar operand.

Returns: Variable result of dividing the variable by the scalar.

Definition at line 110 of file operator_division.hpp.

var stan::agrad::operator/	(	const double	a,
		const var &	b
	)

inline

Division operator for dividing a scalar by a variable (C++).

The derivative with respect to the variable is

$\frac{d}{d y} (c/y) = -c / y^2$ .

Parameters

a	Scalar operand.
b	Variable operand.

Returns: Variable result of dividing the scalar by the variable.

Definition at line 127 of file operator_division.hpp.

template<typename T >

bool stan::agrad::operator<	(	const fvar< T > &	x,
		double	y
	)

inline

Definition at line 12 of file operator_less_than.hpp.

template<typename T >

bool stan::agrad::operator<	(	double	x,
		const fvar< T > &	y
	)

inline

Definition at line 18 of file operator_less_than.hpp.

template<typename T >

bool stan::agrad::operator<	(	const fvar< T > &	x,
		const fvar< T > &	y
	)

inline

Definition at line 24 of file operator_less_than.hpp.

bool stan::agrad::operator<	(	const var &	a,
		const var &	b
	)

inline

Less than operator comparing variables' values (C++).

$\mbox{operator\textless}(x,y) = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: True if first variable's value is less than second's.

Definition at line 24 of file operator_less_than.hpp.

bool stan::agrad::operator<	(	const var &	a,
		const double	b
	)

inline

Less than operator comparing variable's value and a double (C++).

Parameters

a	First variable.
b	Second value.

Returns: True if first variable's value is less than second value.

Definition at line 36 of file operator_less_than.hpp.

bool stan::agrad::operator<	(	const double	a,
		const var &	b
	)

inline

Less than operator comparing a double and variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns: True if first value is less than second variable's value.

Definition at line 48 of file operator_less_than.hpp.

template<typename T >

bool stan::agrad::operator<=	(	const fvar< T > &	x,
		const fvar< T > &	y
	)

inline

Definition at line 14 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::agrad::operator<=	(	const fvar< T > &	x,
		double	y
	)

inline

Definition at line 21 of file operator_less_than_or_equal.hpp.

bool stan::agrad::operator<=	(	const var &	a,
		const var &	b
	)

inline

Less than or equal operator comparing two variables' values (C++).

$\mbox{operator\textless=}(x,y) = \begin{cases} 0 & \mbox{if } x > y\\ 1 & \mbox{if } x \leq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: True if first variable's value is less than or equal to the second's.

Definition at line 26 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::agrad::operator<=	(	double	x,
		const fvar< T > &	y
	)

inline

Definition at line 28 of file operator_less_than_or_equal.hpp.

bool stan::agrad::operator<=	(	const var &	a,
		const double	b
	)

inline

Less than or equal operator comparing a variable's value and a scalar (C++).

Parameters

a	First variable.
b	Second value.

Returns: True if first variable's value is less than or equal to the second value.

Definition at line 39 of file operator_less_than_or_equal.hpp.

bool stan::agrad::operator<=	(	const double	a,
		const var &	b
	)

inline

Less than or equal operator comparing a double and variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns: True if first value is less than or equal to the second variable's value.

Definition at line 52 of file operator_less_than_or_equal.hpp.

template<typename T >

bool stan::agrad::operator==	(	const fvar< T > &	x,
		const fvar< T > &	y
	)

inline

Definition at line 14 of file operator_equal.hpp.

template<typename T >

bool stan::agrad::operator==	(	const fvar< T > &	x,
		double	y
	)

inline

Definition at line 21 of file operator_equal.hpp.

bool stan::agrad::operator==	(	const var &	a,
		const var &	b
	)

inline

Equality operator comparing two variables' values (C++).

$\mbox{operator==}(x,y) = \begin{cases} 0 & \mbox{if } x \neq y\\ 1 & \mbox{if } x = y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: True if the first variable's value is the same as the second's.

Definition at line 26 of file operator_equal.hpp.

template<typename T >

bool stan::agrad::operator==	(	double	x,
		const fvar< T > &	y
	)

inline

Definition at line 28 of file operator_equal.hpp.

bool stan::agrad::operator==	(	const var &	a,
		const double	b
	)

inline

Equality operator comparing a variable's value and a double (C++).

Parameters

a	First variable.
b	Second value.

Returns: True if the first variable's value is the same as the second value.

Definition at line 39 of file operator_equal.hpp.

bool stan::agrad::operator==	(	const double	a,
		const var &	b
	)

inline

Equality operator comparing a scalar and a variable's value (C++).

Parameters

a	First scalar.
b	Second variable.

Returns: True if the variable's value is equal to the scalar.

Definition at line 51 of file operator_equal.hpp.

template<typename T >

bool stan::agrad::operator>	(	const fvar< T > &	x,
		const fvar< T > &	y
	)

inline

Definition at line 14 of file operator_greater_than.hpp.

template<typename T >

bool stan::agrad::operator>	(	const fvar< T > &	x,
		double	y
	)

inline

Definition at line 21 of file operator_greater_than.hpp.

bool stan::agrad::operator>	(	const var &	a,
		const var &	b
	)

inline

Greater than operator comparing variables' values (C++).

$\mbox{operator\textgreater}(x,y) = \begin{cases} 0 & \mbox{if } x \leq y\\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: True if first variable's value is greater than second's.

Definition at line 25 of file operator_greater_than.hpp.

template<typename T >

bool stan::agrad::operator>	(	double	x,
		const fvar< T > &	y
	)

inline

Definition at line 28 of file operator_greater_than.hpp.

bool stan::agrad::operator>	(	const var &	a,
		const double	b
	)

inline

Greater than operator comparing variable's value and double (C++).

Parameters

a	First variable.
b	Second value.

Returns: True if first variable's value is greater than second value.

Definition at line 37 of file operator_greater_than.hpp.

bool stan::agrad::operator>	(	const double	a,
		const var &	b
	)

inline

Greater than operator comparing a double and a variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns: True if first value is greater than second variable's value.

Definition at line 49 of file operator_greater_than.hpp.

template<typename T >

bool stan::agrad::operator>=	(	const fvar< T > &	x,
		const fvar< T > &	y
	)

inline

Definition at line 14 of file operator_greater_than_or_equal.hpp.

template<typename T >

bool stan::agrad::operator>=	(	const fvar< T > &	x,
		double	y
	)

inline

Definition at line 21 of file operator_greater_than_or_equal.hpp.

bool stan::agrad::operator>=	(	const var &	a,
		const var &	b
	)

inline

Greater than or equal operator comparing two variables' values (C++).

$\mbox{operator\textgreater=}(x,y) = \begin{cases} 0 & \mbox{if } x < y\\ 1 & \mbox{if } x \geq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

a	First variable.
b	Second variable.

Returns: True if first variable's value is greater than or equal to the second's.

Definition at line 27 of file operator_greater_than_or_equal.hpp.

template<typename T >

bool stan::agrad::operator>=	(	double	x,
		const fvar< T > &	y
	)

inline

Definition at line 28 of file operator_greater_than_or_equal.hpp.

bool stan::agrad::operator>=	(	const var &	a,
		const double	b
	)

inline

Greater than or equal operator comparing variable's value and double (C++).

Parameters

a	First variable.
b	Second value.

Returns: True if first variable's value is greater than or equal to second value.

Definition at line 40 of file operator_greater_than_or_equal.hpp.

bool stan::agrad::operator>=	(	const double	a,
		const var &	b
	)

inline

Greater than or equal operator comparing double and variable's value (C++).

Parameters

a	First value.
b	Second variable.

Returns: True if the first value is greater than or equal to the second variable's value.

Definition at line 53 of file operator_greater_than_or_equal.hpp.

template<typename T >

fvar<T> stan::agrad::owens_t	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 17 of file owens_t.hpp.

template<typename T >

fvar<T> stan::agrad::owens_t	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 40 of file owens_t.hpp.

template<typename T >

fvar<T> stan::agrad::owens_t	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 57 of file owens_t.hpp.

var stan::agrad::owens_t	(	const var &	h,
		const var &	a
	)

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	var parameter.
a	var parameter.

Returns: The Owen's T function.

Definition at line 82 of file owens_t.hpp.

var stan::agrad::owens_t	(	const var &	h,
		const double &	a
	)

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	var parameter.
a	double parameter.

Returns: The Owen's T function.

Definition at line 98 of file owens_t.hpp.

var stan::agrad::owens_t	(	const double &	h,
		const var &	a
	)

inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters

h	double parameter.
a	var parameter.

Returns: The Owen's T function.

Definition at line 114 of file owens_t.hpp.

template<typename T , typename F >

void stan::agrad::partial_derivative	(	const F &	f,
		const Eigen::Matrix< T, Eigen::Dynamic, 1 > &	x,
		int	n,
		T &	fx,
		T &	dfx_dxn
	)

Return the partial derivative of the specified multiivariate function at the specified argument.

Template Parameters

T	Argument type
F	Function type

Parameters

	f	Function
[in]	x	Argument vector
[in]	n	Index of argument with which to take derivative
[out]	fx	Value of function applied to argument
[out]	dfx_dxn	Value of partial derivative

Definition at line 49 of file autodiff.hpp.

template<typename T >

fvar<T> stan::agrad::Phi ( const fvar< T > & x )

inline

Definition at line 14 of file Phi.hpp.

var stan::agrad::Phi ( const stan::agrad::var & a )

inline

The unit normal cumulative density function for variables (stan).

See stan::math::Phi() for the double-based version.

The derivative is the unit normal density function,

$\frac{d}{dx} \Phi(x) = \mbox{\sf Norm}(x|0,1) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{1}{2} x^2)$ .

$\mbox{Phi}(x) = \begin{cases} 0 & \mbox{if } x < -37.5 \\ \Phi(x) & \mbox{if } -37.5 \leq x \leq 8.25 \\ 1 & \mbox{if } x > 8.25 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{Phi}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } x < -27.5 \\ \frac{\partial\,\Phi(x)}{\partial x} & \mbox{if } -27.5 \leq x \leq 27.5 \\ 0 & \mbox{if } x > 27.5 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{0}^{x} e^{-t^2/2} dt$

$\frac{\partial \, \Phi(x)}{\partial x} = \frac{e^{-x^2/2}}{\sqrt{2\pi}}$

Parameters

a	Variable argument.

Returns: The unit normal cdf evaluated at the specified argument.

Definition at line 66 of file Phi.hpp.

var stan::agrad::Phi_approx ( const stan::agrad::var & a )

inline

Approximation of the unit normal CDF for variables (stan).

http://www.jiem.org/index.php/jiem/article/download/60/27

$\mbox{Phi\_approx}(x) = \begin{cases} \Phi_{\mbox{\footnotesize approx}}(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{Phi\_approx}(x)}{\partial x} = \begin{cases} \frac{\partial\,\Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\Phi_{\mbox{\footnotesize approx}}(x) = \mbox{logit}^{-1}(0.07056 \, x^3 + 1.5976 \, x)$

$\frac{\partial \, \Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} = -\Phi_{\mbox{\footnotesize approx}}^2(x) e^{-0.07056x^3 - 1.5976x}(-0.21168x^2-1.5976)$

Parameters

a	Variable argument.

Returns: The corresponding unit normal cdf approximation.

Definition at line 48 of file Phi_approx.hpp.

template<typename T >

fvar<T> stan::agrad::pow	(	const fvar< T > &	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 17 of file pow.hpp.

template<typename T >

fvar<T> stan::agrad::pow	(	const double	x1,
		const fvar< T > &	x2
	)

inline

Definition at line 29 of file pow.hpp.

template<typename T >

fvar<T> stan::agrad::pow	(	const fvar< T > &	x1,
		const double	x2
	)

inline

Definition at line 39 of file pow.hpp.

var stan::agrad::pow	(	const var &	base,
		const var &	exponent
	)

inline

Return the base raised to the power of the exponent (cmath).

The partial derivatives are

$\frac{\partial}{\partial x} \mbox{pow}(x,y) = y x^{y-1}$ , and

$\frac{\partial}{\partial y} \mbox{pow}(x,y) = x^y \ \log x$ .

$\mbox{pow}(x,y) = \begin{cases} x^y & \mbox{if } -\infty\leq x,y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{pow}(x,y)}{\partial x} = \begin{cases} yx^{y-1} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{pow}(x,y)}{\partial y} = \begin{cases} x^y\ln x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases}$

Parameters

base	Base variable.
exponent	Exponent variable.

Returns: Base raised to the exponent.

Definition at line 106 of file pow.hpp.

var stan::agrad::pow	(	const var &	base,
		const double	exponent
	)

inline

Return the base variable raised to the power of the exponent scalar (cmath).

The derivative for the variable is

$\frac{d}{dx} \mbox{pow}(x,c) = c x^{c-1}$ .

Parameters

base	Base variable.
exponent	Exponent scalar.

Returns: Base raised to the exponent.

Definition at line 122 of file pow.hpp.

var stan::agrad::pow	(	const double	base,
		const var &	exponent
	)

inline

Return the base scalar raised to the power of the exponent variable (cmath).

The derivative for the variable is

$\frac{d}{d y} \mbox{pow}(c,y) = c^y \log c$ .

Parameters

base	Base scalar.
exponent	Exponent variable.

Returns: Base raised to the exponent.

Definition at line 144 of file pow.hpp.

var stan::agrad::precomputed_gradients	(	const double	value,
		const std::vector< var > &	vars,
		const std::vector< double > &	gradients
	)

This function is provided for Stan users that want to compute gradients without using Stan's auto-diff.

Users need to provide the value, the independent variables, and the gradients of this expression with respect to the indepedent variables.

(For advanced users, a faster version that doesn't involve copying vectors exists can be written.)

Parameters

value	The value of the resulting dependent variable.
vars	The independent variables.
gradients	The value of the gradients of the dependent variable with respect to the independent variables.

Returns: An auto-diff variable that uses the precomputed gradients provided.

Definition at line 86 of file precomputed_gradients.hpp.

double stan::agrad::primitive_value ( const agrad::var & v )

inline

Return the primitive double value for the specified auto-diff variable.

Parameters

v	input variable.

Returns: value of input.

Definition at line 17 of file primitive_value.hpp.

template<typename T >

double stan::agrad::primitive_value ( const fvar< T > & v )

inline

Return the primitive value of the specified forward-mode autodiff variable.

This function applies recursively to higher-order autodiff types to return a primitive double value.

Template Parameters

T	scalar type for autodiff variable.

Parameters

v	input variable.

Returns: primitive value of input.

Definition at line 22 of file primitive_value.hpp.

void stan::agrad::print_stack ( std::ostream & o )

inline

Prints the auto-dif variable stack.

This function is used for debugging purposes.

Only works if all members of stack are vari* as it casts to vari*.

Parameters

o	ostream to modify

Definition at line 20 of file print_stack.hpp.

template<typename T >

Eigen::Matrix<fvar<T>,Eigen::Dynamic,Eigen::Dynamic> stan::agrad::qr_Q ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > & m )

Definition at line 15 of file qr_Q.hpp.

template<typename T >

Eigen::Matrix<fvar<T>,Eigen::Dynamic,Eigen::Dynamic> stan::agrad::qr_R ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > & m )

Definition at line 15 of file qr_R.hpp.

template<typename TA , int RA, int CA, typename TB , int RB, int CB>

boost::enable_if_c< boost::is_same<TA,var>::value \|\| boost::is_same<TB,var>::value, Eigen::Matrix<var,CB,CB> >::type stan::agrad::quad_form	(	const Eigen::Matrix< TA, RA, CA > &	A,
		const Eigen::Matrix< TB, RB, CB > &	B
	)

inline

Definition at line 129 of file quad_form.hpp.

template<typename TA , int RA, int CA, typename TB , int RB>

boost::enable_if_c< boost::is_same<TA,var>::value \|\| boost::is_same<TB,var>::value, var >::type stan::agrad::quad_form	(	const Eigen::Matrix< TA, RA, CA > &	A,
		const Eigen::Matrix< TB, RB, 1 > &	B
	)

inline

Definition at line 145 of file quad_form.hpp.

template<typename TA , int RA, int CA, typename TB , int RB, int CB>

boost::enable_if_c< boost::is_same<TA,var>::value \|\| boost::is_same<TB,var>::value, Eigen::Matrix<var,CB,CB> >::type stan::agrad::quad_form_sym	(	const Eigen::Matrix< TA, RA, CA > &	A,
		const Eigen::Matrix< TB, RB, CB > &	B
	)

inline

Definition at line 162 of file quad_form.hpp.

template<typename TA , int RA, int CA, typename TB , int RB>

boost::enable_if_c< boost::is_same<TA,var>::value \|\| boost::is_same<TB,var>::value, var >::type stan::agrad::quad_form_sym	(	const Eigen::Matrix< TA, RA, CA > &	A,
		const Eigen::Matrix< TB, RB, 1 > &	B
	)

inline

Definition at line 179 of file quad_form.hpp.

static void stan::agrad::recover_memory ( )

inlinestatic

Recover memory used for all variables for reuse.

Exceptions

std::logic_error if empty_nested() returns false

Definition at line 52 of file var_stack.hpp.

static void stan::agrad::recover_memory_nested ( )

inlinestatic

Recover only the memory used for the top nested call.

If there is nothing on the nested stack, then a std::logic_error exception is thrown.

Exceptions

std::logic_error if empty_nested() returns true

Definition at line 72 of file var_stack.hpp.

template<typename T >

fvar<T> stan::agrad::rising_factorial	(	const fvar< T > &	x,
		const fvar< T > &	n
	)

inline

Definition at line 16 of file rising_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::rising_factorial	(	const fvar< T > &	x,
		const double	n
	)

inline

Definition at line 28 of file rising_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::rising_factorial	(	const double	x,
		const fvar< T > &	n
	)

inline

Definition at line 40 of file rising_factorial.hpp.

var stan::agrad::rising_factorial	(	const var &	a,
		const double &	b
	)

inline

Definition at line 48 of file rising_factorial.hpp.

var stan::agrad::rising_factorial	(	const var &	a,
		const var &	b
	)

inline

Definition at line 53 of file rising_factorial.hpp.

var stan::agrad::rising_factorial	(	const double &	a,
		const var &	b
	)

inline

Definition at line 58 of file rising_factorial.hpp.

template<typename T >

fvar<T> stan::agrad::round ( const fvar< T > & x )

inline

Definition at line 15 of file round.hpp.

var stan::agrad::round ( const stan::agrad::var & a )

inline

Returns the rounded form of the specified variable (C99).

See boost::math::round() for the double-based version.

The derivative is zero everywhere but numbers half way between whole numbers, so for convenience the derivative is defined to be everywhere zero,

$\frac{d}{dx} \mbox{round}(x) = 0$ .

$\mbox{round}(x) = \begin{cases} \lceil x \rceil & \mbox{if } x-\lfloor x\rfloor \geq 0.5 \\ \lfloor x \rfloor & \mbox{if } x-\lfloor x\rfloor < 0.5 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{round}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Specified variable.

Returns: Rounded variable.

Definition at line 57 of file round.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, 1> stan::agrad::rows_dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2
	)

inline

Definition at line 19 of file rows_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, 1> stan::agrad::rows_dot_product	(	const Eigen::Matrix< double, R1, C1 > &	v1,
		const Eigen::Matrix< fvar< T >, R2, C2 > &	v2
	)

inline

Definition at line 35 of file rows_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>

Eigen::Matrix<fvar<T>, R1, 1> stan::agrad::rows_dot_product	(	const Eigen::Matrix< fvar< T >, R1, C1 > &	v1,
		const Eigen::Matrix< double, R2, C2 > &	v2
	)

inline

Definition at line 51 of file rows_dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>

boost::enable_if_c<boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value, Eigen::Matrix<var, R1, 1> >::type stan::agrad::rows_dot_product	(	const Eigen::Matrix< T1, R1, C1 > &	v1,
		const Eigen::Matrix< T2, R2, C2 > &	v2
	)

inline

Definition at line 277 of file dot_product.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,1> stan::agrad::rows_dot_self ( const Eigen::Matrix< fvar< T >, R, C > & x )

inline

Definition at line 15 of file rows_dot_self.hpp.

var stan::agrad::sd ( const std::vector< var > & v )

Return the sample standard deviation of the specified standard vector.

Raise domain error if size is not greater than zero.

Parameters

[in] v a vector

Returns: sample standard deviation of specified vector

Definition at line 63 of file sd.hpp.

template<int R, int C>

var stan::agrad::sd ( const Eigen::Matrix< var, R, C > & m )

Definition at line 80 of file sd.hpp.

static void stan::agrad::set_zero_all_adjoints ( )

static

Reset all adjoint values in the stack to zero.

Definition at line 87 of file chainable.hpp.

template<typename T >

fvar<T> stan::agrad::sin ( const fvar< T > & x )

inline

Definition at line 14 of file sin.hpp.

var stan::agrad::sin ( const var & a )

inline

Return the sine of a radian-scaled variable (cmath).

The derivative is defined by

$\frac{d}{dx} \sin x = \cos x$ .

$\mbox{sin}(x) = \begin{cases} \sin(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{sin}(x)}{\partial x} = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable for radians of angle.

Returns: Sine of variable.

Definition at line 50 of file sin.hpp.

template<typename T >

fvar<T> stan::agrad::sinh ( const fvar< T > & x )

inline

Definition at line 14 of file sinh.hpp.

var stan::agrad::sinh ( const var & a )

inline

Return the hyperbolic sine of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \sinh x = \cosh x$ .

$\mbox{sinh}(x) = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{sinh}(x)}{\partial x} = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a Variable.

Returns: Hyperbolic sine of variable.

Definition at line 50 of file sinh.hpp.

template<typename T >

Eigen::Matrix<fvar<T>,Eigen::Dynamic,1> stan::agrad::softmax ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > & alpha )

inline

Definition at line 14 of file softmax.hpp.

Eigen::Matrix<var,Eigen::Dynamic,1> stan::agrad::softmax ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > & alpha )

inline

Return the softmax of the specified Eigen vector.

Softmax is guaranteed to return a simplex.

The gradient calculations are unfolded.

Parameters

alpha Unconstrained input vector.

Returns: Softmax of the input.

Exceptions

std::domain_error If the input vector is size 0.

Definition at line 58 of file softmax.hpp.

template<typename T >

std::vector< fvar<T> > stan::agrad::sort_asc ( std::vector< fvar< T > > xs )

inline

Definition at line 18 of file sort.hpp.

std::vector<var> stan::agrad::sort_asc ( std::vector< var > xs )

inline

Return the specified standard vector in ascending order with gradients kept.

Parameters

xs	Standard vector to order.

Returns: Standard vector ordered.

Template Parameters

T	Type of elements of the vector.

Definition at line 24 of file sort.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::sort_asc ( Eigen::Matrix< fvar< T >, R, C > xs )

inline

Definition at line 34 of file sort.hpp.

template<int R, int C>

Eigen::Matrix<var,R,C> stan::agrad::sort_asc ( Eigen::Matrix< var, R, C > xs )

inline

Return the specified eigen vector in ascending order with gradients kept.

Parameters

xs	Eigen vector to order.

Returns: Eigen vector ordered.

Template Parameters

T	Type of elements of the vector.

Definition at line 49 of file sort.hpp.

template<typename T >

std::vector< fvar<T> > stan::agrad::sort_desc ( std::vector< fvar< T > > xs )

inline

Definition at line 26 of file sort.hpp.

std::vector<var> stan::agrad::sort_desc ( std::vector< var > xs )

inline

Return the specified standard vector in descending order with gradients kept.

Parameters

xs	Standard vector to order.

Returns: Standard vector ordered.

Template Parameters

T	Type of elements of the vector.

Definition at line 36 of file sort.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,C> stan::agrad::sort_desc ( Eigen::Matrix< fvar< T >, R, C > xs )

inline

Definition at line 42 of file sort.hpp.

template<int R, int C>

Eigen::Matrix<var,R,C> stan::agrad::sort_desc ( Eigen::Matrix< var, R, C > xs )

inline

Return the specified eigen vector in descending order with gradients kept.

Parameters

xs	Eigen vector to order.

Returns: Eigen vector ordered.

Template Parameters

T	Type of elements of the vector.

Definition at line 62 of file sort.hpp.

template<typename T >

fvar<T> stan::agrad::sqrt ( const fvar< T > & x )

inline

Definition at line 15 of file sqrt.hpp.

var stan::agrad::sqrt ( const var & a )

inline

Return the square root of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}$ .

$\mbox{sqrt}(x) = \begin{cases} \textrm{NaN} & x < 0 \\ \sqrt{x} & \mbox{if } x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{sqrt}(x)}{\partial x} = \begin{cases} \textrm{NaN} & x < 0 \\ \frac{1}{2\sqrt{x}} & x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable whose square root is taken.

Returns: Square root of variable.

Definition at line 51 of file sqrt.hpp.

template<typename T >

fvar<T> stan::agrad::square ( const fvar< T > & x )

inline

Definition at line 15 of file square.hpp.

var stan::agrad::square ( const var & x )

inline

Return the square of the input variable.

Using square(x) is more efficient than using x * x.

$\mbox{square}(x) = \begin{cases} x^2 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{square}(x)}{\partial x} = \begin{cases} 2x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

x	Variable to square.

Returns: Square of variable.

Definition at line 47 of file square.hpp.

template<int R1, int C1, int R2, int C2>

var stan::agrad::squared_distance	(	const Eigen::Matrix< var, R1, C1 > &	v1,
		const Eigen::Matrix< var, R2, C2 > &	v2
	)

inline

Definition at line 112 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>

var stan::agrad::squared_distance	(	const Eigen::Matrix< var, R1, C1 > &	v1,
		const Eigen::Matrix< double, R2, C2 > &	v2
	)

inline

Definition at line 121 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>

var stan::agrad::squared_distance	(	const Eigen::Matrix< double, R1, C1 > &	v1,
		const Eigen::Matrix< var, R2, C2 > &	v2
	)

inline

Definition at line 130 of file squared_distance.hpp.

void stan::agrad::stan_print	(	std::ostream *	o,
		const var &	x
	)

Definition at line 10 of file stan_print.hpp.

static void stan::agrad::start_nested ( )

inlinestatic

Record the current position so that recover_memory_nested() can find it.

Definition at line 96 of file var_stack.hpp.

var stan::agrad::step ( const stan::agrad::var & a )

inline

Return the step, or heaviside, function applied to the specified variable (stan).

See stan::math::step() for the double-based version.

The derivative of the step function is zero everywhere but at 0, so for convenience, it is taken to be everywhere zero,

$\mbox{step}(x) = 0$ .

Parameters

a	Variable argument.

Returns: The constant variable with value 1.0 if the argument's value is greater than or equal to 0.0, and value 0.0 otherwise.

Definition at line 25 of file step.hpp.

template<typename T , int R, int C>

fvar<T> stan::agrad::sum ( const Eigen::Matrix< fvar< T >, R, C > & m )

inline

Definition at line 14 of file sum.hpp.

template<int R, int C>

var stan::agrad::sum ( const Eigen::Matrix< var, R, C > & m )

inline

Returns the sum of the coefficients of the specified matrix, column vector or row vector.

Parameters

m	Specified matrix or vector.

Returns: Sum of coefficients of matrix.

Definition at line 68 of file sum.hpp.

template<typename T >

fvar<T> stan::agrad::tan ( const fvar< T > & x )

inline

Definition at line 14 of file tan.hpp.

var stan::agrad::tan ( const var & a )

inline

Return the tangent of a radian-scaled variable (cmath).

The derivative is defined by

$\frac{d}{dx} \tan x = \sec^2 x$ .

$\mbox{tan}(x) = \begin{cases} \tan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{tan}(x)}{\partial x} = \begin{cases} \sec^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Variable for radians of angle.

Returns: Tangent of variable.

Definition at line 50 of file tan.hpp.

template<typename T >

fvar<T> stan::agrad::tanh ( const fvar< T > & x )

inline

Definition at line 14 of file tanh.hpp.

var stan::agrad::tanh ( const var & a )

inline

Return the hyperbolic tangent of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \tanh x = \frac{1}{\cosh^2 x}$ .

$\mbox{tanh}(x) = \begin{cases} \tanh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{tanh}(x)}{\partial x} = \begin{cases} \mbox{sech}^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a Variable.

Returns: Hyperbolic tangent of variable.

Definition at line 51 of file tanh.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>,R,R> stan::agrad::tcrossprod ( const Eigen::Matrix< fvar< T >, R, C > & m )

inline

Definition at line 17 of file tcrossprod.hpp.

matrix_v stan::agrad::tcrossprod ( const matrix_v & M )

inline

Returns the result of post-multiplying a matrix by its own transpose.

Parameters

M	Matrix to multiply.

Returns: M times its transpose.

Definition at line 24 of file tcrossprod.hpp.

template<typename T >

fvar<T> stan::agrad::tgamma ( const fvar< T > & x )

inline

Definition at line 15 of file tgamma.hpp.

var stan::agrad::tgamma ( const stan::agrad::var & a )

inline

Return the Gamma function applied to the specified variable (C99).

See boost::math::tgamma() for the double-based version.

The derivative with respect to the argument is

$\frac{d}{dx} \Gamma(x) = \Gamma(x) \Psi^{(0)}(x)$

where $\Psi^{(0)}(x)$ is the digamma function.

See boost::math::digamma() for the double-based version.

$\mbox{tgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots,-3,-2,-1,0\}\\ \Gamma(x) & \mbox{if } x\not\in \{\dots,-3,-2,-1,0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{tgamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots,-3,-2,-1,0\}\\ \frac{\partial\, \Gamma(x)}{\partial x} & \mbox{if } x\not\in \{\dots,-3,-2,-1,0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\Gamma(x)=\int_0^{\infty} u^{x - 1} \exp(-u) \, du$

$\frac{\partial \, \Gamma(x)}{\partial x} = \Gamma(x)\Psi(x)$

Parameters

a	Argument to function.

Returns: The Gamma function applied to the specified argument.

Definition at line 66 of file tgamma.hpp.

template<typename T >

fvar<T> stan::agrad::to_fvar ( const T & x )

inline

Definition at line 17 of file to_fvar.hpp.

template<typename T >

fvar<T> stan::agrad::to_fvar ( const fvar< T > & x )

inline

Definition at line 24 of file to_fvar.hpp.

matrix_fd stan::agrad::to_fvar ( const stan::math::matrix_d & m )

inline

Definition at line 30 of file to_fvar.hpp.

matrix_fd stan::agrad::to_fvar ( const matrix_fd & m )

inline

Definition at line 40 of file to_fvar.hpp.

matrix_fv stan::agrad::to_fvar ( const matrix_fv & m )

inline

Definition at line 46 of file to_fvar.hpp.

matrix_ffd stan::agrad::to_fvar ( const matrix_ffd & m )

inline

Definition at line 52 of file to_fvar.hpp.

matrix_ffv stan::agrad::to_fvar ( const matrix_ffv & m )

inline

Definition at line 58 of file to_fvar.hpp.

vector_fd stan::agrad::to_fvar ( const stan::math::vector_d & v )

inline

Definition at line 64 of file to_fvar.hpp.

vector_fd stan::agrad::to_fvar ( const vector_fd & v )

inline

Definition at line 73 of file to_fvar.hpp.

vector_fv stan::agrad::to_fvar ( const vector_fv & v )

inline

Definition at line 79 of file to_fvar.hpp.

vector_ffd stan::agrad::to_fvar ( const vector_ffd & v )

inline

Definition at line 85 of file to_fvar.hpp.

vector_ffv stan::agrad::to_fvar ( const vector_ffv & v )

inline

Definition at line 91 of file to_fvar.hpp.

row_vector_fd stan::agrad::to_fvar ( const stan::math::row_vector_d & rv )

inline

Definition at line 97 of file to_fvar.hpp.

row_vector_fd stan::agrad::to_fvar ( const row_vector_fd & rv )

inline

Definition at line 106 of file to_fvar.hpp.

row_vector_fv stan::agrad::to_fvar ( const row_vector_fv & rv )

inline

Definition at line 112 of file to_fvar.hpp.

row_vector_ffd stan::agrad::to_fvar ( const row_vector_ffd & rv )

inline

Definition at line 118 of file to_fvar.hpp.

row_vector_ffv stan::agrad::to_fvar ( const row_vector_ffv & rv )

inline

Definition at line 124 of file to_fvar.hpp.

template<typename T , int R, int C>

Eigen::Matrix<fvar<T>, R, C> stan::agrad::to_fvar	(	const Eigen::Matrix< T, R, C > &	val,
		const Eigen::Matrix< T, R, C > &	deriv
	)

inline

Definition at line 131 of file to_fvar.hpp.

var stan::agrad::to_var ( const double & x )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] x A scalar value

Returns: An automatic differentiation variable with the input value.

Definition at line 21 of file to_var.hpp.

var stan::agrad::to_var ( const var & x )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] x An automatic differentiation variable.

Returns: An automatic differentiation variable with the input value.

Definition at line 32 of file to_var.hpp.

matrix_v stan::agrad::to_var ( const stan::math::matrix_d & m )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] m A Matrix with scalars

Returns: A Matrix with automatic differentiation variables

Definition at line 43 of file to_var.hpp.

matrix_v stan::agrad::to_var ( const matrix_v & m )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] m A Matrix with automatic differentiation variables.

Returns: A Matrix with automatic differentiation variables.

Definition at line 58 of file to_var.hpp.

vector_v stan::agrad::to_var ( const stan::math::vector_d & v )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] v A Vector of scalars

Returns: A Vector of automatic differentiation variables with values of v

Definition at line 70 of file to_var.hpp.

vector_v stan::agrad::to_var ( const vector_v & v )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] v A Vector of automatic differentiation variables

Returns: A Vector of automatic differentiation variables with values of v

Definition at line 85 of file to_var.hpp.

row_vector_v stan::agrad::to_var ( const stan::math::row_vector_d & rv )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] rv A row vector of scalars

Returns: A row vector of automatic differentation variables with values of rv.

Definition at line 97 of file to_var.hpp.

row_vector_v stan::agrad::to_var ( const row_vector_v & rv )

inline

Converts argument to an automatic differentiation variable.

Returns a stan::agrad::var variable with the input value.

Parameters

[in] rv A row vector with automatic differentiation variables

Returns: A row vector with automatic differentiation variables with values of rv.

Definition at line 112 of file to_var.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename T3 , int R3, int C3>

boost::enable_if_c<boost::is_same<T1,var>::value \|\| boost::is_same<T2,var>::value \|\| boost::is_same<T3,var>::value, var>::type stan::agrad::trace_gen_inv_quad_form_ldlt	(	const Eigen::Matrix< T1, R1, C1 > &	D,
		const stan::math::LDLT_factor< T2, R2, C2 > &	A,
		const Eigen::Matrix< T3, R3, C3 > &	B
	)

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(D B^T A^-1 B) where D is a square matrix and the LDLT_factor of A is provided.

Definition at line 25 of file trace_gen_inv_quad_form_ldlt.hpp.

template<int RD, int CD, int RA, int CA, int RB, int CB, typename T >

fvar<T> stan::agrad::trace_gen_quad_form	(	const Eigen::Matrix< fvar< T >, RD, CD > &	D,
		const Eigen::Matrix< fvar< T >, RA, CA > &	A,
		const Eigen::Matrix< fvar< T >, RB, CB > &	B
	)

inline

Definition at line 15 of file trace_gen_quad_form.hpp.

template<typename TD , int RD, int CD, typename TA , int RA, int CA, typename TB , int RB, int CB>

boost::enable_if_c< boost::is_same<TD,var>::value \|\| boost::is_same<TA,var>::value \|\| boost::is_same<TB,var>::value, var >::type stan::agrad::trace_gen_quad_form	(	const Eigen::Matrix< TD, RD, CD > &	D,
		const Eigen::Matrix< TA, RA, CA > &	A,
		const Eigen::Matrix< TB, RB, CB > &	B
	)

inline

Definition at line 111 of file trace_gen_quad_form.hpp.

template<typename T2 , int R2, int C2, typename T3 , int R3, int C3>

boost::enable_if_c<boost::is_same<T2,var>::value \|\| boost::is_same<T3,var>::value, var>::type stan::agrad::trace_inv_quad_form_ldlt	(	const stan::math::LDLT_factor< T2, R2, C2 > &	A,
		const Eigen::Matrix< T3, R3, C3 > &	B
	)

inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(B^T A^-1 B) where the LDLT_factor of A is provided.

Definition at line 162 of file trace_inv_quad_form_ldlt.hpp.

template<int RA, int CA, int RB, int CB, typename T >

stan::agrad::fvar<T> stan::agrad::trace_quad_form	(	const Eigen::Matrix< stan::agrad::fvar< T >, RA, CA > &	A,
		const Eigen::Matrix< stan::agrad::fvar< T >, RB, CB > &	B
	)

inline

Definition at line 19 of file trace_quad_form.hpp.

template<typename TA , int RA, int CA, typename TB , int RB, int CB>

boost::enable_if_c< boost::is_same<TA,var>::value \|\| boost::is_same<TB,var>::value, var >::type stan::agrad::trace_quad_form	(	const Eigen::Matrix< TA, RA, CA > &	A,
		const Eigen::Matrix< TB, RB, CB > &	B
	)

inline

Definition at line 99 of file trace_quad_form.hpp.

template<typename T >

fvar<T> stan::agrad::trunc ( const fvar< T > & x )

inline

Definition at line 15 of file trunc.hpp.

var stan::agrad::trunc ( const stan::agrad::var & a )

inline

Returns the truncatation of the specified variable (C99).

See boost::math::trunc() for the double-based version.

The derivative is zero everywhere but at integer values, so for convenience the derivative is defined to be everywhere zero,

$\frac{d}{dx} \mbox{trunc}(x) = 0$ .

$\mbox{trunc}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

$\frac{\partial\,\mbox{trunc}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$

Parameters

a	Specified variable.

Returns: Truncation of the variable.

Definition at line 55 of file trunc.hpp.

template<int R, int C>

const Eigen::Matrix<double,R,C>& stan::agrad::value_of ( const Eigen::Matrix< double, R, C > & M )

inline

Convert a matrix to a matrix of doubles.

When the input is already a matrix of doubles a reference is returned.

This is used as a convenience function for implementing varis of matrix operations.

Definition at line 16 of file value_of.hpp.

template<typename T >

T stan::agrad::value_of ( const fvar< T > & v )

inline

Return the value of the specified variable.

Parameters

v Variable.

Returns: Value of variable.

Definition at line 16 of file value_of.hpp.

double stan::agrad::value_of ( const agrad::var & v )

inline

Return the value of the specified variable.

This function is used internally by auto-dif functions along with stan::math::value_of(T x) to extract the double value of either a scalar or an auto-dif variable. This function will be called when the argument is a stan::agrad::var even if the function is not referred to by namespace because of argument-dependent lookup.

Parameters

v Variable.

Returns: Value of variable.

Definition at line 22 of file value_of.hpp.

template<int R, int C>

Eigen::Matrix<double,R,C> stan::agrad::value_of ( const Eigen::Matrix< var, R, C > & M )

inline

Convert a matrix to a matrix of doubles.

When the input is already a matrix of vars a new matrix is returned with the value of the constituent vars.

This is used as a convenience function for implementing varis of matrix operations.

Definition at line 27 of file value_of.hpp.

var stan::agrad::variance ( const std::vector< var > & v )

Return the sample variance of the specified standard vector.

Raise domain error if size is not greater than zero.

Parameters

[in] v a vector

Returns: sample variance of specified vector

Definition at line 51 of file variance.hpp.

template<int R, int C>

var stan::agrad::variance ( const Eigen::Matrix< var, R, C > & m )

Definition at line 68 of file variance.hpp.

Variable Documentation

memory::stack_alloc stan::agrad::memalloc_

Definition at line 16 of file var_stack.cpp.

std::vector< size_t > stan::agrad::nested_var_alloc_stack_starts_

Definition at line 20 of file var_stack.cpp.

std::vector< size_t > stan::agrad::nested_var_nochain_stack_sizes_

Definition at line 19 of file var_stack.cpp.

std::vector< size_t > stan::agrad::nested_var_stack_sizes_

Definition at line 18 of file var_stack.cpp.

std::vector< chainable_alloc * > stan::agrad::var_alloc_stack_

Definition at line 15 of file var_stack.cpp.

std::vector< chainable * > stan::agrad::var_nochain_stack_

Definition at line 14 of file var_stack.cpp.

std::vector< chainable * > stan::agrad::var_stack_

Definition at line 13 of file var_stack.cpp.

Classes

Typedefs

Functions

Variables

Detailed Description

Typedef Documentation

Function Documentation

Variable Documentation