OpenCL

Detailed Description

Stan's OpenCL backend allows for computation to be executed in parallel on a GPU or in multithreaded CPUs.

It is meant to easily conform with Eigen such that you can create and read from a matrix_cl by doing

Eigen::MatrixXd A_eig = Eigen::MatrixXd::Random(10, 10);
matrix_cl<double> A(A_eig);
matrix_cl<double> B = to_matrix_cl(A_eig);
matrix_cl<double> C = cholesky_decompose(A * B);
// Read back to eigen matrix.
Eigen::MatrixXd C_eig = from_matrix_cl(C);
 
// Also for vectors and raw pointers of pointers
std::vector<var> A_vec(10, 0);
matrix_cl<var> B_var(A_vec, 10, 1);
 
vari** A_vari= // fill
matrix_cl<var> B_vari(A_vari, 10, 1);

Execution is performed in async and Kernel operations are compounded and compiled Just In Time. This allows for a low amount of overhead when passing data to and from the device and executing computations.

For more details see the paper on Arvix. https://arxiv.org/pdf/1907.01063.pdf

Modules
	OpenCL Kernel Generator
	The OpenCL kernel generator is used to combine multiple matrix operations into a single OpenCL kernel.
	Matrix
	The matrix_cl class - allocates memory space on the OpenCL device.
	OpenCL Context
	Initialization for OpenCL Context:
	Error Checks
	Kernel Executor
	The kernel executor allows OpenCL kernels to be executed in async.
	Custom OpenCL kernels
	OpenCL overloads of stan/math/prim functions

Functions
template<typename T , typename = require_floating_point_t<T>>
void	stan::math::opencl::cholesky_decompose (matrix_cl< T > &A)
	Performs an in-place computation of the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix.
template<typename T , require_st_arithmetic< T > * = nullptr>
matrix_cl< scalar_type_t< T > >	stan::math::to_matrix_cl (T &&src)
	Copies the source Eigen matrix, std::vector or scalar to the destination matrix that is stored on the OpenCL device.
template<typename T_ret , typename T , require_eigen_t< T_ret > * = nullptr, require_matrix_cl_t< T > * = nullptr, require_st_same< T_ret, T > * = nullptr>
auto	stan::math::from_matrix_cl (const T &src)
	Copies the source matrix that is stored on the OpenCL device to the destination Eigen matrix.
template<typename T_ret , typename T , require_all_kernel_expressions_t< T > * = nullptr, require_not_matrix_cl_t< T > * = nullptr>
auto	stan::math::from_matrix_cl (const T &src)
	Copies result of a kernel generator expression to the specified destination type.
template<typename T_dst , typename T , require_arithmetic_t< T > * = nullptr, require_same_t< T_dst, T > * = nullptr>
T_dst	stan::math::from_matrix_cl (const matrix_cl< T > &src)
	Copy A 1 by 1 source matrix from the Device to the host.
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	stan::math::from_matrix_cl (const T &src)
	Copies the source kernel generator expression or matrix that is stored on the OpenCL device to the destination Eigen matrix.
template<typename T , require_matrix_cl_t< T > * = nullptr>
auto	stan::math::packed_copy (const T &src)
	Packs the square flat triangular matrix on the OpenCL device and copies it to the std::vector.
template<matrix_cl_view matrix_view, typename Vec , typename Vec_scalar = scalar_type_t<Vec>, require_vector_vt< std::is_arithmetic, Vec > * = nullptr>
matrix_cl< Vec_scalar >	stan::math::packed_copy (Vec &&src, int rows)
	Copies the packed triangular matrix from the source std::vector to an OpenCL buffer and unpacks it to a flat matrix on the OpenCL device.
template<typename T , require_matrix_cl_t< T > * = nullptr>
plain_type_t< T >	stan::math::copy_cl (const T &src)
	Copies the source matrix to the destination matrix.
const matrix_cl_view	stan::math::either (const matrix_cl_view left_view, const matrix_cl_view right_view)
	Determines which parts are nonzero in any of the input views.
const matrix_cl_view	stan::math::both (const matrix_cl_view left_view, const matrix_cl_view right_view)
	Determines which parts are nonzero in both input views.
bool	stan::math::contains_nonzero (const matrix_cl_view view, const matrix_cl_view part)
	Check whether a view contains certain nonzero part.
const matrix_cl_view	stan::math::transpose (const matrix_cl_view view)
	Transposes a view - swaps lower and upper parts.
const matrix_cl_view	stan::math::invert (const matrix_cl_view view)
	Inverts a view.
matrix_cl_view	stan::math::from_eigen_uplo_type (Eigen::UpLoType eigen_type)
	Creates a view from Eigen::UpLoType.
template<typename T , typename = require_arithmetic_t<T>>
matrix_cl< T >	stan::math::multiply_transpose (const matrix_cl< T > &A)
	Computes the product of a square OpenCL matrix with its transpose.
template<bool propto, typename T_x_cl , typename T_y_cl , typename T_alpha_cl , typename T_beta_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_x_cl, T_alpha_cl, T_beta_cl > * = nullptr>
return_type_t< T_x_cl, T_alpha_cl, T_beta_cl >	stan::math::bernoulli_logit_glm_lpmf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Bernoulli distribution and logit link function.
template<bool propto, typename T_n_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_prob_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	stan::math::bernoulli_lpmf (const T_n_cl &n, const T_prob_cl &theta)
	Returns the log PMF of the Bernoulli distribution.
template<bool propto, typename T_n_cl , typename T_N_cl , typename T_size1_cl , typename T_size2_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_size1_cl, T_size2_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_size1_cl, T_size2_cl > * = nullptr>
return_type_t< T_n_cl, T_size1_cl, T_size2_cl >	stan::math::beta_binomial_lpmf (const T_n_cl &n, const T_N_cl N, const T_size1_cl &alpha, const T_size2_cl &beta)
	Returns the log PMF of the Beta-Binomial distribution with given population size, prior success, and prior failure parameters.
template<bool propto, typename T_y_cl , typename T_scale_succ_cl , typename T_scale_fail_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_succ_cl, T_scale_fail_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_succ_cl, T_scale_fail_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_succ_cl, T_scale_fail_cl >	stan::math::beta_lpdf (const T_y_cl &y, const T_scale_succ_cl &alpha, const T_scale_fail_cl &beta)
	The log of the beta density for the specified scalar(s) given the specified sample stan::math::size(s).
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_prec_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_prec_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_prec_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_prec_cl >	stan::math::beta_proportion_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_prec_cl &kappa)
	The log of the beta density for specified y, location, and precision: beta_proportion_lpdf(y | mu, kappa) = beta_lpdf(y | mu * kappa, (1 - mu) * kappa).
template<bool propto, typename T_n_cl , typename T_N_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr, require_any_nonscalar_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	stan::math::binomial_logit_lpmf (const T_n_cl &n, const T_N_cl N, const T_prob_cl &alpha)
	Binomial log PMF in logit parametrization.
template<bool propto, typename T_n_cl , typename T_N_cl , typename T_prob_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr, require_any_nonscalar_prim_or_rev_kernel_expression_t< T_n_cl, T_N_cl, T_prob_cl > * = nullptr>
return_type_t< T_prob_cl >	stan::math::binomial_lpmf (const T_n_cl &n, const T_N_cl N, const T_prob_cl &theta)
	Returns the log PMF for the binomial distribution evaluated at the specified success, population size, and chance of success.
template<bool propto, typename T_y , typename T_x , typename T_alpha , typename T_beta , require_all_prim_or_rev_kernel_expression_t< T_y, T_x, T_alpha, T_beta > * = nullptr>
return_type_t< T_x, T_alpha, T_beta >	stan::math::categorical_logit_glm_lpmf (const T_y &y, const T_x &x, const T_alpha &alpha, const T_beta &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with categorical distribution and logit (softmax) link function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::cauchy_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the cauchy cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::cauchy_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the cauchy log complementary cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::cauchy_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the cauchy cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::cauchy_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s).
template<bool propto, typename T_y_cl , typename T_dof_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl >	stan::math::chi_square_lpdf (const T_y_cl &y, const T_dof_cl &nu)
	The log of a chi-squared density for y with the specified degrees of freedom parameter.
template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>
auto	stan::math::col (T_x &&x, size_t j)
	Return the specified column of the specified kernel generator expression using start-at-1 indexing.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
int	stan::math::cols (const T_x &x)
	Returns the number of columns in the specified kernel generator expression.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
void	stan::math::dims (const T_x &x, std::vector< int > &result)
	matrix_cl overload of the dims helper function in prim/fun/dims.hpp.
template<bool propto, typename T_prob_cl , typename T_prior_size_cl , require_all_prim_or_rev_kernel_expression_t< T_prob_cl, T_prior_size_cl > * = nullptr, require_any_not_stan_scalar_t< T_prob_cl, T_prior_size_cl > * = nullptr>
return_type_t< T_prob_cl, T_prior_size_cl >	stan::math::dirichlet_lpdf (const T_prob_cl &theta, const T_prior_size_cl &alpha)
	The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha.
template<typename T_a , typename = require_all_kernel_expressions_and_none_scalar_t<T_a>>
auto	stan::math::divide (T_a &&a, double d)
	Returns the elementwise division of the kernel generator expression.
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
void	stan::math::divide_columns (matrix_cl< T1 > &A, const matrix_cl< T2 > &B)
	Divides each column of a matrix by a vector.
template<typename T1 , typename T2 , typename = require_all_arithmetic_t<T1, T2>>
void	stan::math::divide_columns (matrix_cl< T1 > &A, const T2 &divisor)
	Divides each column of a matrix by a scalar.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::double_exponential_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::double_exponential_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential log complementary cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::double_exponential_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential log cumulative density function.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::double_exponential_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the double exponential log probability density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	stan::math::exp_mod_normal_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the double exponential cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	stan::math::exp_mod_normal_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the exp mod normal log complementary cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	stan::math::exp_mod_normal_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the exp mod normal log cumulative density function.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_inv_scale_cl >	stan::math::exp_mod_normal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_inv_scale_cl &lambda)
	Returns the log PMF of the exp mod normal distribution.
template<typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	stan::math::exponential_cdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	Calculates the exponential cumulative distribution function for the given y and beta.
template<typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	stan::math::exponential_lccdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	Calculates the log exponential cumulative distribution function for the given y and beta.
template<typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	stan::math::exponential_lcdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	Calculates the log exponential cumulative distribution function for the given y and beta.
template<bool propto, typename T_y_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_inv_scale_cl >	stan::math::exponential_lpdf (const T_y_cl &y, const T_inv_scale_cl &beta)
	The log of an exponential density for y with the specified inverse scale parameter.
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::frechet_cdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the frechet cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::frechet_lccdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the frechet log complementary cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::frechet_lcdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the frechet log cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::frechet_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	The log of the frechet density for the specified scalar(s) given the specified sample stan::math::size(s).
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_inv_scale_cl >	stan::math::gamma_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_inv_scale_cl &beta)
	The log of a gamma density for y with the specified shape and inverse scale parameters.
template<typename T_x , typename T_sigma , require_all_prim_or_rev_kernel_expression_t< T_x > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>
auto	stan::math::gp_dot_prod_cov (const T_x &x, const T_sigma sigma)
	Dot product kernel on the GPU.
template<typename T_x , typename T_y , typename T_sigma , require_all_prim_or_rev_kernel_expression_t< T_x, T_y > * = nullptr, require_stan_scalar_t< T_sigma > * = nullptr>
auto	stan::math::gp_dot_prod_cov (const T_x &x, const T_y &y, const T_sigma sigma)
	Dot product kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , typename = require_all_arithmetic_t<T1, T2, T3>>
matrix_cl< return_type_t< T1, T2, T3 > >	stan::math::gp_exp_quad_cov (const matrix_cl< T1 > &x, const T2 sigma, const T3 length_scale)
	Squared exponential kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , typename T4 , typename = require_all_arithmetic_t<T1, T2, T3, T4>>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	stan::math::gp_exp_quad_cov (const matrix_cl< T1 > &x, const matrix_cl< T2 > &y, const T3 sigma, const T4 length_scale)
	Squared exponential kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , require_all_kernel_expressions_and_none_scalar_t< T1 > * = nullptr, require_all_arithmetic_t< T2, T3 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3 > >	stan::math::gp_exponential_cov (const T1 &x, const T2 sigma, const T3 length_scale)
	Matern exponential kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , typename T4 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr, require_all_arithmetic_t< T3, T4 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	stan::math::gp_exponential_cov (const T1 &x, const T2 &y, const T3 sigma, const T4 length_scale)
	Matern exponential kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , require_all_kernel_expressions_and_none_scalar_t< T1 > * = nullptr, require_all_arithmetic_t< T2, T3 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3 > >	stan::math::gp_matern32_cov (const T1 &x, const T2 sigma, const T3 length_scale)
	Matern 3/2 kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , typename T4 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr, require_all_arithmetic_t< T3, T4 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	stan::math::gp_matern32_cov (const T1 &x, const T2 &y, const T3 sigma, const T4 length_scale)
	Matern 3/2 kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , require_all_kernel_expressions_and_none_scalar_t< T1 > * = nullptr, require_all_arithmetic_t< T2, T3 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3 > >	stan::math::gp_matern52_cov (const T1 &x, const T2 sigma, const T3 length_scale)
	Matern 5/2 kernel on the GPU.
template<typename T1 , typename T2 , typename T3 , typename T4 , require_all_kernel_expressions_and_none_scalar_t< T1, T2 > * = nullptr, require_all_arithmetic_t< T3, T4 > * = nullptr>
matrix_cl< return_type_t< T1, T2, T3, T4 > >	stan::math::gp_matern52_cov (const T1 &x, const T2 &y, const T3 sigma, const T4 length_scale)
	Matern 5/2 kernel on the GPU.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::gumbel_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the gumbel cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::gumbel_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the Gumbel log complementary cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::gumbel_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the Gumbel log complementary cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::gumbel_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &beta)
	Returns the Gumbel log probability density for the given location and scale.
template<bool propto, typename T_y_cl , typename T_dof_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl >	stan::math::inv_chi_square_lpdf (const T_y_cl &y, const T_dof_cl &nu)
	The log of an inverse chi-squared density for y with the specified degrees of freedom parameter.
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::inv_gamma_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &beta)
	The log of an inverse gamma density for y with the specified shape and scale parameters.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::logistic_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the logistic cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::logistic_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the logistic cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::logistic_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the logistic cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::logistic_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of a logistic density for y with the specified location and scale parameters.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::lognormal_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the loghormal cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::lognormal_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the lognormal log complementary cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::lognormal_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the lognormal log cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::lognormal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the lognormal density for the specified scalar(s) given the specified sample stan::math::size(s).
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_covar_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_covar_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_covar_cl >	stan::math::multi_normal_cholesky_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_covar_cl &L)
	The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix.
template<typename T1 , typename T2 , typename = require_all_kernel_expressions_and_none_scalar_t<T1, T2>>
matrix_cl< return_type_t< T1, T2 > >	stan::math::multiply (const T1 &A, const T2 &B)
	Computes the product of the specified matrices with the option of specifying the triangularity of either input matrices.
template<bool propto, typename T_y_cl , typename T_x_cl , typename T_alpha_cl , typename T_beta_cl , typename T_phi_cl , require_all_prim_or_rev_kernel_expression_t< T_x_cl, T_y_cl, T_alpha_cl, T_beta_cl, T_phi_cl > * = nullptr>
return_type_t< T_x_cl, T_alpha_cl, T_beta_cl, T_phi_cl >	stan::math::neg_binomial_2_log_glm_lpmf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta, const T_phi_cl &phi)
	Returns the log PMF of the Generalized Linear Model (GLM) with Negative-Binomial-2 distribution and log link function.
template<bool propto, typename T_n_cl , typename T_log_location_cl , typename T_precision_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_log_location_cl, T_precision_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_log_location_cl, T_precision_cl > * = nullptr>
return_type_t< T_n_cl, T_log_location_cl, T_precision_cl >	stan::math::neg_binomial_2_log_lpmf (const T_n_cl &n, const T_log_location_cl &eta, const T_precision_cl &phi)
	The log of the log transformed negative binomial density for the specified scalars given the specified mean(s) and deviation(s).
template<bool propto, typename T_n_cl , typename T_location_cl , typename T_precision_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_location_cl, T_precision_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_location_cl, T_precision_cl > * = nullptr>
return_type_t< T_n_cl, T_location_cl, T_precision_cl >	stan::math::neg_binomial_2_lpmf (const T_n_cl &n, const T_location_cl &mu, const T_precision_cl &phi)
	The log of the negative binomial density for the specified scalars given the specified mean(s) and deviation(s).
template<bool propto, typename T_n_cl , typename T_shape_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_shape_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_shape_cl, T_inv_scale_cl > * = nullptr>
return_type_t< T_n_cl, T_shape_cl, T_inv_scale_cl >	stan::math::neg_binomial_lpmf (const T_n_cl &n, const T_shape_cl &alpha, const T_inv_scale_cl &beta)
	The log of the negative binomial density for the specified scalars given the specified mean(s) and deviation(s).
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::normal_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the normal cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_x_cl , typename T_alpha_cl , typename T_beta_cl , typename T_sigma_cl , require_all_prim_or_rev_kernel_expression_t< T_x_cl, T_y_cl, T_alpha_cl, T_beta_cl, T_sigma_cl > * = nullptr>
return_type_t< T_y_cl, T_x_cl, T_alpha_cl, T_beta_cl, T_sigma_cl >	stan::math::normal_id_glm_lpdf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta, const T_sigma_cl &sigma)
	Returns the log PDF of the Generalized Linear Model (GLM) with Normal distribution and id link function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::normal_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the normal log complementary cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::normal_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	Returns the normal log complementary cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl >	stan::math::normal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s).
template<bool propto, typename T_y , typename T_x , typename T_beta , typename T_cuts , require_all_prim_or_rev_kernel_expression_t< T_y, T_x, T_beta, T_cuts > * = nullptr>
return_type_t< T_x, T_beta, T_cuts >	stan::math::ordered_logistic_glm_lpmf (const T_y &y, const T_x &x, const T_beta &beta, const T_cuts &cuts)
	Returns the log PMF of the ordinal regression Generalized Linear Model (GLM).
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_cuts_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_cuts_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_cuts_cl >	stan::math::ordered_logistic_lpmf (const T_y_cl &y, const T_loc_cl &lambda, const T_cuts_cl &cuts)
	Returns the (natural) log probability of the specified array of integers given the vector of continuous locations and specified cutpoints in an ordered logistic model.
template<typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_cdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	Returns the Pareto cumulative density function.
template<typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_lccdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	Returns the Pareto cumulative density function.
template<typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_lcdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	Returns the Pareto cumulative density function.
template<bool propto, typename T_y_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_lpdf (const T_y_cl &y, const T_scale_cl &y_min, const T_shape_cl &alpha)
	The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s).
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_type_2_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the pareto type 2 cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_type_2_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the pareto type 2 log complementaty cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_type_2_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the pareto type 2 log cumulative density function.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	stan::math::pareto_type_2_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &lambda, const T_shape_cl &alpha)
	Returns the log PMF of the Pareto type 2 distribution.
template<bool propto, typename T_y_cl , typename T_x_cl , typename T_alpha_cl , typename T_beta_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_x_cl, T_alpha_cl, T_beta_cl > * = nullptr>
return_type_t< T_x_cl, T_alpha_cl, T_beta_cl >	stan::math::poisson_log_glm_lpmf (const T_y_cl &y, const T_x_cl &x, const T_alpha_cl &alpha, const T_beta_cl &beta)
	Returns the log PMF of the Generalized Linear Model (GLM) with Poisson distribution and log link function.
template<bool propto, typename T_n_cl , typename T_log_rate_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_log_rate_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_log_rate_cl > * = nullptr>
return_type_t< T_log_rate_cl >	stan::math::poisson_log_lpmf (const T_n_cl &n, const T_log_rate_cl &alpha)
	Returns the log PMF of the Poisson log distribution.
template<bool propto, typename T_n_cl , typename T_rate_cl , require_all_prim_or_rev_kernel_expression_t< T_n_cl, T_rate_cl > * = nullptr, require_any_not_stan_scalar_t< T_n_cl, T_rate_cl > * = nullptr>
return_type_t< T_rate_cl >	stan::math::poisson_lpmf (const T_n_cl &n, const T_rate_cl &lambda)
	Returns the log PMF of the Poisson distribution.
template<typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	stan::math::rayleigh_cdf (const T_y_cl &y, const T_scale_cl &sigma)
	Returns the Rayleigh cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	stan::math::rayleigh_lccdf (const T_y_cl &y, const T_scale_cl &sigma)
	Returns the Rayleigh log complementary cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	stan::math::rayleigh_lcdf (const T_y_cl &y, const T_scale_cl &sigma)
	Returns the Rayleigh log cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_scale_cl >	stan::math::rayleigh_lpdf (const T_y_cl &y, const T_scale_cl &sigma)
	The log of an Rayleigh density for y with the specified scale parameter.
template<typename T , require_any_t< is_matrix_cl< T >, math::conjunction< is_var< T >, is_matrix_cl< value_type_t< T > > > > * = nullptr>
auto	stan::math::rep_array (const scalar_type_t< T > &x, int n)
	Creates a matrix_cl representing an array by replicating the input value.
template<typename T , require_matrix_cl_t< T > * = nullptr>
auto	stan::math::rep_matrix (const value_type_t< T > &x, int n, int m)
	Creates a matrix_cl by replicating the given value of arithmetic type.
template<typename T , require_arithmetic_t< T > * = nullptr>
auto	stan::math::rep_matrix (const matrix_cl< T > &x, int m)
	Creates a matrix_cl by replicating the input vector or row_vector.
template<typename T , require_any_t< is_matrix_cl< T >, math::conjunction< is_var< T >, is_matrix_cl< value_type_t< T > > > > * = nullptr>
auto	stan::math::rep_row_vector (const scalar_type_t< T > &x, int n)
	Creates a matrix_cl representing a row vector by replicating the input value.
template<typename T , require_any_t< is_matrix_cl< T >, math::conjunction< is_var< T >, is_matrix_cl< value_type_t< T > > > > * = nullptr>
auto	stan::math::rep_vector (const scalar_type_t< T > &x, int n)
	Creates a matrix_cl representing a vector by replicating the input value.
template<typename T_x , typename = require_nonscalar_prim_or_rev_kernel_expression_t<T_x>>
auto	stan::math::row (T_x &&x, size_t j)
	Return the specified row of the specified kernel generator expression using start-at-1 indexing.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
int	stan::math::rows (const T_x &x)
	Returns the number of rows in the specified kernel generator expression.
template<bool propto, typename T_y_cl , typename T_dof_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl, T_scale_cl >	stan::math::scaled_inv_chi_square_lpdf (const T_y_cl &y, const T_dof_cl &nu, const T_scale_cl &s)
	The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	stan::math::skew_double_exponential_cdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the skew double exponential cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	stan::math::skew_double_exponential_lccdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the skew double exponential cumulative density function.
template<typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	stan::math::skew_double_exponential_lcdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the skew double exponential cumulative density function.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_skewness_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_skewness_cl >	stan::math::skew_double_exponential_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_skewness_cl &tau)
	Returns the log PMF of the skew double exponential distribution.
template<bool propto, typename T_y_cl , typename T_loc_cl , typename T_scale_cl , typename T_shape_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl > * = nullptr>
return_type_t< T_y_cl, T_loc_cl, T_scale_cl, T_shape_cl >	stan::math::skew_normal_lpdf (const T_y_cl &y, const T_loc_cl &mu, const T_scale_cl &sigma, const T_shape_cl &alpha)
	The log of the skew normal density for the specified scalar(s) given the specified mean(s), deviation(s) and shape(s).
template<typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	stan::math::std_normal_cdf (const T_y_cl &y)
	Returns the standard normal cumulative distribution function.
template<typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	stan::math::std_normal_lccdf (const T_y_cl &y)
	Returns the log standard normal complementary cumulative distribution function.
template<typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	stan::math::std_normal_lcdf (const T_y_cl &y)
	Returns the log standard normal complementary cumulative distribution function.
template<bool propto, typename T_y_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl > * = nullptr>
return_type_t< T_y_cl >	stan::math::std_normal_lpdf (const T_y_cl &y)
	The log of the normal density for the specified scalar(s) given a location of 0 and a scale of 1.
template<bool propto, typename T_y_cl , typename T_dof_cl , typename T_loc_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_dof_cl, T_loc_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_dof_cl, T_loc_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_dof_cl, T_loc_cl, T_scale_cl >	stan::math::student_t_lpdf (const T_y_cl &y, const T_dof_cl &nu, const T_loc_cl &mu, const T_scale_cl &sigma)
	The log of the Student-t density for the given y, nu, mean, and scale parameter.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	stan::math::to_array_1d (T_x &&x)
	Returns input matrix reshaped into a vector.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
T_x	stan::math::to_array_2d (T_x &&x)
	Returns input matrix converted into a nested std vector.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
T_x	stan::math::to_matrix (T_x &&x)
	Returns input matrix.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	stan::math::to_row_vector (T_x &&x)
	Returns input matrix reshaped into a row vector.
template<typename T_x , require_nonscalar_prim_or_rev_kernel_expression_t< T_x > * = nullptr>
auto	stan::math::to_vector (T_x &&x)
	Returns input matrix reshaped into a vector.
template<typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	stan::math::uniform_cdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	Returns the uniform cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	stan::math::uniform_lccdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	Returns the log uniform complementary cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	stan::math::uniform_lcdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	Returns the log uniform cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_low_cl , typename T_high_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_low_cl, T_high_cl > * = nullptr>
return_type_t< T_y_cl, T_low_cl, T_high_cl >	stan::math::uniform_lpdf (const T_y_cl &y, const T_low_cl &alpha, const T_high_cl &beta)
	The log of a uniform density for the given y, lower, and upper bound.
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::weibull_cdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the weibull cumulative distribution function for the given location, and scale.
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::weibull_lccdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the weibull log cumulative complementary distribution function for the given location, and scale.
template<typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::weibull_lcdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the weibull log cumulative distribution function for the given location, and scale.
template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_scale_cl > * = nullptr>
return_type_t< T_y_cl, T_shape_cl, T_scale_cl >	stan::math::weibull_lpdf (const T_y_cl &y, const T_shape_cl &alpha, const T_scale_cl &sigma)
	Returns the Weibull log probability density for the given location and scale.
template<typename T >
var_value< matrix_cl< value_type_t< T > > >	stan::math::to_matrix_cl (const var_value< T > &a)
	Copies the source var containing Eigen matrices to destination var that has data stored on the OpenCL device.
template<typename T , require_stan_scalar_t< T > * = nullptr>
var_value< matrix_cl< value_type_t< T > > >	stan::math::to_matrix_cl (const std::vector< var_value< T > > &a)
	Copies the source std::vector of vars to a destination var that has data stored on the OpenCL device.
template<typename T , require_eigen_vt< is_var, T > * = nullptr>
var_value< matrix_cl< value_type_t< value_type_t< T > > > >	stan::math::to_matrix_cl (const T &src)
	Copies the source Eigen matrix of vars to the destination matrix that is stored on the OpenCL device.
template<typename T , require_eigen_vt< is_var, T > * = nullptr>
var_value< matrix_cl< value_type_t< value_type_t< T > > > >	stan::math::to_matrix_cl (const std::vector< T > &src)
	Copies the source vector of Eigen matrices of vars to the destination matrix that is stored on the OpenCL device.
template<typename T_dst , typename T , require_var_vt< is_eigen, T_dst > * = nullptr, require_all_kernel_expressions_t< T > * = nullptr>
T_dst	stan::math::from_matrix_cl (const var_value< T > &a)
	Copies the source var that has data stored on the OpenCL device to destination var containing Eigen matrix.
template<typename T_dst , typename T , require_std_vector_vt< is_var, T_dst > * = nullptr, require_all_stan_scalar_t< value_type_t< T_dst > > * = nullptr, require_all_kernel_expressions_t< T > * = nullptr>
T_dst	stan::math::from_matrix_cl (const var_value< T > &a)
	Copies the source var that has data stored on the OpenCL device to destination std::vector containing vars.
template<typename T , require_all_kernel_expressions_t< T > * = nullptr>
auto	stan::math::from_matrix_cl (const var_value< T > &src)
	Copies the source var that has data stored on the OpenCL device to destination Eigen matrix containing vars.
template<typename T_ret , require_var_vt< is_matrix_cl, T_ret > * = nullptr>
var_value< matrix_cl< double > >	stan::math::rep_matrix (const var &A, int n, int m)
	Creates a matrix_cl by replicating the given value of arithmetic type.
template<typename T , require_all_kernel_expressions_and_none_scalar_t< T > * = nullptr>
var_value< matrix_cl< double > >	stan::math::rep_matrix (const var_value< T > &A, int m)
	Creates a matrix_cl by replicating the input vector or row_vector.
template<matrix_cl_view matrix_view = matrix_cl_view::Entire, typename T , require_matrix_cl_st< std::is_floating_point, T > * = nullptr>
plain_type_t< T >	stan::math::tri_inverse (const T &A)
	Computes the inverse of a triangular matrix.