Computational

Factor, forward and back solve, invert. More...

Functions
template<typename scalar_t >
int64_t	slate::potrf (HermitianMatrix< scalar_t > &A, Options const &opts)
	Distributed parallel Cholesky factorization.
template<typename scalar_t >
void	slate::potri (HermitianMatrix< scalar_t > &A, Options const &opts)
	Distributed parallel Cholesky inversion.
template<typename scalar_t >
void	slate::potrs (HermitianMatrix< scalar_t > &A, Matrix< scalar_t > &B, Options const &opts)
	Distributed parallel Cholesky solve.

Detailed Description

Factor, forward and back solve, invert.

Function Documentation

potrf()

template<typename scalar_t >

int64_t slate::potrf	(	HermitianMatrix< scalar_t > &	A,
		Options const &	opts
	)

Distributed parallel Cholesky factorization.

Performs the Cholesky factorization of a Hermitian positive definite matrix \(A\).

The factorization has the form

\[ A = L L^H, \]

if \(A\) is stored lower, where \(L\) is a lower triangular matrix, or

\[ A = U^H U, \]

if \(A\) is stored upper, where \(U\) is an upper triangular matrix.

Complexity (in real): \(\approx \frac{1}{3} n^{3}\) flops.

Template Parameters

scalar_t

One of float, double, std::complex<float>, std::complex<double>.

Parameters

[in,out]	A	On entry, the n-by-n Hermitian positive definite matrix \(A\). On exit, if return value = 0, the factor \(U\) or \(L\) from the Cholesky factorization \(A = U^H U\) or \(A = L L^H\). If scalar_t is real, \(A\) can be a SymmetricMatrix object.
[in]	opts	Additional options, as map of name = value pairs. Possible options: Option::Lookahead: Number of panels to overlap with matrix updates. lookahead >= 0. Default 1. Option::Target: Implementation to target. Possible values: HostTask: OpenMP tasks on CPU host [default]. HostNest: nested OpenMP parallel for loop on CPU host. HostBatch: batched BLAS on CPU host. Devices: batched BLAS on GPU device.

Returns

0: successful exit

i > 0: the leading minor of order \(i\) of \(A\) is not positive definite, so the factorization could not be completed.

potri()

template<typename scalar_t >

void slate::potri	(	HermitianMatrix< scalar_t > &	A,
		Options const &	opts
	)

Distributed parallel Cholesky inversion.

Computes the inverse of a complex Hermitian positive definite matrix \(A\) using the Cholesky factorization \(A = U^H U\) or \(A = L L^H\) computed by potrf.

Complexity (in real): \(\approx \frac{2}{3} n^{3}\) flops.

Template Parameters

scalar_t

One of float, double, std::complex<float>, std::complex<double>.

Parameters

[in,out]	A	On entry, the triangular factor \(U\) or \(L\) from the Cholesky factorization \(A = U^H U\) or \(A = L L^H\), as computed by potrf. On exit, the upper or lower triangle of the (Hermitian) inverse of \(A\), overwriting the input factor \(U\) or \(L\). If scalar_t is real, \(A\) can be a SymmetricMatrix object.
[in]	opts	Additional options, as map of name = value pairs. Possible options: Option::Lookahead: Number of panels to overlap with matrix updates. lookahead >= 0. Default 1. Option::Target: Implementation to target. Possible values: HostTask: OpenMP tasks on CPU host [default]. HostNest: nested OpenMP parallel for loop on CPU host. HostBatch: batched BLAS on CPU host. Devices: batched BLAS on GPU device.

TODO: return value

Return values

0	successful exit
>0	for return value = \(i\), \(A(i,i)\) is exactly zero. The triangular matrix is singular and its inverse can not be computed.

potrs()

template<typename scalar_t >

void slate::potrs	(	HermitianMatrix< scalar_t > &	A,
		Matrix< scalar_t > &	B,
		Options const &	opts
	)

Distributed parallel Cholesky solve.

Solves a system of linear equations

\[ A X = B \]

with a Hermitian positive definite matrix \(A\) using the Cholesky factorization \(A = U^H U\) or \(A = L L^H\) computed by potrf.

Complexity (in real): \(2 n^{2} r\) flops.

Template Parameters

scalar_t

One of float, double, std::complex<float>, std::complex<double>.

Parameters

[in]	A	The n-by-n triangular factor \(U\) or \(L\) from the Cholesky factorization \(A = U^H U\) or \(A = L L^H\), computed by potrf. If scalar_t is real, \(A\) can be a SymmetricMatrix object.
[in,out]	B	On entry, the n-by-nrhs right hand side matrix \(B\). On exit, the n-by-nrhs solution matrix \(X\).
[in]	opts	Additional options, as map of name = value pairs. Possible options: Option::Lookahead: Number of panels to overlap with matrix updates. lookahead >= 0. Default 1. Option::Target: Implementation to target. Possible values: HostTask: OpenMP tasks on CPU host [default]. HostNest: nested OpenMP parallel for loop on CPU host. HostBatch: batched BLAS on CPU host. Devices: batched BLAS on GPU device.