Factor, forward and back solve. More...

Functions
template<typename scalar_t >
int64_t	slate::hetrf (HermitianMatrix< scalar_t > &A, Pivots &pivots, BandMatrix< scalar_t > &T, Pivots &pivots2, Matrix< scalar_t > &H, Options const &opts)
	Distributed parallel Hermitian indefinite \(LTL^T\) factorization.

template<typename scalar_t >
void	slate::hetrs (HermitianMatrix< scalar_t > &A, Pivots &pivots, BandMatrix< scalar_t > &T, Pivots &pivots2, Matrix< scalar_t > &B, Options const &opts)
	Distributed parallel Hermitian indefinite \(LTL^T\) solve.

Detailed Description

Factor, forward and back solve.

Function Documentation

◆ hetrf()

template<typename scalar_t >

int64_t slate::hetrf	(	HermitianMatrix< scalar_t > &	A,
		Pivots &	pivots,
		BandMatrix< scalar_t > &	T,
		Pivots &	pivots2,
		Matrix< scalar_t > &	H,
		Options const &	opts
	)

Distributed parallel Hermitian indefinite \(LTL^T\) factorization.

Computes the factorization of a Hermitian matrix \(A\) using Aasen's 2-stage algorithm. The form of the factorization is

\[ A = L T L^H, \]

if \(A\) is stored lower, where \(L\) is a product of permutation and unit lower triangular matrices, or

\[ P A P^H = U^H T U, \]

if \(A\) is stored upper, where \(U\) is a product of permutation and unit upper triangular matrices. \(T\) is a Hermitian band matrix that is LU factorized with partial pivoting.

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 matrix \(A\). On exit, if return value = 0, overwritten by the factor \(U\) or \(L\) from the factorization \(A = U^H T U\) or \(A = L T L^H\). If scalar_t is real, \(A\) can be a SymmetricMatrix object.
[out]	pivots	On exit, details of the interchanges applied to \(A\), i.e., row and column k of \(A\) were swapped with row and column pivots(k).
[out]	T	On exit, details of the LU factorization of the band matrix.
[out]	pivots2	On exit, details of the interchanges applied to \(T\), i.e., row and column k of \(T\) were swapped with row and column pivots2(k).
[out]	H	Auxiliary matrix used during the factorization. TODO: can this be made internal?
[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 band LU factorization failed on the \(i\)-th column.

◆ hetrs()

template<typename scalar_t >

void slate::hetrs	(	HermitianMatrix< scalar_t > &	A,
		Pivots &	pivots,
		BandMatrix< scalar_t > &	T,
		Pivots &	pivots2,
		Matrix< scalar_t > &	B,
		Options const &	opts
	)

Distributed parallel Hermitian indefinite \(LTL^T\) solve.

Solves a system of linear equations \(A X = B\) with a Hermitian matrix \(A\) using the factorization \(A = U^H T U\) or \(A = L T L^H\) computed by hetrf.

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