Implementation of Linear solvers

monolish switches between MKL, NVIDIA and OSS at build time (see here).

This chapter explains what libraries are called by matrix and vector operations.

Implementation of Linear Solvers

LAPACK is a complete direct solver for dense matrices. monolish calls LAPACK for direct solver for dense matrices.

The direct solver for sparse matrices is implemented in paradiso/mumps/cuSOLVER. monolish calls these libraries. Currently, only MKL and cuSOLVER is implemented.

Iterative solver for sparse matrix is implemented in MKL and CUDA libraries. However, the sparse matrix storage formats implemented by these libraries are different. monolish does not use these libraries. monolish has and provides an iterative solver implementation for sparse matrices.

In the future, we plan to implement a switch to call these libraries.

Implementation status (solving / preconditioning)

	Dense / MKL	Dense / NVIDIA	Dense / OSS	Sparse / MKL	Sparse / NVIDIA	Sparse / OSS	LinearOperator / MKL	LinearOperator / NVIDIA	LinearOperator / OSS
monolish::equation::LU	Y / N	Y / N	Y / N	N / N	N / N	N / N	N / N	N / N	N / N
monolish::equation::QR	N / N	N / N	N / N	N / N	Y / N	N / N	N / N	N / N	N / N
monolish::equation::Cholesky	Y / N	N / N	Y / N	N / N	Y / N	N / N	N / N	N / N	N / N
monolish::equation::CG	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	N / N	Y / N
monolish::equation::BiCGSTAB	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	N / N	Y / N
monolish::equation::Jacobi	Y / Y	Y / Y	Y / Y	Y / Y	Y / Y	Y / Y	Y / Y	N / N	Y / Y
monolish::equation::SOR*	Y / Y	Y / Y	Y / Y	Y / Y	Y / Y	Y / Y	N / N	N / N	N / N
monolish::equation::ILU	N / N	N / N	N / N	N / N	Y / Y	N / N	N / N	N / N	N / N
monolish::equation::IC	N / N	N / N	N / N	N / N	Y / Y	N / N	N / N	N / N	N / N
monolish::standard_eigen::LOBPCG	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	N / N	Y / N
monolish::standard_eigen::DC	Y / N	Y / N	Y / N	N / N	N / N	N / N	N / N	N / N	N / N
monolish::generalized_eigen::LOBPCG	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	Y / N	N / N	Y / N
monolish::generalized_eigen::DC	Y / N	Y / N	Y / N	N / N	N / N	N / N	N / N	N / N	N / N

• *) SOR is not parallelization.

What libraries are called by solvers

Direct Solvers for for Dense matrix

func	MKL	NVIDIA	OSS
LU	MKL	cuSOLVER	OpenBLAS+LAPACK
Cholesky	MKL	not impl.*	OpenBLAS+LAPACK
QR	todo) not impl.->MKL	todo) not impl.->cuSOLVER	todo) not impl. -> OpenBLAS+LAPACK

• *) Cholesky is not impl. in cuSOLVER 11.1

Direct Solvers for sparse matrix

func	MKL	NVIDIA	OSS
LU	todo) not impl.->MKL	not impl.*	todo) not impl. -> MUMPS
ILU	todo) not impl.->monolish	cuSPARSE	todo) not impl. -> ????
IC	todo) not impl.->monolish	cuSPARSE	todo) not impl. -> ????
Cholesky	todo) not impl.->MKL	cuSOLVER	todo) not impl. -> ????
QR	todo) not impl.->MKL	cuSOLVER	todo) not impl. -> ????

• *) sparse LU is not impl. in cuSOLVER 11.1

Iterative solvers (for Dense, Sparse)

func	MKL	NVIDIA	OSS
CG	monolish	monolish	monolish
BiCGSTAB	monolish	monolish	monolish
Jacobi	monolish	monolish	monolish
SOR	monolish	monolish	monolish

Preconditioners (for Dense)

func	MKL	NVIDIA	OSS
Jacobi	monolish	monolish	monolish
SOR	monolish	monolish	monolish

Preconditioners (for Sparse)

func	MKL	NVIDIA	OSS
Jacobi	monolish	monolish	monolish
SOR	monolish	monolish	monolish
ILU	not impl.	cuSPARSE	not impl.
IC	not impl.	cuSPARSE	not impl.

Standard Eigen Solvers

For dense matrix

func	MKL	NVIDIA	OSS
Devide and Conquer	MKL	cuSOLVER	OpenBLAS+LAPACK

For sparse matrix

func	MKL	NVIDIA	OSS
LOBPCG	monolish	monolish	monolish

Generalized Eigen Solvers

For dense matrix

func	MKL	NVIDIA	OSS
Devide and Conquer	MKL	cuSOLVER	OpenBLAS+LAPACK

For sparse matrix

func	MKL	NVIDIA	OSS
LOBPCG	monolish	monolish	monolish