scholarly journals NSPCG (Nonsymmetric Preconditioned Conjugate Gradient) user's guide: Version 1. 0: A package for solving large sparse linear systems by various iterative methods

1988 ◽  
Author(s):  
T Oppe ◽  
W Joubert ◽  
D Kincaid
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Conjugate Gradient ◽  
Preconditioned Conjugate Gradient ◽  
Sparse Linear Systems

scholarly journals Parallel Preconditioned Conjugate Gradient Square Method Based on Normalized Approximate Inverses

2005 ◽  
Vol 13 (2) ◽  
pp. 79-91 ◽  
Author(s):  
George A. Gravvanis ◽  
Konstantinos M. Giannoutakis
Keyword(s):  
Linear Systems ◽  
Conjugate Gradient ◽  
Message Passing ◽  
Message Passing Interface ◽  
Distributed Memory ◽  
Inverse Matrix ◽  
Preconditioned Conjugate Gradient ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Approximate Inverse Matrix Techniques

A new class of normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. A new parallel normalized explicit preconditioned conjugate gradient square method in conjunction with normalized approximate inverse matrix techniques for solving efficiently sparse linear systems on distributed memory systems, using Message Passing Interface (MPI) communication library, is also presented along with theoretical estimates on speedups and efficiency. The implementation and performance on a distributed memory MIMD machine, using Message Passing Interface (MPI) is also investigated. Applications on characteristic initial/boundary value problems in three dimensions are discussed and numerical results are given.

ON THE RATE OF CONVERGENCE AND COMPLEXITY OF NORMALIZED IMPLICIT PRECONDITIONING FOR SOLVING FINITE DIFFERENCE EQUATIONS IN THREE SPACE VARIABLES

2004 ◽  
Vol 01 (02) ◽  
pp. 367-386 ◽  
Author(s):  
GEORGE A. GRAVVANIS ◽  
KONSTANTINOS M. GIANNOUTAKIS
Keyword(s):  
Finite Difference ◽  
Rate Of Convergence ◽  
Linear Systems ◽  
Conjugate Gradient ◽  
Three Dimensional ◽  
Preconditioned Conjugate Gradient ◽  
Sparse Linear Systems ◽  
Approximate Factorization ◽  
Dimensional Boundary ◽  
Three Space

Normalized approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference method of partial differential equations in three space variables, are presented. Normalized implicit preconditioned conjugate gradient-type schemes in conjunction with normalized approximate factorization procedures are presented for the efficient solution of sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized implicit preconditioned conjugate gradient method are also given. Application of the proposed method on characteristic three dimensional boundary value problems is discussed and numerical results are given.

scholarly journals Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

2012 ◽  
Vol 20 (3) ◽  
pp. 241-255 ◽  
Author(s):  
Eric Bavier ◽  
Mark Hoemmen ◽  
Sivasankaran Rajamanickam ◽  
Heidi Thornquist
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
Direct Methods ◽  
Iterative Solvers ◽  
Software Project ◽  
Sparse Linear Systems ◽  
Scalar Type ◽  
Mixed Precision

Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.

scholarly journals Analysis of Monte Carlo accelerated iterative methods for sparse linear systems

2017 ◽  
Vol 24 (3) ◽  
pp. e2088 ◽  
Author(s):  
Michele Benzi ◽  
Thomas M. Evans ◽  
Steven P. Hamilton ◽  
Massimiliano Lupo Pasini ◽  
Stuart R. Slattery
Keyword(s):  
Monte Carlo ◽  
Linear Systems ◽  
Iterative Methods ◽  
Sparse Linear Systems
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