pracma (version 1.8.8)

gmres: Generalized Minimal Residual Method

Description

gmres(A,b) attempts to solve the system of linear equations A*x=b for x.

Usage

gmres(A, b, x0 = rep(0, length(b)), 
          errtol = 1e-6, kmax = length(b)+1, reorth = 1)

Arguments

A

square matrix.

b

numerical vector or column vector.

x0

initial iterate.

errtol

relative residual reduction factor.

kmax

maximum number of iterations

reorth

reorthogonalization method, see Details.

Value

• Returns a list with components x the solution, error the vector of residual norms, and niter the number of iterations.

Details

Iterative method for the numerical solution of a system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector.

Reorthogonalization method: 1 -- Brown/Hindmarsh condition (default) 2 -- Never reorthogonalize (not recommended) 3 -- Always reorthogonalize (not cheap!)

References

C. T. Kelley (1995). Iterative Methods for Linear and Nonlinear Equations. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, USA.

See Also

solve

Examples

A <- matrix(c(0.46, 0.60, 0.74, 0.61, 0.85,
              0.56, 0.31, 0.80, 0.94, 0.76,
              0.41, 0.19, 0.15, 0.33, 0.06,
              0.03, 0.92, 0.15, 0.56, 0.08,
              0.09, 0.06, 0.69, 0.42, 0.96), 5, 5)
x <- c(0.1, 0.3, 0.5, 0.7, 0.9)
b <- A %*% x
gmres(A, b)
# $x
#      [,1]
# [1,]  0.1
# [2,]  0.3
# [3,]  0.5
# [4,]  0.7
# [5,]  0.9
# 
# $error
# [1] 2.37446e+00 1.49173e-01 1.22147e-01 1.39901e-02 1.37817e-02 2.81713e-31
# 
# $niter
# [1] 5