Generalized Minimal Residual Method
gmres(A,b) attempts to solve the system of linear equations
A*x=b for x.
gmres(A, b, x0 = rep(0, length(b)),
errtol = 1e-6, kmax = length(b)+1, reorth = 1)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. |
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!)
Returns a list with components x the solution, error the
vector of residual norms, and niter the number of iterations.
Based on Matlab code from C. T. Kelley's book, see references.
C. T. Kelley (1995). Iterative Methods for Linear and Nonlinear Equations. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, USA.
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] 5Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.