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)

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.

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!)

Value

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

References

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

Author

Based on Matlab code from C. T. Kelley's book, see references.

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)
#> Warning: Recycling array of length 1 in array-vector arithmetic is deprecated.
#>   Use c() or as.vector() instead.
#> Warning: Recycling array of length 1 in array-vector arithmetic is deprecated.
#>   Use c() or as.vector() instead.
#> Warning: Recycling array of length 1 in array-vector arithmetic is deprecated.
#>   Use c() or as.vector() instead.
#> Warning: Recycling array of length 1 in array-vector arithmetic is deprecated.
#>   Use c() or as.vector() instead.
#> Warning: Recycling array of length 1 in array-vector arithmetic is deprecated.
#>   Use c() or as.vector() instead.
#> $x
#>      [,1]
#> [1,]  0.1
#> [2,]  0.3
#> [3,]  0.5
#> [4,]  0.7
#> [5,]  0.9
#> 
#> $error
#> [1] 2.374455e+00 1.491729e-01 1.221472e-01 1.399005e-02 1.378165e-02
#> [6] 9.243376e-32
#> 
#> $niter
#> [1] 5
#> 
# $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