gmres

Generalized Minimum RESidual method

Calling Sequence

[x,flag,err,iter,res] = gmres(A,b,[rstr,[tol,[maxi,[M,[x0]]]]])

Arguments

:A n-by-n matrix or function returning A*x. If A is a function, it must have the following header :

function y=A(x)

: :b right hand side vector : :x0 initial guess vector (default: zeros(n,1)) : :M preconditioner: matrix of size n-by-n or function returning M*x (In the first case, default: eye(n,n)). If M is a function, it must have the following header :

function y=M(x)

: :rstr number of iterations between restarts (default: 10) : :maxi maximum number of iterations (default: n) : :tol error tolerance (default: 1e-6) : :x solution vector : :err final residual norm : :iter number of iterations performed : :flag

:0 = gmres converged to the desired tolerance within maxi

iterations

: :1 = no convergence given maxi :

: :res residual vector :

Description

:GMRES solves the linear system Ax=b using the Generalized Minimal

residual method with restarts.

: :Details of this algorithm are described in : “Templates for the

Solution of Linear Systems: Building Blocks for Iterative Methods”, Barrett, Berry, Chan, Demmel, Donato, Dongarra, Eijkhout, Pozo, Romine, and Van der Vorst, SIAM Publications, 1993 (ftp netlib2.cs.utk.edu; cd linalg; get templates.ps). “Iterative Methods for Sparse Linear Systems, Second Edition” Saad, SIAM Publications, 2003 (ftp ftp.cs.umn.edu; cd dept/users/saad/PS; get all_ps.zip).

:

Examples

// If A and M are matrices
A=[ 94  0   0   0    0   28  0   0   32  0
    0   59  13  5    0   0   0   10  0   0
    0   13  72  34   2   0   0   0   0   65
    0   5   34  114  0   0   0   0   0   55
    0   0   2   0    70  0   28  32  12  0
    28  0   0   0    0   87  20  0   33  0
    0   0   0   0    28  20  71  39  0   0
    0   10  0   0    32  0   39  46  8   0
    32  0   0   0    12  33  0   8   82  11
    0   0   65  55   0   0   0   0   11  100];
b=`ones`_(10,1);
[x,flag,err,iter,res] = gmres(A, b)

M = `eye`_(10, 10);

[x,flag,err,iter,res] = gmres(A, b, 10, 1d-12, 20, M, `zeros`_(10, 1))

// If A is a matrix and M is a function
A=[ 94  0   0   0    0   28  0   0   32  0
    0   59  13  5    0   0   0   10  0   0
    0   13  72  34   2   0   0   0   0   65
    0   5   34  114  0   0   0   0   0   55
    0   0   2   0    70  0   28  32  12  0
    28  0   0   0    0   87  20  0   33  0
    0   0   0   0    28  20  71  39  0   0
    0   10  0   0    32  0   39  46  8   0
    32  0   0   0    12  33  0   8   82  11
    0   0   65  55   0   0   0   0   11  100];
b=`ones`_(10,1);

function y=Mtimesx(x)
M = `eye`_(10,10);
y = M*x;
endfunction

[x,flag,err,iter,res] = gmres(A, b, 10, 1d-12, 20, Mtimesx, `zeros`_(10, 1))

// If A is a function and M is a matrix
function y=Atimesx(x)
A=[ 94  0   0   0    0   28  0   0   32  0
    0   59  13  5    0   0   0   10  0   0
    0   13  72  34   2   0   0   0   0   65
    0   5   34  114  0   0   0   0   0   55
    0   0   2   0    70  0   28  32  12  0
    28  0   0   0    0   87  20  0   33  0
    0   0   0   0    28  20  71  39  0   0
    0   10  0   0    32  0   39  46  8   0
    32  0   0   0    12  33  0   8   82  11
    0   0   65  55   0   0   0   0   11  100];
 y = A * x;
 endfunction

 b = `ones`_(10,1);
 M = `eye`_(10, 10);

 [x,flag,err,iter,res] = gmres(Atimesx, b)

 [x,flag,err,iter,res] = gmres(Atimesx, b, 10, 1d-12, 20, M, `zeros`_(10,1))

 // If A and M are functions
 function y=Atimesx(x)
A=[ 94  0   0   0    0   28  0   0   32  0
    0   59  13  5    0   0   0   10  0   0
    0   13  72  34   2   0   0   0   0   65
    0   5   34  114  0   0   0   0   0   55
    0   0   2   0    70  0   28  32  12  0
    28  0   0   0    0   87  20  0   33  0
    0   0   0   0    28  20  71  39  0   0
    0   10  0   0    32  0   39  46  8   0
    32  0   0   0    12  33  0   8   82  11
    0   0   65  55   0   0   0   0   11  100];
 y = A * x;
 endfunction

 function y=Mtimesx(x)
M = `eye`_(10,10);
y = M*x;
endfunction

 [x,flag,err,iter,res] = gmres(Atimesx, b, 10, 1d-12, 20, Mtimesx, `zeros`_(10,1))

See Also

• pcg precondioned conjugate gradient
• qmr quasi minimal resiqual method with preconditioning