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See the recommended documentation of this function

# gmres

Generalized Minimum RESidual method

### Syntax

[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))