# 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

- conjgrad
- qmr