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Aide de Scilab >> Matrices creuses > Linear Equations (Iterative Solvers) > qmr

qmr

quasi minimal residual method with preconditioning

Syntax

[x, flag, err, iter, res] = qmr(A, b, x0, M1, M2, maxi, tol)
[x, flag, err, iter, res] = qmr(A,Ap,b,x0,M1,M1p,M2,M2p,maxi,tol) // deprecated

Arguments

A
Square dense or sparse matrix of size n-by-n, or function:

  • If A is a function which returns A*x, it must have the following header :

    function y = A ( x )
  • If A is a function which returns A*x or A'*x depending on a option t, it must have the following header :

    function y = A(x, t)

    • If t = "notransp": the function returns A*x.
    • If t = "transp": the function returns A'*x.

Ap
function returning A'*x. It must have the following header :

function y = Ap(x)

b
right hand side vector

x0
initial guess vector (default: zeros(n,1)).

M1
left preconditioner : matrix or function (In the first case, default: eye(n,n)). If M1 is a function, it returns either,
  • only M1*x, or
  • M1*x or M1'*x, depending on t.

M1p
must only be provided when M1 is a function returning M1*x. In this case M1p is the function which returns M1'*x.

M2
right preconditioner : matrix or function (In the first case, default: eye(n,n)). If M2 is a function, it returns either
  • only M2*x, or
  • M2*x or M2'*x depending on an option t.

M2p
must only be provided when M2 is a function returning M2*x. In this case M2p is the function which returns M2'*x

maxi
maximum number of iterations (default: n)

tol
error tolerance (default: 1000*%eps)

x
solution vector.

flag
  • flag=0: qmr converged to the desired tolerance within maxi iterations.
  • flag=1: no convergence up to maxi iterations,
  • -7 < flag < 0: A breakdown occurred because one of the scalar quantities calculated was equal to zero.

res
residual vector.

err
final residual norm.

iter
number of iterations performed.

Description

Solves the linear system Ax=b using the Quasi Minimal Residual Method with preconditioning.

Examples

If A is a matrix:

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] = qmr(A, b)

[x,flag,err,iter,res] = qmr(A, b, zeros(10,1), eye(10,10), eye(10,10), 10, 1d-12)

If A is a function:

function y=Atimesx(x, t)
    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];
     if (t == 'notransp') then
        y = A*x;
    elseif (t ==  'transp') then
        y = A'*x;
    end
endfunction
b = ones(10,1);

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

[x,flag,err,iter,res] = qmr(Atimesx, b, zeros(10,1), eye(10,10), eye(10,10), 10, 1d-12)

OR

function y=funA(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=funAp(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);

[x,flag,err,iter,res] = qmr(funA, funAp, b)

[x,flag,err,iter,res] = qmr(funA, funAp, b, zeros(10,1), eye(10,10), eye(10,10), 10, 1d-12)

If A is a matrix, M1 and M2 are functions:

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=M1timesx(x, t)
    M1 = eye(10,10);
    if(t=="notransp") then
        y = M1*x;
    elseif (t=="transp") then
        y = M1'*x;
    end
endfunction

function y=M2timesx(x, t)
    M2 = eye(10,10);
    if(t=="notransp") then
        y = M2*x;
    elseif (t=="transp") then
        y = M2'*x;
    end
endfunction

[x, flag, err, iter, res] = qmr(A, b, zeros(10,1), M1timesx, M2timesx, 10, 1d-12)

OR

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=funM1(x)
    M1 = eye(10,10);
    y = M1*x;
endfunction

function y=funM1p(x)
    M1 = eye(10,10);
    y = M1'*x;
endfunction

function y=funM2(x)
    M2 = eye(10,10);
    y = M2*x;
endfunction

function y=funM2p(x)
    M2 = eye(10,10);
    y = M2'*x;
endfunction

[x,flag,err,iter,res] = qmr(A, b, zeros(10,1), funM1, funM1p, funM2, funM2p, 10, 1d-12)

If A, M1, M2 are functions:

// See functions defined above in previous examples. Then,

[x,flag,err,iter,res] = qmr(funA, funAp, b, zeros(10,1), funM1, funM1p, funM2, funM2p, 10, 1d-12)
// or
[x,flag,err,iter,res] = qmr(Atimesx, b, zeros(10,1), M1timesx, M2timesx, 10, 1d-12)

See also

  • gmres — Generalized Minimum RESidual method
  • conjgrad — conjugate gradient solvers

History

VersionDescription
5.4.0 Calling qmr(A, Ap) is deprecated. qmr(A) should be used instead.
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