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qmr

quasi minimal residual method with preconditioning

Syntax

[x, flag, err, iter, res] = qmr(A, b, x0, M1, M2, maxi, tol)

Arguments

A

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

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.

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: M1*x or M1'*x, depending on t.

M2

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

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)

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)

If A, M1, M2 are functions:

// See functions defined above in previous examples. Then,
[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.
2023.0.0 Calling qmr(A, Ap) is removed.
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Last updated:
Mon Mar 27 11:52:44 GMT 2023