qmr

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)

History

Version	Description
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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qmr

Syntax

Arguments

Description

Examples

See also

History