qmr

quasi minimal resiqual method with preconditioning

Calling Sequence

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

Arguments

:A matrix of size n-by-n or function.

• `matrix.`If A is a matrix, it can be dense or sparse
• 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 t. If `t =
"notransp"`, the function returns `A*x`. If `t = "transp"`, the
function returns `A'*x`. It must have the following header :

function y=A(x, t)

: :Ap function returning A’*x. It must have the followinf 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, she returns either,

• only M1*x or
• M1*x or M1’*x depending 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, she returns either

• only M2*x or
• M2*x or M2’*x depending 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

:0 = gmres converged to the desired tolerance within maxi

iterations

: :1 = no convergence given maxi :

: :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

 [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

 [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
 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

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

See Also

• gmres Generalized Minimum RESidual method
• pcg precondioned conjugate gradient

History

Version Description 5.4.0 Calling qmr(A, Ap) is deprecated. qmr(A) should be used instead. The following function is an example :

function y=A(x, t)
Amat = `eye`_(2,2);
if ( t== "notransp") then
y = Amat*x;
elseif (t == "transp") then
y = Amat'*x;
end
endfunction