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# qmr

quasi minimal residual method with preconditioning

### Syntax

[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 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, 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
• flag=0: qmr converged to the desired tolerance within maxi iterations,

• flag=1: no convergence given maxi,

• -7 < flag < 0: A breakdown occurred because one of the scalar quantities calculated during qmr 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

[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)