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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)
[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)```

• gmres — Generalized Minimum RESidual method

### History

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