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
orA'*x
depending on a option t, it must have the following header:function y = A(x, t)
- If
t = "notransp"
: the function returnsA*x
. - If
t = "transp"
: the function returnsA'*x
.
- If
- 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
orM1'*x
, depending ont
.- M2
right preconditioner: matrix or function (In the first case, default: eye(n,n)). If
M2
is a function, it returns:M2*x
orM2'*x
depending ont
.- 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 withinmaxi
iterations.flag
=1: no convergence up tomaxi
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. |
Report an issue | ||
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