quasi minimal residual method with preconditioning
[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)
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 :
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 returning A'*x
. It must have the following header :
right hand side vector
initial guess vector (default: zeros(n,1))
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
.
must only be provided when M1
is a function returning M1*x
.
In this case M1p
is the function which returns M1'*x
.
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
.
must only be provided when M2
is a function returning M2*x
.
In this case M2p
is the function which returns M2'*x
maximum number of iterations (default: n)
error tolerance (default: 1000*%eps)
solution vector
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.
residual vector
final residual norm
number of iterations performed
Solves the linear system Ax=b
using the Quasi Minimal Residual Method with preconditioning.
// 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) | ![]() | ![]() |
Version | Description | ||||
5.4.0 | Calling qmr(A, Ap) is deprecated. qmr(A) should be used instead. The following function is an example :
|