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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 sparsefunction.If A is a function which returnsA*x, it must have the following header :function y=A(x)
If A is a function which returns
A*xorA'*xdepending t. Ift = "notransp", the function returnsA*x. Ift = "transp", the function returnsA'*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
M1is a function, she returns either,only
M1*xM1*xorM1'*xdependingt.
or
- M1p
must only be provided when
M1is a function returningM1*x. In this caseM1pis the function which returnsM1'*x.- M2
right preconditioner : matrix or function (In the first case, default: eye(n,n)). If
M2is a function, she returns eitheronly
M2*xM2*xorM2'*xdependingt.
or
- M2p
must only be provided when
M2is a function returningM2*x. In this caseM2pis the function which returnsM2'*x- maxi
maximum number of iterations (default: n)
- tol
error tolerance (default: 1000*%eps)
- x
solution vector
- flag
flag=0:qmrconverged to the desired tolerance withinmaxiiterations,flag=1: no convergence givenmaxi,-7 < flag < 0: A breakdown occurred because one of the scalar quantities calculated duringqmrwas 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)
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 |
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