quasi minimal residual method with preconditioning
[x, flag, err, iter, res] = qmr(A, b, x0, M1, M2, maxi, tol)
Square dense or sparse matrix of size n-by-n, or function.
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)
t = "notransp"
: the function returns
A*x
.t = "transp"
: the function returns
A'*x
.left preconditioner: matrix or function (In the first case, default: eye(n,n)).
If M1
is a function, it returns: M1*x
or M1'*x
, depending on t
.
right preconditioner: matrix or function (In the first case, default: eye(n,n)).
If M2
is a function, it returns: M2*x
or M2'*x
depending on t
.
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.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 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) | ![]() | ![]() |
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. |