x = qmr(A, b)
x = qmr(A, b, tol, maxit)
x = qmr(A, b, tol, maxit, M1, M2, x0)
[x, flag, relres, iter, resvec] = qmr(...)
| Parameter | Description |
|---|---|
| A | sparse square coefficient matrix. |
| b | right-hand side vector. |
| tol | relative residual tolerance. Default is 1e-6. |
| maxit | maximum number of iterations. |
| M1, M2 | optional preconditioners: sparse or full square matrices, diagonal vectors, or function handles that apply the preconditioner or its transpose to one vector. |
| x0 | initial guess. |
| Parameter | Description |
|---|---|
| x | computed solution. |
| flag | 0 if convergence was reached, 1 if maxit was reached, 4 on numerical breakdown. |
| relres | relative residual norm. |
| iter | number of iterations performed. |
| resvec | residual norm history. |
qmr solves A*x = b using the quasi-minimal residual method.
The method is intended for sparse nonsymmetric systems. It supports sparse and full matrix preconditioners, diagonal vector preconditioners, and function handle preconditioners.
When M1 or M2 is a matrix, the solver applies it through an internal linear solve. A vector preconditioner is interpreted as the diagonal of a square preconditioner. A function handle preconditioner must accept a vector and a transpose flag, and return a vector with the same length.
Sparse single and sparse single complex matrices are supported. If M1, M2, or x0 is complex, the computation uses the matching complex solver path.
A = sparse([4 1 0; 2 3 1; 0 1 2]);
b = [1; 2; 3];
[x, flag, relres, iter, resvec] = qmr(A, b, 1e-12, 20)A = sparse([4 1 0; 2 3 1; 0 1 2]);
b = [1; 2; 3];
M = diag(diag(full(A)));
[x, flag] = qmr(A, b, 1e-12, 20, M)A = sparse([4 1; 2 3]);
b = [5; 5];
M1 = [2 0; 0 1];
M2 = [2 0.5; 2 3];
[x, flag, relres, iter] = qmr(A, b, 1e-12, 10, M1, M2)A = sparse(single([4 1i; 2 3]));
b = single([1; 2]);
[x, flag] = qmr(A, b, 1e-6, 20)| Version | Description |
|---|---|
| 2.0.0 | initial version |
| 2.0.0 | sparse single and sparse single complex inputs, matrix preconditioners, and function handle preconditioners supported. |