## Documentation Center |

Quasi-minimal residual method

`x = qmr(A,b)qmr(A,b,tol)qmr(A,b,tol,maxit)qmr(A,b,tol,maxit,M)qmr(A,b,tol,maxit,M1,M2)qmr(A,b,tol,maxit,M1,M2,x0)[x,flag] = qmr(A,b,...)[x,flag,relres] = qmr(A,b,...)[x,flag,relres,iter] = qmr(A,b,...)[x,flag,relres,iter,resvec] = qmr(A,b,...)`

`x = qmr(A,b)` attempts to
solve the system of linear equations `A*x=b` for `x`.
The `n`-by-`n` coefficient matrix `A` must
be square and should be large and sparse. The column vector `b` must
have length `n`. You can specify `A` as
a function handle, `afun`, such that `afun(x,'notransp')` returns `A*x` and `afun(x,'transp')` returns `A'*x`.

Parameterizing Functions explains how to provide additional
parameters to the function `afun`, as well as the
preconditioner function `mfun` described below, if
necessary.

If `qmr` converges, a message to that effect
is displayed. If `qmr` fails to converge after the
maximum number of iterations or halts for any reason, a warning message
is printed displaying the relative residual `norm(b-A*x)/norm(b)` and
the iteration number at which the method stopped or failed.

`qmr(A,b,tol)` specifies
the tolerance of the method. If `tol` is `[]`,
then `qmr` uses the default, `1e-6`.

`qmr(A,b,tol,maxit)` specifies
the maximum number of iterations. If `maxit` is `[]`,
then `qmr` uses the default, `min(n,20)`.

`qmr(A,b,tol,maxit,M)` and `qmr(A,b,tol,maxit,M1,M2)` use
preconditioners `M` or `M = M1*M2` and
effectively solve the system `inv(M)*A*x = inv(M)*b` for `x`.
If `M` is `[]` then `qmr` applies
no preconditioner. `M` can be a function handle `mfun` such
that `mfun(x,'notransp')` returns `M\x` and `mfun(x,'transp')` returns `M'\x`.

`qmr(A,b,tol,maxit,M1,M2,x0)` specifies
the initial guess. If `x0` is `[]`,
then `qmr` uses the default, an all zero vector.

`[x,flag] = qmr(A,b,...)` also
returns a convergence flag.

Flag | Convergence |
---|---|

| |

| |

Preconditioner | |

The method stagnated. (Two consecutive iterates were the same.) | |

One of the scalar quantities calculated during |

Whenever `flag` is not `0`,
the solution `x` returned is that with minimal norm
residual computed over all the iterations. No messages are displayed
if the `flag` output is specified.

`[x,flag,relres] = qmr(A,b,...)`
also returns the relative residual `norm(b-A*x)/norm(b)`.
If `flag` is `0`, `relres
<= tol`.

`[x,flag,relres,iter] = qmr(A,b,...)` also
returns the iteration number at which `x` was computed,
where `0 <= iter <= maxit`.

`[x,flag,relres,iter,resvec] = qmr(A,b,...)` also
returns a vector of the residual norms at each iteration, including `norm(b-A*x0)`.

This example shows how to use `qmr` with
a matrix input. The code:

n = 100; on = ones(n,1); A = spdiags([-2*on 4*on -on],-1:1,n,n); b = sum(A,2); tol = 1e-8; maxit = 15; M1 = spdiags([on/(-2) on],-1:0,n,n); M2 = spdiags([4*on -on],0:1,n,n); x = qmr(A,b,tol,maxit,M1,M2);

displays the message:

qmr converged at iteration 9 to a solution... with relative residual 5.6e-009

This example replaces the matrix `A` in the
previous example with a handle to a matrix-vector product function `afun`.
The example is contained in a file `run_qmr` that

Calls

`qmr`with the function handle`@afun`as its first argument.Contains

`afun`as a nested function, so that all variables in`run_qmr`are available to`afun`.

The following shows the code for `run_qmr`:

function x1 = run_qmr n = 100; on = ones(n,1); A = spdiags([-2*on 4*on -on],-1:1,n,n); b = sum(A,2); tol = 1e-8; maxit = 15; M1 = spdiags([on/(-2) on],-1:0,n,n); M2 = spdiags([4*on -on],0:1,n,n); x1 = qmr(@afun,b,tol,maxit,M1,M2); function y = afun(x,transp_flag) if strcmp(transp_flag,'transp') % y = A'*x y = 4 * x; y(1:n-1) = y(1:n-1) - 2 * x(2:n); y(2:n) = y(2:n) - x(1:n-1); elseif strcmp(transp_flag,'notransp') % y = A*x y = 4 * x; y(2:n) = y(2:n) - 2 * x(1:n-1); y(1:n-1) = y(1:n-1) - x(2:n); end end end

When you enter

x1=run_qmr;

MATLAB^{®} software displays the message

qmr converged at iteration 9 to a solution with relative residual 5.6e-009

This example demonstrates the use of a preconditioner.

Load `A = west0479`, a real 479-by-479 nonsymmetric sparse matrix.

```
load west0479;
A = west0479;
```

Define `b` so that the true solution is a vector of all ones.

b = full(sum(A,2));

Set the tolerance and maximum number of iterations.

tol = 1e-12; maxit = 20;

Use `qmr` to find a solution at the requested tolerance and number of iterations.

[x0,fl0,rr0,it0,rv0] = qmr(A,b,tol,maxit);

`fl0` is 1 because `qmr` does not converge to the requested tolerance `1e-12` within the requested 20 iterations. The seventeenth iterate is the best approximate solution and is the one returned as indicated by `it0 = 17`. MATLAB stores the residual history in `rv0`.

Plot the behavior of `qmr`.

semilogy(0:maxit,rv0/norm(b),'-o'); xlabel('Iteration number'); ylabel('Relative residual');

The plot shows that the solution does not converge. You can use a preconditioner to improve the outcome.

Create the preconditioner with `ilu`, since the matrix `A` is nonsymmetric.

[L,U] = ilu(A,struct('type','ilutp','droptol',1e-5));

Error using ilu There is a pivot equal to zero. Consider decreasing the drop tolerance or consider using the 'udiag' option.

MATLAB cannot construct the incomplete LU as it would result in a singular factor, which is useless as a preconditioner.

You can try again with a reduced drop tolerance, as indicated by the error message.

[L,U] = ilu(A,struct('type','ilutp','droptol',1e-6)); [x1,fl1,rr1,it1,rv1] = qmr(A,b,tol,maxit,L,U);

`fl1` is 0 because `qmr` drives the relative residual to `4.1410e-014` (the value of `rr1`). The relative residual is less than the prescribed tolerance of `1e-12` at the sixth iteration (the value of `it1`) when preconditioned by the incomplete LU factorization with a drop tolerance of `1e-6`. The output `rv1(1)` is `norm(b)`, and the output `rv1(7)` is `norm(b-A*x2)`.

You can follow the progress of `qmr` by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0).

semilogy(0:it1,rv1/norm(b),'-o'); xlabel('Iteration number'); ylabel('Relative residual');

[1] Barrett, R., M. Berry, T. F. Chan, et
al., *Templates for the Solution of Linear Systems: Building
Blocks for Iterative Methods*, SIAM, Philadelphia, 1994.

[2] Freund, Roland W. and Nöel M. Nachtigal,
"QMR: A quasi-minimal residual method for non-Hermitian linear
systems," *SIAM Journal: Numer. Math.* 60,
1991, pp. 315–339.

`bicg` | `bicgstab` | `cgs` | `function_handle` | `gmres` | `ilu` | `lsqr` | `minres` | `mldivide` | `pcg` | `symmlq`

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