MATLAB Function Reference
qmr

Quasi-Minimal Residual method

Syntax

• ```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)
qmr(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...)
[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,...)
```

Description

```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`. `A` can be a function `afun` such that `afun(x)` returns `A*x` and `afun(x,'transp')` returns `A'*x`.

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 `mfun` such that `mfun(x)` 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.

```qmr(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...) ``` passes parameters `p1,p2,...` to functions `afun(x,p1,p2,...)` and `afun(x,p1,p2,...,'transp')` and similarly to the preconditioner functions `m1fun` and `m2fun`.

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

 Flag Convergence `0` `qmr `converged to the desired tolerance `tol` within `maxit `iterations. `1` `qmr` iterated `maxit` times but did not converge. `2` Preconditioner `M` was ill-conditioned. `3` The method stagnated. (Two consecutive iterates were the same.) `4` One of the scalar quantities calculated during `qmr` became too small or too large to continue computing.

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)`.

Examples

Example 1.

• ```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,[]);
```

Alternatively, use this matrix-vector product function

• ```function y = afun(x,n,transp_flag)
if (nargin > 2) & strcmp(transp_flag,'transp')
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);
else
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
```

as input to `qmr`

• ```x1 = qmr(@afun,b,tol,maxit,M1,M2,[],n);
```

Example 2.

• ```load west0479;
A = west0479;
b = sum(A,2);
[x,flag] = qmr(A,b)
```

`flag` is `1` because `qmr` does not converge to the default tolerance `1e-6` within the default 20 iterations.

• ```[L1,U1] = luinc(A,1e-5);
[x1,flag1] = qmr(A,b,1e-6,20,L1,U1)
```

`flag1` is `2` because the upper triangular `U1` has a zero on its diagonal, and `qmr` fails in the first iteration when it tries to solve a system such as `U1*y = r` for `y` using backslash.

• ```[L2,U2] = luinc(A,1e-6);
[x2,flag2,relres2,iter2,resvec2] = qmr(A,b,1e-15,10,L2,U2)
```

`flag2` is `0` because `qmr` converges to the tolerance of `1.6571e-016` (the value of r`elres2`) at the eighth iteration (the value of `iter2`) when preconditioned by the incomplete LU factorization with a drop tolerance of `1e-6`. `resvec2(1) = norm(b)` and `resvec2(9) = 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:iter2,resvec2/norm(b),'-o')
``````xlabel('iteration number')
``````ylabel('relative residual')
```

```

`bicg`, `bicgstab`, `cgs`, `gmres`, `lsqr`, `luinc`, `minres`, `pcg`, `symmlq`
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