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# symmlq

Symmetric LQ method

## Syntax

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

## Description

`x = symmlq(A,b)` attempts to solve the system of linear equations `A*x=b` for `x`. The `n`-by-`n` coefficient matrix `A` must be symmetric but need not be positive definite. It should also 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)` 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 `symmlq` converges, a message to that effect is displayed. If `symmlq` 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.

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

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

`symmlq(A,b,tol,maxit,M)` and `symmlq(A,b,tol,maxit,M1,M2)` use the symmetric positive definite preconditioner `M` or ```M = M1*M2``` and effectively solve the system ```inv(sqrt(M))*A*inv(sqrt(M))*y = inv(sqrt(M))*b``` for `y` and then return``` x = in(sqrt(M))*y```. If `M` is `[]` then `symmlq` applies no preconditioner. `M` can be a function handle `mfun` such that `mfun(x)` returns `M\x`.

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

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

Flag

Convergence

`0`

`symmlq` converged to the desired tolerance `tol` within `maxit `iterations.

`1`

`symmlq` iterated `maxit` times but did not converge.

`2`

Preconditioner `M` was ill-conditioned.

`3`

`symmlq` stagnated. (Two consecutive iterates were the same.)

`4`

One of the scalar quantities calculated during `symmlq` became too small or too large to continue computing.

`5`

Preconditioner `M` was not symmetric positive definite.

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

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

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

```[x,flag,relres,iter,resvec,resveccg] = symmlq(A,b,...)``` also returns a vector of estimates of the conjugate gradients residual norms at each iteration.

## Examples

### Example 1

```n = 100; on = ones(n,1); A = spdiags([-2*on 4*on -2*on],-1:1,n,n); b = sum(A,2); tol = 1e-10; maxit = 50; M1 = spdiags(4*on,0,n,n); x = symmlq(A,b,tol,maxit,M1); symmlq converged at iteration 49 to a solution with relative residual 4.3e-015```

### Example 2

This example replaces the matrix `A` in Example 1 with a handle to a matrix-vector product function `afun`. The example is contained in the function `run_symmlq` that:

• Calls `symmlq` with the function handle `@afun` as its first argument.

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

The following shows the code for `run_symmlq`:

```function x1 = run_symmlq 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 = symmlq(@afun,b,tol,maxit,M1); function y = afun(x) y = 4 * x; y(2:n) = y(2:n) - 2 * x(1:n-1); y(1:n-1) = y(1:n-1) - 2 * x(2:n); end end```

When you enter

`x1=run_symmlq;`

MATLAB® software displays the message

```symmlq converged at iteration 49 to a solution with relative residual 4.3e-015```

### Example 3

Use a symmetric indefinite matrix that fails with `pcg`.

```A = diag([20:-1:1,-1:-1:-20]); b = sum(A,2); % The true solution is the vector of all ones. x = pcg(A,b); % Errors out at the first iteration. pcg stopped at iteration 1 without converging to the desired tolerance 1e-006 because a scalar quantity became too small or too large to continue computing. The iterate returned (number 0) has relative residual 1```

However, `symmlq` can handle the indefinite matrix `A`.

```x = symmlq(A,b,1e-6,40); symmlq converged at iteration 39 to a solution with relative residual 1.3e-007```

## References

[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] Paige, C. C. and M. A. Saunders, "Solution of Sparse Indefinite Systems of Linear Equations." SIAM J. Numer. Anal., Vol.12, 1975, pp. 617-629.