MATLAB Function Reference
cgs

Syntax

• ```x = cgs(A,b)
cgs(A,b,tol)
cgs(A,b,tol,maxit)
cgs(A,b,tol,maxit,M)
cgs(A,b,tol,maxit,M1,M2)
cgs(A,b,tol,maxit,M1,M2,x0)
cgs(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...)
[x,flag] = cgs(A,b,...)
[x,flag,relres] = cgs(A,b,...)
[x,flag,relres,iter] = cgs(A,b,...)
[x,flag,relres,iter,resvec] = cgs(A,b,...)
```

Description

```x = cgs(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`.

If `cgs` converges, a message to that effect is displayed. If `cgs` 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.

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

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

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

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

```cgs(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...) ``` passes parameters `p1,p2,...` to functions `afun(x,p1,p2,...)`, `m1fun(x,p1,p2,...)`, and `m2fun(x,p1,p2,...)`

```[x,flag] = cgs(A,b,...) ``` returns a solution `x` and a flag that describes the convergence of `cgs`.

 Flag Convergence `0` `cgs` converged to the desired tolerance `tol` within `maxit `iterations. `1` `cgs` iterated `maxit` times but did not converge. `2` Preconditioner `M` was ill-conditioned. `3` `cgs` stagnated. (Two consecutive iterates were the same.) `4` One of the scalar quantities calculated during `cgs` 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] = cgs(A,b,...) ``` also returns the relative residual `norm(b-A*x)/norm(b)`. If `flag` is `0`, then `relres <= tol`.

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

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

Examples

Example 1.

• ```A = gallery('wilk',21);
b = sum(A,2);
tol = 1e-12;  maxit = 15;
M1 = diag([10:-1:1 1 1:10]);
x = cgs(A,b,tol,maxit,M1,[],[]);
```

Alternatively, use this matrix-vector product function

• ```function y = afun(x,n)
y = [ 0;
x(1:n-1)] + [((n-1)/2:-1:0)';
(1:(n-1)/2)'] .*x + [x(2:n);
0 ];
```

and this preconditioner backsolve function

• ```function y = mfun(r,n)
y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)'];
```

as inputs to `cgs`.

• ```x1 = cgs(@afun,b,tol,maxit,@mfun,[],[],21);
```

Note that both `afun` and `mfun` must accept `cgs`'s extra input `n=21`.

Example 2.

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

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

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

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

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

`flag2` is `0` because `cgs` converges to the tolerance of `6.344e-16` (the value of `relres2`) at the fifth 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(6) = norm(b-A*x2)`. You can follow the progress of `cgs` by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0) with

• ``````semilogy(0:iter2,resvec2/norm(b),'-o')
```xlabel('iteration number')
ylabel('relative residual')

```

`bicg`, `bicgstab`, `gmres`, `lsqr`, `luinc`, `minres`, `pcg`, `qmr`, `symmlq`
`@` (function handle), `\` (backslash)