MATLAB Function Reference
bicgstab

Syntax

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

Description

```x = bicgstab(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 `bicgstab` converges, a message to that effect is displayed. If `bicgstab` 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.

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

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

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

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

```bicgstab(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] = bicgstab(A,b,...) ``` also returns a convergence flag.

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

```[x,flag,relres,iter] = bicgstab(A,b,...) ``` also returns the iteration number at which `x` was computed, where `0 <= iter <= maxit`. `iter` can be an integer `+` 0.5, indicating convergence half way through an iteration.

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

Example

Example 1. This example first solves `Ax = b` by providing `A` and the preconditioner `M1` directly as arguments. It then solves the same system using functions that return `A` and the preconditioner.

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

x = bicgstab(A,b,tol,maxit,M1,[],[]);
```

displays this message

• ```bicgstab converged at iteration 12.5 to a solution with relative
residual 2.9e-014
```

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 `bicgstab`

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

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

Example 2. This examples demonstrates the use of a preconditioner. Start with `A = west0479`, a real 479-by-479 sparse matrix, and define `b` so that the true solution is a vector of all ones.

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

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

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

`flag1` is `2` because the upper triangular `U1` has a zero on its diagonal. This causes `bicgstab` to fail in the first iteration when it tries to solve a system such as `U1*y = r` using backslash.

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

`flag2` is `0` because `bicgstab` converges to the tolerance of `3.1757e-016` (the value of `relres2`) at the sixth 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(13) = norm(b-A*x2)`. You can follow the progress of `bicgstab` by plotting the relative residuals at the halfway point and end of each iteration starting from the initial estimate (iterate number 0).

• ``````semilogy(0:0.5:iter2,resvec2/norm(b),'-o')
```xlabel('iteration number')
ylabel('relative residual')

```

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