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Biconjugate gradients stabilized method

`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)[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,...)`

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

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

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

| |

| |

Preconditioner | |

| |

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] = 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 halfway 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)`.

This example first solves `Ax = b` by providing `A` and
the preconditioner `M1` directly as arguments.

The code:

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 the message:

bicgstab converged at iteration 12.5 to a solution with relative residual 2e-014.

This example replaces the matrix `A` in the
previous example with a handle to a matrix-vector product function `afun`,
and the preconditioner `M1` with a handle to a backsolve
function `mfun`. The example is contained in a file `run_bicgstab` that

Calls

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

`afun`and`mfun`as nested functions, so that all variables in`run_bicgstab`are available to`afun`and`mfun`.

The following shows the code for `run_bicgstab`:

function x1 = run_bicgstab n = 21; b = afun(ones(n,1)); tol = 1e-12; maxit = 15; x1 = bicgstab(@afun,b,tol,maxit,@mfun); function y = afun(x) y = [0; x(1:n-1)] + ... [((n-1)/2:-1:0)'; (1:(n-1)/2)'].*x + ... [x(2:n); 0]; end function y = mfun(r) y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)']; end end

When you enter

x1 = run_bicgstab;

MATLAB^{®} software displays the message

bicgstab converged at iteration 12.5 to a solution with relative residual 2e-014.

This example demonstrates the use of a preconditioner.

Load `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 `bicgstab` to find a solution at the requested tolerance and number of iterations.

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

`fl0` is 1 because `bicgstab` does not converge to the requested tolerance `1e-12` within the requested 20 iterations. In fact, the behavior of `bicgstab` is so bad that the initial guess (`x0 = zeros(size(A,2),1)`) is the best solution and is returned as indicated by `it0 = 0`. MATLAB® stores the residual history in `rv0`.

Plot the behavior of `bicgstab`.

semilogy(0:0.5: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 a preconditioner with `ilu`, since `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] = bicgstab(A,b,tol,maxit,L,U);

`fl1` is 0 because `bicgstab` drives the relative residual to `5.9829e-014` (the value of `rr1`). The relative residual is less than the prescribed tolerance of `1e-12` at the third 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)` since `bicgstab` uses half iterations.

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

semilogy(0:0.5: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] van der Vorst, H.A., "BI-CGSTAB: A fast
and smoothly converging variant of BI-CG for the solution of nonsymmetric
linear systems," *SIAM J. Sci. Stat. Comput.*,
March 1992, Vol. 13, No. 2, pp. 631–644.

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

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