bicgstab (MATLAB Functions)

Biconjugate gradients stabilized method

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)
[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 handle afun such that afun(x) returns A*x. See Function Handles in the MATLAB Programming documentation for more information.

Parameterizing Functions Called by Function Functions, in the MATLAB Mathematics documentation, 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

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.

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.

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 6.7e-014

Example 2.

This example replaces the matrix A in Example 1 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 an M-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;
A = gallery('wilk',n);
b = sum(A,2);
tol = 1e-12;  
maxit = 15; 
M1 = diag([10:-1:1 1 1:10]);
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 displays the message

bicgstab converged at iteration 12.5 to a solution with relative 
residual 6.7e-014

Example 3. 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')

See Also

bicg, cgs, gmres, lsqr, luinc, minres, pcg, qmr, symmlq

function_handle (@), mldivide (\)

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] 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 bin2dec