MATLAB Function Reference

bicgstab - 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.

, 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.

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).

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 software 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')

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.

See Also

bicg, cgs, gmres, lsqr, luinc, minres, pcg, qmr, symmlq, function_handle (@), mldivide (\)

bicg

bin2dec