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.

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
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 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).

Examples

Using bicgstab with a Matrix Input

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.

Using bicgstab with a Function Handle

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.

Using bicgstab with a Preconditioner

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');

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.