Biconjugate gradients stabilized method


x = bicgstab(A,b)
[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.




bicgstab converged to the desired tolerance tol within maxit iterations.


bicgstab iterated maxit times but did not converge.


Preconditioner M was ill-conditioned.


bicgstab stagnated. (Two consecutive iterates were the same.)


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).


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];
    function y = mfun(r)
        y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)'];

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.

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).

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.

Extended Capabilities

Introduced before R2006a