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)
bicgstab(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...)
[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 afun such that afun(x) returns A*x.

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 that 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.

bicgstab(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...) passes parameters p1,p2,... to functions afun(x,p1,p2,...), m1fun(x,p1,p2,...), and m2fun(x,p1,p2,...).

[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. It then solves the same system using functions that return A and the preconditioner.

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 this message

bicgstab converged at iteration 12.5 to a solution with relative
residual 2.9e-014

Alternatively, use this matrix-vector product function

function y = afun(x,n)
y = [0; 
     x(1:n-1)] + [((n-1)/2:-1:0)'; 
     (1:(n-1)/2)'] .*x + [x(2:n); 
     0];

and this preconditioner backsolve function

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

as inputs to bicgstab

x1 = bicgstab(@afun,b,tol,maxit,@mfun,[],[],21);

Note that both afun and mfun must accept bicgstab's extra input n=21.

Example 2. 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), \ (backslash)

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