Biconjugate gradients stabilized method
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,...)
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 |
---|---|
|
|
|
|
| Preconditioner |
|
|
| One of the scalar quantities calculated during |
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)
.
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.
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.
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');
[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.