Biconjugate gradients stabilized (l) method
x = bicgstabl(A,b)
x = bicgstabl(afun,b)
x = bicgstabl(A,b,tol)
x = bicgstabl(A,b,tol,maxit)
x = bicgstabl(A,b,tol,maxit,M)
x
= bicgstabl(A,b,tol,maxit,M1,M2)
x = bicgstabl(A,b,tol,maxit,M1,M2,x0)
[x,flag] = bicgstabl(A,b,...)
[x,flag,relres] = bicgstabl(A,b,...)
[x,flag,relres,iter] = bicgstabl(A,b,...)
[x,flag,relres,iter,resvec] = bicgstabl(A,b,...)
x = bicgstabl(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 the right-hand side column vector b
must have length
n
.
x = bicgstabl(afun,b)
accepts a function handle afun
instead of the matrix A
. afun(x)
accepts a vector input
x
and returns the matrix-vector product A*x
. In all of the
following syntaxes, you can replace A
by afun
.
x = bicgstabl(A,b,tol)
specifies the tolerance of the method. If
tol
is [] then bicgstabl
uses the default, 1e-6.
x = bicgstabl(A,b,tol,maxit)
specifies the maximum number of
iterations. If maxit
is [] then bicgstabl
uses the
default, min(N,20)
.
x = bicgstabl(A,b,tol,maxit,M)
and
x
= bicgstabl(A,b,tol,maxit,M1,M2)
use preconditioner M
or M=M1*M2
and effectively solve the system A*inv(M)*x = b
for x. If M
is [] then a preconditioner is not applied. M
may be a function handle returning M\x
.
x = bicgstabl(A,b,tol,maxit,M1,M2,x0)
specifies the initial guess. If
x0
is [] then bicgstabl
uses the default, an all zero
vector.
[x,flag] = bicgstabl(A,b,...)
also returns a
convergence flag
:
Flag | Convergence |
---|---|
| bicgstabl converged to the desired tolerance
tol within maxit iterations. |
| bicgstabl iterated maxit times but did not
converge. |
| Preconditioner M was ill-conditioned. |
|
|
| One of the scalar quantities calculated during |
[x,flag,relres] = bicgstabl(A,b,...)
also returns
the relative residual norm(b-A*x)/norm(b)
. If flag
is
0
, relres <= tol
.
[x,flag,relres,iter] = bicgstabl(A,b,...)
also
returns the iteration number at which x
was computed, where 0 <=
iter <= maxit
. iter
can be k/4
where
k
is some integer, indicating convergence at a given quarter
iteration.
[x,flag,relres,iter,resvec] = bicgstabl(A,b,...)
also returns a vector of the residual norms at each quarter iteration, including
norm(b-A*x0)
.
You can pass inputs directly to bicgstabl
:
n = 21; A = gallery('wilk',n); b = sum(A,2); tol = 1e-12; maxit = 15; M = diag([10:-1:1 1 1:10]); x = bicgstabl(A,b,tol,maxit,M);
You can also use a 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 a preconditioner backsolve function:
function y = mfun(r,n) y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)'];
as inputs to bicgstabl
:
x1 = bicgstabl(@(x)afun(x,n),b,tol,maxit,@(x)mfun(x,n));
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 bicgstabl
to find a solution at the requested tolerance and number of iterations.
[x0,fl0,rr0,it0,rv0] = bicgstabl(A,b,tol,maxit);
fl0
is 1 because bicgstabl
does not converge to the requested tolerance 1e-12
within the requested 20 iterations. In fact, the behavior of bicgstabl
is so poor 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 bicgstabl
.
semilogy(0:0.25: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] = bicgstabl(A,b,tol,maxit,L,U);
fl1
is 0 because bicgstabl
drives the relative residual to 1.0257e-015
(the value of rr1
). The relative residual is less than the prescribed tolerance of 1e-12
at the sixth 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(9)
is norm(b-A*x2)
since bicgstabl
uses quarter iterations.
You can follow the progress of bicgstabl
by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0).
semilogy(0:0.25:it1,rv1/norm(b),'-o'); h = gca; h.XTick = 0:0.25:it1; xlabel('Iteration number'); ylabel('Relative residual');