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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. | |
bicgstabl stagnated. (Two consecutive iterates were the same.) | |
One of the scalar quantities calculated during bicgstabl became too small or too large to continue computing. |
[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');