使用bicgstabl与输入或一个函数

You can pass inputs directly tobicgstabl:

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 tobicgstabl:

x1 = bicgstabl(@(x)afun(x,n),b,tol,maxit,@(x)mfun(x,n));

Using bicgstabl with a Preconditioner

This example demonstrates the use of a preconditioner.

Loadwest0479, a real 479-by-479 nonsymmetric sparse matrix.

loadwest0479; A = west0479;

Definebso 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;

Usebicgstablto find a solution at the requested tolerance and number of iterations.

[x0,fl0,rr0,it0,rv0] = bicgstabl(A,b,tol,maxit);

fl0is 1 becausebicgstabldoes not converge to the requested tolerance1e-12within the requested 20 iterations. In fact, the behavior ofbicgstablis so poor that the initial guess (x0 = zeros(size(A,2),1)) is the best solution and is returned as indicated byit0 = 0. MATLAB® stores the residual history inrv0.

Plot the behavior ofbicgstabl.

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 withilu, sinceAis 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);

fl1is 0 becausebicgstabldrives the relative residual to1.0257e-015(the value ofrr1). The relative residual is less than the prescribed tolerance of1e-12at the sixth iteration (the value ofit1) when preconditioned by the incomplete LU factorization with a drop tolerance of1e-6. The outputrv1(1)isnorm(b), and the outputrv1(9)isnorm(b-A*x2)sincebicgstabluses quarter iterations.

You can follow the progress ofbicgstablby 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');