pcg (MATLAB Functions)

Preconditioned Conjugate Gradients method

Syntax

x = pcg(A,b)
pcg(A,b,tol)
pcg(A,b,tol,maxit)
pcg(A,b,tol,maxit,M)
pcg(A,b,tol,maxit,M1,M2)
pcg(A,b,tol,maxit,M1,M2,x0)
pcg(A,b,tol,maxit,M1,M2,x0,p1,p2,...)
[x,flag] = pcg(A,b,tol,maxit,M1,M2,x0,p1,p2,...) 
[x,flag,relres] = pcg(A,b,tol,maxit,M1,M2,x0,p1,p2,...)
[x,flag,relres,iter] = pcg(A,b,tol,maxit,M1,M2,x0,p1,p2,...)
[x,flag,relres,iter,resvec] = pcg(A,b,tol,maxit,M1,M2,x0,p1,p2,...)

Description

x = pcg(A,b) attempts to solve the system of linear equations A*x=b for x. The n-by-n coefficient matrix A must be symmetric and positive definite, and should also 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 pcg converges, a message to that effect is displayed. If pcg 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.

pcg(A,b,tol) specifies the tolerance of the method. If tol is [], then pcg uses the default, 1e-6.

pcg(A,b,tol,maxit) specifies the maximum number of iterations. If maxit is [], then pcg uses the default, min(n,20).

pcg(A,b,tol,maxit,M) and pcg(A,b,tol,maxit,M1,M2) use symmetric positive definite preconditioner M or M = M1*M2 and effectively solve the system inv(M)*A*x = inv(M)*b for x. If M is [] then pcg applies no preconditioner. M can be a function that returns M\x.

pcg(A,b,tol,maxit,M1,M2,x0) specifies the initial guess. If x0 is [], then pcg uses the default, an all-zero vector.

pcg(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] = pcg(A,b,tol,maxit,M1,M2,x0) also returns a convergence flag.

Flag
Convergence

0
pcg converged to the desired tolerance tol within maxit iterations.

1
pcg iterated maxit times but did not converge.

2
Preconditioner M was ill-conditioned.

3
pcg stagnated. (Two consecutive iterates were the same.)

4

One of the scalar quantities calculated during pcg became too small or too large to continue computing.

Flag	Convergence
`0`	`pcg` converged to the desired tolerance `tol` within `maxit` iterations.
`1`	`pcg` iterated `maxit` times but did not converge.
`2`	Preconditioner `M` was ill-conditioned.
`3`	`pcg` stagnated. (Two consecutive iterates were the same.)
`4`	One of the scalar quantities calculated during `pcg` 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] = pcg(A,b,tol,maxit,M1,M2,x0) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, relres <= tol.

[x,flag,relres,iter] = pcg(A,b,tol,maxit,M1,M2,x0) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.

[x,flag,relres,iter,resvec] = pcg(A,b,tol,maxit,M1,M2,x0) also returns a vector of the residual norms at each iteration including norm(b-A*x0).

Examples

Example 1.

A = gallery('wilk',21);  
b = sum(A,2);
tol = 1e-12;  
maxit = 15;  
M = diag([10:-1:1 1 1:10]);
[x,flag,rr,iter,rv] = pcg(A,b,tol,maxit,M);

Alternatively, use this one-line 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 one-line preconditioner backsolve function

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

as inputs to pcg

[x1,flag1,rr1,iter1,rv1] = pcg(@afun,b,tol,maxit,@mfun,...
                               [],[],21);

Example 2.

A = delsq(numgrid('C',25));
b = ones(length(A),1);
[x,flag] = pcg(A,b)

flag is 1 because pcg does not converge to the default tolerance of 1e-6 within the default 20 iterations.

R = cholinc(A,1e-3);
[x2,flag2,relres2,iter2,resvec2] = pcg(A,b,1e-8,10,R',R)

flag2 is 0 because pcg converges to the tolerance of 1.2e-9 (the value of relres2) at the sixth iteration (the value of iter2) when preconditioned by the incomplete Cholesky factorization with a drop tolerance of 1e-3. resvec2(1) = norm(b) and resvec2(7) = norm(b-A*x2). You can follow the progress of pcg by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0).

semilogy(0:iter2,resvec2/norm(b),'-o')
xlabel('iteration number')
ylabel('relative residual')

See Also

bicg, bicgstab, cgs, cholinc, gmres, lsqr, minres, 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.

pbaspect pchip