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Preconditioned conjugate gradients method
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)
[x,flag] = pcg(A,b,...)
[x,flag,relres] = pcg(A,b,...)
[x,flag,relres,iter] = pcg(A,b,...)
[x,flag,relres,iter,resvec] = pcg(A,b,...)
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. You also can specify A to 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 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 handle mfun such that mfun(x) 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.
[x,flag] = pcg(A,b,...) also returns a convergence flag.
Flag | Convergence |
---|---|
pcg converged to the desired tolerance tol within maxit iterations. | |
pcg iterated maxit times but did not converge. | |
Preconditioner M was ill-conditioned. | |
pcg stagnated. (Two consecutive iterates were the same.) | |
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,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, relres <= tol.
[x,flag,relres,iter] = pcg(A,b,...) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.
[x,flag,relres,iter,resvec] = pcg(A,b,...) also returns a vector of the residual norms at each iteration including norm(b-A*x0).
This example shows how to use pcg with a matrix input and with a function handle.
n1 = 21; A = gallery('moler',n1); b1 = sum(A,2); tol = 1e-6; maxit = 15; M1 = spdiags((1:n1)',0,n1,n1); [x1,flag1,rr1,iter1,rv1] = pcg(A,b1,tol,maxit,M1);
Alternatively, you can use the following function in place of the matrix A:
function y = applyMoler(x) y = x; y(end-1:-1:1) = y(end-1:-1:1) - cumsum(y(end:-1:2)); y(2:end) = y(2:end) - cumsum(y(1:end-1));
By using this function, you can solve larger systems more efficiently as there is no need to store the entire matrix A:
n2 = 21; b2 = applyMoler(ones(n2,1)); tol = 1e-6; maxit = 15; M2 = spdiags((1:n2)',0,n2,n2); [x2,flag2,rr2,iter2,rv2] = pcg(@applyMoler,b2,tol,maxit,M2);
This example demonstrates how to use a preconditioner matrix with pcg.
Create an input matrix and try to solve the system with pcg.
A = delsq(numgrid('S',100));
b = ones(size(A,1),1);
[x0,fl0,rr0,it0,rv0] = pcg(A,b,1e-8,100);
fl0 is 1 because pcg does not converge to the requested tolerance of 1e-8 within the requested maximum 100 iterations. A preconditioner can make the system converge more quickly.
Use ichol with only one input argument to construct an incomplete Cholesky factorization with zero fill.
L = ichol(A); [x1,fl1,rr1,it1,rv1] = pcg(A,b,1e-8,100,L,L');
fl1 is 0 because pcg drives the relative residual to 9.8e-09 (the value of rr1) which is less than the requested tolerance of 1e-8 at the seventy-seventh iteration (the value of it1) when preconditioned by the zero-fill incomplete Cholesky factorization. rv1(1) = norm(b) and rv1(78) = norm(b-A*x1).
The previous matrix represents the discretization of the Laplacian on a 100x100 grid with Dirichlet boundary conditions. This means that a modified incomplete Cholesky preconditioner might perform even better.
Use the michol option to create a modified incomplete Cholesky preconditioner.
L = ichol(A,struct('michol','on')); [x2,fl2,rr2,it2,rv2] = pcg(A,b,1e-8,100,L,L');
In this case you attain convergence in only forty-seven iterations.
You can see how the preconditioners affect the rate of convergence of pcg by plotting each of the residual histories starting from the initial estimate (iterate number 0).
figure; semilogy(0:it0,rv0/norm(b),'b.'); hold on; semilogy(0:it1,rv1/norm(b),'r.'); semilogy(0:it2,rv2/norm(b),'k.'); legend('No Preconditioner','IC(0)','MIC(0)'); xlabel('iteration number'); ylabel('relative residual'); hold off;