This example shows how to use an incomplete Cholesky factorization as a preconditioner to improve convergence.

Create a symmetric positive definite matrix, `A`.

N = 100;
A = delsq(numgrid('S',N));

Create an incomplete Cholesky factorization as a preconditioner for `pcg`. Use a constant vector as the right hand side. As a baseline, execute `pcg` without a preconditioner.

b = ones(size(A,1),1);
tol = 1e-6; maxit = 100;
[x0,fl0,rr0,it0,rv0] = pcg(A,b,tol,maxit);

Note that `fl0 = 1` indicating that `pcg` did not drive the relative residual to the requested tolerance in the maximum allowed iterations. Try the no-fill incomplete Cholesky factorization as a preconditioner.

L1 = ichol(A);
[x1,fl1,rr1,it1,rv1] = pcg(A,b,tol,maxit,L1,L1');

`fl1 = 0`, indicating that `pcg` converged to the requested tolerance and did so in 59 iterations (the value of `it1`). Since this matrix is a discretized Laplacian, however, using modified incomplete Cholesky can create a better preconditioner. A modified incomplete Cholesky factorization constructs an approximate factorization that preserves the action of the operator on the constant vector. That is, `norm(A*e � L*(L'*e))` will be approximately zero for `e = ones(size(A,2),1)` even though `norm(A-L*L','fro')/norm(A,'fro')` is not close to zero. It is not necessary to specify type for this syntax since `nofill` is the default, but it is good practice.

opts.type = 'nofill'; opts.michol = 'on';
L2 = ichol(A,opts);
e = ones(size(A,2),1);
norm(A*e-L2*(L2'*e))
[x2,fl2,rr2,it2,rv2] = pcg(A,b,tol,maxit,L2,L2');

ans =
3.7983e-14

`pcg` converges (`fl2 = 0`) but in only 38 iterations. Plotting all three convergence histories shows the convergence.

semilogy(0:maxit,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)');

The plot shows that the modified incomplete Cholesky preconditioner creates a much faster convergence.

You can also try incomplete Cholesky factorizations with threshold dropping. The following plot shows convergence of `pcg` with preconditioners constructed with various drop tolerances.

L3 = ichol(A, struct('type','ict','droptol',1e-1));
[x3,fl3,rr3,it3,rv3] = pcg(A,b,tol,maxit,L3,L3');
L4 = ichol(A, struct('type','ict','droptol',1e-2));
[x4,fl4,rr4,it4,rv4] = pcg(A,b,tol,maxit,L4,L4');
L5 = ichol(A, struct('type','ict','droptol',1e-3));
[x5,fl5,rr5,it5,rv5] = pcg(A,b,tol,maxit,L5,L5');
figure; semilogy(0:maxit,rv0./norm(b),'b-','linewidth',2);
hold on;
semilogy(0:it3,rv3./norm(b),'b-.','linewidth',2);
semilogy(0:it4,rv4./norm(b),'b--','linewidth',2);
semilogy(0:it5,rv5./norm(b),'b:','linewidth',2);
legend('No Preconditioner','ICT(1e-1)','ICT(1e-2)', ...
'ICT(1e-3)','Location','SouthEast');

Note the incomplete Cholesky preconditioner constructed with drop tolerance `1e-2` is denoted as `ICT(1e-2)`.

As with the zero-fill incomplete Cholesky, the threshold dropping factorization can benefit from modification (i.e. `opts.michol = 'on'`) since the matrix arises from an elliptic partial differential equation. As with `MIC(0)`, the modified threshold based dropping incomplete Cholesky will preserve the action of the preconditioner on constant vectors, that is `norm(A*e-L*(L'*e))` will be approximately zero.