`ichol`

Incomplete Cholesky factorization of a sparse symmetric positive definite matrix.

Syntax

L = ichol(A)

Returns a sparse lower triangular L with the same nonzero pattern as tril(A) such that L*L' approximates A.

L = ichol(A, opts)

Chooses the variant with a struct: type is 'nofill' or 'ict', plus options such as droptol and michol.

Description

A full Cholesky factorization of a sparse matrix creates many new nonzeros (fill-in). The incomplete version discards most of them, keeping a factor that is sparse but only approximately equal to A. It is not used to solve the system directly: L*L' serves as a preconditioner for pcg, which then converges in far fewer iterations.

The factorization can fail with an error if a nonpositive pivot appears.

Input arguments

A: Sparse, symmetric, positive definite matrix. Only its lower triangle is read.
opts: Struct with fields type, droptol (for 'ict') and michol ('on' or 'off').

Output arguments

L: Sparse lower triangular incomplete Cholesky factor.

Example

Preconditioning pcg

A = sparse(diag(4*ones(5,1)) + diag(ones(4,1),1) + diag(ones(4,1),-1));
L = ichol(A);
full(L)
b = (1:5)';
[x, flag] = pcg(A, b, 1e-10, 50, L, L');
flag

Output

ans = 5×5

    2.0000         0         0         0         0
    0.5000    1.9365         0         0         0
         0    0.5164    1.9322         0         0
         0         0    0.5175    1.9319         0
         0         0         0    0.5176    1.9319
flag = 0

ichol