Incomplete Cholesky factorization

In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method.

The Cholesky factorization of a positive definite matrix A is A = LL* where L is a lower triangular matrix. An incomplete Cholesky factorization is given by a sparse lower triangular matrix K that is in some sense close to L. The corresponding preconditioner is KK*.

One popular way to find such a matrix K is to use the algorithm for finding the exact Cholesky decomposition, except that any entry is set to zero if the corresponding entry in A is also zero. This gives an incomplete Cholesky factorization which is as sparse as the matrix A.

Algorithm[]

For ${\displaystyle i}$ from ${\displaystyle 1}$ to ${\displaystyle N}$:

${\displaystyle L_{ii}=\left({a_{ii}-\sum \limits _{k=1}^{i-1}{L_{ik}^{2}}}\right)^{1 \over 2}}$

For ${\displaystyle j}$ from ${\displaystyle i+1}$ to ${\displaystyle N}$:

${\displaystyle L_{ji}={1 \over {L_{ii}}}\left({a_{ji}-\sum \limits _{k=1}^{i-1}{L_{ik}L_{jk}}}\right)}$

Implementation[]

Implementation of the incomplete Cholesky factorization in the Octave scripting language. The factorization is stored as a lower triangular matrix, with the elements in the upper triangle set to zero.

function a = ichol(a)
	n = size(a,1);

	for k=1:n
		a(k,k) = sqrt(a(k,k));
		for i=(k+1):n
		    if (a(i,k)!=0)
		        a(i,k) = a(i,k)/a(k,k);            
		    endif
		endfor
		for j=(k+1):n
		    for i=j:n
		        if (a(i,j)!=0)
		            a(i,j) = a(i,j)-a(i,k)*a(j,k);  
		        endif
		    endfor
		endfor
	endfor

    for i=1:n
        for j=i+1:n
            a(i,j) = 0;
        endfor
    endfor            
endfunction

References[]

• Incomplete Cholesky factorization at CFD Online wiki
• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9. See Section 10.3.2.

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