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Choleski Factorization

If the matrix A is an symmetric positive definite sparse matrix, then a special form of LU factorization can be used that exploits the symmetry and inherently numerical stable characteristics of this matrix form [9]. A symmetric positive definite sparse matrix A can be numerically factored into a single lower triangular matrix L:

 

Equation 3 is solved by setting , and substituting y for . The numerical solution for Ly=b is found by forward reduction, and the numerical solution for x is calculated by backward substitution in the equation . Our analysis of the available parallelism in block-diagonal-bordered LU factorization, presented in section 4, can be extended to an analysis of available parallelism in block-diagonal-bordered Choleski factorization by simply substituting for U. Additional discussions on the state of the literature for Choleski factorization are presented below.

We present a general sequential sparse factorization algorithm based upon the column Choleski factorization algorithm [20], which is similar to the factorization algorithms commonly attributed to Crout and Doolittle, and similar to the LU algorithm presented in figure 3. A sequential sparse factorization algorithm is presented in figure 6, and we present sequential sparse forward reduction and backward substitution algorithms for Choleski factorization in figures 7 and 8 respectively. In the backward substitution algorithm, the calculations are performed by implicitly transposing L.

 
Figure 6: Sparse Choleski Factorization 

 
Figure 7: Sparse Forward Reduction for Choleski Factorization 

 
Figure 8: Sparse Backward Substitution for Choleski Factorization 



David P. Koester
Sun Oct 22 15:31:10 EDT 1995