Interactive Educational Modules in
Scientific Computing

Cholesky Factorization

This module illustrates Cholesky factorization of a symmetric positive definite matrix. This factorization expresses the initial matrix A as a product of a lower triangular matrix and its transpose, A = LLT. Because the matrix is symmetric, the algorithm accesses only the lower triangular portion. Each successive column is divided by the square root of its diagonal entry, and then a multiple of the scaled column is subtracted from each remaining column.

The user first selects a matrix size (n = 2, 3, or 4), then selects a matrix by choosing a preset example, a random matrix, or typing in desired entries. Since the matrix is symmetric, only its lower triangle is shown in the computational display. The successive steps of Cholesky factorization are carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step. The current column is indicated by an arrow. When Cholesky factorization is complete, the resulting lower triangular Cholesky factor is shown. Formats provided for displaying numeric entries include exponential (e), fixed (f), and generic (g), with g the default.

Reference: Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Section 2.5.1, especially Algorithm 2.7 and Example 2.21.

Developers: Jessica Schoen and Michael Heath