A modified Cholesky algorithm based on a symmetric indefinite factorization

Given a symmetric and not necessarily positive definite matrix A, a modified Cholesky algorithm computes a Cholesky factorization P (A + E)PT = RTR, where P is a permutation matrix and E is a perturbation chosen to make A+ E positive definite. The aims include producing a small-normed E and making A+E reasonably well conditioned. Modified Cholesky factorizations are widely used in optimization. We propose a new modified Cholesky algorithm based on a symmetric indefinite factorization computed using a new pivoting strategy of Ashcraft, Grimes, and Lewis. We analyze the effectiveness of the algorithm, both in theory and practice, showing that the algorithm is competitive with the existing algorithms of Gill, Murray, and Wright and Schnabel and Eskow. Attractive features of the new algorithm include easy-to-interpret inequalities that explain the extent to which it satisfies its design goals, and the fact that it can be implemented in terms of existing software.

[1]  I. Duff,et al.  Direct Solution of Sets of Linear Equations whose Matrix is Sparse, Symmetric and Indefinite , 1979 .

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  G. Stewart.  The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators , 1980 .

[4]  I. Duff,et al.  The factorization of sparse symmetric indefinite matrices , 1991 .

[5]  Tamar Schlick,et al.  Modified Cholesky Factorizations for Sparse Preconditioners , 1993, SIAM J. Sci. Comput..

[6]  J. Bunch,et al.  Some stable methods for calculating inertia and solving symmetric linear systems , 1977 .

[7]  Philip E. Gill,et al.  Practical optimization , 1981 .

[8]  Jack Dongarra,et al.  LINPACK Users' Guide , 1987 .

[9]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[10]  Danny C. Sorensen,et al.  On the use of directions of negative curvature in a modified newton method , 1979, Math. Program..

[11]  Elizabeth Eskow,et al.  A New Modified Cholesky Factorization , 1990, SIAM J. Sci. Comput..

[12]  Nicholas J. Higham,et al.  FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation , 1988, TOMS.

[13]  N. Higham.  COMPUTING A NEAREST SYMMETRIC POSITIVE SEMIDEFINITE MATRIX , 1988 .

[14]  Philip E. Gill,et al.  Newton-type methods for unconstrained and linearly constrained optimization , 1974, Math. Program..

[15]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[16]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[17]  M. SIAMJ..  STABILITY OF THE DIAGONAL PIVOTING METHOD WITH PARTIAL PIVOTING , 1995 .

[18]  J. L’Excellent,et al.  On the use of element-by-element preconditioners to solve large scale partially separable optimization problems , 1995 .

[19]  Nicholas J. Higham,et al.  The Test Matrix Toolbox for Matlab (Version 3.0) , 2013 .