Computing modified Newton directions using a partial Cholesky factorization
SIAM Journal on Scientific Computing
ACM Transactions on Mathematical Software (TOMS)
Computing an Eigenvector with Inverse Iteration
SIAM Review
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
SIAM Journal on Matrix Analysis and Applications
On the numerical solution of the definite generalized eigenvalue problem.
On the numerical solution of the definite generalized eigenvalue problem.
An Extension of MATLAB to Continuous Functions and Operators
SIAM Journal on Scientific Computing
Convex Optimization
On nonsymmetric saddle point matrices that allow conjugate gradient iterations
Numerische Mathematik
Numerical Methods in Scientific Computing: Volume 1
Numerical Methods in Scientific Computing: Volume 1
Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Definite Matrix Polynomials and their Linearization by Definite Pencils
SIAM Journal on Matrix Analysis and Applications
LAPACK-style codes for pivoted Cholesky and QR updating
PARA'06 Proceedings of the 8th international conference on Applied parallel computing: state of the art in scientific computing
| Hi-index | 0.00 |
A 25-year-old and somewhat neglected algorithm of Crawford and Moon attempts to determine whether a given Hermitian matrix pair $(A,B)$ is definite by exploring the range of the function $f(x) = x^*(A+iB)x / | x^*(A+iB)x |$, which is a subset of the unit circle. We revisit the algorithm and show that with suitable modifications and careful attention to implementation details it provides a reliable and efficient means of testing definiteness. A clearer derivation of the basic algorithm is given that emphasizes an arc expansion viewpoint and makes no assumptions about the definiteness of the pair. Convergence of the algorithm is proved for all $(A,B$), definite or not. It is shown that proper handling of three details of the algorithm is crucial to the efficiency and reliability: how the midpoint of an arc is computed, whether shrinkage of an arc is permitted, and how directions of negative curvature are computed. For the latter, several variants of Cholesky factorization with complete pivoting are explored and the benefits of pivoting demonstrated. The overall cost of our improved algorithm is typically just a few Cholesky factorizations. Three applications of the algorithm are described: testing the hyperbolicity of a Hermitian quadratic matrix polynomial, constructing conjugate gradient methods for sparse linear systems in saddle point form, and computing the Crawford number of the pair $(A,B)$ via a quasiconvex univariate minimization problem.