An Improved Arc Algorithm for Detecting Definite Hermitian Pairs

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.

[1]  Nicholas J. Higham,et al.  Analysis of the Cholesky Method with Iterative Refinement for Solving the Symmetric Definite Generalized Eigenproblem , 2001, SIAM J. Matrix Anal. Appl..

[2]  Chun-Hua Guo,et al.  Algorithms for hyperbolic quadratic eigenvalue problems , 2005, Math. Comput..

[3]  Nicholas J. Higham,et al.  LAPACK-Style Codes for Pivoted Cholesky and QR Updating , 2006, PARA.

[4]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[5]  G. Stewart.  Pertubation bounds for the definite generalized eigenvalue problem , 1979 .

[6]  Jochem Zowe,et al.  A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results , 1992, SIAM J. Optim..

[7]  Nicholas J. Higham,et al.  Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems , 2008, SIAM J. Matrix Anal. Appl..

[8]  Y. Moon,et al.  Finding a positive definite linear combination of two Hermitian matrices , 1983 .

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

[10]  Ilse C. F. Ipsen.  Computing an Eigenvector with Inverse Iteration , 1997, SIAM Rev..

[11]  Yik-Hoi Au-Yeung.  A theorem on a mapping from a sphere to the circle and the simultaneous diagonalization of two hermitian matrices , 1969 .

[12]  Charles R. Crawford.  Algorithm 646: PDFIND: a routine to find a positive definite linear combination of two real symmetric matrices , 1986, TOMS.

[13]  K. Veselié.  A Jacobi eigenreduction algorithm for definite matrix pairs , 1993 .

[14]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[15]  Germund Dahlquist,et al.  Numerical methods in scientific computing , 2008 .

[16]  Yik-hoi Au-yeung.  SOME THEOREMS ON THE REAL PENCIL AND SIMULTANEOUS DIAGONALIZATION OF TWO HERMITIAN BILINEAR FUNCTIONS , 1969 .

[17]  Nicholas J. Higham,et al.  Definite Matrix Polynomials and their Linearization by Definite Pencils , 2009, SIAM J. Matrix Anal. Appl..

[18]  Lloyd N. Trefethen,et al.  An Extension of MATLAB to Continuous Functions and Operators , 2004, SIAM J. Sci. Comput..

[19]  Yiu-Sang Moon.  On the numerical solution of the definite generalized eigenvalue problem. , 1979 .

[20]  Anders Forsgren,et al.  Computing Modified Newton Directions Using a Partial Cholesky Factorization , 1995, SIAM J. Sci. Comput..

[21]  N. Higham,et al.  Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems , 2002 .

[22]  J. Cullum,et al.  Real Symmetric Matrices , 1985 .

[23]  Beresford N. Parlett,et al.  On nonsymmetric saddle point matrices that allow conjugate gradient iterations , 2008, Numerische Mathematik.

[24]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.