The ubiquitous Kronecker product

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The Kronecker Product

Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Introduction to Matrix Analysis. By Richard Bellman. 2nd Edition. Pp. xxiii, 403. £7·50. (McGraw-Hill.)

The transformation of multi-dimensional potential energy surfaces (PESs) from a grid-based multimode representation to an analytical one is a standard procedure in quantum chemical programs. Within the framework of linear least squares fitting, a simple and highly efficient algorithm is presented, which relies on a direct product representation of the PES and a repeated use of Kronecker products. It shows the same scalings in computational cost and memory requirements as the potfit approach. In comparison to customary linear least squares fitting algorithms, this corresponds to a speed-up and memory saving by several orders of magnitude. Different fitting bases are tested, namely, polynomials, B-splines, and distributed Gaussians. Benchmark calculations are provided for the PESs of a set of small molecules.

Efficient generation of sum-of-products representations of high-dimensional potential energy surfaces based on multimode expansions

Linearized aeroelastic stability and response analyses in the state domain may require the identification of an asymptotically stable finite-state aerodynamic subsystem from available aerodynamic transfer matrices, related to structural motions and gusts. To such an aim, the paper develops an improved rational matrix approximation, combining three nonlinear least squares identification techniques with a system reduction based on a double dynamic residualization. An alternative gust formulation is also presented that, by reconstructing generalized gust forces through the use of special structural motion-like modes called gust modes, makes it possible to determine a gust response even without the usual gust-penetration model in the frequency domain. Examples are presented to demonstrate the behavior of the proposed approaches applied to sample flutter and gust/turbulence response analyses.

Hany Hashish

Papers

Improved parallel QR method for large least squares problems involving Kronecker products