Axel Ruhe

The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses

The use of partial least squares (PLS) for handling collinearities among the independent variables X in multiple regression is discussed. Consecutive estimates

Templates for the solution of algebraic eigenvalue problems: a practical guide

({\text{rank }}1,2,\cdots )

The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems

are obtained using the residuals from previous rank as a new dependent variable y. The PLS method is equivalent to the conjugate gradient method used in Numerical Analysis for related problems.To estimate the “optimal” rank, cross validation is used. Jackknife estimates of the standard errors are thereby obtained with no extra computation.The PLS method is compared with ridge regression and principal components regression on a chemical example of modelling the relation between the measured biological activity and variables describing the chemical structure of a set of substituted phenethylamines.

Rational Krylov sequence methods for eigenvalue computation

List of symbols and acronyms List of iterative algorithm templates List of direct algorithms List of figures List of tables 1: Introduction 2: A brief tour of Eigenproblems 3: An introduction to iterative projection methods 4: Hermitian Eigenvalue problems 5: Generalized Hermitian Eigenvalue problems 6: Singular Value Decomposition 7: Non-Hermitian Eigenvalue problems 8: Generalized Non-Hermitian Eigenvalue problems 9: Nonlinear Eigenvalue problems 10: Common issues 11: Preconditioning techniques Appendix: of things not treated Bibliography Index .

Algorithms for the Nonlinear Eigenvalue Problem

A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in the intervals between them. It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices. For large problems the operation counts are about five times smaller than for traditional subspace iteration methods. Tests on a numerical example, arising from a finite element computation of a nuclear power piping system, are reported, and it is shown how the performance predicted bears out in a practical situation.

Algorithms for Separable Nonlinear Least Squares Problems

Abstract Algorithms to solve large sparse eigenvalue problems are considered. A new class of algorithms which is based on rational functions of the matrix is described. The Lanczos method, the Arnoldi method, the spectral transformation Lanczos method, and Rayleigh quotient iteration all are special cases, but there are also new algorithms which correspond to rational functions with several poles. In the simplest case a basis of a rational Krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of Hessenberg matrices.

Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. matrix pairs

The following nonlinear eigenvalue problem is studied : Let

Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

T(\lambda )

Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices

be an

The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices

n \times n