A Jacobi–Davidson type method for the generalized singular value problem

AbstractWe discuss a new method for the iterative computation of some of the generalized singular values and vectors of a large sparse matrix. Our starting point is the augmented matrix formulation of the GSVD. The subspace expansion is performed by (approximately) solving a Jacobi–Davidson type correction equation, while we give several alternatives for the subspace extraction. Numerical experiments illustrate the performance of the method

Fields of values and inclusion regions for matrix pencils

We are interested in (approximate) eigenvalue inclusion regions for matrix pencils (A;B), in particular of large dimension, based on certain fields of values. We show how the usual field of values may be efficiently approximated for large Hermitian positive definite B, but also point out limitations of this set. We introduce four field of values based inclusion regions, which may effectively be approximated, also for large pencils. Furthermore, we show that these four sets are special members of two families of inclusion regions, of which we study several properties. Connections with the usual harmonic Rayleigh–Ritz method and a new variant are shown, and we propose an automated algorithm which gives an approximated inclusion region. The results are illustrated by several numerical examples

Probabilistic upper bounds for the matrix two-norm

We develop probabilistic upper bounds for the matrix two-norm, the largest singular value. These bounds, which are true upper bounds with a user-chosen high probability, are derived with a number of different polynomials that implicitly arise in the Lanczos bidiagonalization process. Since these polynomials are adaptively generated, the bounds typically give very good results. They can be computed efficiently. Together with an approximation that is a guaranteed lower bound, this may result in a small probabilistic interval for the matrix norm of large matrices within a fraction of a second

On the numerical range of a matrix

This is an English translation of the paper "Über den Wertevorrat einer Matrix" by Rudolf Kippenhahn, Mathematische Nachrichten 6 (1951), 193–228. This paper is often cited by mathematicians who work in the area of numerical ranges, thus it is hoped that this translation may be useful. Some notation and wording has been changed to make the paper more in line with present papers on the subject written in English. In Part 1 of this paper Kippenhahn characterized the numerical range of a matrix as being the convex hull of a certain algebraic curve that is associated to the matrix. More than 55 years later this "boundary generating curve" is still a topic of current research, and "¨Uber den Wertevorrat einer Matrix" is almost always present in the bibliographies of papers on this topic. In Part 2, the author initiated the study of a generalization of the numerical range to matrices with quaternion entries. The translators note that in Theorem 36, it is stated incorrectly that this set of points in 4-dimensional space is convex. A counterexample to this statement was given in 1984.[ I ] In the notes at the end of this paper the translators pinpoint the flaw in the argument. In the opinion of the translators, this error does not significantly detract from the overall value and significance of this paper. In the translation, footnotes in the original version are indicated by superscript Arabic numerals, while superscript Roman numerals in brackets are used to indicate that the translators have a comment about the original paper. All of these comments appear at the end of this paper, and the translators also have corrected some minor misprints in the original without comment

Roots of bivariate polynomial systems via determinantal representations

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a twoparameter eigenvalue problem. The first determinantal representation is suitable for polynomials with scalar or matrix coecients and consists of matrices with asymptotic order n2=4, where n is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order n2=6. The resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.</p

Regularization parameter determination for discrete ill-posed problems

Straightforward solution of discrete ill-posed linear systems of equations or least-squares problems with error contaminated data does not, in general, give meaningful results, because the propagated error destroys the computed solution. The problems have to be modified to reduce their sensitivity to the error in the data. The amount of modification is determined by a regularization parameter. It can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available. This paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation. It is based on comparing computed solutions determined by Tikhonov regularization and truncated singular value decomposition. Analogous comparisons are proposed for large-scale problems. The technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data. Keywords: Ill-posed problem; Regularization; Noise level estimation; TSVD; Tikhonov regularization; Heuristic parameter choice rul