Separable Type Representations of Matrices and Fast Algorithms

Volume 2 Eigenvalue Method

  • Yuli Eidelman
  • Israel Gohberg
  • Iulian Haimovici
Part of the Operator Theory: Advances and Applications book series (OT, volume 235)

Table of contents

  1. Front Matter
    Pages i-xi
  2. The Eigenvalue Structure of Order One Quasiseparable Matrices

    1. Front Matter
      Pages 1-2
    2. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 3-12
    3. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 13-31
    4. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 33-49
    5. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 51-71
  3. Divide and Conquer Method for the Eigenproblem

    1. Front Matter
      Pages 73-74
    2. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 75-93
    3. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 95-101
    4. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 103-115
    5. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 117-131
  4. Algorithms for QR Iterations and for Reduction to Hessenberg Form

    1. Front Matter
      Pages 133-134
    2. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 135-162
    3. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 163-205
    4. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 207-255
  5. QR Iterations for Companion Matrices

    1. Front Matter
      Pages 257-258
    2. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 259-279
    3. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 281-294
    4. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 295-303
    5. Yuli Eidelman, Israel Gohberg, Iulian Haimovici
      Pages 305-321

About this book

Introduction

This two-volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The primary focus is on fast algorithms (many of linear complexity) for matrices in semiseparable, quasiseparable, band and companion form. The work examines algorithms of multiplication, inversion and description of eigenstructure and includes a wealth of illustrative examples throughout the different chapters.

The second volume, consisting of four parts, addresses the eigenvalue problem for matrices with quasiseparable structure and applications to the polynomial root finding problem. In the first part the properties of the characteristic polynomials of principal leading submatrices, the structure of eigenspaces and the basic methods for computing eigenvalues are studied in detail for matrices with quasiseparable representation of the first order. The second part is devoted to the divide and conquer method, with the main algorithms also being derived for matrices with quasiseparable representation of order one. The QR iteration method for some classes of matrices with quasiseparable representations of any order is studied in the third part. This method is then used in the last part in order to provide a fast solver for the polynomial root finding problem. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and accessible style, the text will be a valuable resource for engineers, scientists, numerical analysts, computer scientists and mathematicians alike.

Keywords

Eigenvalue problems Fast algorithms Polynomial root finder problem Quasiseparable representations Structured matrices

Authors and affiliations

  • Yuli Eidelman
    • 1
  • Israel Gohberg
    • 2
  • Iulian Haimovici
    • 3
  1. 1.School of Mathematical SciencesTel Aviv University Raymond & Beverly Sackler Faculty of ExaTel AvivIsrael
  2. 2.School of Mathematical SciencesTel Aviv University Raymond & Beverly Sackler Faculty of ExaTel AvivIsrael
  3. 3.School of Mathematical SciencesTel Aviv University Raymond & Beverly Sackler Faculty of ExaTel AvivIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0612-1
  • Copyright Information Springer Basel 2014
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0611-4
  • Online ISBN 978-3-0348-0612-1
  • Series Print ISSN 0255-0156
  • Series Online ISSN 2296-4878
  • About this book