Separable type representations of matrices and fast algorithms
- Yuli Eidelman, Israel Gohberg, Iulian Haimovici.
- Basel : Birkhäuser/Springer, 
- Copyright notice
- Physical description
- 2 volumes ; 25 cm.
- Operator theory, advances and applications ; 234-235.
Math & Statistics Library
Library has: v.1-
|QA188 .E376 2014 V.1||Unknown|
|QA188 .E376 2014 V.2||Unknown|
- Includes bibliographical references and index.
- Part 1. Basics on separable, semiseparable and quasiseparable representations of matrices.- 1. Matrices with separable representation and low complexity algorithms.- 2. The minimal rank completion problem.- 3. Matrices in diagonal plus semiseparable form.- 4. Quasiseparable representations: the basics.- 5. Quasiseparable generators.- 6. Rank numbers of pairs of mutually inverse matrices, Asplund theorems.- 7. Unitary matrices with quasiseparable representations.- Part 2. Completion of matrices with specified bands.- 8. Completion to Green matrices.- 9. Completion to matrices with band inverses and with minimal ranks.- 10. Completion of special types of matrices.- 11. Completion of mutually inverse matrices.- 12. Completion to unitary matrices.- Part 3. Quasiseparable representations of matrices, descriptor systems with boundary conditions and first applications.- 13. Quasiseparable representations and descriptor systems with boundary conditions.- 14. The first inversion algorithms.- 15. Inversion of matrices in diagonal plus semiseparable form.- 16. Quasiseparable/semiseparable representations and one-direction systems.- 17. Multiplication of matrices.- Part 4. Factorization and inversion.- 18. The LDU factorization and inversion.- 19. Scalar matrices with quasiseparable order one.- 20. The QR factorization based method.
- (source: Nielsen Book Data)9783034806053 20160618
- Part 5. The eigenvalue structure of order one quasiseparable matrices.- 21. Quasiseparable of order one matrices. Characteristic polynomials.- 22. Eigenvalues with geometric multiplicity one.- 23. Kernels of quasiseparable of order one matrices.- 24. Multiple eigenvalues.- Part 6. Divide and conquer method for eigenproblems.- 25. Divide step.- 26. Conquer step and rational matrix functions eigenproblem.- 27. Complete algorithm for Hermitian matrices.- 28. Complete algorithm for unitary Hessenberg matrices.- Part 7. Algorithms for qr iterations and for reduction to Hessenberg form.- 29. The QR iteration method for eigenvalues.- 30. The reduction to Hessenberg form.- 31. The implicit QR iteration method for eigenvalues of upper Hessenberg matrices.- Part 8. QR iterations for companion matrices.- 32. Companion and unitary matrices.- 33. Explicit methods.- 34. Implicit methods with compression.- 35. The factorization based implicit method.- 36. Implicit algorithms based on the QR representation.- Bibliography.
- (source: Nielsen Book Data)9783034806114 20160618
- Publisher's Summary
- This two-volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The main attention is paid to fast algorithms (many of linear complexity) for matrices in semiseparable, quasiseparable, band and companion form. The work is focused on algorithms of multiplication, inversion and description of eigenstructure and includes a large number 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 to compute 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 being derived also for matrices with quasiseparable representation of order one. The QR iteration method for some classes of matrices with quasiseparable of any order representations is studied in the third part. This method is then used in the last part in order to get 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 the accessible style the text will be useful to engineers, scientists, numerical analysts, computer scientists and mathematicians alike.
(source: Nielsen Book Data)9783034806114 20160618
- Publication date
- Copyright date
- Operator theory: advances and applications, 0255-0156 ; 234-235
- Current copyright fee: GBP12.00 0.
- 9783034806053 (v. 1)
- 3034806051 (v. 1)
- 3034806116 (v. 2)
- 9783034806114 (v. 2)
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