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020 _a9783034806121
_9978-3-0348-0612-1
024 7 _a10.1007/978-3-0348-0612-1
_2doi
035 _ato000541796
040 _aSpringer
_cSpringer
_dRU-ToGU
050 4 _aQA184-205
072 7 _aPBF
_2bicssc
072 7 _aMAT002050
_2bisacsh
082 0 4 _a512.5
_223
100 1 _aEidelman, Yuli.
_eauthor.
_9333923
245 1 0 _aSeparable Type Representations of Matrices and Fast Algorithms
_helectronic resource
_bVolume 2 Eigenvalue Method /
_cby Yuli Eidelman, Israel Gohberg, Iulian Haimovici.
260 _aBasel :
_bSpringer Basel :
_bImprint: Birkhäuser,
_c2014.
300 _aXI, 359 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
490 1 _aOperator Theory: Advances and Applications,
_x0255-0156 ;
_v235
505 0 _aPart 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.  .
520 _aThis 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.
650 0 _amathematics.
_9566183
650 0 _aMatrix theory.
_9303236
650 0 _aNumerical analysis.
_9566288
650 1 4 _aMathematics.
_9566184
650 2 4 _aLinear and Multilinear Algebras, Matrix Theory.
_9303237
650 2 4 _aNumerical Analysis.
_9566289
700 1 _aGohberg, Israel.
_eauthor.
_9331543
700 1 _aHaimovici, Iulian.
_eauthor.
_9447741
710 2 _aSpringerLink (Online service)
_9143950
773 0 _tSpringer eBooks
830 0 _aOperator Theory: Advances and Applications,
_9566535
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-0348-0612-1
912 _aZDB-2-SMA
999 _c399764