메뉴 건너뛰기




Volumn 302-303, Issue , 1999, Pages 265-293

A fast parallel Björck-Pereyra-type algorithm for solving Cauchy linear equations

Author keywords

Backward error analysis; Bj rck Pereyra algorithm; Cauchy matrix; Effective well conditioning; Fast algorithms; Forward error analysis; Hubert matrix; Ordering of interpolation points; Skeel condition number; Total positivity; Vandermonde matrix

Indexed keywords


EID: 0033260853     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(99)00115-9     Document Type: Article
Times cited : (29)

References (60)
  • 1
    • 0345788507 scopus 로고
    • Observations sur des théoremes de Géométrie, énoncées page 160 de ce volume et page 222 du volume et page 222 du volume précédent
    • J.P.M. Binet, Observations sur des théoremes de Géométrie, énoncées page 160 de ce volume et page 222 du volume et page 222 du volume précédent, J. (de Liouville) de Math, ii. (1837) 248-252.
    • (1837) J. (De Liouville) de Math , vol.2 , pp. 248-252
    • Binet, J.P.M.1
  • 4
    • 84968468488 scopus 로고
    • Solution of Vandermonde Systems of Equations
    • A. Björck, V. Pereyra, Solution of Vandermonde Systems of Equations, Math. Comp. 24 (1970) 893-903.
    • (1970) Math. Comp. , vol.24 , pp. 893-903
    • Björck, A.1    Pereyra, V.2
  • 5
    • 0347680470 scopus 로고
    • Mémorie sur les fonctions alternées et sur les sommes alternées
    • A.L. Cauchy, Mémorie sur les fonctions alternées et sur les sommes alternées, Exercices d'analyse et de Phys. Math, ii (1841) 151-159.
    • (1841) Exercices d'Analyse et de Phys. Math , vol.2 , pp. 151-159
    • Cauchy, A.L.1
  • 8
    • 0042045909 scopus 로고    scopus 로고
    • On the solution of Cauchy systems of equations
    • D. Calvetti, L. Reichel, On the solution of Cauchy systems of equations, Elec. Trans. Numer. Anal. 4 (1996) 125-136.
    • (1996) Elec. Trans. Numer. Anal. , vol.4 , pp. 125-136
    • Calvetti, D.1    Reichel, L.2
  • 9
    • 0043223564 scopus 로고    scopus 로고
    • Factorizations of Cauchy matrices
    • D. Calvetti, L. Reichel, Factorizations of Cauchy matrices, J. Comput. Applied Math. 1 (86) (1997) 103-123.
    • (1997) J. Comput. Applied Math. , vol.1 , Issue.86 , pp. 103-123
    • Calvetti, D.1    Reichel, L.2
  • 12
    • 0012087402 scopus 로고
    • Backward error analysis for totally positive linear systems
    • C. de Boor, A. Pincus, Backward error analysis for totally positive linear systems, Numer. Math. 27 (1977) 485-490.
    • (1977) Numer. Math. , vol.27 , pp. 485-490
    • De Boor, C.1    Pincus, A.2
  • 14
    • 0040731675 scopus 로고
    • Inversion d'une matrice generalisant la matrice de hilbert
    • N. Gastinel, Inversion d'une matrice generalisant la matrice de hilbert, Chiffres 3 (1960) 149-152.
    • (1960) Chiffres , vol.3 , pp. 149-152
    • Gastinel, N.1
  • 15
    • 0000835513 scopus 로고
    • Lower bounds for the condition number of Vandermonde matrix
    • W. Gautschi, G. Inglese, Lower bounds for the condition number of Vandermonde matrix, Numer. Math. 52 (1988) 241-250.
    • (1988) Numer. Math. , vol.52 , pp. 241-250
    • Gautschi, W.1    Inglese, G.2
  • 17
    • 0039937652 scopus 로고
    • On the inversion of Cauchy matrices
    • Proceedings of the MTNS-89 M.A. Kaashoek, J.H. van Schuppen, A.C.M. Ran (Eds.), Birkhäuser, Boston, MA
    • I. Gohberg, I. Koltracht, On the inversion of Cauchy matrices, In Signal processing, Scattering and Operator Theory, and Numerical methods. Proceedings of the MTNS-89 M.A. Kaashoek, J.H. van Schuppen, A.C.M. Ran (Eds.), Birkhäuser, Boston, MA, 1990, pp. 381-392.
    • (1990) Signal Processing, Scattering and Operator Theory, and Numerical Methods , pp. 381-392
    • Gohberg, I.1    Koltracht, I.2
  • 18
    • 21144461298 scopus 로고
    • Mixed, componentwise and structured condition numbers
    • I. Gohberg, I. Koltracht, Mixed, componentwise and structured condition numbers, SIAM J. Matrix Anal. Appl. 14 (1993) 688-704.
    • (1993) SIAM J. Matrix Anal. Appl. , vol.14 , pp. 688-704
    • Gohberg, I.1    Koltracht, I.2
  • 19
    • 0001926832 scopus 로고
    • Fast state space algorithms for matrix Nehari and Nehari-Takagi interpolation problems
    • I. Gohberg, V. Olshevsky, Fast state space algorithms for matrix Nehari and Nehari-Takagi interpolation problems, Integral Equations and Operator Theory 20 (1) (1994) 44-83.
    • (1994) Integral Equations and Operator Theory , vol.20 , Issue.1 , pp. 44-83
    • Gohberg, I.1    Olshevsky, V.2
  • 20
    • 84968505008 scopus 로고
    • Fast Gaussian elimination with partial pivoting for matrices with displacement structure
    • I. Gohberg, T. Kailath, V. Olshevsky, Fast Gaussian elimination with partial pivoting for matrices with displacement structure, Math. Comp. 64 (1995) 1557-1576.
    • (1995) Math. Comp. , vol.64 , pp. 1557-1576
    • Gohberg, I.1    Kailath, T.2    Olshevsky, V.3
  • 22
    • 0001005483 scopus 로고
    • Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems
    • N.J. Higham, Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems, Numer. Math. 50 (1987) 613-632.
    • (1987) Numer. Math. , vol.50 , pp. 613-632
    • Higham, N.J.1
  • 23
    • 0012063613 scopus 로고
    • Fast solution of Vandermonde-like systems, involving orthogonal polynomials
    • N.J. Higham, Fast solution of Vandermonde-like systems, involving orthogonal polynomials, IMA J. Numer. Anal. 8 (1988) 473-486.
    • (1988) IMA J. Numer. Anal. , vol.8 , pp. 473-486
    • Higham, N.J.1
  • 24
    • 0001939091 scopus 로고
    • Stability analysis of algorithms for solving confluent Vandermonde like systems
    • N.J. Higham, Stability analysis of algorithms for solving confluent Vandermonde like systems, SIAM J. Matrix Anal. Appl. 11 (1) (1990) 23-41.
    • (1990) SIAM J. Matrix Anal. Appl. , vol.11 , Issue.1 , pp. 23-41
    • Higham, N.J.1
  • 27
    • 0346419535 scopus 로고
    • April
    • W. Kahan, I. Farkas, Algorithms 167-169 Comm. ACM, 6 (4) April 1963, 164-165 see also the certification, isid, 6 (9) September 1963, 523.
    • (1963) Algorithms 167-169 Comm. ACM , vol.6 , Issue.4 , pp. 164-165
    • Kahan, W.1    Farkas, I.2
  • 28
    • 84945708840 scopus 로고
    • September
    • W. Kahan, I. Farkas, Algorithms 167-169 Comm. ACM, 6 (4) April 1963, 164-165 see also the certification, isid, 6 (9) September 1963, 523.
    • (1963) Algorithms 167-169 Comm. ACM , vol.6 , Issue.9 , pp. 523
  • 30
    • 0004167764 scopus 로고
    • Stanford University Press, Stanford
    • S. Karlin, Total Positivity, vol. 1, Stanford University Press, Stanford, 1972.
    • (1972) Total Positivity , vol.1
    • Karlin, S.1
  • 31
    • 0002301645 scopus 로고
    • Displacement structure approach to Chebyshev-Vandermonde and related matrices
    • T. Kailath, V. Olshevsky, Displacement structure approach to Chebyshev-Vandermonde and related matrices, Integral Equations and Operator Theory 22 (1995) 65-92.
    • (1995) Integral Equations and Operator Theory , vol.22 , pp. 65-92
    • Kailath, T.1    Olshevsky, V.2
  • 32
    • 0029715230 scopus 로고    scopus 로고
    • Displacement structure approach to discrete transform based preconditioners of G. Strang type and of T. Chan type
    • T. Kailth, V. Olshevsky, Displacement structure approach to discrete transform based preconditioners of G. Strang type and of T. Chan type, Calcolo 33 (1996) 191-208.
    • (1996) Calcolo , vol.33 , pp. 191-208
    • Kailth, T.1    Olshevsky, V.2
  • 33
    • 0039972012 scopus 로고    scopus 로고
    • Diagonal pivoting for partially reconstructible cauchy-like matrices with applications to Toeplitz-like linear equations and to boundary rational matrix interpolation problems
    • T. Kailath, V. Olshevsky, Diagonal pivoting for partially reconstructible cauchy-like matrices with applications to Toeplitz-like linear equations and to boundary rational matrix interpolation problems, Linear Algebra Appl. 254 (1997) 251-302.
    • (1997) Linear Algebra Appl. , vol.254 , pp. 251-302
    • Kailath, T.1    Olshevsky, V.2
  • 34
    • 0000665794 scopus 로고
    • Displacement structure: Theory and applications
    • T. Kailath, A.H. Sayed, Displacement structure: theory and applications, SIAM Review 37 (3) (1995) 297-386.
    • (1995) SIAM Review , vol.37 , Issue.3 , pp. 297-386
    • Kailath, T.1    Sayed, A.H.2
  • 37
    • 0041577164 scopus 로고    scopus 로고
    • Factorizations of Cauchy-Vandermonde matrices
    • J.J. Martnez, J.M. Peña, Factorizations of Cauchy-Vandermonde matrices, Linear Algebra Appl. 1-3 (284) (1998) 229-237.
    • (1998) Linear Algebra Appl. , vol.1-3 , Issue.284 , pp. 229-237
    • Martnez, J.J.1    Peña, J.M.2
  • 39
    • 0032319031 scopus 로고    scopus 로고
    • A unified superfast algorithm for boundary rational tangential interpolation problems and for inversion and factorization of dense structured matrices
    • IEEE Computer Society, Los Alamitos, CA
    • V. Olshevsky, V. Pan, A unified superfast algorithm for boundary rational tangential interpolation problems and for inversion and factorization of dense structured matrices, Proceedings of 39th Annual Symposium on Foundations of Computer Science, FOCS'98, IEEE Computer Society, Los Alamitos, CA, 1998, 192-201.
    • (1998) Proceedings of 39th Annual Symposium on Foundations of Computer Science, FOCS'98 , pp. 192-201
    • Olshevsky, V.1    Pan, V.2
  • 42
    • 0012536950 scopus 로고
    • Inverses of Vandermonde matrices
    • F. Parker, Inverses of Vandermonde matrices, Amer. Math. Monthly 71 (1964) 410-411.
    • (1964) Amer. Math. Monthly , vol.71 , pp. 410-411
    • Parker, F.1
  • 43
    • 0007428490 scopus 로고
    • A matrix problem with application to rapid solution of integral equations
    • L. Reichel, A matrix problem with application to rapid solution of integral equations, SIAM J. Sci. Stat. Comput. 11 (1990) 263-280.
    • (1990) SIAM J. Sci. Stat. Comput. , vol.11 , pp. 263-280
    • Reichel, L.1
  • 44
    • 0012089844 scopus 로고
    • Newton interpolation at Leja points
    • L. Reichel, Newton interpolation at Leja points, BIT 30 (1990) 23-41.
    • (1990) BIT , vol.30 , pp. 23-41
    • Reichel, L.1
  • 45
    • 46549096170 scopus 로고
    • Rapid solution of integral equations of classic potential theory
    • V. Rokhlin, Rapid solution of integral equations of classic potential theory, J. Comput. Phys. 60 (1985) 187-207.
    • (1985) J. Comput. Phys. , vol.60 , pp. 187-207
    • Rokhlin, V.1
  • 46
    • 0001334220 scopus 로고
    • Chebyshev-Vandermonde systems
    • L. Reichel, G. Opfer, Chebyshev-Vandermonde systems, Math. Comp. 57 (1991) 703-721.
    • (1991) Math. Comp. , vol.57 , pp. 703-721
    • Reichel, L.1    Opfer, G.2
  • 49
    • 84968490755 scopus 로고
    • Error analysis of the algorithm for shifting the zeros of a polynomial by synthetic division
    • G. Stewart, Error analysis of the algorithm for shifting the zeros of a polynomial by synthetic division, Math. Comp. 19 (1971) 135-139.
    • (1971) Math. Comp. , vol.19 , pp. 135-139
    • Stewart, G.1
  • 50
    • 0018492360 scopus 로고
    • Scaling for numerical stability in Gaussian elimination
    • R. Skeel, Scaling for numerical stability in Gaussian elimination, J. Assoc. Comput. Mach. 26 (1979) 494-526.
    • (1979) J. Assoc. Comput. Mach. , vol.26 , pp. 494-526
    • Skeel, R.1
  • 53
    • 0012096423 scopus 로고
    • Associated polynomials and uniform methods for the solution of linear problems
    • J. Traub, Associated polynomials and uniform methods for the solution of linear problems, SIAM Review 8 (3) (1966) 277-301.
    • (1966) SIAM Review , vol.8 , Issue.3 , pp. 277-301
    • Traub, J.1
  • 54
    • 0022769618 scopus 로고
    • An efficient implementation of conformal mapping method using the Szegö kernel
    • M. Trummer, An efficient implementation of conformal mapping method using the Szegö kernel, SIAM J. Numer. Anal. 23 (1986) 853-872.
    • (1986) SIAM J. Numer. Anal. , vol.23 , pp. 853-872
    • Trummer, M.1
  • 55
    • 0019689784 scopus 로고
    • The block decomposition of a Vandermonde matrix and its applications
    • W. Tang, G. Golub, The block decomposition of a Vandermonde matrix and its applications, BIT 21 (1981) 505-517.
    • (1981) BIT , vol.21 , pp. 505-517
    • Tang, W.1    Golub, G.2
  • 56
    • 21344486577 scopus 로고
    • How bad are Hankel matrices
    • E. Tyrtyshnikov, How bad are Hankel matrices, Numer. Math. 67 (2) (1994) 261-269.
    • (1994) Numer. Math. , vol.67 , Issue.2 , pp. 261-269
    • Tyrtyshnikov, E.1
  • 57
    • 21344496299 scopus 로고
    • Errors and perturbations in vandermonde systems
    • J.M. Varah, Errors and perturbations in vandermonde systems, IMA J. of Numer. Anal. 13 (1993) 1-12.
    • (1993) IMA J. of Numer. Anal. , vol.13 , pp. 1-12
    • Varah, J.M.1
  • 59
    • 0012066994 scopus 로고
    • A priori error analysis of algebraic processes
    • Moskow, Moskow, Mir
    • J. Wilkinson, A priori error analysis of algebraic processes, Proc. Intern. Congr. Math. (Moskow 1966), Moskow, Mir 1968, pp. 629-639.
    • (1966) Proc. Intern. Congr. Math. , pp. 629-639
    • Wilkinson, J.1
  • 60
    • 0001167652 scopus 로고
    • Modern error analysis
    • J. Wilkinson, Modern error analysis, SIAM Review 14 (1971) 548-568.
    • (1971) SIAM Review , vol.14 , pp. 548-568
    • Wilkinson, J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.