A fast parallel Cholesky decomposition algorithm for tridiagonal symmetric matrices

SIAM Journal on Matrix Analysis and Applications

Volumn 18, Issue 2, 1997, Pages 403-418

  Bar On, Ilan ^a,b   Codenotti, Bruno ^c   Leoncini, Mauro ^d  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b UNIVERSITY OF TEXAS OF THE PERMIAN BASIN   (United States)

^c CNR   (Italy)

^d UNIVERSITY OF PISA   (Italy)

Author keywords

Cholesky decomposition; CM5; Eigenvalues; LR and QR algorithms; Parallel algorithm; Symmetric, tridiagonal, and band matrices

Indexed keywords

EID: 0031531915     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479895282623     Document Type: Article

References (30)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. DU CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, 2nd ed., SIAM, Philadelphia, PA, 1992.
2. I. BAR-ON, A practical parallel algorithm for solving band symmetric positive definite systems of linear equations, ACM Trans. Math. Software, 13 (1987), pp. 323-332.
3. I. BAR-ON, Fast Parallel LR and QR Algorithms for Symmetric Band Matrices, Tech. report 749, Technion. Computer Science Department, Haifa, Israel, 1992.
4. I. BAR-ON, A New Divide and Conquer Parallel Algorithm for Computing the Eigenvalues of a Symmetric Tridiagonal Matrix, Tech. report 832, Technion, Computer Science Department, Haifa, Israel, 1994.
5. I. BAR-ON, Interlacing properties for tridiagonal symmetric matrices with applications to parallel computing, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 548-562.
6. I. BAR-ON AND B. CODENOTTI, A fast and stable parallel QR algorithm for symmetric tridiagonal matrices, Linear Algebra Appl., 220 (1995), pp. 63-96.
7. I. BAR-ON, B. CODENOTTI, AND M. LEONCINI, A Fast Parallel Cholesky Decomposition Algorithm for Tridiagonal Symmetric Matrices, Tech. report TR-95-006, International Computer Science Institute, Berkeley, CA, 1995.
8. I. BAR-ON AND M. LEONCINI, Fast and reliable parallel solution of bidiagonal systems, SIAM J. Numer. Anal., 1995, submitted.
9. J. BARLOW AND J. DEMMEL, Computing accurate eigensystems of scaled diagonally dominant matrices, SIAM J. Numer. Anal., 27 (1990), pp. 762-791.
10. J. CUPPEN, A divide and conquer method for the symmetric eigenproblem, Numer. Math., 36 (1981), pp. 177-195.
11. J. DEMMEL AND W. KAHAN, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 873-912.
12. J. J. DONGARRA AND A. H. SAMEH, On some parallel banded system solvers, Parallel Comput., 1 (1984), pp. 223-235.
13. J. J. DONGARRA AND D. SORENSEN, A fully parallel algorithm for the symmetric eigenvalue problem, SIAM J. Sci. Stat. Comput., 8 (1987), pp. s139-s154.
14. Z. DRMAČ, M. OMLADIČ, AND K. VESELIĆ, On the perturbation of the Cholesky factorization, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1319-1332.
15. K. V. FERNANDO AND B. PARLETT, Accurate singular values and differential qd algorithms, Numer. Math., 67 (1994), pp. 191-229.
16. J. FRANCIS, The QR transformation, part I, Comput. J., 4 (1961), pp. 265-271.
17. J. FRANCIS, The QR transformation, part II, Comput. J., 4 (1962), pp. 332-345.
18. J. GRAD AND E. ZAKRAJŠEK, LR algorithm with Laguerre shift for symmetric tridiagonal matrices, Comput. J., 15 (1972), pp. 268-270.
19. S. L. JOHNSSON, Solving tridiagonal systems on ensemble architectures, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 354-392.
20. B. N. PARLETT, Laguerre's method applied to the matrix eigenvalue problem, Math. Comput., 18 (1964), pp. 464-485.
21. H. RUTISHAUSER, Solution of eigenvalue problems with the LR transformation, Nat. Bur. Standards, AMS, 49 (1958), pp. 47-81.
22. H. RUTISHAUSER AND H. SCHWARZ, The LR transformation method for symmetric matrices, Numer. Math., 5 (1963), pp. 273-289.
23. A. SAMEH AND D. KUCK, On stable parallel linear system solver, J. Assoc. Comput. Mach., 25 (1978), pp. 81-91.
24. H. S. STONE, An efficient parallel algorithm for the solution of a tridiagonal system of equations, J. Assoc. Comput. Mach., 20 (1975), pp. 27-38.
25. H. S. STONE, Parallel tridiagonal equation solver, ACM Trans. Math. Software, 1 (1975), pp. 289-307.
26. P. SWARZTRAUBER, A parallel algorithm for solving general tridiagonal equations, Math. Comput., 33 (1979), pp. 185-199.
27. THINKING MACHINE CORPORATION, The Connection Machine CM-5 Technical Summary, 1991.
28. H. WANG, A parallel method for tridiagonal equations, ACM Trans. Math. Software, 7 (1981), pp. 170-183.
29. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Oxford University Press, London, UK, 1965. Reprinted in Oxford Science Publications, 1988.
30. J. H. WILKINSON, A priori error analysis of algebraic processes, in Proc. International Congress Math., Moscow: Izdat, 1968, pp. 629-639.