Parallel complexity of computations with general and Toeplitz-like matrices filled with integers and extensions

SIAM Journal on Computing

Volumn 30, Issue 4, 2000, Pages 1080-1125

  Pan, Victor Y ^a  

^a CITY UNIVERSITY OF NEW YORK   (United States)

Author keywords

Block Gauss Jordan decomposition; Computational complexity; Displacement rank; Newton Hensel's lifting; p adic lifting; Parallel algorithms; Polynomial gcd; Randomized algorithms; Toeplitz matrix computations; Toeplitz like matrices

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; COMPUTATIONAL COMPLEXITY; DIGITAL ARITHMETIC; NUMERICAL ANALYSIS; PARALLEL ALGORITHMS; POLYNOMIALS; PROBLEM SOLVING;

BLOCK GAUSS-JORDAN DECOMPOSITION; INTEGERS; PARALLEL COMPLEXITY; TOEPLITZ-LIKE MATRICES;

MATRIX ALGEBRA;

EID: 0034843051     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539797349959     Document Type: Article

References (95)

1. G. S. AMMAR AND W. B. GRAGG, Superfast solution of real positive definite Toeplitz systems, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 61-76.
2. A. V. AHO, J. E. HOPCROFT, AND J. D. ULLMAN, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.
3. E. H. BAREISS, Sylvester's identity and multistep integer-preserving Gaussian elimination, Math. Comp., 22 (1968), pp. 565-578.
4. R. R. BITMEAD AND B. D. O. ANDERSON, Asymptotically fast solution of Toeplitz and related systems of linear equations, Linear Algebra Appl., 34 (1980), pp. 103-116.
5. E. R. BERLEKAMP, Algebraic Coding Theory, McGraw-Hill, New York, 1968.
6. S. BERKOWITZ, On computing the determinant in small parallel time using a small number of processors, Inform. Process. Lett., 18 (1984), pp. 147-150.
7. A. BORODIN, J. VON ZUR GATHEN, AND J. HOPCROFT, Fast parallel matrix and GCD computation, Inform. and Control, 52 (1982), pp. 241-256.
8. R. P. BRENT, F. G. GUSTAVSON, AND D. Y. Y. YUN, Fast solution of Toeplitz systems of equations and computation of Padé approximations, J. Algorithms, 1 (1980), pp. 259-295.
9. A. BRUCHSTEIN AND T. KAILATH, An inverse scattering framework for several problems in signal processing, IEEE Acoustics, Speech and Signal Processing (ASSP) Magazine, January 1987, pp. 6-20.
10. D. BINI AND G. LOTTI, Stability of fast algorithms for matrix multiplication, Numer. Math., 36 (1980), pp. 63-72.
11. D. BONDYFALAT, B. MOURRAIN, AND V. Y. PAN, Controlled iterative methods for solving polynomial systems, in Proceedings of the Annual ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 1998, pp. 252-259.
12. D. BINI AND V. Y. PAN, Parallel complexity of tridiagonal symmetric eigenvalue problem, in Proceedings of the 2nd Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 1991, pp. 384-393.
13. D. BINI AND V. Y. PAN, Improved parallel computation with Toeplitz-like and Hankel-like matrices, Linear Algebra Appl., 188/189 (1993), pp. 3-29.
14. D. BINI AND V. Y. PAN, Polynomial and Matrix Computations, Fundamental Algorithms 1, Birkhäuser, Boston, 1994.
15. W. S. BROWN AND J. F. TRAUB, On Euclid's algorithm and the theory of subresultants, J. ACM, 18 (1971), pp. 505-514.
16. M. BEN-OR AND P. TIWARI, Simple algorithm for approximating all roots of a polynomial with real roots, J. Complexity, 6 (1990), pp. 417-442.
17. J. R. BUNCH, Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 349-364.
18. S. CHANDRASEKHAR, On the radiative equilibrium of a stellar atmosphere, Astrophys. J., 106 (1947), pp. 152-216,
19. S. CHANDRASEKHAR, Astrophys. J., 107 (1948), pp. 48-72.
20. R. W. COTTLE, Manifestation of the Schur complement, Linear Algebra Appl., 8 (1974), pp. 189-211.
21. A. L. CHISTOV, Fast parallel calculation of the rank of matrices over a field of arbitrary characteristics, in Fundamentals of Computation Theory (Cottbus, 1985), Lecture Notes in Comput. Sci. 199, Springer, Berlin, 1985, pp 63-69.
22. D. G. CANTOR AND E. KALTOFEN, On fast multiplication of polynomials over arbitrary rings, Acta Inform., 28 (1991), pp. 697-701.
23. J. CHUN, T. KAILATH, AND H. LEV-ARI, Fast parallel algorithm for QR-factorization of structured matrices, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 899-913.
24. L. CSANKY, Fast parallel matrix inversion algorithms, SIAM J. Comput., 5 (1976), pp. 618-623.
25. F. R. DE HOOG, On the solution of Toeplitz systems, Linear Algebra Appl., 88/89 (1987), pp. 123-138.
26. J. EDMONDS, Systems of distinct representatives and linear algebra, J. Res. Nat. Bur. Standards, 71B (1967), pp. 241-245.
27. D. EPPSTEIN AND Z. GALIL, Parallel algorithmic techniques for combinatorial computation, Annual Rev. Comput. Sci., 3 (1988), pp. 233-283.
28. I. Z. EMIRIS AND V. Y. PAN, The structure of sparse resultant matrices, in Proceedings of the Annual ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 1997, pp. 189-196.
29. L. FOX, An Introduction to Numerical Linear Algebra, Oxford University Press, Oxford, UK, 1964.
30. J. VON ZUR GATHEN, Parallel algorithms for algebraic problems, SIAM J. Comput., 13 (1984), pp. 802-824.
31. J. VON ZUR GATHEN, Parallel arithmetic computations: A survey, in Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 233, Springer, Berlin, 1986, pp. 93-112.
32. I. GOHBERG, T. KAILATH, AND V. OLSHEVSKY, Fast Gaussian elimination with partial pivoting for matrices with displacement structure, Math. Comp., 64 (1995), pp. 1557-1576.
33. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1989 (2nd ed.), 1996 (3rd ed.).
34. S. HAYKIN, Adaptive Filter Theory, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1991.
35. G. HEINIG, Inversion of generalized Cauchy matrices and other classes of structured matrices, in Linear Algebra for Signal Processing, IMA Vol. Math. Appl. 69, Springer, New York, 1995, pp. 95-114.
36. K. IRELAND AND M. ROSEN, A Classical Introduction to Modern Number Theory, Springer, Berlin, 1982.
37. J. JÀ JÀ, An Introduction to Parallel Algorithms, Addison-Wesley, Reading, MA, 1992.
38. T. KAILATH, A view of three decades of linear filtering theory, IEEE Trans. Inform. Theory, 20 (1974), pp. 146-181.
39. T. KAILATH, Signal processing applications of some moment problems, in Moments in Mathematics, Proc. Sympos. App. Math. 37, AMS, Providence, RI, 1987, pp. 71-100.
40. E. KALTOFEN, Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems, Math. Comput., 64 (1995), pp. 777-806.
41. R. KING, M. AHMADI, R. GORGUI-NAGUIB, A. KWABWE, AND M. AZIMI-SADJADI, Digital Filtering in One and Two Dimensions: Design and Applications, Plenum Press, New York, 1989.
42. T. KAILATH, S.-Y. KUNG, AND M. MORF, Displacement ranks of matrices and linear equations, J. Math. Anal. Appl., 68 (1979), pp. 395-407.
43. T. KAILATH, L. LJUNG, AND M. MORF, A new approach to the dertemination of Fredholm resolvents of nondisplacement kernels, in Topics in Functional Analysis, I. Gohberg and M. Kac, eds., Academic Press, New York, 1978, pp. 169-184.
44. E. KALTOFEN AND V. Y. PAN, Processor efficient parallel solution of linear systems over an abstract field, in Proceedings of the 3rd Annual ACM Symposium on Parallel Algorithms and Architectures, ACM, New York, 1991, pp. 180-191.
45. E. KALTOFEN AND V. Y. PAN, Processor-efficient parallel solution of linear systems II. The positive characteristic and singular cases, in Proceedings of 33rd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 1992, pp. 714-723.
46. E. KALTOFEN AND V. Y. PAN, Parallel solution of Toeplitz and Toeplitz-like linear systems over fields of small positive characteristic, in Proceedings of the First International Symposium on Parallel Symbolic Computation, Lecture Notes Ser. Comput. 5, World Scientific, Singapore, 1994, pp. 225-233.
47. R. KARP AND V. RAMACHANDRAN, A survey of parallel algorithms for shared memory machines, in Handbook for Theoretical Computer Science, J. van Leeuwen, ed., North-Holland, Amsterdam, 1990, pp. 869-941.
48. E. KALTOFEN AND B. D. SAUNDERS, On Wiedemann's method for solving sparse linear systems, Proc. AAECC-9, Lecture Notes in Comput. Sci. 539, Springer, Berlin, 1991, pp. 29-38.
49. T. KAILATH, A. VIEIRA, AND M. MORF, Inverses of Toeplitz operators, innovations, and orthogonal polynomials, SIAM Rev., 20 (1978), pp. 106-119.
50. H. LEV-ARI AND T. KAILATH, Lattice filter parametrization and modelling of nonstationary processes, IEEE Trans. Inform. Theory, IT-30 (1984), pp. 2-16.
51. H. LEV-ARI, T. KAILATH, AND J. CIOFFI, Least squares adaptive lattice and transversal filters; a unified geometrical theory, IEEE Trans. Inform. Theory, IT-30 (1984), pp. 222-236.
52. F. T. LEIGHTON, Introduction to Parallel Algorithms and Architectures: Arrays, Trees and Hypercubes, Morgan Kaufmann, San Mateo, CA, 1992.
53. R. J. LIPTON, D. ROSE, AND R. E. TARJAN, Generalized nested dissection, SIAM J. Numer. Anal., 16 (1979), pp. 346-358.
54. J. MAKHOUL, Linear prediction: A tutorial review, Proc. IEEE, 63 (1975), pp. 561-580.
55. R. T. MOENCK AND J. H. CARTER, Approximate algorithms to derive exact solutions to systems of linear equations, in Symbolic and Algebraic Computation, Lecture Notes in Comput. Sci. 72, Springer, Berlin, 1979, pp. 63-73.
56. M. MORF, Fast Algorithms for Multivariable Systems, Ph.D. Thesis, Stanford University, Stanford, CA, 1974.
57. M. MORF, Doubling algorithms for Toeplitz and related equations, in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, IEEE Computer Society, Los Alamitos, CA, 1980, pp. 954-959.
58. B. MOURRAIN AND V. Y. PAN, Asymptotic acceleration of solving multivariate polynomial systems of equations, in Proceedings of the ACM Symposium on Theory of Computing, ACM, New York, 1998, pp. 488-496.
59. G. L. MILLER, V. RAMACHANDRAN, AND E. KALTOFEN, Efficient parallel evaluation of straight-line code and arithmetic circuits, SIAM J. Comput., 17 (1988), pp. 687-695.
60. B. R. MUSICUS, Levinson and Fast Choleski Algorithms for Toeplitz and Almost Toeplitz Matrices, Internal Report, Lab. of Electronics, M.I.T., Cambridge, MA, 1981.
61. V. OLSHEVSKY AND V. Y. PAN, A unified superfast algorithm for boundary rational tangential interpolation problem and for inversion and factorization of dense structured matrices, in Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 1998, pp. 192-201.
62. V. OLSHEVSKY AND V. Y. PAN, Polynomial and rational interpolation and multipoint evaluation (with structured matrices), in Proceedings of the 26th International Colloquium on Automata, Languages and Programming (ICALP 99), Lecture Notes in Comput. Sci. 1644, J. Wiedermann, P. van Emde Boas, and M. Nielsen, eds., Springer, Berlin, 1999, pp. 585-594.
63. V. Y. PAN, Fast and efficient parallel algorithms for the exact inversion of integer matrices, in Proceedings of the 5th Annual Conference on Foundations of Software Technology and Theoretical Compututer Science, Lecture Notes in Comput. Sci. 206, Springer-Verlag, New York, 1985, pp. 504-521.
64. V. Y. PAN, Complexity of parallel matrix computations, Theoret. Comput. Sci., 54 (1987), pp. 65-85.
65. V. Y. PAN, Computations with dense structured matrices, Math. Comp., 55 (1990), pp. 179-190.
66. V. Y. PAN, Complexity of algorithms for linear systems of equations, in Computer Algorithms for Solving Linear Algebraic Equations (The State of the Art), E. Spedicato, ed., NATO Adv. Sci. Inst. Ser. F Comput. and Systems Sci. 77, Springer, Berlin, 1991, pp. 27-56.
67. V. Y. PAN, Parametrization of Newton's iteration for computations with structured matrices and applications, Comput. Math. Appl., 24 (1992), pp. 61-75.
68. V. Y. PAN, Complexity of computations with matrices and polynomials, SIAM Rev., 34 (1992), pp. 225-262.
69. V. Y. PAN, Parallel solution of Toeplitz-like linear systems, J. Complexity, 8 (1992), pp. 1-21.
70. V. Y. PAN, Decreasing the displacement rank of a matrix, SIAM J. on Matrix Anal., Appl. 14 (1993), pp. 118-121.
71. V. Y. PAN, Concurrent iterative algorithm for Toeplitz-like linear systems, IEEE Trans. Parallel and Distributed Systems, 4 (1993), pp. 592-600.
72. V. Y. PAN, Parallel solution of sparse linear and path systems, in Synthesis of Parallel Algorithms, J.H. Reif, ed., Morgan Kaufmann, San Mateo, CA, 1993, pp. 621-678.
73. V. Y. PAN, Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros, in Proceedings of the 27th Annual ACM Symposium on Theory of Computing, ACM, New York, 1995, pp. 741-750.
74. V. Y. PAN, A new approach to parallel computation of polynomial GCD and to related parallel computations over abstract fields, in Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, PA, 1996, pp. 518-527.
75. V. Y. PAN, Optimal and nearly optimal algorithms for approximating polynomial zeros, Comput. Math. Appl., 31 (1996), pp. 97-138.
76. V. Y. PAN, Parallel computation of polynomial GCD and some related parallel computations over abstract fields, Theoret. Comput. Sci., 162 (1996), pp. 173-223.
77. V. Y. PAN, Effective parallel computations with Toeplitz and Toeplitz-like matrices filled with integers, in The Mathematics of Numerical Analysis (Park City, Utah, 1995), Lectures in Appl. Math. 32, J. Renegar, M. Shub, and S. Smale, eds., Amer. Math. Soc., Providence, RI, 1996, pp. 591-641.
78. V. Y. PAN, Solving a polynomial equation: Some history and recent progress, SIAM Rev., 39 (1997), pp. 187-220.
79. V. Y. PAN, Nearly optimal computations with structured matrices, in Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2000, pp. 953-962.
80. V. Y. PAN, Matrix structure, polynomial arithmetic, and erasure-resilient encoding/decoding, to appear in Proceedings of the ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2000.
81. V. Y. PAN, S. BRANHAM, R. ROSHOLT, AND A. ZHENG, Newton's iteration for structured matrices, in Fast Reliable Algorithms for Matrices with Structure, SIAM, Philadelphia, PA, 1999, pp. 189-210.
82. V. Y. PAN, F. P. PREPARATA, Work-preserving speed-up of parallel matrix computations, SIAM J. Comput., 24 (1995), pp. 811-821.
83. V. Y. PAN AND J. REIF, Fast and efficient parallel solution of dense linear systems, Comput. Math. Appl., 17 (1989), pp. 1481-1491.
84. V. Y. PAN AND J. REIF, The parallel computation of the minimum cost paths in graphs by stream contraction, Inform. Process. Lett., 40 (1991), pp. 79-83.
85. V. Y. PAN AND J. REIF, Fast and efficient parallel solution of sparse linear systems, SIAM J. Comput., 22 (1993), pp. 1227-1250.
86. V. Y. PAN, A. SADIKOU, E. LANDOWNE, AND O. TIGA, A new approach to fast polynomial interpolation and multipoint evaluation, Comput. Math. Appl., 25 (1993), pp. 25-30.
87. V. Y. PAN, A. ZHENG, X. HUANG, AND Y. YU, Fast multipoints polynomial evaluation and interpolation via computations with structured matrices, Ann. Numer. Math., 4 (1997), pp. 483-510.
88. M.J. QUINN, Parallel Computing: Theory and Practice, McGraw-Hill, New York, 1994.
89. 3 n) algorithm for the real root problem, in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 1993, pp. 626-635.
90. J. REIF, Work efficient parallel solution of Toeplitz systems and polynomial GCD, in Proceedings of the 27th Annual ACM Symposium on Theory of Computing, ACM, New York, 1995, pp. 751-761.
91. J. REIF AND S.R. TATE, Optimal size integer division circuits, SIAM J. Comput., 19 (1990), pp. 912-924.
92. V. STRASSEN, Gaussian elimination is not optimal, Numer. Math., 13 (1969), pp. 354-356.
93. W. F. STEWART, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994.
94. L. VALIANT, S. SKYUM, S. BERKOWITZ, AND C. RACKOFF, Fast parallel computation of polynomials using few processors, SIAM J. Comput., 12 (1983), pp. 641-644.
95. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, UK, 1965.