Processor-efficient parallel solution of linear systems. II. The positive characteristic and singular cases

For pt.I see Proc. 3rd Ann. ACM Symp. Parallel Algms. Architecture, p. 180-91 (1991). The authors show that over any field, the solution set to a system of n linear equations in n unknowns can be computed in parallel with randomization simultaneously in poly-logarithmic time in n and with only as many processors as are utilized to multiply two n * n matrices. A time unit represents an arithmetic operation in the field. For singular systems the parallel timings are asymptotically as fast as those for non-singular systems, due to the avoidance of binary search in the matrix rank problem, except when the field has small positive characteristic; in that case, binary search is avoided at a somewhat higher processor count measure.<<ETX>>

[1]  Erich Kaltofen,et al.  On Wiedemann's Method of Solving Sparse Linear Systems , 1991, AAECC.

[2]  Victor Y. Pan,et al.  Parallel Evaluation of the Determinant and of the Inverse of a Matrix , 1989, Inf. Process. Lett..

[3]  Erich Kaltofen,et al.  Greatest common divisors of polynomials given by straight-line programs , 1988, JACM.

[4]  Walter Baur,et al.  The Complexity of Partial Derivatives , 1983, Theor. Comput. Sci..

[5]  Allan Borodin,et al.  Fast parallel matrix and GCD computations , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[6]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[7]  Stuart J. Berkowitz,et al.  On Computing the Determinant in Small Parallel Time Using a Small Number of Processors , 1984, Inf. Process. Lett..

[8]  Victor Y. Pan,et al.  Improved Parallel Computations with Matrices and Polynomials , 1991, ICALP.

[9]  M. Morf,et al.  Displacement ranks of matrices and linear equations , 1979 .

[10]  L. Csanky,et al.  Fast parallel matrix inversion algorithms , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[11]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[12]  Douglas H. Wiedemann.  Solving sparse linear equations over finite fields , 1986, IEEE Trans. Inf. Theory.

[13]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[14]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[15]  Sanjeev Saxena,et al.  On Parallel Prefix Computation , 1994, Parallel Process. Lett..

[16]  Le Verrier,et al.  Mémoire sur les variations séculaires des éléments des orbites : pour les sept planètes principales, Mercure, Vénus, la Terre, Mars, Jupiter, Saturne et Uranus , 1976 .

[17]  Victor Y. Pan,et al.  Processor efficient parallel solution of linear systems over an abstract field , 1991, SPAA '91.

[18]  Ketan Mulmuley,et al.  A fast parallel algorithm to compute the rank of a matrix over an arbitrary field , 1986, STOC '86.

[19]  V. Pan.  PARAMETRIZATION OF NEWTON'S ITERATION FOR COMPUTATIONS WITH STRUCTURED MATRICES AND APPLICATIONS , 1992 .

[20]  Alexander L. Chistov,et al.  Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic , 1985, FCT.

[21]  Franco P. Preparata,et al.  An Improved Parallel Processor Bound in Fast Matrix Inversion , 1978, Inf. Process. Lett..

[22]  S. Linnainmaa.  Taylor expansion of the accumulated rounding error , 1976 .

[23]  W. F. Trench.  An Algorithm for the Inversion of Finite Toeplitz Matrices , 1964 .

[24]  T. Kailath,et al.  Fast Parallel Algorithms for QR and Triangular Factorization , 1987 .