Concurrent Iterative Algorithm for Toeplitz-like Linear Systems

A nonsingular n*n matrix A is given with its short displacement generator. It has small displacement rank bounded by a fixed constant. The class of such matrices generalizes Toeplitz matrices. A good initial approximation to a short displacement generator for A/sup -1/ is readily available. Ways to refine this approximation and numerically compute a displacement generator of A/sup -1/ and the solution vector x=A/sup -1/b to a linear system Ax=b by using O(log/sup 2/n) parallel arithmetic steps and n processors are presented. These results are extended to some other important classes of dense structure matrices. >

[1]  Marshall C. Pease,et al.  An Adaptation of the Fast Fourier Transform for Parallel Processing , 1968, JACM.

[2]  M. Morf,et al.  Inverses of Toeplitz operators, innovations, and orthogonal polynomials , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[3]  Elliot L. Linzer,et al.  On the stability of solution methods for band Toeplitz systems , 1992 .

[4]  James R. Bunch,et al.  Stability of Methods for Solving Toeplitz Systems of Equations , 1985 .

[5]  D. Eppstein,et al.  Parallel Algorithmic Techniques for Combinatorial Computation , 1988 .

[6]  Franklin T. Luk,et al.  Canonical correlations and generalized SVD: Applications and new algorithms , 1989 .

[7]  Erich Kaltofen,et al.  Solving systems of nonlinear polynomial equations faster , 1989, ISSAC '89.

[8]  Robert T. Moenck,et al.  Approximate algorithms to derive exact solutions to systems of linear equations , 1979, EUROSAM.

[9]  David Y. Y. Yun,et al.  Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants , 1980, J. Algorithms.

[10]  Gene H. Golub,et al.  Matrix computations , 1983 .

[11]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[12]  Joachim von zur Gathen,et al.  Parallel algorithms for algebraic problems , 1983, SIAM J. Comput..

[13]  V. Pan.  On computations with dense structured matrices , 1990 .

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

[15]  Bruce Ronald. Musicus,et al.  Levinson and fast Choleski algorithms for Toeplitz and almost Toeplitz matrices , 1988 .

[16]  Victor Y. Pan,et al.  Complexity of Computations with Matrices and Polynomials , 1992, SIAM Rev..

[17]  G. Strang.  A proposal for toeplitz matrix calculations , 1986 .

[18]  A. Laub,et al.  Computing the singular value decompostion of a product of two matrices , 1986 .

[19]  L. Ljung,et al.  New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices , 1979 .

[20]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[21]  ByoungSeon Choi.  A Decomposition of the Inverse of a Toeplitz Matrix , 1989 .

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

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

[24]  David Eppstein,et al.  Parallel Algorithmic Techniques for Combinatorial Computation , 1988, ICALP.

[25]  W. Gragg,et al.  Superfast solution of real positive definite toeplitz systems , 1988 .

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

[27]  Victor Y. Pan,et al.  An Improved Newton Iteration for the Generalized Inverse of a Matrix, with Applications , 1991, SIAM J. Sci. Comput..

[28]  G. Cybenko,et al.  Hyperbolic Householder algorithms for factoring structured matrices , 1990 .

[29]  J. Hopcroft,et al.  Fast parallel matrix and GCD computations , 1982, FOCS 1982.

[30]  Dario Bini,et al.  On the Euclidean scheme for polynomials having interlaced real zeros , 1990, SPAA '90.

[31]  L. Csanky,et al.  Fast Parallel Matrix Inversion Algorithms , 1976, SIAM J. Comput..

[33]  T. Chan.  Rank revealing QR factorizations , 1987 .

[34]  Dario Bini,et al.  A new preconditioner for the parallel solution of positive definite Toeplitz systems , 1990, SPAA '90.

[35]  Richard P. Brent,et al.  The Parallel Evaluation of General Arithmetic Expressions , 1974, JACM.

[36]  Victor Y. Pan,et al.  Fast and Efficient Parallel Inversion of Toeplitz and Block Toeplitz Matrices , 1989 .

[37]  William F. Trench,et al.  A note on a Toeplitz inversion formula , 1990 .

[38]  B. Anderson,et al.  Asymptotically fast solution of toeplitz and related systems of linear equations , 1980 .

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

[40]  Thomas Kailath,et al.  Application of fast subspace decomposition to signal processing and communication problems , 1994, IEEE Trans. Signal Process..

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

[42]  Michael J. Quinn,et al.  Designing Efficient Algorithms for Parallel Computers , 1987 .

[43]  Victor Y. Pan,et al.  Parallel least-squares solution of general and Toeplitz systems , 1990, SPAA '90.

[44]  Victor Y. Pan.  On some computations with dense structured matrices , 1989, ISSAC '89.

[45]  Victor Y. Pan,et al.  Some polynomial and Toeplitz matrix computations , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[46]  Thomas Kailath,et al.  Efficient solution of linear systems of equations with recursive structure , 1986 .