Parallel Algorithms for Toeplitz Systems

We describe some parallel algorithms for the solution of Toeplitz linear systems and Toeplitz least squares problems. First we consider the parallel implementation of the Bareiss algorithm (which is based on the classical Schur algorithm). The alternative Levinson algorithm is less suited to parallel implementation because it involves inner products.

[1]  Al Young Providence, Rhode Island , 1975 .

[2]  T. Kailath A Theorem of I. Schur and Its Impact on Modern Signal Processing , 1986 .

[3]  N. Wiener The Wiener RMS (Root Mean Square) Error Criterion in Filter Design and Prediction , 1949 .

[4]  James Durbin,et al.  The fitting of time series models , 1960 .

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

[6]  Philippe Delsarte,et al.  A polynomial approach to the generalized Levinson algorithm based on the Toeplitz distance , 1983, IEEE Trans. Inf. Theory.

[7]  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.

[8]  Richard P. Brent,et al.  Parallel solution of certain Toeplitz least-squares problems , 1986 .

[9]  Richard P. Brent,et al.  A Note on Downdating the Cholesky Factorization , 1987 .

[10]  J. Jain An efficient algorithm for a large Toeplitz set of linear equations , 1979 .

[11]  Adam W. Bojanczyk,et al.  Linearly Connected Arrays for Toeplitz Least-Squares Problems , 1990, J. Parallel Distributed Comput..

[12]  N. Levinson The Wiener (Root Mean Square) Error Criterion in Filter Design and Prediction , 1946 .

[13]  Jean-Marc Delosme,et al.  Highly concurrent computing structures for matrix arithmetic and signal processing , 1982, Computer.

[14]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  George Cybenko,et al.  Fast toeplitz orthogonalization using inner decompositions , 1987 .

[16]  H. T. Kung Why systolic architectures? , 1982, Computer.

[17]  G. Stewart Introduction to matrix computations , 1973 .

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

[19]  J. Schur,et al.  Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. , 1917 .

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

[21]  D. Sweet Fast Toeplitz orthogonalization , 1984 .

[22]  Charles E. Leiserson,et al.  Optimizing synchronous systems , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

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

[24]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

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

[26]  George Cybenko The numerical stability of the lattice algorithm for least squares linear prediction problems , 1984 .

[27]  Jorma Rissanen,et al.  Solution of linear equations with Hankel and Toeplitz matrices , 1974 .

[28]  Franklin T. Luk,et al.  A fast but unstable orthogonal triangularization technique for Toeplitz matrices , 1987 .

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

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

[31]  G. Cybenko A general orthogonalization technique with applications to time series analysis and signal processing , 1983 .

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

[33]  George Cybenko,et al.  The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations , 1980 .

[34]  Shalhav Zohar,et al.  The Solution of a Toeplitz Set of Linear Equations , 1974, JACM.

[35]  G. Alistair Watson,et al.  An Algorithm for the Inversion of Block Matrices of Toeplitz Form , 1973, JACM.

[36]  Jeffrey D Ullma Computational Aspects of VLSI , 1984 .

[37]  F. Hoog A new algorithm for solving Toeplitz systems of equations , 1987 .

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

[39]  Franklin T. Luk,et al.  Some Linear-Time Algorithms for Systolic Arrays , 2010, IFIP Congress.

[40]  R. Brent,et al.  QR factorization of Toeplitz matrices , 1986 .

[41]  W. E. Gentleman Least Squares Computations by Givens Transformations Without Square Roots , 1973 .

[42]  Richard P. Brent,et al.  Old And New Algorithms For Toeplitz Systems , 1988, Optics & Photonics.

[43]  Shalhav Zohar,et al.  Toeplitz Matrix Inversion: The Algorithm of W. F. Trench , 1969, JACM.

[44]  J. Makhoul,et al.  Linear prediction: A tutorial review , 1975, Proceedings of the IEEE.

[45]  Marilyn Bohl,et al.  Information processing , 1971 .

[46]  Rajendra Kumar,et al.  A fast algorithm for solving a Toeplitz system of equations , 1983, IEEE Trans. Acoust. Speech Signal Process..

[47]  I. Gohberg,et al.  Convolution Equations and Projection Methods for Their Solution , 1974 .

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

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

[50]  J. L. Hock,et al.  An exact recursion for the composite nearest‐neighbor degeneracy for a 2×N lattice space , 1984 .

[51]  M. Shensa,et al.  Remarks on a displacement-rank inversion method for Toeplitz systems , 1982 .

[52]  E. Bareiss Numerical solution of linear equations with Toeplitz and Vector Toeplitz matrices , 1969 .

[53]  Sun-Yuan Kung,et al.  A highly concurrent algorithm and pipeleined architecture for solving Toeplitz systems , 1983 .

[54]  Franklin T. Luk,et al.  A Systolic Array for the Linear-Time Solution of Toeplitz Systems of Equations , 1982 .

[55]  G. Stewart The Effects of Rounding Error on an Algorithm for Downdating a Cholesky Factorization , 1979 .

[56]  Adam W. Bojanczyk,et al.  Stability analysis of fast Toeplitz linear system solvers , 1991 .

[57]  I Gohberg I. Schur methods inoperator theory and signal processing , 1986 .

[58]  Lennart Ljung,et al.  The Factorization and Representation of Operators in the Algebra Generated by Toeplitz Operators , 1979 .

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

[60]  Martin Morf,et al.  Doubling algorithms for Toeplitz and related equations , 1980, ICASSP.