Parallel least-squares solution of general and Toeplitz systems

We use O(log2 n) parallel arithmetic steps and n2 processors to compute the least-squares solution x = A+b to a linear system of equations, Ax = b, given a g x h matrix A and a vector b, both filled with integers or with rational numbers, provided that g + h 5 2n and that A is given with its displacement generator of length T = O(1) and thus has displacement rank O(1). For a vector b and for a general p x Q matrix A (with p + Q 5 n), we compute A+ and A+b by using O(log2 n) parallel arithmetic steps and n2.851 processors, and we may also compute A+b by using O(n2.376) arithmetic operations.

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

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

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

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

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

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

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

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

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

[10]  Joachim von zur Gathen.  Parallel algorithms for algebraic problems , 1983, STOC '83.

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

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

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

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

[15]  Thomas Kailath.  Signal-processing applications of some moment problems , 1987 .

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

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