This Article 
 Bibliographic References 
 Add to: 
Concurrent Iterative Algorithm for Toeplitz-like Linear Systems
May 1993 (vol. 4 no. 5)
pp. 592-600

A nonsingular n*n matrix A is given with its short displacement generator. It has smalldisplacement rank bounded by a fixed constant. The class of such matrices generalizesToeplitz matrices. A good initial approximation to a short displacement generator forA/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 arepresented. These results are extended to some other important classes of densestructure matrices.

[1] A. V. Aho, J. E. Hopcroft, and J. D. Ullman,The Design and Analysis of Computer Algorithms. Menlo Park, CA: Addison-Wesley, 1974.
[2] G. S. Ammar and W. B. Gragg, "Superfast solution of real positive definite Toeplitz systems,"SIAM J. Matrix Analysis and Appl., vol. 9, no. 1, pp. 61-76, 1988.
[3] G. S. Ammar and P. Gader, "New decomposition of the inverse of a Toeplitz matrix," inProc. 1989 Int. Symp. Math. Theory of Network and Syst. (MTNS-89), Amsterdam, 1989.
[4] D. Bini and F. di Benedetto, "A new preconditioner for parallel solution of positive definite Toeplitz systems," inProc. 2nd Annu. ACM Symp. Parallel. Algorithms and Architecture(SPAA-90), 1990, pp. 220-223;SIAM J. Sci. Stat. Comput., to be published.
[5] D. Bini and L. Gemignani, "On the Euclidian scheme for polynomials having interlaced real roots," inProc. 2nd Annu. ACM Symp. Parallel Algorithms and Architecture(SPAA-90), 1990, pp. 254-258.
[6] D. Bini and V. Pan,Numerical and Algebraic Computations with Matrices and Polynomials. Boston, MA: Birkhauser, 1993.
[7] R. R. Bitmead and B. D. O. Anderson, "Asymptotically fast solution of Toeplitz and related systems of linear equations,"Linear Algebra and its Appl., vol. 34, pp. 103-116, 1980.
[8] A. Borodin, J. von zur Gathen, and J. Hopcroft, "Fast parallel matrix and GCD computation,"Inform. Contr., vol. 52, no. 3, pp. 241-256, 1982.
[9] R. P. Brent, "The parallel evalution of general arithmetic expressions,"J. ACM, vol. 21, pp. 201-206, 1974.
[10] R. P. Brent, F. G. Gustavson, and D. Y. Y. Yun, "Fast solution of Toeplitz systems of equations and computation of Padéapproximations,"J. Algorithms, vol. 1, pp. 259-295, 1980.
[11] J. R. Bunch, "Stability of methods for solving Toeplitz systems of equations,"SIAM J. Sci. Statist. Comput., vol. 6, no. 2, pp. 349-364, 1985.
[12] J. F. Canny, E. Kaltofen, and L. Yagati, "Solving systems of nonlinear polynomial equations faster," inProc. ACM-SIGSAM Symp. Symb. and Alg. Comp. (ISSAC-89), pp. 121-128, 1989.
[13] T. F. Chan, "Rank revealing QR-factorization,"Linear Alg. and its Appl., vol. 88/89, pp. 67-82, 1987.
[14] J. Chun, T. Kailath, and H. Lev-Ari, "Fast parallel algorithm for QR-factorization of structured matrices,"SIAM J. Scient. Statist. Comput., vol. 8, no. 6, pp. 899-913, 1987.
[15] L. Csanky, "Fast parallel matrix inversion algorithm,"SIAM J. Comput., vol. 5, no. 4, pp. 618-623, 1976.
[16] G. Cybenko and M. Berry, "Hyperbolic Householder algorithms, for factoring structured matrices,"SIAM J. Matrix Anal and Appl., vol. 11, no. 4, pp. 494-520, 1990.
[17] F. R. deHoog, "On the solution of Toeplitz systems,"Linear Algebra and its Appl., vol. 88/89, pp. 123-128, 1987.
[18] D. Eppstein and Z. Galil, "Parallel algorithmic techniques for combinatorial computation,"Annu. Rev. Comput. Sci., vol. 3, pp. 233-283, 1988.
[19] M. Ewerbring and F. T. Luk, "Canonical correlations and generalized SVD: Applications and new algorithms,"J. Comput. Appl. Math., vol. 27, pp. 37-52, 1989.
[20] K. V. Fernando and S. J. Hammarling, "A product induced singular value decomposition for two matrices and balanced realization,"Linear Algebra in Signals, Systems and Control, B. N. Dattaet al.Eds. Philadelphia, PA: SIAM, 1988, pp. 128-140.
[21] B. Friedlander, M. Morf, T. Kailath, and L. Ljung, "New inversion formulas for matrices classified in terms of their distances from Toeplitz matrices,"Linear Algebra and its Appl., vol. 27, pp. 31-60, 1979.
[22] Z. Galil and V. Pan, "Parallel evaluation of the determinant and of the inverse of a matrix,"Inform. Processing Lett., vol. 30, pp. 41-45, 1989.
[23] J. von zur Gathen, "Parallel algorithms for algebraic problems,"SIAM J. Comput., vol. 13, no. 4, pp. 802-824, 1984.
[24] I. C. Gohberg and A. A. Semencul, "On the inversion of finite Toeplitz matrices and their continuous analogs,"Mat. Issled., vol. 2, pp. 201-233 (in Russian), 1972.
[25] I. C. Gohberg, T. Kailath, and I. Koltracht, "Efficient solution of linear systems of equations with recursive structure,"Linear Algebra and its Appl., vol. 80, pp. 81-113, 1986.
[26] G. H. Golub and C. F. van Loan,Matrix Computations. Baltimore, MD: Johns Hopkins Univ. Press, 1989.
[27] G. Heinig, "Beitrage zur spektraltheorie von Operatorbuschen und zur algebraischen Theorie von Toeplitzmatrizen," Diss. B, TH Karl-Marx-Stadt, 1979.
[28] M. T. Heath, A. J. Laub, C. C. Paige, and R. C. Ward, "Computing the SVD of a product of two matrices,"SIAM J. Sci. Statist. Comput., vol. 7, pp. 1147-1159, 1986.
[29] I. S. Iohvidov,Hankel and Toeplitz Matrices and Forms. Boston, MA: Birkhauser, 1982.
[30] T. Kailath, "Signal processing applications of some moment problems,"Proc. AMS Symp. Appl. Math., vol. 37, pp. 71-100, 1987.
[31] T. Kaliath, A. Viera, and M. Morf, "Inverses of Toeplitz operators, innovations and orthogonal polynomials,"SIAM Rev., vol. 20, no. 1, pp. 106-119, 1978.
[32] T. Kailath, S. Y. Kung, and M. Morf, "Displacement ranks of matrices and linear equations,"J. Math. Anal. Appl., vol. 68, no. 2, pp. 395-407, 1979.
[33] R. M. Karp and V. Ramachandran, "Parallel algorithms for shared-memory machines," inHandbook of Theoretical Computer Science, J. van Leeuwen, Ed. Cambridge, MA: M.I.T. Press, 1990, ch. 17, pp. 869-941.
[34] E. Linzer, "On the stability of solution methods for band Toeplitz systems,"Linear Algebra and its Appl., to be published.
[35] E. Linzer and M. Vetterli, "Iterative Toeplitz solvers with local quadratic convergence,"Computing, submitted for publication.
[36] R. T. Moenck and J. H. Carter, "Approximate algorithms to derive exact solutions to systems of linear equations," inProc. EUROSAM, Lecture Notes in Computer Science, vol. 72, Springer, 1979, pp. 63-73.
[37] B. R. Musicus, "Levinson and fast Choleski algorithms for Toeplitz and almost Toeplitz matrices," Internal Rep., Lab. of Electronics, M.I.T., 1981.
[38] V. Pan, "New effective methods for computations with Toeplitz-like matrices," Tech. Rep. 88-28, Computer Sci. Dep., S.U.N.Y. Albany, 1988.
[39] V. Pan, "Fast and efficient parallel inversion of Toeplitz and block Toeplitz matrices,"Operator Theory: Advances and Appl., vol. 40, pp. 359-389, 1989.
[40] V. Pan, "Parallel least-squares solution of general and Toeplitz-like linear systems," inProc. 2nd Annu. ACM Symp. Parallel Algorithms and Architecture (SPAA-90), 1990, pp. 244-253.
[41] V. Pan, "On computations with dense structured matrices,"Math. of Comp., vol. 55, 191, pp. 179-190, 1990 (preliminary version,Proc. ACM-SIGSAM Int. Symp. Symb. and Alg. Comp. (ISSAC-89), 1989, pp. 34-42).
[42] V. Pan, "Parametrization of Newton's iteration for computations with structured matrices and applications,"Computers and Mathematics (with Applications), vol. 24, no. 2, pp. 61-75, 1992.
[43] V. Pan, "Complexity of computations with matrices and polynomials,"SIAM Rev., vol. 34, no. 9, pp. 225-262, 1992.
[44] V. Pan, "Complexity of algorithms for linear systems of equations," inComputer Algorithms for Solving Linear Algebraic Equations (The State of the Art), E. Spedicato, Ed., NATO ASI Series, Series F, Computer and Systems Sciences, vol. 77, 1991, pp. 27-56.
[45] V. Pan and J. Reif, "Some polynomial and Toeplitz matrix computations," inProc. 28th Annu. IEEE Symp. FOCS, 1987, pp. 173-181.
[46] V. Pan and R. Schreiber, "An improved Newton iteration for the generalized inverse of a matrix, with applications,"SIAM J. Sci. Statist. Comput., vol. 12, no. 5, 1109-1131, 1991.
[47] M. C. Pease, "An adaptation of the fast Fourier transform for parallel processing,"J. ACM, vol. 15, pp. 252-264, 1968.
[48] F. P. Preparata and D. V. Sarwate, "An improved parallel processor bound in fast matrix inversion,"Inform. Processing Lett., vol. 7, no. 3, pp. 148-150, 1978.
[49] M. J. Quinn,Designing Efficient Algorithms for Parallel Computers. New York: McGraw-Hill, 1989.
[50] G. Strang, "A proposal for Toeplitz matrix calculations,"Study of Appl. Math., vol. 74, pp. 171-176, 1986.
[51] W. F. Trench, "An algorithm for inversion of finite Toeplitz matrices,"J. SIAM, vol. 12, no. 3, pp. 515-522, 1964.
[52] W. F. Trench, "A note on a Toeplitz inversion formula,"Linear Alg. and its Appl., vol. 129, pp. 55-61, 1990.
[53] L. Valiant, "General purpose parallel architectures," inHandbook of Theoretical Computer Science. Amsterdam: North-Holland, 1990, pp. 943-971.
[54] J. H. Wilkinson,The Algebraic Eigenvalue Problem. London, England: Oxford University Press, 1965.

Index Terms:
Index Termsparallel algorithm; iterative algorithm; Toeplitz-like linear systems; parallel arithmeticsteps; computational complexity; matrix algebra; parallel algorithms
V.Y. Pan, "Concurrent Iterative Algorithm for Toeplitz-like Linear Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 5, pp. 592-600, May 1993, doi:10.1109/71.224221
Usage of this product signifies your acceptance of the Terms of Use.