The symmetric-Toeplitz linear system problem in parallel

Abstract

[EN] Many algorithms exist that exploit the special structure of Toeplitz matrices for solving linear systems. Nevertheless, these algorithms are difficult to parallelize due to its lower computational cost and the great dependency of the operations involved that produces a great communication cost. The foundation of the parallel algorithm presented in this paper consists of transforming the Toeplitz matrix into a another structured matrix called Cauchy¿like. The particular properties of Cauchy¿like matrices are exploited in order to obtain two levels of parallelism that makes possible to highly reduce the execution time. The experimental results were obtained in a cluster of PC¿s.Supported by Spanish MCYT and FEDER under Grant TIC 2003-08238-C02-02Alonso-Jordá, P.; Vidal Maciá, AM. (2005). The symmetric-Toeplitz linear system problem in parallel. Computational Science -- ICCS 2005,Pt 1, Proceedings. 3514:220-228. https://doi.org/10.1007/11428831_28S2202283514Sweet, D.R.: The use of linear-time systolic algorithms for the solution of toeplitz problems. k Technical Report JCU-CS-91/1, Department of Computer Science, James Cook University, Tue, 23 April 1996 15, 17, 55 GMT (1991)Evans, D.J., Oka, G.: Parallel solution of symmetric positive definite Toeplitz systems. Parallel Algorithms and Applications 12, 297–303 (1998)Gohberg, I., Koltracht, I., Averbuch, A., Shoham, B.: Timing analysis of a parallel algorithm for Toeplitz matrices on a MIMD parallel machine. Parallel Computing 17, 563–577 (1991)Gallivan, K., Thirumalai, S., Dooren, P.V.: On solving block toeplitz systems using a block schur algorithm. In: Proceedings of the 23rd International Conference on Parallel Processing, Boca Raton, FL, USA, vol. 3, pp. 274–281. CRC Press, Boca Raton (1994)Thirumalai, S.: High performance algorithms to solve Toeplitz and block Toeplitz systems. Ph.d. th., Grad. College of the U. of Illinois at Urbana–Champaign (1996)Alonso, P., Badía, J.M., Vidal, A.M.: Parallel algorithms for the solution of toeplitz systems of linear equations. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds.) PPAM 2004. LNCS, vol. 3019, pp. 969–976. Springer, Heidelberg (2004)Anderson, E., et al.: LAPACK Users’ Guide. SIAM, Philadelphia (1995)Blackford, L., et al.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)Alonso, P., Badía, J.M., González, A., Vidal, A.M.: Parallel design of multichannel inverse filters for audio reproduction. In: Parallel and Distributed Computing and Systems, IASTED, Marina del Rey, CA, USA, vol. II, pp. 719–724 (2003)Loan, C.V.: Computational Frameworks for the Fast Fourier Transform. SIAM Press, Philadelphia (1992)Heinig, G.: Inversion of generalized Cauchy matrices and other classes of structured matrices. Linear Algebra and Signal Proc., IMA, Math. Appl. 69, 95–114 (1994)Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Mathematics of Computation 64, 1557–1576 (1995)Alonso, P., Vidal, A.M.: An efficient and stable parallel solution for symmetric toeplitz linear systems. TR DSIC-II/2005, DSIC–Univ. Polit. Valencia (2005)Kailath, T., Sayed, A.H.: Displacement structure: Theory and applications. SIAM Review 37, 297–386 (1995