The symmetric-Toeplitz linear system problem in parallel

[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

On Solving Block Toeplitz Systems Using a Block Schur Algorithm

This paper presents a block Schur algorithm to obtain a factorization of a symmetric block Toeplitz matrix. It is inspired by the various block Schur algorithms that have appeared in the literature but which have not considered the influence of performance tradeoffs on implementation choices. We develop a version based on block hyperbolic Householder reflectors by adapting the representation schemes for block Householder reflectors in the literature to the hyperbolic case. The basic algorithm is applicable to symmetric positive definite Toeplitz matrices. Leading evidence is presented that, under certain circumstances, performance gains can be obtained by foregoing some of the Toeplitz structure by using have a block size larger than the actual block size given by the structure of the matrix. This allows the block algorithm to also be used to factor efficiently standard symmetric positive definite Toeplitz matrices. An extension to the algorithm that can be used to solve symmetric Toep..

A multicore solution to Block-Toeplitz linear systems of equations

There exist algorithms, also called "fast" algorithms, which exploit the special structure of Toeplitz matrices so that, e.g., allow to solve a linear system of equations in O(n2) flops. However, some implementations of classical algorithms that do not use this structure (O(n3) flops) highly reduce the time to solution when several cores are available. That is why it is necessary to work on "fast" algorithms so that they do not lose track of the benefits of new hardware/software. In this work, we propose a new approach to the Generalized Schur Algorithm, a very known al- gorithm for the solution of Toeplitz systems, to work on a Block-Toeplitz matrix. Our algorithm is based on matrix-matrix multiplications, thus allowing to exploit an efficient implementation of this operation if it exists. Our algorithm also makes use of the thread level parallelism featured by multicores to decrease execution time.PROMETEO/2009/013, Generalitat Valenciana. Projects TEC2009-13741, TIN2010-14971 and TIN2011-15734-E of the Ministerio Espanol de Ciencia e Innovacion, and TEC2012-38142-C04 of the Ministerio Espanol de Economia y Competitividad.Alonso-Jordá, P.; Argüelles, D.; Ranilla, J.; Vidal Maciá, AM. (2013). A multicore solution to Block-Toeplitz linear systems of equations. Journal of Supercomputing. 65(3):999-1009. https://doi.org/10.1007/s11227-012-0824-4S9991009653Alonso P, Badía JM, Vidal AM (2005) An efficient parallel algorithm to solve block–Toeplitz systems. J Supercomput 32:251–278Alonso P, Argüeso F, Cortina R, Ranilla J, Vidal AM Non-linear parallel solver for detecting point sources in CMB maps using Bayesian techniques. J Math Chem. doi: 10.1007/s10910-012-0078-7Anderson E et al (1999) LAPACK users’ guide, 3rd edn. SIAM, PhiladelphiaBischof C, van Loan C (1987) The WY representation for products of householder matrices. SIAM J Sci Stat Comput 8(1):2–13Chun J, Kailath T, Lev-Ari H (1987) Fast parallel algorithms for QR and triangular factorization. SIAM J Sci Stat Comput 8(6):899–913Cybenko G, Berry M (1990) Hyperbolic householder algorithms for factoring structured matrices. SIAM J Matrix Anal Appl 11(4):499–520Gallivan K, Thirumalai S, Van Dooren P (1994) On solving block Toeplitz systems using a block Schur algorithm. In: Proceedings of the 23rd international conference on parallel processing, vol 3. CRC Press, Boca Raton, pp 274–281Gustavson FG (1997) Recursion leads to automatic variable blocking for dense linear-algebra algorithms. IBM J Res Dev 41(6):737–755Intel MKL (2012) http://software.intel.com/en-us/articles/intel-mklJin XQ (2002) Developments and applications of Block Toeplitz iterative solvers. Combinatorics and computer science. Science Press, BeijingKailath T, Sayed AH (1995) Displacement structure: theory and applications. SIAM Rev 37(3):297–386PLASMA Project (2012) The parallel linear algebra for scalable multi-core architectures. http://icl.cs.utk.edu/plasmaStructPack (2012) A high performance computing library for structured matrices. http://www.inco2.upv.es/structpack.htm