
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
X.H. Sun, H. Zhang, L.M. Ni, "Efficient Tridiagonal Solvers on Multicomputers," IEEE Transactions on Computers, vol. 41, no. 3, pp. 286296, March, 1992.  
BibTex  x  
@article{ 10.1109/12.127441, author = {X.H. Sun and H. Zhang and L.M. Ni}, title = {Efficient Tridiagonal Solvers on Multicomputers}, journal ={IEEE Transactions on Computers}, volume = {41}, number = {3}, issn = {00189340}, year = {1992}, pages = {286296}, doi = {http://doi.ieeecomputersociety.org/10.1109/12.127441}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  Efficient Tridiagonal Solvers on Multicomputers IS  3 SN  00189340 SP286 EP296 EPD  286296 A1  X.H. Sun, A1  H. Zhang, A1  L.M. Ni, PY  1992 KW  parallel partition LU algorithm; computation complexity; tridiagonal solvers; multicomputers; parallel algorithms; parallel partition hybrid; parallel diagonal dominant; tridiagonal linear systems; divideandconquer parallel computation model; pivoting; nonpivoting; approximate solution; communication complexities; 64node nCUBE1 multicomputer; computational complexity; linear algebra; mathematics computing. VL  41 JA  IEEE Transactions on Computers ER   
Three parallel algorithms, namely, the parallel partition LU (PPT) algorithm, the parallel partition hybrid (PPH) algorithm, and the parallel diagonal dominant (PDD) algorithm, are proposed for solving tridiagonal linear systems on multicomputers. These algorithms are based on the divideandconquer parallel computation model. The PPT and PPH algorithms support both pivoting and nonpivoting. The PPT algorithm is good when the number of processors is small; otherwise, the PPH algorithm is better. When the system is diagonal dominant, the PDD algorithm is highly parallel and provides an approximate solution which equals the exact solution within machine accuracy. Computation and communication complexities of the three algorithms are presented. All three methods have been implemented on a 64node nCUBE1 multicomputer. The analytic results closely match the results measured from the nCUBE1 machine.
[1] W. C. Athas, and C. L. Seitz, "Multicomputers: Messagepassing concurrent computers,"IEEE Comput. Mag., pp. 925, Aug. 1988.
[2] L. Bomans and D. Roose, "Benchmarking the iPSC/2 hypercube multiprocessor,"Concurrency: Practice and Experience, pp. 318, Sept. 1989.
[3] S. Demko, W. F. Moss, and P. W. Smith, "Decay rates for inverses of hand matrices,"Math. Computat., vol. 43, no. 168, pp. 491499, Oct. 1984.
[4] J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart,Linpack Users' Guide. SIAM, Philadelphia, PA, 1979.
[5] I. S. Duff, A. M. Erisman, and J. K. Reid,Direct Methods for Sparse Matrices. Oxford, England: Clarendon, 1986.
[6] P. Dubois and G. Rodrigue, "An analysis of the recursive doubling algorithm," inHigh Speed Computer and Algorithm Organization, Kucket al., Eds. New York: Academic, 1977.
[7] O. Egecioglu, C. K. Koc, and Laub, A. J., "A recursive doubling algorithm for solution of tridiagonal systems on hypercube multiprocessors,"J. Comput. Appl. Math., vol. 27, 1989.
[8] M. R. Grunwald and D. A. Reed, "Benchmarking hypercube hardware and software," Tech. Rep., UIUCDCSR861303, Dep. Comput. Sci., Univ. of Illinois at UrbanaChampaign, 1986.
[9] D. Heller, "A survey of parallel algorithms in numerical algebra,"SIAM Rev., vol. 20, pp. 740777, Oct. 1978.
[10] R. W. Hockney, "A fast direct solution of Poisson's equation using Fourier analysis,"J. ACM, vol. 12, pp. 95113, 1965.
[11] C. T. Ho and S. L. Johnsson, "Optimizing tridiagonal solvers for alternating direction methods on Boolean cube multiprocessors,"SIAM J. Sci. Statist. Comput., vol. 11, no. 3, pp. 563592, May 1990.
[12] K. Hwang, "Advanced parallel processing with supercomputer architectures,"Proc. IEEE, pp. 3347, Oct. 1987.
[13] S. L. Johnson, "Solving tridiagonal systems on ensemble architectures,"SIAM J. Sci. Stat. Comput., vol. 8, pp. 345392, 1987.
[14] S. L. Johnsson, "Communication efficient basic linear algebra computations on hypercube architectures,"J. Parallel Distributed Comput., pp. 133172, 1987.
[15] S. L. Johnsson and C. T. Ho, "Spanning graphs for optimum broadcasting and personalized communication in hypercubes,"IEEE Trans. Comput., vol. 38, Sept. 1989.
[16] D. H. Lawrie and A. H. Sameh, "The computation and communication complexity of a parallel banded system solver,"ACM Trans. Math. Software, vol. 10, no. 2, pp. 185195, June 1984.
[17] P. H. Michielse and H. A. Vorst, "Data transport in Wang's partition method," inParallel Computing, 7. New York: NorthHolland, 1988, pp. 8795.
[18] J. M. Ortega and R. G. Voigt, "Solution of partial differential equations on vector and parallel computers,"SIAM Rev., pp. 149240, June 1985.
[19] Y. Saad and M. H. Schultz, "Data communication in hypercube," Res. Rep., YALEU/DCS/RR428, Oct. 1985.
[20] J. Sherman and W. J. Morrison, "Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix,"Ann. Math. Statist., vol. 20, 621, 1949.
[21] G. D. Smith,Numerical Solution of Partial Differential Equations. Oxford, England: Oxford University Press, 1985.
[22] H. S. Stone, "An efficient parallel algorithm for the solution of a tridiagonal linear system of equations,"J. ACM, vol. 20, no. 1, pp. 2738, Jan. 1973.
[23] H. H. Wang, "A parallel method for tridiagonal equations,"ACM Trans. Math. Software, vol. 7, pp. 170183, June 1981.
[24] M. Woodbury, "Inverting modified matrices," Memo. 42, Statistics Research Group, Princeton, NJ, 1950.
[25] H. Zhang, "On the accuracy of the parallel diagonal dominant algorithm,"Parallel Comput., pp. 265272, 1991.