A parallel algorithm for solving general tridiagonal equations

Author:
Paul N. Swarztrauber

Journal:
Math. Comp. **33** (1979), 185-199

MSC:
Primary 65F05; Secondary 68C25

DOI:
https://doi.org/10.1090/S0025-5718-1979-0514818-5

MathSciNet review:
514818

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Abstract | References | Similar Articles | Additional Information

Abstract: A parallel algorithm for the solution of the general tridiagonal system is presented. The method is based on an efficient implementation of Cramer's rule, in which the only divisions are by the determinant of the matrix. Therefore, the algorithm is defined without pivoting for any nonsingular system. storage is required for *n* equations and operations are required on a parallel computer with *n* processors. operations are required on a sequential computer. Experimental results are presented from both the CDC 7600 and CRAY-1 computers.

**[1]**O. BUNEMAN,*A Compact Non-Iterative Poisson Solver*, Rep. 294, Stanford Univ. Institute for Plasma Research, Stanford, Calif., 1969.**[2]**B. L. Buzbee, G. H. Golub, and C. W. Nielson,*On direct methods for solving Poisson’s equations*, SIAM J. Numer. Anal.**7**(1970), 627–656. MR**0287717**, https://doi.org/10.1137/0707049**[3]**D. E. HELLER, D. K. STEVENSON & J. F. TRAUB,*Accelerated Iterative Methods for the Solution of Tridiagonal Systems on Parallel Computers*, Dept. Computer Sci. Rep., Carnegie-Mellon Univ., Pittsburgh, Pa., 1974.**[4]**Don Heller,*A determinant theorem with applications to parallel algorithms*, SIAM J. Numer. Anal.**11**(1974), 559–568. MR**0348993**, https://doi.org/10.1137/0711048**[5]**R. W. Hockney,*A fast direct solution of Poisson’s equation using Fourier analysis*, J. Assoc. Comput. Mach.**12**(1965), 95–113. MR**0213048**, https://doi.org/10.1145/321250.321259**[6]**R. W. HOCKNEY, "The potential calculation and some applications,"*Methods of Computational Physics*, B. Adler, S. Fernback and M. Rotenberg (Eds.), Academic Press, New York, 1969, pp. 136-211.**[7]**P. M. KOGGE,*Parallel Algorithms for the Efficient Solution of Recurrence Problems*, Rep. 43, Digital Systems Laboratory, Stanford Univ., Stanford, Calif., 1972.**[8]**Jules J. Lambiotte Jr. and Robert G. Voigt,*The solution of tridiagonal linear systems on the CDC STAR-100 computer*, ACM Trans. Math. Software**1**(1975), no. 4, 308–329. MR**0388843**, https://doi.org/10.1145/355656.355658**[9]**C. B. MOLER, "Cramer's rule on 2-by-2 systems,"*ACM SIGNUM Newsletter*, v. 9, 1974, pp. 13-14.**[10]**Harold S. Stone,*An efficient parallel algorithm for the solution of a tridiagonal linear system of equations*, J.Assoc. Comput. Mach.**20**(1973), 27–38. MR**0334473**, https://doi.org/10.1145/321738.321741**[11]**Harold S. Stone,*Parallel tridiagonal equation solvers*, ACM Trans. Math. Software**1**(1975), no. 4, 289–307. MR**0388842**, https://doi.org/10.1145/355656.355657**[12]**Paul N. Swarztrauber,*A direct method for the discrete solution of separable elliptic equations*, SIAM J. Numer. Anal.**11**(1974), 1136–1150. MR**0368399**, https://doi.org/10.1137/0711086**[13]**J. F. Traub,*Iterative solution of tridiagonal systems on parallel or vector computers*, Complexity of sequential and parallel numerical algorithms (Proc. Sympos., Carnegie-Mellon Univ., Pittsburgh, Pa., 1973) Academic Press, New York, 1973, pp. 49–82. MR**0371146**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0514818-5

Keywords:
Tridiagonal matrices,
parallel algorithms,
linear equations

Article copyright:
© Copyright 1979
American Mathematical Society