Matrix formulation of the Picard method for parallel computation
Tóm tắt
The increasing availability of computing machines capable of parallel computation has accelerated interest in numerical methods that exhibit natural parallel structures. In particular, the parallel structure of the Picard method of successive approximations for the numerical solution of ordinary differential equations allows straightforward adaptation of the method for use on parallel computers. A matrix formulation of the Picard method for parallel computation is presented here in which the numerical solution is obtained in truncated Chebyshev series. The application of the formulation to parallel processing computing machines is discussed.
Tài liệu tham khảo
Feagin, T.: 1972, ‘The Numerical Solution of Two Point Boundary Value Problems’, Doctoral Dissertation, University of Texas. Available from University Microfilms, 300 N. Zeeb Road, Ann Arbor, Michigan.
Fox, L. and Parker, I. B.: 1968,Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London.
Franklin, M. A.: 1978, ‘Parallel Solution of Ordinary Differential Equations’,IEEE Transactions on Computers C-27, No. 5, 413–420.
Kascic, M. J.: 1979, ‘Vector Processing on the CYBER 200’, published in theINFOTECH State of the Art Report ‘Super Computers’ INFOTECH International, Limited, Maidenhead, United Kingdom.
Keller, H. B.: 1968,Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell, Waltham, Massachusetts, pp. 61–68.
Lanczos, C.: 1957,Applied Analysis, Prentice-Hall, New York.
Miranker, W. L. and Liniger, W.: 1967, ‘Parallel Methods for the Numerical Solution of Ordinary Differential Equations’,Mathematics of Computation 21, No. 99, 303–320.
Nacozy, P. and Feagin, T.: 1971, ‘Chebyshev Series-Solutions of Swing-By Trajectories’, AIAA Paper No. 71-192. Available from Technical Information Service, 750 3rd Avenue, New York, New York.
Nacozy, P. and Feagin, T.: 1972, ‘Approximations of Interplanetary Trajectories by Chebyshev Series’,AIAA Journal 10, No. 3, 243–244.
Nacozy, P.: 1972, in G. Chebotarevet al. (eds.), ‘The Motion, Evolution of Orbits, and Origin of Comets’,IAU Symp. 45, 43.
Nievergelt, J.: 1964, ‘Parallel Methods for Integration Ordinary Differential Equations’,Communications of the ACM 7, No. 12, 731–733.
Picard, E.: 1893, ‘Sur l'application des Methodes d'Approximations successives à l'étude de certaines équations différentialles ordinaire’,J. Math. 9, 217–271.
Warland, P. B.: 1976, ‘Parallel Methods for the Numerical Solution of Ordinary Differential Equations’,IEEE Transactions on Computers, pp. 1045–1048.