Parallel Algorithms to Evaluate Orthogonal Polynomial Series

In this paper four parallel algorithms for the evaluation of finite series of orthogonal polynomials are introduced. The algorithms are based on the Forsythe and Clenshaw sequential algorithms. Several tests carried out on a Cray T3D are presented.

[1]  G. Forsythe Generation and Use of Orthogonal Polynomials for Data-Fitting with a Digital Computer , 1957 .

[2]  Francis J. Smith,et al.  An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation , 1965 .

[3]  Ayse Kiper,et al.  Modified Dorn's Algorithm with Improved Speed-Up , 1996, PARA.

[4]  Jack J. Dongarra,et al.  Solving linear systems on vector and shared memory computers , 1990 .

[5]  U. Meier A parallel partition method for solving banded systems of linear equations , 1985 .

[6]  Lei Li,et al.  A Simple Parallel Algorithm for Polynomial Evaluation , 1996, SIAM J. Sci. Comput..

[7]  David J. Kuck,et al.  Practical Parallel Band Triangular System Solvers , 1978, TOMS.

[8]  Richard E. Overill,et al.  Performance of Parallel Algorithms for the Evaluation of Power Series , 1994, Parallel Comput..

[9]  R. Barrio,et al.  Parallel evaluation of Chebyshev and trigonometric series , 1999 .

[10]  S. Lennart Johnsson,et al.  Solving narrow banded systems on ensemble architectures , 1985, TOMS.

[11]  Kiyoshi M. Maruyama,et al.  On the Parallel Evaluation of Polynomials , 1973, IEEE Transactions on Computers.

[12]  P. Swarztrauber A parallel algorithm for solving general tridiagonal equations , 1979 .

[13]  Ayse Kiper,et al.  Parallel Polynomial Evaluation by Decoupling Algorithm , 1996, Parallel Algorithms Appl..

[14]  Plamen Y. Yalamov Stability of a Partitioning Algorithm for Bidiagonal Systems , 1997, Parallel Comput..

[15]  Plamen Y. Yalamov Stability of Parallel Bidiagonal Solvers , 1994, PARA.

[16]  R. P. Soni,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1967 .

[17]  Roberto Barrio,et al.  A Parallel Algorithm to Evaluate Chebyshev Series on a Message Passing Environment , 1998, SIAM J. Sci. Comput..

[18]  Harold S. Stone,et al.  Parallel Tridiagonal Equation Solvers , 1975, TOMS.

[19]  R. Askey Orthogonal Polynomials and Special Functions , 1975 .

[20]  H. H. Wang,et al.  A Parallel Method for Tridiagonal Equations , 1981, TOMS.

[21]  J. Ian Munro,et al.  Optimal Algorithms for Parallel Polynomial Evaluation , 1971, J. Comput. Syst. Sci..

[22]  Nicholas J. Higham,et al.  The Accuracy of Floating Point Summation , 1993, SIAM J. Sci. Comput..

[23]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[24]  C. W. Clenshaw A note on the summation of Chebyshev series , 1955 .

[25]  S. Johnsson Solving tridiagonal systems on ensemble architectures , 1987 .

[26]  Jack J. Dongarra,et al.  Solving banded systems on a parallel processor , 1987, Parallel Comput..

[27]  Roberto Barrio,et al.  Perturbation Simulations of Rounding Errors in the Evaluation of Chebyshev Series , 1998, J. Univers. Comput. Sci..