Representations and Parallel Computations for Rational Functions
Abstract
Representations of univariate rational functions over a given base of polynomials are considered, and a fast parallel algorithm for converting from one base representation to another is given. Special cases of this conversion include the following symbolic manipulation problems: Taylor expansion, partial fraction decomposition, Chinese remainder algorithm, elementary symmetric functions, Padé approximation, and various interpolation problems. If n is the input size, then all algorithms run in parallel time $O(\log ^2 n)$ and use $n^{O(1)} $ processors. They work over an arbitrary field.
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References
1.
G. A. Baker, P. Graves-Morris, Padé approximants, Encyclopedia of Mathematics and Its Applications, vols. 13 and 14, Addison-Wesley, Reading, MA, 1981
2.
Stuart J. Berkowitz, On computing the determinant in small parallel time using a small number of processors, Inform. Process. Lett., 18 (1984), 147–150
3.
A. Borodin, S. Cook, N. Pippenger, Parallel computation for well-endowed rings and space-bounded probabilistic machines, Inform. and Control, 58 (1983), 113–136
4.
Allan Borodin, Joachim von zur Gathen, John Hopcroft, Fast parallel matrix and GCD computations, Inform. and Control, 52 (1982), 241–256
5.
Richard P. Brent, Fred G. Gustavson, David Y. Y. Yun, Fast solution of Toeplitz systems of equations and computation of Padé approximants, J. Algorithms, 1 (1980), 259–295
6.
W. S. Brown, On Euclid's algorithm and the computation of polynomial greatest common divisors, J. Assoc. Comput. Mach., 18 (1971), 478–504
7.
A. Cauchy, Cours d'analyse de l'école royale polytechnique (Analyse algébrique), Oeuvres completes, IIe série, III (1821), 429–433
8.
George E. Collins, Subresultants and reduced polynomial remainder sequences, J. Assoc. Comput. Mach., 14 (1967), 128–142
9.
L. Csanky, Fast parallel matrix inversion algorithms, SIAM J. Comput., 5 (1976), 618–623
10.
W. Eberly, Very fast parallel matrix and polynomial arithmetic, Proc. 25th Annual IEEE Symposium on Foundations of Computer Science, Singer Island FL, 1984, 21–30
11.
Joachim von zur Gathen, Hensel and Newton methods in valuation rings, Math. Comp., 42 (1984), 637–661
12.
Joachim von zur Gathen, Parallel algorithms for algebraic problems, SIAM J. Comput., 13 (1984), 802–824
13.
K. O. Geddes, Symbolic computation of Padé approximants, ACM Trans. Math. Software, 5 (1979), 218–233
14.
W. B. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev., 14 (1972), 1–16
15.
P. R. Graves-Morris, Efficient reliable rational interpolationPadé approximation and its applications, Amsterdam 1980 (Amsterdam, 1980), Lecture Notes in Math., Vol. 888, Springer, Berlin, 1981, 28–63
16.
Fred G. Gustavson, David Y. Y. Yun, Fast algorithms for rational Hermite approximation and solution of Toeplitz systems, IEEE Trans. Circuits and Systems, 26 (1979), 750–755
17.
C. G. J. Jacobi, Ueber die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function, J. Reine Angew. Math., 30 (1846), 127–156
18.
Donald E. Knuth, The art of computer programming. Vol. 2, Addison-Wesley Publishing Co., Reading, Mass., 1981xiii+688, 2nd ed.
19.
E. Kronecker, Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen, Monatsberichte der Akademie der Wissenschaften, Berlin, 1881, 535–600
20.
A. M. Miola, The conversion of Hensel codes to their rational equivalents, SIGSAM Bull, 16 (1992), 24–26
21.
H. Padé, Sur la représentation approchée d'une fonction par des fractions rationnelles, Annales Scientifiques de l'Ecole Normale Supérieure, 3e série, 9 (1892), S3–S93, Supplément
22.
J. Reif, Logarithmic depth circuits for algebraic functions, Proc. 24th Annual IEEE Symposium on Foundations of Computer Science, Tucson AZ, 1983, 138–145
23.
Volker Strassen, Some results in algebraic complexity theory, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 497–501
24.
V. Strassen, The computational complexity of continued fractions, SIAM J. Comput., 12 (1983), 1–27
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Copyright © 1986 Society for Industrial and Applied Mathematics.
History
Submitted: 28 June 1983
Accepted: 10 December 1984
Published online: 13 July 2006
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