On the risch-norman integration method and its implementation in MAPLE

ABSTRACT

Unlike the Recursive Risch Algorithm for the integration of transcendental elementary functions, the Risch-Norman Method processes the tower of field extensions directly in one step. In addition to logarithmic and exponential field extensions, this method can handle extensions in terms of tangents. Consequently, it allows trigonometric functions to be treated without converting them to complex exponential form. We review this method and describe its implementation in MAPLE. A heuristic enhancement to this method is also presented.

REFERENCES

• 1: W. H. Beget. CRC Standard Mathemalical Tables, ~5th Edition. CRC Press, Inc., Boca Raton, Florida, 1978.
• 2: M. Bronstein, Simplification of real elementary functions, Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation, p.207-211, July 17-19, 1989, Portland, Oregon, United States [doi>10.1145/74540.74566]
• 3: B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan and S. M. Watt. Maple Reference Manual, 5th Edition. WATCOM Publications Limited, Waterloo, Ontario, Canada, 1988.
• 4: James H. Davenport, The Parallel Risch Algorithm (I), Proceedings of the European Computer Algebra Conference on Computer Algebra, p.144-157, April 05-07, 1982
• 5: James H. Davenport, Integration - What do we want from the theory?, Proceedings of the European Computer Algebra Conference on Computer Algebra, p.2-11, March 28-30, 1983
• 6: 3. H. Davenport. Integration Formelle. Research Report No. 375, Laboratoire d'lnformatique et de Mathematiques Appliquees de Grenoble, March 1983.
• 7: J. H. Davenport , B. M. Trager, On the parallel Risch Algorithm (II), ACM Transactions on Mathematical Software (TOMS), v.11 n.4, p.356-362, Dec. 1985 [doi>10.1145/6187.6189]
• 8: James H. Davenport, On the parallel Risch algorithm (III): use of tangents, ACM SIGSAM Bulletin, v.16 n.3, p.3-6, August 1982 [doi>10.1145/1089302.1089303]
• 9: John Fitch, User-based integration software, Proceedings of the fourth ACM symposium on Symbolic and algebraic computation, p.245-248, August 05-07, 1981, Snowbird, Utah, United States [doi>10.1145/800206.806404]
• 10: K. O. Geddes. Algebraic Algorithms fo, Symbolic Computation. Course Notes, Department of Compurer Science, University of Waterloo, 1988.
• 11: G. H. Gonnet and M. B. Monagan. Solving Systerns of Algebraic Equations, or the interface between Software and Mathematics. Res. Rep. CS- 89-13, Dept. of Computer Science, Univ. of Weterloo, Waterloo, Ontario, Canada.
• 12: S. J. Harrington. A New Symbolic Integration System in Reduce. Computer Journal, Vol. 22, No. 2, 1979, pp. 127-131.
• 13: A. C. Norman and P. M. A. Moore. Implementing the New Risch Integration Algorithm. Proc. 4th Int. Colloquium on Advanced Computing Methods in Theoretical Physics, St. Maxlmin, 1977, pp. 99-110.
• 14: R. H. Risch. Algebraic Proi~erties of the Elementary Functions of Analysis. American Journal of Mathematics 101, 1979, pp. 743-7'59.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

Additional Classification:
  F. Theory of Computation
  F.1 COMPUTATION BY ABSTRACT DEVICES

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

General Terms:
Algorithms, Theory