# Symbolic-numeric interface: A review

9. Symbolic-Numeric Interface

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## Keywords

Symbolic System Numerical Mathematic High Level Language Mathematical Software Extend Euclidean Algorithm
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## References

- 1.A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, chap. 8, Addison-Wesley: Menlo Park, California (1974).Google Scholar
- 2.C. M. Andersen and A. Noor, "Symbolic Manipulation Techniques for Vibration Analysis of Laminated Elliptic Plates," Proc. of 1977 MACSYMA User's Conference, NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 3.Y. Avgoutis, "Symbolic Laplace Transforms of Special Functions," Proc. of 1977 MACSYMA Users' Conference, NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 4.D. Barton, I. M. Willers, R. Zahar, "Taylor Series Methods for ODE," in Mathematical Software, ed. J. Rice, Academic Press, N. Y. (1971).Google Scholar
- 5.G. Birkhoff, "Mathematics and Psychology," SIAM Rev. Vol. 11, pp. 429–469 (1969).Google Scholar
- 6.A. Borodin and I. Munro, The Computational Complexity of Algebraic and Numeric Problems, American Elsevier, N. Y. (1975).Google Scholar
- 7.R. P. Brent and H. T. Kung, "Fast Algorighms for Manipulating Formal Power Series," Dept. of Comp. Sci. Rept., Carnegie-Mellon Univ. (January 1976).Google Scholar
- 8.R. P. Brent and J. F. Traub, "On the Complexity of Composition and Generalized Composition of Power Series," Dept. of Comp. Sci. Rept., Carnegie-Mellon Univ. (May 1978).Google Scholar
- 9.W. S. Brown and A. C. Hearn, "Applications of Symbolic Algebraic Computation," Comp. Phys. Comm. to appear.Google Scholar
- 10.S. Cabay and T. P. L. Lam, "Congruence Techniques for the Solution of Integer Systems," ACM Trans. Mathematical Software, Vol. 3, pp. 386–397 (1977).Google Scholar
- 11.J. A. Cambell, ACM/SIGSAM Bulletin #25, pp. 13–14 (1973).Google Scholar
- 12.W. J. Cody, Rose M. Motley, and L. Wayne Fullerton, "The Computation of Real Fractional Order Bessel Functions of the Second King," ACM Transactions on Mathematical Software 3, pp. 232–239 (September 1977).Google Scholar
- 13.J. Cohen and J. Katcoff, "Symbolic Solution of Finite-Difference Equations," ACM Trans. Mathematical Software, Vol. 3, pp. 261–271 (1977).Google Scholar
- 14.G. E. Collins, "Infallible Calculation of Polynomial Zeros to Specified Precision," Mathematical Software III, ed. J. R. Rice, Academic Press, N. Y. (1977).Google Scholar
- 15.G. E. Collins and S. C. Schaller, SAC1 User's Guide and SAC1 Implementation Guide, Comp. Sci. Tech. Repts. #268 and #269, Univ. of Wisconsin (1976). (A SAC2 manual is forthcoming in the same series of reports.)Google Scholar
- 16.F. D. Crary, "The Augment Precompiler: I. User Information," U. of Wis., Math. Res. Ctr. Rept. #1469 (December 1974).Google Scholar
- 17.E. Cuthill, "Rational Approximations to e
^{−x}with Negative Real Poles," Proc. of 1977 MACSYMA User's Conference, NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar - 18.J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart, LINPACK Users' Guide, Soc. Indus. Appl. Math. Publications, Philadelphia (1979).Google Scholar
- 19.EUROSAM '74, ACM/SIGSAM Bulletin, vol. 8 #3 (1974).Google Scholar
- 20.R. J. Fateman, "An Approach to Automatic Asymtotic Expansions," Proc. of 1976 ACM Symp. on Sym. & Alg. Comp., pp. 365–371, ACM:NY (1976).Google Scholar
- 21.R. J. Fateman, "An Improved Algorithm for the Isolation of Polynomial Real Zeros," Proc. of 1977 MACSYMA Users' Conference, NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 22.S. I. Feldman, "An Application of Symbolic Computation to Crystal Physics," Computing Science Technical Report No. 15, Bell Laboratories, Murray Hill, N. J. (May 1974).Google Scholar
- 23.Stuart I. Feldman and Julia Ho, "A rational Expression Evaluation Package," Computing Science Technical Report No. 34, Bell Laboratories, Murray Hill, N. J. (September 1975).Google Scholar
- 24.J. A. Fox and A. C. Hearn, "Analytic Computation of Some Integrals in Fourth Order Quantum Electrodynamics," J. Comp. Physics 14, pp. 301–317 (March 1974).Google Scholar
- 25.L. Wayne Fullerton and G. A. Rinker, Jr., "Accurate and Efficient Methods for the Evaluation of Vacuum-Polarization Potentials of Order Zα and Zα
^{2}," Physical Review A 13, pp. 1283–1287 (March 1976).Google Scholar - 26.W. M. Gentleman, "A New Problem Section for the Bulletin," ACM/SIGSAM Bulletin, pp. 4–6 (April 1971).Google Scholar
- 27.R. W. Gosper, Jr., "Indefinite Hypergeometric Sums in MACSYMA," Proc. of 1977 MACSYMA Users' Conference, NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 28.M. Griss, "The Algebraic Solution of Large Sparse Systems of Linear Equations Using REDUCE2," Proc. of ACM National Conference, Vol. 1, pp. 105–111 (1974).Google Scholar
- 29.M. L. Griss, "The Algebraic Solution of Sparse Linear Systems via Minor Expansion," ACM Trans. Mathematical Software, Vol. 2, pp. 31–49 (1976).Google Scholar
- 30.K. Gupta and E. W. Ng, "Symbolic Calculations in a Finite Dynamic Element Analysis," Proc. of 1977 MACSYMA Users' Conference, NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 31.F. G. Gustavson and D. Y. Y. Yun, "Fast Computation of the Rational Interpolation Table and Toeplitz Systems of Equations via the Fast Extended Euclidean Algorithm," these Proceedings (1979).Google Scholar
- 32.A. C. Hearn, ed., REDUCE Newsletter, Univ. of Utah, Symbolic Comp. Grp., Dept. of Comp. Sci. (1978).Google Scholar
- 33.J. Ivie, "Some MACSYMA Programs for Solving Difference Equations," MACSYMA Users' Conf. Proc., pp. 317–326, NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 34.M. A. Jenkins and J. F. Traub, "Principles for Testing Polynomial Zerofinding Programs," ACM Trans. on Mathematical Software, Vol. 1, pp. 26–34 (1975).Google Scholar
- 35.R. d. Jenks, ACM SIGSAM Bulletin #37, pp. 6–7 (1976).Google Scholar
- 36.R. D. Jenks, "SCRATCHPAD/360: Reflections on a Language design," ACM/SIGSAM Bulletin, Vol. 13, #1 (February 1979).Google Scholar
- 37.M. Juncosa, "A General Analytic Approach ...," Information Processing 71, pp. 1356–1362, Proc. of IFIP Congress 1971, North Holland Publishing Co. (1972).Google Scholar
- 38.W. H. Kahan, "One Numerical Analyst's Experience with One Symbol Manipulator," SIAM Rev. Vol. 16, #1, p. 129 (1974).Google Scholar
- 39.G. Kedem, "Auto. Diff. of Comp. Programs," Univ. of Wis. Math. Res. Ctr. Report #1697 (November 1976).Google Scholar
- 40.H. Keller and V. Pereyra, "Symbolic Generation of Finite Difference Formulae," Math. Comp. Vol. 32, pp. 955–971 (1978).Google Scholar
- 41.W. Kerner and R. C. Grimm, "Magneto-hydrodynamic Spectra for Tokamaks with Non-circular cross Sections," Proceedings of the Seventh Conference on Numerical Simulation of Plasmas, Courant Institute, N. Y. (June 1975).Google Scholar
- 42.D. Knuth, "Seminumerical Algorithms," The Art of Computer Programming, Vol. II, pp. 360–443, Addison-Wesley: Reading, Mass. (1969).Google Scholar
- 43.H. T. Kung and J. F. Traub, "All Algebraic Functions Can Be Computed Fast," J. of ACM, Vol. 25, pp. 245–260 (1978).Google Scholar
- 44.J. C. R. Licklider, "Man-Computer Symbiosis," IRE TRANSACTIONS on Human Factors in Electronics (March 1960).Google Scholar
- 45.J. D. Lipson, "Newton's Method: A Great Algebraic Algorithm," SYMSAC 76, pp. 260–270 (1976).Google Scholar
- 46.Rudiger Loos, "Solution of Problem Number 1 by REDUCE2," ACM SIGSAM Bulletin, pp. 32–40 (January 1972).Google Scholar
- 47.MACSYMA Users' Conf. Proc., NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 48.M. T. McClellan, "A Comparison of Algorithms for the Exact Solution of Linear Equations," ACM Trans. Mathematical Software, Vol. 3, pp. 147–158 (1977).Google Scholar
- 49.W. L. Miranker and D. Y. Y. Yun, "A Note on Boundary Layer Elements," IBM Research Note #RC 4990, T. J. Watson Res. Ctr., Yorktown Heights, N.Y. (1974).Google Scholar
- 50.R. Moenck, "Practical Fast Polynomial Multiplication," SYMSAC 76, pp. 136–145 (1976).Google Scholar
- 51.R. Moenck and J. Carter, "Approximate Algorithms to Derive Exact Solutions to Systems of Linear Equations," these Proceedings (1979).Google Scholar
- 52.J. Moses, "The Variety of Variables in Mathematical Expressions," Proc. of 1977 MACSYMA Users' Conf., NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 53.E. W. Ng and B. Char, "Gradient and Jacobian Computation for Numerical Applications," Proc. of Second MACSYMA Users' Conference, Washington, D. C. (1979).Google Scholar
- 54.E. W. Ng and D. Polajnar, "A Study of Alternative Methods for the Symbolic Calculation of Elliptic Integrals," SYMSAC 76, pp. 372–376 (1976).Google Scholar
- 55.E. W. Ng, "Observations on Approximate Integrations," Proc. of 1977 MACSYMA User's Conf., NASA Publication #CP-2012, NASA, Washington, D. C. (1977).Google Scholar
- 56.F. W. J. Olver, "Asymptotics and Special Functions," Academic Press: N. Y. (1974).Google Scholar
- 57.F. W. J. Olver, "A New Approach to Error Arithmetic," SIAM J. of Num. Anal. Vol. 15, pp. 368–393 (1978).Google Scholar
- 58.J. R. Rice, "Prologue," in Mathematical Software, ed. J. Rice, Academic Press, N. Y. (1971).Google Scholar
- 59.J. R. Rice, "On the Present and Future of Scientific Computation," Comm. ACM, Vol. 15, pp. 637–639 (1972).Google Scholar
- 60.M. Rochan and H. Strubbe, ACM/SIGSAM Bulletin #36, pp. 30–38 (1975).Google Scholar
- 61.J. E. Sammet, "Software for Nonnumerical Mathematics," in Mathematical Software, ed. J. R. Rice, Academic Press, N. Y. (1971).Google Scholar
- 62.P. J. Shelus and W. H. Jeffreys, III, "A Note on an Attempt at More Efficient Poisson Series Evaluation," Celestial Mechanics 11, pp. 75–78 (1975).Google Scholar
- 63.SIGSAM Bulletin, Vol. 12, #2, Assoc. for Comp. Mach. (1978).Google Scholar
- 64.G. W. Stewart, Introduction to Matrix Computations, Academic Press, N. Y. (1973).Google Scholar
- 65.D. R. Stoutemyer, "Analytical Optimization Using Computer Algebraic Manipulation," ACM Transactions on Mathematical Software 1, pp. 147–164 (June 1975).Google Scholar
- 66.D. R. Stoutemyer, "Automatic Error Analysis Using Computer Algebraic Manipulation," ACM Transactions on Mathematical Software, 3, pp. 26–43, (March 1977).Google Scholar
- 67.D. R. Stoutemyer, "Automatic Categorization of Optimization Problems," Operations Research, to appear.Google Scholar
- 68.D. R. Stoutemyer, "Qualitative Analysis of Mathematical Expressions ...," Proc. of ACM Symp. on Sym. & Comp., pp. 105–113, ACM, N. Y. (1976).Google Scholar
- 69.D. R. Stoutemyer, "Dimensional Analysis, Using Computer Symbolic Mathematics," J. of Comp. Phys. Vol. 24, pp. 141–149, (1977).Google Scholar
- 70.SYMSAM II, Proc. of 2nd Symp. on Sym. and Alg. Manipulation, ed. S. Petrick, ACM, N. Y. (1971).Google Scholar
- 71.SYMSAC 76, Proc. of A Symp. on Sym. and Alg. Comp., ACM, N. Y. (1976).Google Scholar
- 72.R. G. Tobey, "Symbolic Mathematical Computation — Introduction and Overview," SYMSAM II, p. 10 (1971)Google Scholar
- 73.J. R. Traub, "SIGNUM Panel on the Future of Numerical Mathematics," ACM/SIGNUM Newsletter, Vol. 7, pp. 40–45 (1972).Google Scholar
- 74.J. F. Traub, "Numerical Mathematics and Computer Science," Comm. ACM, Vol. 15, pp. 537–541 (1972).Google Scholar
- 75.J. F. Traub, ed., Analytic Computational Complexity, Academic Press: N. Y. (1975).Google Scholar
- 76.J. F. Traub, ed., Algorithms and Complexity, Academic Press: N. Y. (1976).Google Scholar
- 77.D. L. Ulery and H. M. Khalil, "A Semi-Numeric-Difference Algorithm," these Proceedings (1979).Google Scholar
- 78.J. Verner, "Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error," SIAM J. of Numerical Analysis Vol. 15, pp. 772–790 (1978).Google Scholar
- 79.P. Wang, "Matrix Computations in MACSYMA," MACSYMA Users' Conf. Proc., pp. 435–446, NASA Publication #CP-2012, Washington, D. C. (1977).Google Scholar
- 80.D. D. Warner, "A Partial Derivative Generator," Computing Science Technical Report No. 28, Bell Laboratories, Murray Hill, N. J. (April 1975).Google Scholar
- 81.H. Weyl, "Philosophy of Mathematics and Natural Science," pp. 162–163, Princeton Univ. Press, Princeton (1949).Google Scholar
- 82.S. Winograd, "On Computing Discrete Fourier Transform," these Proceedings (1979).Google Scholar
- 83.D. Y. Y. Yun, "Algebraic Algorithms Using P-adic Constructions," SYMSAC 76, pp. 248–259 (1976).Google Scholar
- 84.D. Y. Y. Yun, "Fast Algorithm for Rational Function Integration," Information Processing 77, North-Holland, Amsterdam (1977).Google Scholar
- 85.D. Y. Y. Yun, "Hansel Meets Newton — Algebraic Construction in an Analytic Setting," Analytic Computational Complexity, ed J. Traub, Academic Press, N. Y. (1975).Google Scholar

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