Symbolic-numeric nonlinear equation solving

ABSTRACT

A numerical equation-solving algorithm employing differentiation and interval arithmetic is presented which finds all solutions of f(z) = 0 on an interval I when f is holomorphic and has simple zeros. A two dimensional generalization of this algorithm is discussed. Finally, aspects of a broader symbolic-numeric algorithm which uses the first algorithm as a foundation are considered.

REFERENCES

• 1: Alefeld, G., and Herzberger, J. (1983) Introduction to Interval Computationa, Academic Press.
• 2: Brent, It. P. (1971), "An Algorithm with Guaranteed Convergence for Finding a Zero of a Function", The Computer Journal, 14, 422-425.
• 3: Bundy, A., and Welham, B. (1981) "Using Meta-Level Inference for Selective Application of Multiple Rewrite Rules in Algebraic Manipulation", Artificial Intelligence 16 (2).
• 4: George E. Collins , Rudiger Loos, Real zeroes of polynomials, Computer algebra: symbolic and algebraic computation (2nd ed.), Springer-Verlag New York, Inc., New York, NY, 1983
• 5: Co.w~y. Joh~ B. (~978). F~tio., o/O~, Co=pl,= Variable, Second Edition, Springer-Verlag.
• 6: Hardy, G. H. (1910), Orders of infinity. Cambridge Tract8 in Mathematics 12.
• 7: B. J. Hoenders , C. H. Slump, On the determination of the number and multiplicity of zeros of a function, Computing, v.47 n.3-4, p.323-336, 1991 [doi>10.1007/BF02320200]
• 8: Ioakimidis, N. I. and Anastasselou, E. G. (1986) "On the Simultaneous Determination of Zeros of Analytic or Sectionally Analytic Functions", Computing 36(3) 239-247.
• 9: R. Baker Kearfott, Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems, Computing, v.47 n.2, p.169-191, 1991 [doi>10.1007/BF02253433]
• 10: Nerinckx, D., and Haegemans, A. (1976), "A Compaxison of Non-Linear Equation Solvers", Journal o.f Computational and Applied Mathematics 2, 145-148.
• 11: Ratschek, H., and Rokne, J. (1984) Computer Methods for the Range of Functions, Ellis Horwood Limited.
• 12: Richardson, Daniel (1968) "Some Vndecidable Problems Involving Elementary Functions of a Real Variable", The Journal o.f Symbolic Logic 33(4), 514-520.
• 13: John Rischard Rice, Numerical Methods, Software, and Analysis, Academic Press, Inc., Orlando, FL, 1992
• 14: Rudin, Walter (1976), Principles of Mathematical Analysis, McGraw-Hill.
• 15: Saaty, Thomas L. (1981), Modern Nonlinear Equations, Dover.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

Additional Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.3 Languages and Systems

          Nouns: Maple

General Terms:
Algorithms, Languages