Chapter 5

A Complex Optimization

Dirk Laurie

Problem 5

5.1 A First Look · 5.2 Optimization by General-Purpose Methods · 5.3 Improving the Lower Bound · 5.4 Discrete Complex Approximation · 5.5 A Necessary Condition for Optimality · 5.6 Implementation of Algorithm C · 5.7 Evaluation of the Gamma Function · 5.8 More Theory and Other Methods · 5.9 A Harder Problem

Solution is known to more than 10,002 digits.

Code

• Mathematica
   
  • Session: p. 105/106
  • Figures: 5.1–5.5
     
• MATLAB
   
  • Session: p. 106 (as alternative to Mathematica)
  • Solution using COCA
  • Gamma and digamma (psi) function (Paul Godfrey's implementation for complex arguments)
  • Commands: maxerror, diffevol, fgamma
     
• Octave (download Octave)
   
  • Sessions: pp. 113/114, 116
  • Implementation of Algorithms R and C, supporting code, code for the numbers in §5.3 and for Figures 5.1-5.5
  • Gamma and digamma (psi) function (Paul Godfrey's implementation for complex arguments)
     
• PARI/GP (download PARI/GP)
   
  • Solving Problem 5 to 10,240 digits

Links

• Paul Godfrey's Special Functions math library (Matlab/Octave).
   
• Bernd Fischer and Jan Modersitzki: COCA, a collection of MATLAB function files for the computation of linear Chebyshev approximations in the complex plane.

• Gamma function · Digamma function. From functions.wolfram.com―A Wolfram Web Resource.
   
• Eric W. Weisstein. Gamma Function · Digamma Function. From MathWorld―A Wolfram Web Resource.

This is a web page for the book:

Folkmar Bornemann, Dirk Laurie, Stan Wagon, Jörg Waldvogel: The SIAM 100-Digit Challenge, A Study in High-Accuracy Numerical Computing. Society of Industrial and Applied Mathematics (SIAM), Philadelphia, 2004.

last modified: 06/10/06 18:16 +0200 (FB)