Multiple Precision Interval Packages: Comparing Different Approaches

We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is a Maple package, intpakX, and two are C/C++ libraries, GMP-XSC and MPFI. We discuss their different features, present timing results and show several applications from various fields, where high precision intervals are fundamental.

[1]  T. Csendes.  Developments in Reliable Computing , 2000 .

[2]  S. Rump.  Fast and Parallel Interval Arithmetic , 1999 .

[3]  Richard P. Brent,et al.  Recent technical reports , 1977, SIGA.

[4]  J. Keiper.  Interval Arithmetic in Mathematica , 1993 .

[5]  Oliver Aberth,et al.  Precise computation using range arithmetic, via C++ , 1992, TOMS.

[6]  Louis B. Rall,et al.  Automatic Differentiation: Techniques and Applications , 1981, Lecture Notes in Computer Science.

[7]  Helmut Brass,et al.  On the Application of the Peano Representation of Linear Functionals in Numerical Analysis , 1998 .

[8]  Oliver Aberth,et al.  Precise Numerical Methods Using C , 1998 .

[9]  P. Zimmermann,et al.  Efficient isolation of polynomial's real roots , 2004 .

[10]  E. Hansen,et al.  An interval Newton method , 1983 .

[11]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[12]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[13]  M. H. van Emden,et al.  Interval arithmetic: From principles to implementation , 2001, JACM.

[14]  Knut Petras.  Numerical Computation of an Integral Representation for Arithmetic-Average Asian Options , 2004, Computing.

[15]  Eldon Hansen,et al.  Global optimization using interval analysis , 1992, Pure and applied mathematics.

[16]  Louis B. Rall,et al.  Automatic differentiation , 1981 .

[17]  Willard L. Miranker,et al.  A new approach to scientific computation , 1983 .

[18]  Markus Grimmer.  Interval Arithmetic in Maple with intpakX , 2003 .

[19]  Y. Akyildiz,et al.  Towards A More Complete Interval Arith-Metic In Mathematica , 1970 .

[20]  Stephen Smale,et al.  Complexity of Bezout's Theorem: III. Condition Number and Packing , 1993, J. Complex..

[21]  G. Corliss,et al.  C-Xsc: A C++ Class Library for Extended Scientific Computing , 1993 .

[22]  Siegfried M. Rump,et al.  Solving Algebraic Problems with High Accuracy , 1983, IMACS World Congress.

[23]  Ulrich W. Kulisch,et al.  Advanced Arithmetic for the Digital Computer, Design of Arithmetic Units , 2002, RealComp.

[24]  J. M. Yohe,et al.  Portable Software for Interval Arithmetic , 1980 .

[25]  G. William Walster,et al.  On stopping criteria in verified nonlinear systems or optimization algorithms , 2000, TOMS.

[26]  Fabrice Rouillier,et al.  Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library , 2005, Reliab. Comput..

[27]  R. B. Kearfott.  Rigorous Global Search: Continuous Problems , 1996 .

[28]  Ulrich W. Kulisch.  Advanced Arithmetic for the Digital Computer , 2002 .

[29]  Michael Schröder,et al.  The Laplace transform approach to valuing exotic options: the case of the Asian option , 2001 .