A Survey of Methods for the Estimation Ranges of Functions Using Interval Arithmetic

Interval arithmetic is a valuable tool in numerical analysis and modeling. Interval arithmetic operates with intervals defined by two real numbers and produces intervals containing all possible results of corresponding real operations with real numbers from each interval. An interval function can be constructed replacing the usual arithmetic operations by interval arithmetic operations in the algorithm calculating values of functions. An interval value of a function can be evaluated using the interval function with interval arguments and determines the lower and upper bounds for the function in the region defined by the vector of interval arguments.

[1]  A. Žilinskas,et al.  On Underestimating in Interval Computations , 2005 .

[2]  Julius Zilinskas,et al.  Balanced random interval arithmetic , 2004, Comput. Chem. Eng..

[3]  Eldon Hansen,et al.  A globally convergent interval method for computing and bounding real roots , 1978 .

[4]  Ramon E. Moore.  A Test for Existence of Solutions to Nonlinear Systems , 1977 .

[5]  E. Hansen.  Global optimization using interval analysis — the multi-dimensional case , 1980 .

[6]  Julius Zilinskas,et al.  Evaluation Ranges of Functions using Balanced Random Interval Arithmetic , 2003, Informatica.

[7]  Vladik Kreinovich,et al.  Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic) , 1996, Reliab. Comput..

[8]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[9]  Antanas Zilinskas,et al.  On Efficiency of Tightening Bounds in Interval Global Optimization , 2004, PARA.

[10]  R. Mathar.  A Hybrid Global Optimization Algorithm for Multidimensional Scaling , 1997 .

[11]  William H. Press,et al.  Numerical recipes in C , 2002 .

[12]  S. Skelboe.  Computation of rational interval functions , 1974 .

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

[14]  Julius Zilinskas.  Comparison of Packages for Interval Arithmetic , 2005, Informatica.

[15]  Julius Zilinskas,et al.  Estimation of Functional Ranges Using Standard and Inner Interval Arithmetic , 2006, Informatica.

[16]  Otto Opitz,et al.  Classification and Knowledge Organization , 1997 .

[17]  D. Darling.  The Kolmogorov-Smirnov, Cramer-von Mises Tests , 1957 .

[18]  R. Alt,et al.  Experiments on the evaluation of functional ranges using a random interval arithmetic , 2001 .

[19]  E. Kaucher.  Über Eigenschaften und Anwendungsmöglichkeiten der erweiterten Intervallrechnung und des hyperbolischen Fastkörpers über ℝ , 1977 .