#interval arithmetic

Read the full paper at: http://www.scirp.org/journal/PaperInformation.aspx?PaperID=49675 DOI: 10.4236/ajor.2014.45028 Author(s) Orkia Derkaoui, Ahmed Lehireche Affiliation(s) Computer Science Department, University Dr Moulay Tahar of Saida, Saida, Algeria. Computer Science Department, University Djillali Liabes of SidiBel Abbes, SidiBel Abbes, Algeria. ABSTRACT Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underestimate the very global optima. The purpose of this article is to introduce an efficient and safe procedure to rigorously bound the global optima of semidefinite program. This work shows how, using interval arithmetic, rigorous error bounds for the optimal value can be computed by carefully post processing the output of a semidefinite programming solver. A lower bound is computed on a semidefinite relaxation of the constraint system and the objective function. Numerical results are presented using the SDPA (SemiDefinite Programming Algorithm), solver to compute the solution of semidefinite programs. This rigorous bound is injected in a branch and bound algorithm to solve the optimisation problem.gjreww140917 KEYWORDS Semidefinite Programming, Interval Arithmetic, Rigorous Error, Bounds, SDPA Solver, Branch and Bound Algorithm

C-Xsc: A C Class Library for Extended Scientific Computing C-XSC is a tool for the development of numerical algorithms delivering highly accurate and automatically verified results. It provides a large number of predefined numerical data types and operators. These types are implemented as C classes. Thus, C-XSC allows high-level programming of numerical applications in C and C . The most important features of C-XSC are: real, complex, interval, and complex interval arithmetic; dynamic vectors and matrices; subarrays of vectors and matrices; dotprecision data types, predefined arithmetic operators with maximum accuracy; standard functions of high accuracy; multiple precision arithmetic and standard functions; rounding control for I/O data; error handling, and library of problem solving routines with automatic result verification. Thus, C-XSC makes the computer more powerful concerning the arithmetic. C-XSC is immediately usable by C programmers, easy to learn, user-extendable, and may also be combined with other tools. The book can be used as a textbook and as a reference manual. It consists of an introduction to advanced computer arithmetic, a chapter describing the programming languages C and C , the major chapter "C-XSC Reference", sample programs, and indices.