Numerics with automatic result verification
| Autor: | L. B. Rall, Ulrich Kulisch |
|---|---|
| Rok vydání: | 1993 |
| Předmět: |
Numerical Analysis
Correctness General Computer Science Computer science Heuristic (computer science) Applied Mathematics Numerical analysis Carry (arithmetic) Computation Symbolic computation computer.software_genre Field (computer science) Theoretical Computer Science Modeling and Simulation Compiler Arithmetic computer Algorithm |
| Zdroj: | Mathematics and Computers in Simulation. 35:435-450 |
| ISSN: | 0378-4754 |
| DOI: | 10.1016/0378-4754(93)90042-s |
| Popis: | Floating-point arithmetic is the fast way to perform scientific and engineering calculations. Today, individual floating-point operations are maximally accurate as a rule. However, after only two or just a few operations, the result can be completely wrong. Computers now carry out up to 10 11 floating-point operations in a second. Thus, particular attention must be paid to the reliability of the computed results. In recent years, techniques have been developed in numerical analysis which make it possible for the computer itself to verify the correctness of computed results for numerous problems and applications. Moreover, the computer frequently establishes the existence and uniqueness of the solution in this way. For example, a verified solution of a system of ordinary differential equations is just as valid as a solution obtained by a computer algebra system, which as a rule still requires a valid formula evaluation. Furthermore, the numerical routine remains applicable even if the problem does not have a closed-form solution. This paper first illustrates some problems with ordinary floating-point arithmetic by means of some simple examples. Then, necessary extensions of computer arithmetic are entered into. Accumulation must be performed in fixed-point arithmetic. The continuum will be brought into the computer in the form of intervals. In order to have simple access to the new techniques, programming languages themselves have to be extended. ACRITH-XSC, Pascal-XSC and C-XSC are examples of such language extensions, which have been implemented with varying amounts of compiler effort. Result verification is carried out by means of mathematical fixed-point theorems such as Brouwer's fixed-point theorem and its generalizations. Numerical computation with result verification is of fundamental significance for many applications, for example, for simulation or mathematical modeling. Models which are frequently developed by means of heuristic methods can only be refined systematically if computational errors can be essentially excluded. This paper gives an introduction to the whole field of numerical computation with automatic result verification, written with nonspecialists in mind. It extends from the design of hardware for arithmetic operations in VLSI to compilers, algorithms, numerical analysis, and deep into applications. A symbiosis of these new techniques with a computer system for symbolic computation would be highly desirable. |
| Databáze: | OpenAIRE |
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