The Error Estimate And Exception Detection Of Floating-Point Calculation Based On Interval Analysis

Because of the limitation of storage spaces, a real number in computer is only stored as a binary floating-point number with limited numbers of bits. Thus, rounding error will be produced when computer is used for scientific computing and numerical solution. And such rounding error will be accumulated when these floating-point numbers with errors are involved in computing again. Especially for the complex expression or continuous numerical calculation process, the accumulation of such rounding error will lead to noneffective results. It can be seen that error exist inevitably in the process of numerical calculation. Therefore, the description of the accuracy and stability of the numerical algorithm has attracted tremendous attentions. To calculate the error efficiently and exactly is vital for the accuracy and stability assessment of a program algorithm. Interval arithmetic is a controlled and credible arithmetic and can obtain the interval containing the real solution after any kind of operation. It has already been formed a basic theoretical system of interval arithmetic. Many software packages contain the realization of interval arithmetic operations, like INTLAB, Boost and so on.Based on the basic theory of the floating-point arithmetic and interval arithmetic, this paper focuses on the error estimate and exceptions detection of floating-point operations in numerical calculation. Main work includes the following aspects:1. This paper introduces the basic theory of floating-point arithmetic and interval arithmetic. The elementary functions algorithms realized in GNU are studied, too. At the same time, theoretical error bound of the assignment programs of elementary functions is obtained based on the error estimate methods of floating-point arithmetic and the elementary functions algorithms realized in GNU.2. The elementary functions have been rewrote based on the mature interval arithmetic package in Boost library to calculate the corresponding error bound. The error bounds obtained from the interval calculation program are less than the theoretical error bounds, indicating the feasibility and correctness of the method. This paper also gives the error variation of elementary functions in different input values by a large number of experiments.3. Floating-point exceptions such as invalid operation, division by zero, overflow and underflow are researched. Also, the generation conditions of the exceptions in different floating-point operations are summarized. At the same time, a method for exceptions detection of floating-point operations based on the rewriting of codes and detection functions of the state flag is provided in this paper.4. Using the Flex&Bison tool in GNU, an interactive computing tool for interval arithmetic based on the theory of lexical analysis and syntactic analysis is implemented, which can make an interval calculation of complex expression and obtain the corresponding numerical error bounds quickly.