Error Analysis

Abstract

Several error sources are important for numerical data processing. Input data from an experiment have only limited precision. The arithmetic unit of a computer uses only a subset of the real numbers. Input data as well as the results of elementary operations have to be represented by such machine numbers whereby rounding errors can be generated. Results from more complex operations like square roots or trigonometric functions can have even larger errors since iterations and series expansions have to be truncated after a finite number of steps. Different floating point formats defined by IEEE are discussed and in a computer experiment the corresponding machine precision is determined. Rounding errors of elementary arithmetic operations, especially numerical extinction, and of more complex algorithms are studied. The stability criterion for iterative algorithms is introduced. Truncation errors are demonstrated for the time evolution of a simple rotor and are studied in a computer experiment for the approximation of the cosine function by a truncated Taylor series.