Floating-point Arithmetic Operations Research Articles

Error analysis of digital filters realized with floating-point arithmetic

This paper calculates the error at the output of a digital filter using floating-point arithmetic operations due to roundoff accumulation and input quantization. Expressions are derived for the mean square error for each of the three canonical forms of realization: direct, cascade, and parallel. Upper bounds that are useful for a special class of filters are given. The theoretical result is found to be in good agreement with the experimental result.

Significance arithmetic on a digital computer

The 7090 at NYU has been modified to include a “Significance Mode” of operation which is intended to facilitate the identification of significant bits in the results of floating-point arithmetic operations. The manner in which floating-point arithmetic is handled in this mode is discussed. Several numerical experiments using this mode are described and comparisons are made with the ordinary “normalized mode.” Examples include power series evaluation, linear equations solution, determinant evaluation and matrix inversion.

Error analysis of floating-point computation

This paper consists of two main sections. In the first the bounds are derived for the rounding errors made in the fundamental floating-point arithmetic operations. In the second, these results are applied in the analysis of a number of computing techniques for the calculation of the eigenvalues of matrices. In each case thecomputed solution is expressed as the exact solution of a perturbed version of the original matrix and bounds are found for the perturbations. For one of the techniques, an a priori bound is derived for the errors in the eigenvalues themselves.