A Posteriori Error Bounds for Gaussian Elimination

Explicit bounds are constructed for the error in the solutionof a system of linear algebraic equations obtained by Gaussianelimination using floating-point arithmetic. The bounds takeaccount of inherent errors in the data and all abbreviations(choppings or roundings) introduced during the process of solution.The bounds are strict and agree with the estimate for the maximumerror obtained by linearized perturbation theory. The formulationof the bounds avoids the need for specially directed roundingprocedures in the hardware or software; in consequence the boundscan be evaluated on most existing computers. The cost of computingthe bounds is comparable with the cost of computing the originalsolution.     

Further Developments of Rp and Ap Error Analysis

Extensions are made of a recently developed theory of floating-pointerror analysis to facilitate the construction and computationof strict error bounds of a posteriori type. Inter alia, theseextensions obviate the need for specially-directed roundingprocedures in computer hardware or software, and reduce computationalcost by enabling error bounds to be computed to a low workingprecision. Applications are made to the evaluation of products, quotients,powers, sums and inner products, and also to the processes ofinput and output.     

Error Bounds for Polynomial Evaluation and Complex Arithmetic

Explicit and computable strict bounds of a posteriori type areconstructed for the evaluation of a polynomial and its derivativesby nested multiplication, using floating-point arithmetic. Thepolynomial may be real or complex, and may contain errors inits coefficients and argument. Some new error bounds for arithmetic operations with complexnumbers are included.     

Error Bounds for Arithmetic Operations on Computers Without Guard Digits

For computers having no form of guard digit in the accumulatorregister it is not possible to bound the relative error of floating-pointsubtraction processes in a satisfactory manner. This paper describessome modifications of recent error analyses to cover this situation,including the evaluation of sums and inner products, and indicatesthe corresponding modifications for the solution of systemsof linear algebraic equations.