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**The arithmetic of the digital computer: A new approach.**
*(English)*
Zbl 0597.65037

This paper is rather an explanation and a recommendation of two books published by the present authors [Computer arithmetic in theory and practice (1981; Zbl 0487.65026) and A new approach to scientific computation (1983; Zbl 0538.00023)].

A great part of the paper deals with the pitfalls that may occur when simply one uses the floating-point arithmetic. There are also some interesting historic remarks. The main idea of the new approach is to use more systematically the intervals where may take place the results given by this arithmetic. Beside the five fundamental operations (the fifth one being the scalar product), the authors propose other ten operations which give the bounds of the respective intervals. Further we reproduce from the authors’ Introduction: ”In § 5 a process of automatic error control called validation is described. Validation delivers high accuracy with guarantees for scientific computations. High accuracy is furnished by coupling the scalar product with a special numerical process called defect correction. Guarantees and error bounds are obtained by interval techniques”.

A great part of the paper deals with the pitfalls that may occur when simply one uses the floating-point arithmetic. There are also some interesting historic remarks. The main idea of the new approach is to use more systematically the intervals where may take place the results given by this arithmetic. Beside the five fundamental operations (the fifth one being the scalar product), the authors propose other ten operations which give the bounds of the respective intervals. Further we reproduce from the authors’ Introduction: ”In § 5 a process of automatic error control called validation is described. Validation delivers high accuracy with guarantees for scientific computations. High accuracy is furnished by coupling the scalar product with a special numerical process called defect correction. Guarantees and error bounds are obtained by interval techniques”.

Reviewer: Gh.Marinescu