Download A New Approach to Scientific Computation by Ulrich W. Kulisch, Willard L. Miranker PDF

By Ulrich W. Kulisch, Willard L. Miranker

ISBN-10: 0124286607

ISBN-13: 9780124286603

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Then A AQB and A±B, where the #— sign is to be understood componentwise. A denotes the topological interior of A and dA denotes the topological boundary of A. The absolute value of a vector, resp. matrix, over IR or 0 is defined to be the vector, resp. matrix, of the absolute values of the compo­ nents. For A = \a,b\eHT with a,beT the diameter d(A)eT and the absolute value \A\eT defined by d(A):=\b - a\ and \A\:= max ( \a | , | b \ ), where the maximum is to be understood componentwise.

For the latter purpose a precise arithmetic and Bohlender's algorithm (cf. [3]) are required. sponding to U then EU^UT etc. are defined compo­ If T is the set corre­ Solving Algebraic Problems 57 If T is the set corresponding to U, then the monotone downwardly and monotone upwardly directed roundings V: T^U and A: T^U are defined by AeT: 0 {[AM) = [VA,AA]effU. 1). Finally we consider a rounding \J: T^U with the properties (Rl), (R2) and (R3). For the rounding □ and the corresponding operations 0 : Ux U^U defined by a \*\ b:= U(a*b) it can be shown (cf.

This necessary condition turns out to be sufficient also to control the accuracy of numerical algorithms by the computer itself for many problems of numerical analysis. 3) axiomatically by the properties (RG), (Rl), (R2) and (R4) respectively by (RG), (Rl), (R2), (R4) and (R3) within the programming language. Then all other operations that occur in the spaces displayed in Figure 1 can be built up with maximum accuracy by a modular technique as shown in [12] and [15]. 24 Ulrich Kulisch The following sentence is a golden rule in mathematics: "If a computation is correct one gets a correct result".

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A New Approach to Scientific Computation by Ulrich W. Kulisch, Willard L. Miranker


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