Introduction to Interval Computation by Gotz Alefeld, Jurgen Herzberger

By Gotz Alefeld, Jurgen Herzberger

This publication is revised and improved model of the unique German textual content. The association of the fabric and the constitution are primarily unchanged. All feedback within the Preface to the German variation concerning naming conventions for formulation, theorems, lemmas, and definitions are nonetheless legitimate as are these about the association and selection of fabric.

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C0 )) (m . . , u; c >m , . . , c >) - /(*< > , . . , x ; c , . . , c ) < 7 max \u - x \ < yd(X). Interval Evaluation and Range of Real Functions 25 The difference of the upper bounds of the ranges of values may be estimated in a similar way. These two estimates together prove the assertion. • 1 ) (n) The statements of Theorem 3 may, as the( proof shows, immediately be generalized to functions of several variables x , . . , x . Instead of y d(X) we have the quantity The following example demonstrates that the approximation of the range ( m ) evaluation is dependent on the choice of of a function /(0 by an interval expression f(x\a \ .

C ) < 7 max \u - x \ < yd(X). Interval Evaluation and Range of Real Functions 25 The difference of the upper bounds of the ranges of values may be estimated in a similar way. These two estimates together prove the assertion. • 1 ) (n) The statements of Theorem 3 may, as the( proof shows, immediately be generalized to functions of several variables x , . . , x . Instead of y d(X) we have the quantity The following example demonstrates that the approximation of the range ( m ) evaluation is dependent on the choice of of a function /(0 by an interval expression f(x\a \ .

We also wish to show how the proof of Theorem 4 can be shortened using Theorem 5 in a manner similar to the proof of Theorem 6. The interval evaluation of the centered form f(X)=f(z) + (X-z)h(X-z) satisfies W(f,X)g:f(X). 21) we may then estimate q{ W{f, X), f(X)) ^ d(f(X)) - d{ W(f, X)). Let now mm\h(x — z)\ = \h(w — z)\. xeX 37 Interval Evaluation and Range of Real Functions Then we get z)h(w - z) c f(z) f(z) + {X- + {(* - z)h{x -z)\xeX}= W(f X). 14), we get from the above inclusion that d(W(fX)) ^ d{{X - z)h(w - z)) = d(X)\h(w - z)|, weX.

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