Download PDF by Peter Kravanja: Computing the Zeros of Analytic Functions

By Peter Kravanja

ISBN-10: 3540671625

ISBN-13: 9783540671626

Computing all of the zeros of an analytic functionality and their respective multiplicities, finding clusters of zeros and analytic fuctions, computing zeros and poles of meromorphic features, and fixing platforms of analytic equations are difficulties in computational advanced research that result in a wealthy mixture of arithmetic and numerical research. This ebook treats those 4 difficulties in a unified manner. It comprises not just theoretical effects (based on formal orthogonal polynomials or rational interpolation) but additionally numerical research and algorithmic features, implementation heuristics, and polished software program (the package deal ZEAL) that's to be had through the CPC software Library. Graduate studets and researchers in numerical arithmetic will locate this ebook very readable.

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With a similar expansion for (tn − β n )r , we see that r r j K1 (t, z) = ((tn − αn )p j=0 r (z n − αn )j (αn − β n )r−j r j − (z n − αn )p j=0 (tn − αn )j (αn − β n )r−j (t − z)(tn − β n )r (tn − αn )p . 3) is a polynomial of degree pn − 1 and the values of K(t, z) and of K1 (t, z) on the zeros of (z n − αn )p are the same. 3) we see that for |t| = R < ρ, we have |K(t, z)| ≤ C max max (|α|n , |β|n )r−j |z|nj Rnr np |z| max (|α|n , |β|n )r−j Rnr+n(p−j) max 1≤j≤p−1 max 1≤j≤p−1 , . From this inequality, we see that the right side will tend to zero when |z| < Rr/j , (max (|α|, |β|)(r−j)/j and |z| ≤ R j = 1, .

1. Hermite Interpolation As a generalization of Lagrange interpolation, we introduce the notion of Hermite interpolation. Let f (z) be a function analytic in an open domain D. Further let n and r be positive integers, and z0 , . . , zn−1 pairwise different points from D. We shall denote the (unique) Hermite polynomial interpolant of degree rn − 1 and order r, of f (z) in these n zeros by hr,rn−1 (f ; z). This polynomial is defined by the properties (j) hr,rn−1 (f ; zk ) = f (j) (zk ), n−1 k=0 (z With the notation ωn (z) := in the form hr,rn−1 (f ; z) = 1 2πi k = 0, .

Problem. Let f0 (z), . . . , fr−1 (z) be r given functions in Aρ (ρ > 1). Let ( = 0, 1, . . , r − 1) be r sets of given real numbers. For each and {p ,j }n−1 j=0 a set of real numbers {p ,j }n−1 j=0 we define a linear operator L on the space of polynomials of degree ≤ n − 1 such that n−1 n−1 cj z j , if Qn (z) = then L (Qn ) = j=0 cj p ,j z j . 1) j=0 The problem is to find the polynomial Pm,n,r (z) which minimizes the sum r−1 m−1 |f (ω k ) − L Qn (ω k )|2 , ω m = 1, =0 k=0 over all polynomials Qn ∈ πn−1 .

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Computing the Zeros of Analytic Functions by Peter Kravanja

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