Order Stars: Theory and Applications
According to Hilbert's dictum, the scaffolding should be invisible in a math ematical edifice. Less kind interpretation of this common principle of writing and presenting mathematics is that we should always strive to do it baek to-front, forever wise after the event. Nobody should be allowed to see the seams in the supposedly seamless robe or eompare authors' intentions with the outeome of their endeavour. In particular, the short pieee of prose oeea sionally labelIed 'Prefaee' or 'Forward' ought to be written after the main body of the book. And so it is, and we, the authors, can refleet (with much trepidation) on an enterprise that for us is finally over. Order stars have been originally introduced in the context of numerical solution of ordinary differential equations and, as far as many numerical an alysts are concerned, they still belong there. It is our case in this book that the seope of order stars ranges considerably wider and that the cornerstone of the order star theory is a function-theoretic interpretation of complex approximation theory. An application to numerical analysis is a matter of serendipity, not of essen ce.
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A-acceptable A-stable A+-regions According adjoin already analysis analytic analytic function application approach argument assume axis barrier belong boundary bounded branch Chapter coefficients complex computational condition Consequently consider constant count course defined definition denote derive differential equations disc discretizations easy error essential singularities exactly Example exists explicit exponential Figure finite follows formula function further given half-plane hence implies important inequality interest interpolation interpolation points interval Iserles kind leads least Lemma linear mathematical maximal means method Moreover multiplicity multistep negative Nørsett Note obeys once optimal order stars origin Padé approximants particular points poles polynomial possible present problem produces proof Proposition prove rational function require respectively restricted root scheme sectors of A+ semi-discretization single solution stability Suppose Theorem theory true underlying unit values yields zeros