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Manuscript Title: A guide to analytic extrapolations. Part I: a program for optimal extrapolation to interior points.
Authors: M. Ciulli, S. Ciulli
Program title: ANALYT
Catalogue identifier: AAUT_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 18(1979)215
Programming language: Fortran.
Computer: IBM 370/3033.
Operating system: OS/VS 2, COMPILER FORTRAN H OPT.
RAM: 212K words
Word size: 8
Keywords: General purpose, Interpolation, Scattering amplitude, Analytic continuation, Inverse problems, Stability, Dispersion relations, Functional methods, Zeros, Poles.
Classification: 4.7, 4.10.

Nature of problem:
In the scattering theory of elementary particles as well as in any other branch of physics in which the analyticity of some function of physical interest has been established, one is often faced with the practical question of continuing data given on parts of the cuts, in an analytic way towards some interior points of the analyticity domain, since this problem is 'ill posed' in the Hadamard sense, in order to get sensible predictions, one has to 'stabilize' the output by means, for instance, of a boundedness condition, given on those parts of the cuts where the actual data is lacking (e.g. a Froissart bound). The present program processes the input data and the stabilizing condition in an optimal function-analytic way, yielding estimates of analytic continuations in any desired interior point, as well as the vaLUe of some important nonlinear functionals which 'measure' the 'analyticity' of the input. These numbers might then be used to correlate in an analytic way data spread on different energy regions, to get informations about the asymptotics, or to find zeros and poles of the scattering amplitude.

Solution method:
Calling once the subroutine ANALYT, all the pertinent information (the complex-valued data and the points where it is given, the form of the error corridor and the function-bounds), is loaded. Then, according to the name of the extrapolation function one calls, one gets either the PoissoN weigthed EXtrapolation, PNEX(s), which optimizes dispersion relations versus the uniform (L infinity) norm, the Cauch Y weighted EXtrapolation, CYEX(s), (optimization versus the L**2 norm, of Cutkosky's modified x**2-test, or the Central Analytic EXtrapolation, CAEX (s), which gives at any point s, the center of gravity of the (complex) values taken there by all the analytic functions compatible with the initial data and boundedness condition. The values of the other four external functions may be computed by calling the corresponding entry points. Further, calling EMZERO/or EPZERO/, one computes a number (= the width of the smallest bound, or smallest error corridor, still compatible with the initial data and with the analyticity), which might be used then to recognize "good" (i.e. "analytical") data from "wrong" ones. A further facility permits to see whether the data and the analyticity condition is compatible or not with the existence of a (or a pair of) zero(s) or pole(s), placed at some given point szero.

Restrictions:
None, provided sufficient data points and suitable dimensions were taken (in the test run these numbers were 181 and 501, respectively).

Running time:
The Cauchy weighted or the Poisson weighted extrapolations need only some 0.2 s per point, but the central analytic or the other special extrapolations are more time consuming (approx. 1 min,). Their subsequent calls need again only 0.2 s per extrapolated point. Some 10 s are necessary for only 0.2 s per extrapolated point. Some 10 s are necessary for EMZERO/EPZERO. Further details might be found in the two test programs descriptions at the end of C.P.C. (18(1979)305).