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A guide to analytic extrapolations
C-589 Computer Physics Communications 18 (1979) 215-244 © North-Holland Publishing Company
A G U I D E TO A N A L Y T I C E X T R A P O L A T I O N S
Part I: A p r o g r a m f o r o p t i m a l e x t r a p o l a t i o n to interior p o i n t s M. C I U L L I and S. CIULLI
Th. Div., CERN, Geneva, Switzerland Received 27 January 1978; in revised form 8 June 1979
PROGRAM SUMMARY
Title of program: ANALYT
Key-words: nuclear physics, scattering amplitude, analytic
Catalogue number: AAUT
continuation, inverse problems, stability, dispersion relations, functional methods, zeros, poles
Computer: IBM 370/3033, Kernforschungszentrum, Karlsruhe, tested also on UNIVAC 1108 and on CDC 6600 and CDC 7600
Operating system: OS/VS 2, Compiler Fortran H opt. (2) Programming language used: FORTRAN IV High speed storage required (in GO step): 212 k * No. of bits in k-word: 8 bits in a byte, 1024 bytes in 1 k Overlay structure: none No. o/magnetic tapes required: none Other peripherals used: card reader, line printer No. o f cards in combined program and test decks: 2500
Nature of the physical 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 [1] 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 f'md zeros and poles of the scattering amplitude.
(No. of cards of the ANALYT deck: 1800) Method of solution
* The memory requirements may be considerably lowered if the dimension statements are changed accordingly (see outlook of ref. [11).
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 weighted EXtrapolation [ 2 ], PNEX(s), which optimizes dispersion relations versus the uniform (L**) norm, the CauchY weighted EXtrapolation, CYEX(s), (optimization versus the L 2 norm, of Cutkosky's modified x2-test, see ref. [3]), 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
C-590 216
M. Ciulli, S. Ciulli / A nalytic extrapolations I
and boundedness condition [4]. The values of the other four external functions described in part 1, may be computed by calling the corresponding entry points. Further, calling EMZERO/or EPZERO/(see part 2 [ 1 ]), 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 on the complexity of the problem None, provided sufficient data points and suitable dimensions were taken (in the test run these numbers were 181 and 501, respectively).
Typical running times 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, see remark 3.1.0). Their subsequent calls need again 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 ref. [ 1 ].
References [1] I. Caprini, M. Ciulli, S. Ciulli, C. Pomponiu, M. Sararu and I.S. Stefanescu, Comput. Phys. Commun. 18 (1979). [2] S. Ciulli and J. Fischer, Nucl. Phys. B24 (1970) 537. [3] R.E. Cutkosky, Ann. Phys. 54 (1969) 110. [4] S. Ciulli and G. Nenciu, J. Math. Phys. 14 (1973) 1675; S. Ciulli, C. Pomponiu and I.S. Stefanescu, Phys. Rept. 17C (1975) 133.
A G U I D E TO A N A L Y T I C E X T R A P O L A T I O N S
Part I: A p r o g r a m f o r o p t i m a l e x t r a p o l a t i o n to interior p o i n t s M. C I U L L I and S. CIULLI
Th. Div., CERN, Geneva, Switzerland Received 27 January 1978; in revised form 8 June 1979
PROGRAM SUMMARY
Title of program: ANALYT
Key-words: nuclear physics, scattering amplitude, analytic
Catalogue number: AAUT
continuation, inverse problems, stability, dispersion relations, functional methods, zeros, poles
Computer: IBM 370/3033, Kernforschungszentrum, Karlsruhe, tested also on UNIVAC 1108 and on CDC 6600 and CDC 7600
Operating system: OS/VS 2, Compiler Fortran H opt. (2) Programming language used: FORTRAN IV High speed storage required (in GO step): 212 k * No. of bits in k-word: 8 bits in a byte, 1024 bytes in 1 k Overlay structure: none No. o/magnetic tapes required: none Other peripherals used: card reader, line printer No. o f cards in combined program and test decks: 2500
Nature of the physical 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 [1] 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 f'md zeros and poles of the scattering amplitude.
(No. of cards of the ANALYT deck: 1800) Method of solution
* The memory requirements may be considerably lowered if the dimension statements are changed accordingly (see outlook of ref. [11).
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 weighted EXtrapolation [ 2 ], PNEX(s), which optimizes dispersion relations versus the uniform (L**) norm, the CauchY weighted EXtrapolation, CYEX(s), (optimization versus the L 2 norm, of Cutkosky's modified x2-test, see ref. [3]), 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
C-590 216
M. Ciulli, S. Ciulli / A nalytic extrapolations I
and boundedness condition [4]. The values of the other four external functions described in part 1, may be computed by calling the corresponding entry points. Further, calling EMZERO/or EPZERO/(see part 2 [ 1 ]), 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 on the complexity of the problem None, provided sufficient data points and suitable dimensions were taken (in the test run these numbers were 181 and 501, respectively).
Typical running times 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, see remark 3.1.0). Their subsequent calls need again 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 ref. [ 1 ].
References [1] I. Caprini, M. Ciulli, S. Ciulli, C. Pomponiu, M. Sararu and I.S. Stefanescu, Comput. Phys. Commun. 18 (1979). [2] S. Ciulli and J. Fischer, Nucl. Phys. B24 (1970) 537. [3] R.E. Cutkosky, Ann. Phys. 54 (1969) 110. [4] S. Ciulli and G. Nenciu, J. Math. Phys. 14 (1973) 1675; S. Ciulli, C. Pomponiu and I.S. Stefanescu, Phys. Rept. 17C (1975) 133.