Programs for the approximation of real and imaginary single- and multi-valued functions by means of Hermite–Padé-approximants

Abstract

We present programs for the calculation and evaluation of special type Hermite–Padé-approximations. They allow the user to either numerically approximate multi-valued functions represented by a formal series expansion or to compute explicit approximants for them. The approximation scheme is based on Hermite–Padé polynomials and includes both Padé and algebraic approximants as limiting cases. The algorithm for the computation of the Hermite–Padé polynomials is based on a set of recursive equations which were derived from a generalization of continued fractions. The approximations retain their validity even on the cuts of the complex Riemann surface which allows for example the calculation of resonances in quantum mechanical problems. The programs also allow for the construction of multi-series approximations which can be more powerful than most summation methods. Program summary Title of program: hp.sr Catalogue identifier: ADSO Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSO Program obtainable from: CPC Program Library, Queen's University Belfast, Northern Ireland Licensing provisions: Persons requesting the program must sign the standard CPC non-profit use license Computer: Sun Ultra 10 Installation: Computing Center, University of Regensburg, Germany Operating System: Sun Solaris 7.0 Program language used: MapleV.5 Distribution format: tar gzip file Memory required to execute with typical data: 32 MB; the program itself needs only about 20 kB Number of bits in a word: 32 No. of processors used: 1 Has the code been vectorized?: no No. of bytes in distributed program, including test data etc.: 38194 No. of lines in distributed program, including test data, etc.: 4258 Nature of physical problem: Many physical and chemical quantum systems lead to the problem of evaluating a function for which only a limited series expansion is known. These functions can be numerically approximated by summation methods even if the corresponding series is only asymptotic. With the help of Hermite–Padé-approximants many different approximation schemes can be realized. Padé and algebraic approximants are just well-known examples. Hermite–Padé-approximants combine the advantages of highly accurate numerical results with the additional advantage of being able to sum complex multi-valued functions. Method of solution: Special type Hermite–Padé polynomials are calculated for a set of divergent series. These polynomials are then used to implicitly define approximants for one of the functions of this set. This approximant can be numerically evaluated at any point of the Riemann surface of this function. For an approximation order not greater than 3 the approximants can alternatively be expressed in closed form and then be used to approximate the desired function on its complete Riemann surface. Restriction on the complexity of the problem: In principle, the algorithm is only limited by the available memory and speed of the underlying computer system. Furthermore the achievable accuracy of the approximation only depends on the number of known series coefficients of the function to be approximated assuming of course that these coefficients are known with enough accuracy. Typical running time: 10 minutes with parameters comparable to the testruns Unusual features of the program: none