2.2 Decipherment of Singularities from Pade Approximants

 

According to the theory of analytic continuation of functions of a complex variable, all the properties of a function, analytic at a point, are contained in its power-series expansion at that point. While in principle the function may be continued by reexpansion about other points, the central practical problem is to decipher the properties of the function from the given series coefficients. Pade approximants can be used very effectively in determining quantitative results about functions when the analytic proper­ties are qualitatively known, and they can also be used to deduce consider­able information about the singularity structure of a function from its Taylor series coefficients. Before embarking on the proper use of Pade

 

 

(2.2)

 Equation (2.1) permits unique categorization of singularities at z = z0. If bn=0, « = 1,2,..., then z = z0 is a regular point and /( z ) is analytic at z = z0.

If bn =0, n = 2,3,..., and b) ^=0, then z = z0 is a simple pole. If bn— 0, n = m+ 1, m + 2,..., and bm^= 0, then z0 is an wth-order pole, and for m> 1, /(z) is said to have a multipole.

If, for all m and some n>m, bn ^=0, then z0 is an essential singularity of f(z).

Provided z0 is not an essential singularity, (2.1) suggests that /(z) may be approximated by Pade approximants. However, even if z0 is an essential singularity, provided it is not approached too closely, the essential singular­ity resembles a finite sum of multipoles. This is because bn given by (2.2) decrease rapidly with n\ in fact \bn\<r" for any r>0 and n sufficiently large.

Secondly,/(2) may have branch points. If /(2) is analytically continued by expansion and reexpansion and remains a single-valued function of 2, there are no branch points. However, if there is any point 20 for which analytic continuation by a clockwise and counterclockwise path of arbi­trarily small radius yields different values at the same point, then /(2) is not single-valued and 20 is a branch point. A single-valued function can be obtained for/(2) in various ways. One may form the Mittag-Leffler star by drawing a straight line from each branch point to infinity along a ray from the origin. It sometimes happens that a single-valued function can be formed by connecting two or more branch points by branch cuts.

 

Since the points {z } are dense on the unit circle, continuation to | z\ > 1 is no longer straightforward. Sometimes such analytic continuation is possible, and sometimes Pade approximants converge to this continued function: references to Pade approximants and quasianalytic functions are given in the selected bibliography.

In order to interpret the results of using Padé approximants on a new function, we consider how Padé approximants represent the singularities in the previous categories. If f(z) has a simple pole, then a simple zero in the denominator of the Padé approximant near the pole is expected. If f(z) has a multiple pole, a cluster of zeros of the Padé denominator is expected; these zeros should tend to coalesce at the multipole with increasing order of approximation. For an essential singularity, we recall Weierstrass's theorem:

[Titchmarsh, 1939, p. 51]. In other words, this theorem shows that f(z) tends to any desired limit casz->z0 through a suitable sequence of z-values. Thus we expect a clustering of poles and zeros of Pade approximants at an essential singularity. In a low order of approximation, of course, a multiple pole and an essential singularity will appear the same.

 

To simulate a branch cut, we expect to find a path delineated by roughly alternating poles and zeros of the Pade approximants. To appreciate this, consider where P is a path from a to b in the complex plane, F is a contour enclosing P, and the second part of (2.5) is a consequence of Cauchy's theorem./(r) is uniformly differentiable and so is analytic except on the path P. f(z) may be approximated by a Riemann sum which consists of a sequence of poles with residues p(ul)8uj. This behavior is what one expects the Pade approximants to reproduce.

Example.

and the poles of the Padé f(z) is defined with a cut on approximant are located on that cut, as shown in rigure 4.

Since there are many ways of defining the cuts to define, in turn, a single-valued function, and the Mittag-Leffler star is but one, it is interest­ing to speculate that the limiting distribution of the poles and zeros of a suitable sequence of Pade approximants delineate a natural cut structure (or principal Riemann surface) from the Maclaurin series. We presently expect

 

Figure 4, The [2/2] Pade approximant of showing poles at lying on the branch cut of that the cuts so defined are such as to minimize their capacity in the inverse z-plane, a point which we discuss briefly in Section 6.7.

 

The foregoing account states what we expect from Pade approximants to analytic functions with various singularities. In ideal circumstances precisely these results are obtained; in practice, whether one expects them or not, defects occur for all but the simplest functions. A defect is the name given to an extraneous pole and a nearby zero. We consider, nonrigorously, a function f(z) which is smooth near z = a and let pole. There are real difficulties in the numerical detection of defects; we refer to Abd-Elall et al. [1970] for details. In short, the addition of an "insignificant" pole to f(z) produces a defect, and this problem has to be expected with Padé approximation. We regard the nearby zero of numerator and denominator as canceling approximately, which puts the defects in their proper perspective. Defects are easily recognized by their transient nature. They tend to appear and disappear as one looks at one approximant and then the next. They contrast strongly with the more stable patterns seen in conjunction with the true singularities of the function. We will return to this topic in the next section; in Section 6.5, we prove that the residue of the pole of a defect is small for high-order Padé approximation of meromorphic functions.

 

The following example shows why it is normally wise to ignore results derived from Padé approximants with defects close to the origin. The function 1 +z2 has no [1/1] Padé approximant (see Section 1.4). Consider the function [Zinn-Justin, 1970]

 

 

The pole of the [1/1] approximant occurs at z — e, and its residue is — e3. If |e| is small, we see that the pole is close to the origin and its residue is small. We understand a Padé approximant with a defect near the origin as a nearly degenerate approximant, and we are suspicious of drawing any implications from the values of such an approximant.

Given a function defined by a formal power series as the starting point, the first step towards deciphering the information contained in the series is formation of the Padé approximants lying in at least a broad band about the central diagonal of the Padé table. Unless there are reasons to the contrary, as much of the Padé table as possible should be constructed. Normally, most computing effort goes into construction of the coefficients rather than formation of the Padé approximants, and so construction of all approximants, and poles receding to infinity for available Padé approximants is relatively inexpensive. The next step is the examination of the distribution of poles and zeros. Are these persistent, or do they form defects? It should then be possible to decide which poles and zeros closest to the origin represent true singularities of the function. It is sometimes possible at this stage to detect that the function has asymptotic behavior zJ" for some Jn and in some half plane by virtue of the stability of

Once the general nature of the structure of the function has been determined, either by the preceding analysis or by other qualitative informa­tion (see Part II, Section 1.3), one is in a position to make a more refined analysis. The presence of poles is normally sufficiently clear. Their influence on the function depends on their residues, their multiplicity, and their proximity to the origin. The type of structure which has been further analysed profitably is principally the branch point and branch cut. An important strategy is the manipulation of the series to a form which can be exactly represented by Padé approximants, except for small corrections. We consider the particular case where is expected to represent the function./(z) has a cut from z = fi~] to oo, and A(z),B(z) are to be analytic at z = [i~] and are expected to have little structure. The following methods have been used to good effect. [Baker, 1961; Baker et al., 1967; Hunter and Baker, 1973; Baker, 1977].

The use of the appropriate Padé approximant of F}(z) determines the pole position z = fi~\ and — y is its residue. The approximants we have defined for f(z) are not necessarily rational. They are commonly called Z)-log Padé approximants (see Section 5.3). The estimates of ju and y are called "unbi­ased" because no assumed values of ju, y, etc. are used: fi,y are determined directly from the series coefficients, (ii) Form Padé approximants to for an assumed value of ju, and obtain a biased estimate of y by evaluating the Padé approximant at by assuming a value for y, and obtain biased estimates of from the roots and residues of the Padé approximants. (iv) Form Padé approximants to by assuming values for a,y, and obtain a biased estimate of evaluating the Padé approximant at (v) Form Padé approximants to and evaluate the Padé approximants at the assumed value of ju-1. This process yields a biased estimate of y, but as a practical matter it is frequently relatively insensitive to the choice of j[Baker et al., 1967].

 

(vi) In cases where there are two or more series with the same branch point, and it is the branch point closest to the origin, one may obtain an unbiased estimate of the difference of the exponents by the method of "critical-point renormalization" as follows: we have.

 (vii) The Baker-Hunter method [1973] sometimes allows one to detect subdominant confluent singularities. This procedure is a subtle one, and sometimes the method is interfered with by strong, nonconfluent singulari­ties. Suppose, instead of (2.6), that near z — fi ', and that y, >y2 > • • • >ym. Then /(z) has m confluent singu­larities at z = ju~'. Make the change of variable so that

(2.17)

A transform of g(£) is defined by

(2.18)

From (2.17), we find

(2.19)

and hence we may compute biased estimates of yj and Aj to determine at least some of the stronger confluent, subdominant singularities.

z= 1. In (2.12) it is possible to choose

By way of a caution, we add that when confluent singularities exist, they can, and often do, bias the results of the other methods (i)-(vi) listed above as well as appreciably slow down the rate of convergence. Furthermore, it has been frequently observed that other significant singularities interfere not only with the confluent-singularity analysis, but dramatically reduce the rate of convergence. Where feasible, steps to suppress their competing influence are rewarding.