1.3 Sequences and Series; Obstacles
In Section 1.1 we showed how Padé approximants are constructed from given power series, and in Section 1.2 we saw how the Padé approximants of exp(z) are obtained. In this section, we preview a few of the techniques and problems to be discussed in later chapters. The Padé method is directly applicable for the improvement of convergence of series and sequences. This application is fully discussed in Chapter 3. For the moment, assuming t the convergence of a sequence of approximants point we define a series from it Likewise, given a sequence using forward differences, and then Padé approximants may be used to extrapolate sequences. It is common practice to use the diagonal sequence of Padé approximants unless there are reasons to the contrary. We take up these points in Chapter 3. From (1.8),
We reduce this by subtracting z times each column from the previous column, to yield
which is a compact and symmetric form. For the numerator, we use (1.9),
and a similar reduction yields
and adding to the
Dividing each column, except the last, by last, we find
We are now led to define the matrices
We expand (3.3) by its last row and then by its last column, using cofactors of W( z), to deduce
This equation is called Nuttall's compact form for a Pade approximant. If L<A/, the polynomial term in (3.6) is understood as zero, because we use the convention that
Reductions such as these lead to interesting forms for the Pade approximants when they are used for acceleration of convergence of sequences.
Given the sequence we dehne
From (1.8), and after several elementary operations, we find symmetric determinant with all difference operators which is an acting on symmetric determinant:
These formulas suggest, but in no way compel, the choice of diagonal approximants for acceleration of convergence. An element of the final sequence is given by which is a remarkably elegant result [Shanks, 1955].
is called the diagonal sequence.
Finally in this section we mention obstacles. We will take note of some examples which illustrate why precision is mandatory in the treatment of formal nower series.
A'here S is arbitrarily small but positive). It is clear that
Of course, this example is contrived, and is based on a function not in a neighborhood of the origin (this statement means a domain
is not determined by its Maclaurin series, and so our theorems are always phrased so as to exclude the possibility that we are representing such functions.
Another notorious function is Euler's function. This function is a more constructive example, and in Chapter 5 we show how its Pade approximants converge. It is defined by the series is really only determined by a convention about the location of the
and we assume that a certain sequence of Pade approximants has been empirically found to converge. The full theory relevant to this example is explained in Sections 5.5 and 5.6. The moot point is whether convergence is to E(z), again begging the question of the extent to which E(z) is defined by (3.9). With the information that (3.9) is an asymptotic series, an entirely satisfactory definition exists for lo be very pragmatic, take
The magnitudes of the terms in the series are shown in Figure 1, and in the sense of the previous definitions, a plausible value for £Y0.1) is reasonably well determined bv truncation at the minimum, namely ]
This procedure is much less satisfactory for large values of z, and also it would seem to work badly for z small and negative. The problem turns out to be that with the natural approach, is defined with a branch cut along determined uniquely in the sense of cut-plane analyticity. To be pedestrian, cut. This point is easy to overlook using Pade approximants, because the approximants "choose" the cut in the natural place in a sense we describe in Section 2.2.
As a final example of mathematical perversity, we refer to Part II, Section 3.9, where we exhibit a nontrivial function which is analytic in an annulus, and which cannot be properly approximated by polynomials in z in the annulus. Another similar function is our demonstration example of Section 1.1. Here we showed the success of low-order Pade approximants in practice, and rapid convergence at high order may be proved using the Stieltjes series methods of Chapter 5. For the most part in this book, we are concerned primarily with how, one way or another, various natural obstacles can be overcome.