3.4 Wynn's Identity and the e-Algorithm

Wynn's identity is an identity connecting neighboring Pade approxi­mants in the Pade table:

It is easy to remember this identity from the identification with compass points in the Pade table, shown in Figure 1. With this mnemonic, the identity is written as

 

It is valid when all the indicated Pade approximants exist and are nonde- generate. This section is mostly devoted to a self-contained proof of Wynn's identity; in Section 3.5 there is a more complete derivation of the various other identities. At the end of this section, we use Wynn's identity to prove that entries in even columns of the e-table are, in fact, Pade approximants.

We will need two Frobenius identities in the course of this proof. Consider the determinant

 

 

We apply Sylvester's identity and consider the deletion of the first and last rows and the first and last columns. Each determinant defined by these deletions is of a standard type, and the identity.

Hitherto in this section, we have assumed the relevant entries of the Padé table to exist and be nondegenerate. A modification of (4.1) which takes explicit account of the presence of blocks is Cordellier's identity. This identity relates extant elements in the Padé table, four of which are at the vertices of a rectangle.

Cordellier's identity.

denote the nondegenerate entry of an nXn block in the Padé table. It is shown in the top left-hand corner of Figure 2. For any k in the ranee 0 /c < n, we define

For a proof of this identity, which is based on the ideas of the Euclidean modification of Kronecker's algorithm, we refer to Cordellier [1979b],

We next show that the entries in the even columns of the e-table are entries in the rows of the Pad£ table. Specifically we will prove that provided the indicated quantities exist. In Section 3.2, we defined the second column of the е-table, namely {ey\j=0,1,2,...}, to be the sequence °f [У/0] approximants, which is the sequence of truncated Taylor series, both evaluated at z = 1. We proved that the fourth column, namely {е(2;>, /=0,1,2,...} is the sequence of [/+1/1] Pade approximants evaluated at z=l. We will prove (4.10) by induction, noting that we have already established the result for k = 0 and k= 1. We use the e-algorithm repeatedly:

к 4-1 к — 1 T\ek         ek } '

as indicated by the rhombus rule in Figure 3. We will prove the connection

 

Figure 3. Part of the f-table.

 

between the \L/M]. \L/M± 11. and \L± 1 /ЛЛ Padé aDDroximants. and

 

which

 

we expect from (4.10) to involve

are the corresponding epsilons. These are indicated in the figure. Note that because the columns of the e-table will be shown to correspond to rows of the Pade table, the compass points do not correspond, and to emphasize this we use small letters in the e-table.

 

 

 

 

Simple manipulation yields

 

Application of the e-algorithm (4.11) gives the formulas

which is Wynn's identity for Padé approximants with the identification (4.10). We have only to observe that the e-algorithm is used to calculate columns of the e-table working from left to right, and that Wynn's algo­rithm calculates Padé approximants working from the first and second rows down. Then we see by direct construction of individual elements by induc­tion that the formula (4.10) is valid whenever the indicated quantities exist.

(4.11)

Notice that odd columns of the e-table are not Padé approximants and also that even columns of the e-table are Padé approximants on and above

the diagonal, evaluated at the particular value z= 1. The connection between the e-algorithm and the Padé table may be made directly using (1.1.8). (1.1.9), and (1.3.8) [Shanks, 1955; Wynn, 1961b], but the extension to Cordellier's identity is obscured.

Exercise For a normal Padé table, prove the inverse crossed rule

 

 

defined with the notation of Figure 1.