1.4 The Baker Definition, the C-Table, and Block Structure

 

To motivate the discussion of uniqueness and the modern definition, of which the analysis is due to Baker, we must consider what can go wrong with the basic approach discussed so far. The classic example which demonstrated the shortcoming of a simple-minded approach is the construction of

For a long time, the accepted solution was to take an analogue of (4.4) as the agreed definition of Pade approximants. Specifically, the classical defini­tion, also called the Frobenius and Pad6 Frobenius definition, is that if are polynomials of orders L, M respectively, and if is a Pade approximant of then Equation (4.5) is the general form of (4.4), and it is remarkable that polynomials ot orders can always be found to satisfy (4.5). However, our specific example emphasizes that it then in this case(4.5)

 

 

Pad6 defined a deficiency index, which is the least integer for which and a>LM is a measure of the shortcoming of the approximation. Quite simply, the rational function through order in certain circumstances, and then we prefer to say that the Pade approximant does not exist.

In the general theory of rational interpolation, it is well known that there at certain points z; are

specified, the specification may be inconsistent, and in such circumstances, the rational interpolants are declared not to exist. Our approach is entirely in line with this attitude.

Because the accuracy-through-order requirement is fundamental, a defini­tion which preserves it is essential, and we use the modern definition which was fully analysed by Baker [1973b], of degrees

 

The notation emphasizes that numerator and denominator depend on both L and M. An entirely equivalent specification of the definition is to replace (4.6) by provided that (4.7) is retained. The notation of (4.6) and (4.7) is exclusively reserved for this purpose throughout the work, and without further explana­tion.

If, with Equation (1.8) and implies that the two definitions correspond up to an unimportant numerical factor. Consequently, and charitably speaking, the distinction between the definitions is sometimes taken for granted. However, if precise terminology is mandatory. Because the vanishing of important, a special symbol is exclusively reserved for this quantity:

Table 1. The C-Table.

 

 

 

 

Summary.

If it is possible that the Pade approximant does not exist. However, polynomials satisfying 1), called the C-table. This is an array of values of determinants, and should be distinguished from the Pade table.

It is convenient to display in a table (Table (4.5) exist, defining a rational fraction which historically was called the Pade approximant.

the classical Pade-Frobenius and the Baker definitions are entirely equivalent. If Example. Let This is given by the power series (4.9) and the C-table of Table 2 results. The most conspicuous features of the table are the square blocks of zeros. In fact, the zero at C(4/4) is the start of an infinite square block. Before proceeding with the proof of these state­ments, let us investigate how Table 1 was actually constructed. A substantial amount of elementary algebra is needed to construct Table 2 from the basic

Equation (4.10) is called a (***) star which is valid if identity, showing how it relates the entries in the C-table. In order fully to understand the consequences of (4.10), it is worthwhile reconstructing Table 1 from the initializing values. We proceed with Sylvester's theorem, of which (4.10) is a corollary.

Theorem 1.4.1. Let A be a matrix, and let Arp denote the matrix with row r and columns p and q deleted. Provided t denote the matrix A with rows r and s Proof. Suppose that A is an matrix, and we consider deletion of its last two rows and columns. Take Write the matrices in block form

 

 

 

 

 

 

Next we consider a by block matrix, with determinant given

 

 

 

 

 

 This relation (4.11) is the key to the block structure of the C-table.

Theorem 1,4.2. Zero entries occur in the C-table in square blocks which are entirely surrounded by nonzero entries (except at infinity).

Proof. We identify the top left-hand corner of the block by requiring that and  It is obvious how to redefine / or m if either of the latter two conditions does not hold. From the star identity, we deduce that Hence and Similarly, the identity yields and the relevant portion of the C-table is shown in Table 3. Suppose that

Then the star identity establishes that star identity establishes that and the Hence the theorem isproved for a unit square block. The only alternative to (4.12b) is that

 Table 3. Consequences of (4.12)

Suppose that

Table 4. The Left Edge of a Block

 

Thus we have a column of zeros bordered on top, bottom and left by nonzero elements, as shown in Table 4. Similarly, we establish that the block is rectangular and, if finite, it is entirely bordered by nonzero elements as shown in Table 5. To establish that the blocks are square, we consider a block with r rows and 5 columns. First we prove that r>s, and to do this we choose a simple example which makes the general case obvious. Suppose we know that

 

These statements are interpreted as implying that a linear combination of Using the implied multipliers, we deduce that where the x denotes an unknown and totally immaterial entry. Likewise, Finally, by interpreting (4.13) and (4.14) as asserting the existence of linearly dependent column vectors, we find that then there are at least 5 rows. In other words, a block with r rows and 5 columns has r^s.

 

Likewise, that if we have a block of the C-table with 5 columns, We claim that it is now obvious

 

To Drove the converse we refer forward to Hadamard's theorem of Section 1.6. We may always reset the problem for a function with We now assume that so that we may consider the C-table for the implies

 

 

 Hence  proved that all blocks of the C-table are square, and are entirely bordered by nonzero entries.

This completes the preliminary to Pade's theorem. This theorem has been modernized by Baker, but the content is essentially unchanged. The style of proof is quite different from that of the previous century.

Theorem 1.4.3. The Pade table consists of uniquely determined entries given by following the definitions (1.8), (1.9), (4.6), and (4.7), provided

 

Otherwise, suppose that in each entry of an rXr block of the C-table. Corresponding to this, blocks of the Padé table are blocks, for which for all i, and

 

An elementary view of a set of linear equations is that either the equations are consistent and have a solution, or they are inconsistent and have no solution. This view is mirrored by the previous theorem, in which linear equations either do or do not determine Pade approximants; ofTable 6. Part of the c-Table Showing a 3 X 3 Block and part of the Pade Table showing the corresponding 4X4 block.

 

Red.

Con.

Inc.

denotes a Padé approximant with a singular Hankel determinant, but which is determined by a consistent set of equations, and also reduces to [X/^i], denotes a nonexistent Padé approximant. The equations for it are inconsistent.

 course, the theorem also considers the question of uniqueness. Table 6 summarizes graphically the link between a block in the C-table and the Padé table.

 

Up until now, we have tended to take the question of infinite blocks of the C-table and the Padé table for granted. This casualness is for the good reason that an infinite block in the Padé table turns out to be uniquely associated with a rational fraction. It is certainly a consequence of Theorem 1.4.2 that if either side of any block in the C-table is of infinite length, then so is the other side of the block.

Theorem 1.4.4. Suppose that a function f(z) is analytic at the origin and is

uniquely determined by its Maclaurin series: istence of an infinite block in the Padé table

sufficient condition for f(z) to be rational; let f(z) be of type [A/fx].

 

Hence, by the hypothesis of the theorem,

AlW(z) f(Z) BM\z)

and X=X', fi^fi'. Let

N i N b(z) = BW"\z)= II 1"£ , where 2M/=M- <=1V zi!           ,= ,

Let

1

7ri(z)= 2 ckzk ^ X>n and 7r,(z) = 0 otherwise.

k = o

Then a polynomial nr2(z), of degree ju, at most, exists such that /(z) decomposes into partial fractions:

f(z)=^(z)+^l=nl(z)+ 2

Hence ck is given by (4.17) for —

The converse, in which we suppose that /(z) is rational, is a consequence of the uniqueness property of Pade approximants, given by Theorem 1.4.4, and the rest is obvious.

Having established that certain Pade approximants do not exist in certain circumstances, the following theorem establishes that, in every row, column, or diagonal, there are an infinite number of extant approximants.

Theorem 1.4.5. Let 1f_.() ctz' be a formal power series. In its Pade table, an infinite number of Pade approximants exist

in any row [L/M], M fixed, L-+ oo,

in any column [L/M], L fixed, M—> oo, and

in any paradiagonal [M+J/M], J fixed, M-^ oo.

 

 

 

 

 

 

 

 

Having established that certain Pade approximants do not exist in certain circumstances, the following theorem establishes that, in every row, column, or diagonal, there are an infinite number of extant approximants.

 

Proof. Consider the rows. In any row, there are either a finite number of blocks and consequently an infinite number of well-defined approximants, or else an infinite number of blocks. In any block, there is at least one extant Pad6 approximant in any row. Hence an infinite number of Pade approximants exist in any row of the Pade table. The argument is the same for columns and paradiagonals.

Note that the theorem makes no mention of convergence of these ap­proximants, but it does mean that the discussion of convergence of rows, columns, and diagonals is more than a rhetorical exercise.

Exercise 1. Find a function for which the [3/2] Pade approximant does not exist.

Exercise 2. Without using a computer, calculate the C-table

Exercise 4. By using continuity of

and the theory of the continuous denominator [Roth, 1965], renormalize

Exercise 5. Show that blocks in the Padé table may be contiguous but cannot overlap.