The Padé Table

3.5.3 The Padé Table

Toward the end of the nineteenth century Frobenius and Padé developed a more general scheme for expanding a formal power series into rational functions, which we now describe. Let f (z) be a formal power series

f (z) =c 0 +c 1 z +c 2 z 2 +···=

c z i i . (3.5.23)

i =0

Consider a complex rational form with numerator of degree at most m and denominator of degree at most n such that its power series expansion agrees with that of f (z) as far as possible. Such a rational form is called an (m, n) Padé 119 approximation of f (z).

118 There are plenty of other notations for this function. 119 Henri Eugène Padé (1863–1953), French mathematician and student of Charles Hermite, gave a systematic

study of Padé forms in his thesis in 1892.

330 Chapter 3. Series, Operators, and Continued Fractions

Definition 3.5.3.

The (m, n) Padé approximation of the formal power series f (z) is, if it exists, defined to be a rational function

that satisfies

f (z) − [m, n] f (z)

= Rz m +n+1 + O(z +n+2 ), z → 0. (3.5.25) The rational fractions [m, n] f , m, n ≥ 0 for f (z) can be arranged in a doubly infinite

array, called a Padé table. m \n

The first column in the table contains the partial sums m

j =0 c j z of f (z).

Example 3.5.4.

The Padé approximants to the exponential function e z are important because of their relation to methods for solving differential equations. The Padé approximants for m, n = z

0 : 2 for the exponential function f (z) = e are as follows. m \n

2 1+z+ z 2 1+ 3 6 1+ 2 12

There may not exist a rational function that satisfies (3.5.25) for all (m, n). We may have to be content with k < 1. However, the closely related problem of finding Q m,n and P m,n (z) such that

f (z) −P m,n (z) = O(z +n+1 ), z → 0, (3.5.26)

m,n

3.5. Continued Fractions and Padé Approximants 331 always has a solution. The corresponding rational expression is called a Padé form of type

(m, n) . Using (3.5.23) and (3.5.24) gives

= m p i z + O(z +n+1 ).

Matching the coefficients of z i , i = 0 : m + n, gives

if i = m + 1 : m + n, where c i = 0 for i < 0. This is m + n + 1 linear equations for the m + n + 2 unknowns

p 0 ,p 1 ,...,p m ,q 0 ,q 1 ,...,q n . Theorem 3.5.4 (Frobenius).

There always exist Padé forms of type (m, n) for f (z). Each such form is a repre- sentation of the same rational function [m, n] f . A reduced representation is possible with

P m,n (z) and Q m,n (z) relatively prime, q 0 = 1, and p 0 =c 0 .

We now consider how to determine Padé approximants. With q 0 = 1 the last n linear equations in (3.5.27) are

c i −j q j +c i = 0, i = m + 1 : m + n, (3.5.28)

j =1

where c i = 0, i < 0. The system matrix of this linear system is

c m  c m −1 ···c m −n+1

···c m −n+2   

 c m +1

C m,n =   .  (3.5.29) 

c m +n−1 c m +n−2 ···

If c m,n = det(C m,n )

1 ,...,q n . The coefficients p 0 ,...,p n of the numerator are then obtained from

min(i,n)

In the regular case k = 1 the error constant R in (3.5.25) is given by

R =p i =

c i −j q j , i = m + n + 1.

j =0

332 Chapter 3. Series, Operators, and Continued Fractions Note that [m, n] f uses c l for l = 0 : m + n only; R uses c m +n+1 also. Thus, if c l is given

for l = 0 : r, then [m, n] f is defined for m + n ≤ r, m ≥ 0, n ≥ 0. If n is large, the heavy part of the computation of a Padé approximant

[m, n] f (z) =P m,n (z)/Q m,n (z)

of f (z) in (3.5.23) is the solution of the linear system (3.5.28). We see that if m or n is decreased by one, most of the equations of the system will be the same. There are therefore recursive relations between the polynomials Q m,n (z) for adjacent values of m, n, which can

be used for computing any sequence of adjacent Padé approximants. These relations have been subject to intensive research that has resulted in several interesting algorithms; see the next section on the epsilon algorithm, as well as the monographs of Brezinski [50, 51] and the literature cited there.

There are situations where the linear system (3.5.28) is singular, i.e.,

c m,n = det(C m,n ) = 0.

We shall indicate how such singular situations can occur. These matters are discussed more thoroughly in Cheney [66, Chap. 5].

Example 3.5.5.

2 z 2 + · · ·, set m = n = 1, and try to find [1, 1] f (z) = (p 0 +p 1 z)/(q 0 +q 1 z), q 0 = 1. The coefficient matching according to (3.5.27) yields the equations

0·q 1 =− q 0 2 . The last equation contradicts the condition that q 0 = 1. This single contradictory equation

is in this case the “system” (3.5.28). If this equation is ignored, we obtain

[1, 1] f (z) = (1 + q 1 z)/( 1+q 1 z) = 1,

1 z with error ≈ 2

2 , in spite of the fact that we asked for an error that is O(z m +n+1 ) = O(z 3 ) . If we instead allow that q 0 = 0, then p 0 = 0, and we obtain a solution

[1, 1] f (z) = z/z

which satisfies (3.5.26) but not (3.5.25). After dividing out the common factor z we get the same result [1, 1] f (z) = 1 as before.

In a sense, this singular case results from a rather stupid request: we ask to approximate the even function cos z by a rational function where the numerator and the denominator end with odd powers of z. One should, of course, ask for the approximation by a rational

function of z 2 . What would you do if f (z) is an odd function? It can be shown that these singular cases occur in square blocks of the Padé table,

where all the approximants are equal. For example, in Example 3.5.5 we will have [0, 0] f =

3.5. Continued Fractions and Padé Approximants 333 [0, 1] f = [1, 0] f = [1, 1] f = 1. This property, investigated by Padé, is known as the block

structure of the Padé table . For a proof of the following theorem, see Gragg [172].

Theorem 3.5.5.

Suppose that a rational function

P (z)

r(z) =

Q(z)

where P (z) and Q(z) are relatively prime polynomials, occurs in the Padé table. Further suppose that the degrees of P (z) and Q(z) are m and n, respectively. Then the set of all places in the Padé table in which r(z) occurs is a square block. If

(3.5.31) then r ≥ 0 and the square block consists of (r + 1) 2 places

Q(z)f (z)

− P (z) = O(z m +n+r+1 ),

(m +r 1 ,n +r 2 ), r 1 ,r 2 = 0, 1, . . . , r.

An (m, n) Padé approximant is said to be normal if the degrees of P m,n and Q m,n are exactly m and n, respectively, and (3.5.31) holds with r = 0. The Padé table is called

normal if every entry in the table is normal. In this case all the Padé approximants are different.

Theorem 3.5.6.

An (m, n) Padé approximant [m, n] f (z) is normal if and only if the determinants

c m,n ,

c m 1,n+1 ,

c m +1,n ,c m +1,n+1

are nonzero.

A Padé table is normal if and only if

c m,n

In particular each Taylor coefficient c m , 1=c m , must be nonzero. Proof. See Gragg [172].

Imagine a case where [m−1, n−1] f (z) happens to be a more accurate approximation to f (z) than usual; say that

f (z) − f (z) = O(z +n+1 ). (For instance, let f (z) be the ratio of two polynomials of degree m−1 and n−1, respectively.)

[m − 1, n − 1] m

Let b be an arbitrary number, and choose Q m,n (z) = (z + b)Q m −1,n−1 (z),

P m,n (z) = (z + b)P m −1,n−1 (z). (3.5.32)

334 Chapter 3. Series, Operators, and Continued Fractions Then

[m, n] f (z) =P m,n (z)/Q m,n (z) =P m −1,n−1 (z)/Q m −1,n−1 (z) = [m − 1, n − 1] f (z),

which is an O(z m +n+1 ) accurate approximation to f (z). Hence our request for this accuracy is satisfied by more than one pair of polynomials, P m,n (z) ,Q m,n (z) , since b is arbitrary. This is impossible, unless the system (3.5.28) (that determines Q m,n ) is singular.

Numerically singular cases can occur in a natural way. Suppose that one wants to approximate f (z) by [m, n] f (z) , although already [m − 1, n − 1] f (z) would represent f (z)

as well as possible with the limited precision of the computer. In this case we must expect the system (3.5.28) to be very close to a singular system. A reasonable procedure for handling this is to compute the Padé forms for a sequence of increasing values of m, n, to estimate the condition numbers and to stop when it approaches the reciprocal of the machine unit. This illustrates a fact of some generality. Unnecessary numerical trouble can be avoided by means of a well-designed termination criterion.

For f (z) = − ln(1 − z), we have c i = 1/i, i > 0. When m = n the matrix of the system (3.5.28) turns out to be the notorious Hilbert matrix (with permuted columns), for which the condition number grows exponentially like 0.014 · 10 1.5n ; see Example 2.4.7.

(The elements of the usual Hilbert matrix are a ij = 1/(i + j − 1).) There is a close connection between continued fractions and Padé approximants. Suppose that in a Padé table the staircase sequence

[0, 0] f , [1, 0] f , [1, 1] f , [2, 1] f , [2, 2] f , [3, 2] f ,... are all normal. Then there exists a regular continued fraction

a 1 z a 2 z a 3 1+ z

n 1+ = 1, 2, 3, . . . , 1+ 1+ with its nth convergent f n satisfying

..., a n

f 2m+1 = [m + 1, m] f , m = 0, 1, 2, . . . , and vice versa. For a proof, see [214, Theorem 5.19].

f 2m = [m, m] f ,

Historically the theory of orthogonal polynomials, to be discussed later in Sec. 4.5.5. originated from certain types of continued fractions.

Theorem 3.5.7.

Let the coefficients of a formal power series (

3.5.23) be the moments

x n w(x) dx,

where w(t) ≥ 0. Let Q m,n

be the denominator polynomial in the corresponding Padé approximation [m, n] f . Then the reciprocal polynomials

Q ∗ n,n +1 (z)

=z n +1 Q

n,n +1 ( 1/z), n ≥ 0,

3.5. Continued Fractions and Padé Approximants 335 are the orthogonal polynomials with respect to the inner product

(f, g) =

f (x)g(x)w(x) dx.

Example 3.5.6.

The successive convergents of the continued fraction expansion in (3.5.3)

= 1−z ··· 1− 3− 5− 7− are even functions and staircase Padé approximants. The first few are

945 − 735z + 64z

15(63 − 70z 2 + 15z 4 ) . These Padé approximants can be used to evaluate ln(1 + x) by setting z = x/(2 + x).

s 3(35 − 30z 22

+ 3z

The diagonal approximants s mm are of most interest. For example, the approximation s 22 matches the Taylor series up to the term z 8 and the error is approximately equal to the term z 10 / 11. Chebyshev proved that the denominators in the above Padé approximants are the Legendre polynomials in 1/z. These polynomials are orthogonal on [−1, 1] with respect to the uniform weight distribution w(x) = 1; see Sec. 4.5.5.

Explicit expressions for the Padé approximants for e z were given by Padé (1892) in his thesis. They are

Q m,n (z) = , (3.5.34)

j =0 (m + n)! (n − j)! j !

with the error P m,n (z)

m !n!

z − m = (−1) +n+1 + O(z m +n+2 ). (3.5.35) Q m,n (z)

(m + n)!(m + n + 1)!

Note that P m,n (z) =Q z n,m ( −z), which reflects the property that e −z = 1/e . Indeed, the numerator and denominator polynomials can be shown to approximate (less accurately) e z/ 2

and e −z/2 , respectively. There are several reasons for preferring the diagonal Padé approximants (m = n) for

which ( p j

2m − j)! m! =

= (−1) j p j , j = 0 : m. (3.5.36) ( 2m)! (m − j)!j!

336Chapter 3. Series, Operators, and Continued Fractions These coefficients satisfy the recursion

(m

− j)p j

2m − j)(j + 1) = 0 : m − 1. For the diagonal Padé approximants the error R m,n (z) satisfies |R m,n (z) | < 1, for

ℜz < 0. This is an important property in applications for solving differential equations. 120 To evaluate a diagonal Padé approximant of even degree we write

2 2m,2m (z) =p 2m z 2m +···+p 2 z +p 0

+···+p 2 3 z +p 1 ) = u(z) + v(z) and evaluate u(z) and v(z) separately. Then Q 2m (z) = u(z) − v(z). A similar splitting can

+ z(p 2m−1 z 2m−2

be used for an odd degree. Recall that in order to compute the exponential function a range reduction should first

be performed. If an integer k is determined such that

(3.5.38) then exp(z) = exp(z ∗ ) k ·2 . Hence only an approximation of exp(z) for |z| ∈ [0, ln 2] is

z ∗ = z − k ln 2, |z ∗ | ∈ [0, ln 2],

needed; see Problem 3.5.6. The problem of convergence of a sequence of Padé approximants when at least one of the degrees tends to infinity is a difficult problem and outside the scope of this book. Padé proved that for the exponential function the poles of the Padé approximants [m i ,n i ] f tend to infinity when m i +n i tends to infinity and

uniformly on any compact set of C. For a survey of other results, see [54].