The Epsilon Algorithm

3.5.4 The Epsilon Algorithm

One extension of the Aitken acceleration uses a comparison series with terms of the form

Here α ′ ν and k ν are 2p parameters, to be determined, in principle, by means of c j ,j=0: 2p − 1. The parameters may be complex. The power series becomes

p ∞ α ′ S(z) =

ν =1 1−k ν z which is a rational function of z, and thus related to Padé approximation. Note, however,

that the poles at k −1 ν should be simple and that m < n for S(z), because S(z) → 0 as z → ∞.

120 Diagonal Padé approximants are also used for the evaluation of the matrix exponential e A ,A∈R n ×n ; see Volume II, Chapter 9.

3.5. Continued Fractions and Padé Approximants 337 Recall that the calculations for the Padé approximation determine the coefficients of S(z)

without calculating the 2n parameters α ′ ν and k ν . It can happen that m becomes larger than n , and if α ν ′ and k ν are afterward determined, by the expansion of S(z) into partial fractions, it can turn out that some of the k ν are multiple poles. This suggests a generalization of this approach, but how?

If we consider the coefficients q j , j = 1 : n, occurring in (3.5.28) as known quantities, then (3.5.28) can be interpreted as a linear difference equation. 121 The general solution of this is given by (3.5.39) if the zeros of the polynomial

are simple. If multiple roots are allowed, the general solution is, by Theorem 3.3.13 (after some change of notation),

c l n = p ν (l)k ν ,

where k ν runs through the different zeros of Q(x) and p ν is an arbitrary polynomial, the degree of which equals the multiplicity −1 of the zero k ν . Essentially the same mathe-

matical relations occur in several areas of numerical analysis, such as interpolation and approximation by a sum of exponentials (Prony’s method), and in the design of quadrature rules with free nodes (see Sec. 5.3.1).

Shanks [322] considered the sequence transformation (

···s n +k+1 ( ( ( (

( s n +k s n +k+1 ··· s n +2k (

e k (s n ) = ( ( , k = 1, 2, 3, . . . , (3.5.40) (

2 4 ( s n ···4 2

s n +2k−2 (

+k−1

and proved that it is exact if and only if the values s n +i satisfy a linear difference equation

(3.5.41) with a 0 a k

a 0 (s n − a) + · · · + a k (s n +k − a) = 0 ∀ n,

0 +· · ·+a k

process (the proof is left as Problem 3.5.7). The Hankel determinants 122 in the definition of e k (s n ) satisfy a five-term recurrence relationship, which can be used for implementing the transformation.

121 This can also be expressed in terms of the z-transform; see Sec. 3.3.5. 122

A matrix with constant elements in the antidiagonals is called a Hankel matrix, after Hermann Hankel (1839– 1873), German mathematician. In his thesis [185] he studied determinants of the class of matrices now named after him.

338 Chapter 3. Series, Operators, and Continued Fractions Here we are primarily interested in the use of Padé approximants as a convergence

accelerator in the numerical computation of values of f (z) for (say) z = e iφ . A natural question is then whether it is possible to omit the calculation of the coefficients p j ,q j and

find a recurrence relation that gives the function values directly. A very elegant solution to this problem, called the epsilon algorithm, was found in 1956 by Wynn [384], after complicated calculations. We shall present the algorithm, but refer to the survey paper by Wynn [386] for proof and more details.

is computed by the nonlinear recurrence relation,

A two-dimensional array of numbers ǫ (n)

(p) ǫ (p +1)

k +1 =ǫ k −1 + (p +1) (p) , p, k = 0, 1, . . . , (3.5.42)

−ǫ k

which involves four quantities in a rhombus:

The sequence transformation of Shanks can be computed by using the boundary conditions

ǫ (p) −1 = 0, ǫ 0 =s p in the epsilon algorithm. Then

(p)

ǫ (p) 2k =e k (s p ), ǫ 2k+1 = 1/e k (4s p ), p = 0, 1, . . . ; i.e., the ǫ’s with even lower index give the sequence transformation (3.5.40) of Shanks. The

(p)

ǫ ’s with odd lower index are auxiliary quantities only. The epsilon algorithm transforms the partial sums of a series into its Padé quotients or, equivalently, is a process by means of which a series may be transformed into the con- vergents of its associated and corresponding continued fractions. It is a quite powerful all-purpose acceleration process for slowly converging sequences and usually fully exploits the numerical precision of the data. For an application to numerical quadrature, see Exam- ple 5.2.3.

If the boundary conditions

ǫ p −1 = 0, ǫ 0 =r p, 0 (z) = j =0 c j z j (3.5.43) are used in the epsilon algorithm, this yields for even subscripts

(p)

(p)

(3.5.44) Thus the epsilon algorithm can be used to compute recursively the lower half of the Padé

ǫ (p)

2n =r p +n,n (z)

table. The upper half can be computed by using the boundary conditions

(3.5.45) The polynomials r 0,n (z) are obtained from the Taylor expansion of 1/f (z). Several proce-

ǫ −n) 2n =r 0,n (z) =

j =0 d j z j

dures for obtaining this were given in Sec. 3.1.

3.5. Continued Fractions and Padé Approximants 339 It seems easier to program this application of the ǫ-algorithm after a slight change of

notation. We introduce an r × 2r matrix A = [a ij ], where

a (p)

ij =ǫ k , k = j − 2, p = i − j + 1.

Conversely, i = k + p + 1, j = k + 2. The ǫ algorithm, together with the boundary conditions, now takes the following form:

for i =1:r

a i, 1 = 0; a i, 2 =r i −1,0 (z) ; a i, 2i =r 0,i−1 (z) ; for j = 2 : 2(i − 1)

a i,j +1 =a i −1,j−1 + 1/(a ij −a i −1,j ).

end end

Results: [m, n] f (z) =a m +n+1,2n+2 , (m, n ≥ 0, m + n + 1 ≤ r). The above program sketch must be improved for practical use. For example, something

should be done about the risk for a division by zero.