o¬ error. It is useful to illustrate how easily these expansions can be

derived to replace the standard divergent asymptotic series.

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78 John P. Boyd

For example, the Stieltjes function S( ) satis¬es the ordinary di¬er-

ential equation

2dS

+ (1 + )S = 1 (138)

d

For paper-and-pencil or symbolic language calculation, the simplest

method is the “Lanczos „ -method”. He observed [177, 284] that if we

perturb the right-hand side of Eq.(138) by a polynomial of degree M ,

multiplied by an unknown constant „ , we can then solve this perturbed

equation exactly by a polynomial of degree M . (Instead of the usual

strategy of approximately solving the exact di¬erential equation, the

„ -method exactly solves an approximate di¬erential equation.) If the

perturbing polynomial is M , then the „ -method yields the ¬rst M

terms of the usual divergent power series in .

However, this is actually a rather stupid choice if the goal is uniform

accuracy on some interval ∈ [0, z] where z is a complex number. The

power function M is extremely non-uniform ” very small near the

origin, but increasing very rapidly away from it. The polynomial of

degree M (and leading coe¬cient of one) which is most uniform on

—

[0, z] is the shifted Chebyshev polynomial TM ( /z) [177, 178, 55].

Table VI is a short code in the symbolic manipulation language

Maple to solve the ordinary di¬erential equation

2dS —

+ (1 + )S = 1 + „ TM ( /z) (139)

d

through a Chebyshev „ ’ method. The most important feature of the

table is simply its brevity: all the necessary algebra is performed in

exact, rational arithmetic under the control of only nine lines of code!

The choice of Maple is arbitrary; the same calculation could be per-

formed with equal brevity in any of the other widely used symbolic alge-

bra languages including Mathematica, MACSYMA and Reduce. Boyd

[55, 63] gives many examples of problem-solving via spectral methods

in algebraic manipulation languages. Note that because all operations

involve polynomials, not transcendentals, the code also executes very

speedily.

The Chebyshev-„ result is rather messy: polynomial in , rational

in z. However, the Chebyshev (or Chebyshev-like) expansion will obvi-

ously converge most rapidly when the expansion interval [0, z] is small

as possible for a given . This implies that for best results, one should

choose z = . Making this substitution not only optimizes accuracy for

a given , but also simpli¬es the result to a rational function of alone.

The M = 8 approximation, which is a polynomial of degree 7 over a

polynomial of degree 8, is

S8 ≡ 16 {4 + 124 + 1336 + 33 7 }/

2 3 4 5 6

+ 6168 + 12173 + 8955 + 1737

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79

Exponential Asymptotics

{256 + 8192 + 93184 2 + 473088 3 + 1108800 4

+1128960 5 + 423360 6 + 40320 7 + 315 8 } (140)

Fig. 18 shows that for small positive , this simple rational approxima-

tion is on the whole a lot more useful than either the superasymptotic

or hyperasymptotic series.

Chebyshev polynomial approximations are usually polynomials rather

than rational functions and are optimized for a particular line segment

in the complex plane. By computing symbolically, we have obtained an

approximation that is more complicated (because it is rational rather

than polynomial) but has the great virtue of being as accurate, for a

given , as the standard Chebyshev approximation of degree M along

the segment [0, ] even when is complex-valued.

In the previous section, we have already given the ordinary asymp-

totics of the Chebyshev coe¬cients of the Stieltjes function. However,

the comparison of convergent Chebyshev series with divergent power

series is not completely straightforward. The asymptotic series uses a

number of terms N which is inversely proportional to . What happens

if we compare the N -term Chebyshev series on the interval ∈ [0, 1/N ]

with the N -term optimally truncated power series for = 1/N ?

Through an elementary steepest descent analysis of the usual inner

product integrals for the coe¬cients of an orthogonal series,, one ¬nds

that the N -th Chebyshev coe¬cient for the series on [0, 1/N ] is (pre-

viously unpublished)

√

aN ∼ 2.98 N exp(’2.723N )

’1/2

∼ 2.98 exp(’2.723/ ) (141)

(One can show that the error in truncating the Chebyshev series after

N terms is proportional to aN [55, 73].) Intriguingly, the errors for Pad´

e

approximants and for hyperasymptotics are of this same form:

|f ’ fN | ¤ exp(’q/ ), N ∼ O(1/ ) (142)

where the constant q > 0 depends on the precise Chebyshev, Pad´, e

or hyperasymptotic scheme used. There are likely deep connections

between these di¬erent families of approximations which are now only

dimly understood [51].

One can make more entertaining approximations by using other

spectral basis sets. For example, the rational Chebyshev functions are a

good basis set for the semi-in¬nite interval, x ∈ [0, ∞]. Boyd [54] gives

three examples in which the usual pair of series “ divergent series in 1/x

for large x and convergent power series for small x “ can be replaced

by a single expansion over the entire semi-in¬nite range. The examples

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80 John P. Boyd

range from the K1 Bessel function, which has a pole at the origin, to

the J0 Bessel function, in which separate series multiply the sine and

cosine in the uniform approximations, to the ground state eigenvalue

of the quantum quartic oscillator as a function of the coupling constant

, which is a Stieltjes function with a factorially divergent power series

about = 0 [47, 19]. These uniform approximations are much compli-

cated and converge more slowly than the pair of Chebyshev series they

replace, but have the advantage of avoiding a conditional statement,

which is needed in the traditional approach to switch between large

and small x approximations.

Lastly, one must not overlook non-series alternatives. Schulten, Ander-

son and Gordon [277] have developed an e¬cient subroutine to evaluate

the Airy functions at arbitrary points in the complex plane. Instead of

using an asymptotic approximation for large | z |, they use a clever

optimized Gaussian quadrature to directly evaluate the integral repre-

sentations for Ai and Bi, even on Stokes™ lines. Their double precision

code, which is accurate to at least 11 decimal places for all | z | (with

use of the power series about z = 0 near the origin) employs a maximum

of just six quadrature points!

Detailed comparisons between high order hyperasymptotics and oth-

er methods of numerical approximation have not yet been carried out.

Still, the examples and illustrations above show that the comparison,

except perhaps for special cases, is likely to be unfavorable to high

order asymptotics.

Although hyperasymptotics look comparable to Chebyshev and Pad´ e

schemes when N ∼ O( ), the Chebyshev and Pad´ have the profound

e

advantage of converging as N ’ ∞ for ¬xed . Furthermore, these

approximations are built from ordinary polynomials whereas hyper-

asymptotic approximations are series of hyperterminants, which in turn

are approximated by series of hypergeometric functions.

There may be a few exceptions: problems where no alternatives

are available. In quantum chaology, Gutzwiller™s divergent series for

the quantum spectrum has been summed using resurgence[182, 39].

The practical result has been greatly improved energies for quantum

mechanics odd-shaped billiard tables “ idealized but popular for test-

ing theories [182, 39]. Another application is computing the zeros of the

Riemann zeta function, where resurgence has proved to be much better

than the best available competitor, the (un-resurgent) Riemann-Siegel

formula [34].

Berry™s amusing quote,“I am not expecting an early call”, is a frank

admission that most extensions of exponential asymptotics beyond the

lowest nontrivial term are arithmurgically useless. The proper use of

exponential asymptotics is to give insight. A sensible application is to

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81

Exponential Asymptotics

0

10

-2

All Methods ev

10

bysh

he

C

&

tics

-4

10

pto

m

sy Chebyshev Only

-6

er

10 a

p

Hy

-8

10

-10

10

Higher Order

Chebyshev Only

-12

10

-14

10

0 0.1 0.2 0.3 0.4 0.5

µ

Figure 18. A comparison of the rational „ -Chebyshev approximation S8 versus the

superasymptotic and hyperasympotic approximations for the Stieltjes function, S( ).

The three solid curves plot the errors for each of the three methods versus . If

acceptable accuracy for a given is a point in the unshaded region in the upper

left corner, all three methods are satisfactory. In the unshaded lower right region,

none of the three approximations is su¬ciently good (although such tiny errors

< 1 — 10E ’12 can be achieved by simply using a Chebyshev „ approximation of

higher order). The vertical shading “ most of the graph “ shows where the Chebysehv

approximation S8 , a polynomial of degree 7 divided by degree 8, is successful, but

the asymptotic series fail because their minimum error is larger than the required

tolerance. In the vertically-and-horizontally shaded area, both hyperasympotics and

S8 ( ) are successful. Finally, there is a tiny region of horizontal shading where only

hyperasymptotics is successful (though a Chebyshev approximation of higher order

than S8 ( ) would succeed). The hyperasymptotic errors were calculated using the

Berry-Howls scheme (where the errors are O(exp(’2.386/ )), but employing the

more accurate hyperasymptotic methods of later authors such as Olde Daalhuis

would not change the theme of the graph.

compute a small term that is also the leading term to approximate

some crucial feature of a problem, perhaps the lifetime of a quantum

bound state or a nonlocal solitary wave.

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82 John P. Boyd

20. Hybridizing Asymptotics with Numerics

The hyperasymptotic scheme of Boyd [68] and the PKKS method

[171, 172] are both blends of analysis and numerics in the sense that

the ¬nal step, the determination of the proportionality constant which

multiplies the exponential of 1/ , requires a computation. However, the

prior analysis has reduced the problem to a very small calculation that

returns an answer as the product of a number with an analytical fac-

tor. This is far di¬erent than a brute force calculation that requires a

hundred times as much computer time to return only a number.

The ¬‚ow past a sphere or cylinder at small Reynolds number Re[264,

159, 161, 160, 98, 283]. has frustrated ¬‚uid dynamicists for over forty

years, but there has been, very recently, a partial breakthrough by

means of a hybrid numerical-asymptotic method [170]. The source

of pain is that these expansions are double series in powers of Re

and 1/ log(Re) or, de¬ning = 1/ log(Re), in powers of exp(’1/ )

and . Formally, one should include an in¬nite number of logarithmic

corrections to the drag coe¬cient before computing the ¬rst correc-

tion proportional to Re. For the ¬‚ow past the circle, however, Re >

(1/ log(3.70/Re))4 for all Re > 1/12000. (Real ¬‚uid ¬‚ows are typical-

ly at much larger Re.). A systematic scheme for the transcendentally

small terms is still an open problem. For the sphere, which is probably

the easier of the two, Chester and Breach conclude sadly “the expan-

sion is of practical value only in the limited range Re < 1/2 and that

in this range there is little point in continuing the expansion further”.

Kropinski, Ward and Keller [170] made the crucial observation that

if the outer (“Oseen”) problem is solved numerically, the numerical

solution will implicitly incorporate an in¬nite number of logarithmic

corrections. Better yet, the outer solution is independent of whether the

body is a cylinder, an elliptic cylinder, or some other smooth shape: a

single numerical solution provides a good answer to a whole spectrum

of body shapes. The inner solution di¬ers from shape to shape, but

is easy to calculate analytically. They have successfully applied this

same idea, outer-numerical/inner-analytical, to other problems with