Developments in the Double Exponential Formulas
for Numerical Integration
Masatake Mori
Department of Applied Physics, Faculty of Engineering,
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
1. Optimality of the Trapezoidal Rule
The double exponential formula, abbreviated as the DE-formula, was first presented by Takahasi and Mori [18] in 1974 as an efficient and robust quadrature
formula to compute integrals with end point singularity, e.g.
1 =
L
(X-2)(1-X)V4(1+J03/4 >
C1)
or over the half infinite interval, e.g.
OO
e~*logxsinxdx .
(2)
- /
The DE-formula is based Jo
on the optimality of the trapezoidal rule over
(—oo, oo) in the following sense. Consider the integral
oo
/ •00
g(u)du
(3)
where g(u) is analytic over (—00,00) and |g(w)| is integrable. We apply the trapezoidal rule, or equivalently the midpoint rule, to (3) with an equal mesh size
h:
00
h = h X g(kh) .
(4)
k=—00
Then the error of (4) is expressed in terms of a contour integral [16]
Ah = Tr-. I ®h(w)g(w)dw ,
(5)
27C7 Jc
wheretf>/,(w)is called the characteristic function of the error and defined by
+2n
1
<P A (w) = <
/
\„<
^"~ \
; lmw>0
1 - e x p ( — — w)
, h
-%,•
;imW<o
l-exp(+—-w)
(6)
h
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
1586
Masatake Mori
and C consists of two infinite curves one of which runs to the left above the real
axis and the other to the right below the real axis in such a way that there exists
no singularity of g(w) between these two curves. Since Ah is a linear functional
over the family of analytic functions on (—00,00), it can be regarded as a Sato's
hyperfunction [10] and — <t>h(w)/(2%i) is nothing but its defining function.
Although there are infinite number of quadrature formulas for the integral
(3), the trapezoidal rule (4) is proved to be optimal in the following sense. Let an
arbitrary quadrature formula for (3) be
00
k=—00
Then its error is expressed also in terms of the contour integral
AIÄ = I-IÄ
= —[
òA(w)g(w)dw .
(8)
Since |<P>i(w)| usually decays exponentially as |Im w\ becomes large for quadrature
formulas of practical use we define the average decay rate r of | ^ ( w ) | for large
|Imw| as follows:
r= lim ( l i m - — /
\ - — log|# A (w)| \dw ) , w = u + iu.
d->co \R->CC 2R J-RHd
{
dv
J
)
(9)
It is easy to see that the error of numerical integration is smaller if the decay rate
r is larger. Then we have
Theorem (Takahasi-Mori, 1970). Suppose that Ak and ak in IA for I satisfy
k=—00
Then, among quadrature formulas IA whose average density ofak's per unit length
is equal (= vp) to each other, the trapezoidal rule h with equal mesh size h = 1/vp
is optimal in the sense that r attains its maximum
2%
rmax = 27CVp = y
(11)
by the trapezoidal rule.
The decay rate in case of the Simpson's rule is r = rcvp = n/h, so that the
Simpson's rule is as twice inefficient as the trapezoidal rule.
Double Exponential Formula
1587
2. The Double Exponential Formula
Now that the trapezoidal rule over (—00,00) is optimal, we can get a new efficient
quadrature formula by means of a variable transformation. Let the given integral
be
/ = f f(x)dx .
(12)
Ja
The variable transformation
x = c/)(u), (j)(—oo) = a, (p(+co) =b
(13)
leads to
oo
f{<j>{u))^'{u)du .
(14)
•00
Since this is an integral over (—00,00) we apply the trapezoidal rule with equal
mesh size h, which results in a quadrature formula
/
00
h = h x mmwim •
as)
fc=—00
This is an infinite summation and in actual computation we need to truncate the
sum
N+
0
tf =
h
Z nmu'ikh),
(i6)
k=-N-
where N = N- + ÌV+ + 1 is the number of function evaluations. Therefore the
overall error of (16) is
AI^=I-I^=I-h
+ Ih-I^=AIh + 8, ,
(17)
where AI h is the discretization error defined by
oo
°°
/
fi^uMMdu - h £ f www(kh)
=
edithWMMWM'1* '
(18^
and et is the truncation error defined by
N)
-N-
00
st=h - 4 = h x fMkhM'm+h £ mkhwm • a«
fc=—00
k=N+
In general if an analytic function g(w) decays rapidly as Rew -^ ±00, then it
grows rapidly asImw-> ±00, and vice versa. Therefore |zU/,| and \et\ cannot be
made small at the same time and there should be an optimal decay rate of |g(w)|
as Re w —> +00.
1588
Masatake Mori
In order to get an optimal quadrature formula Takahasi and Mori [18]
investigated the efficiency of the formulas based on the three kinds of variable
transformations x — (j)(u) which have the following asymptotic behaviors under
the condition that Alh and et are of the same order of magnitude:
(a) | ^ ( I I ) | « e x p H u D , m =1,2,...
(b) |0'(w)| «exp(-cexp|u|)
(c) |0'(u)| « exp(-cexp \u\m), m = 3,5,....
(20)
(21)
(22)
They found that the optimal decay of \g(u)\ or \f(4)(u))^(u)\ is double exponential,
i.e.
\f(<t>(u))<t>\u)\ ~ exp(-cexp \u\), \u\ -* oo ,
(23)
and the quadrature formula obtained based on this optimal transformation is
. called a double exponential formula, abbreviated as DE-formula.
Specifically, for the integral over (—1,1)
1 = j J(x)dx
(24)
the transformation
x = tanh f - sinh u J
(25)
gives a DE-formula, and for the integral
/•oo
/ = / /i(x)exp(—x)dx
Jo
(26)
the transformation
x = exp(w — exp(—u))
(27)
gives a DE-formula over (0,oo). It is also shown that the asymptotic error of the
formula in terms of the mesh size h of the trapezoidal rule is expressed as
M/*|»exp
-
,
(28)
-§)•
T
and that the asymptotic error in terms of the number N of the function evaluations is
l < V e x p ( - c ^ )
.
(29)
Before the DE-formula was developed a quadrature formula also based
on variable transformation called the IMT-formula had been proposed by Iri,
Moriguti and Takasawa in 1969 [3], which was characterized by the fact that
the original finite interval of integration (0,1) was transformed onto itself. The
asymptotic error behavior of the formula was shown to be
AI{N) « Gxp(-cy/N) ,
(30)
Double Exponential Formula
1589
from which we see that asymptotically the efficiency of the DE-formula is superior to that of the IMT-formula. Mori [6] presented a formula based on the
transformation from (—1,1) onto itself having an asymptotic error behavior
which is slightly inferior to the DE-formula. In 1982 Murota and Iri [8] tried to
improve the IMT-formula by means of parameter tuning and repeated application
of the IMT-transformation and it turned out that, although the efficiency is
improved by the repeated application of the IMT-transformation step by step,
its limit does not attain the efficiency of the DE-formula as shown below:
'
IMT-single : AIm « e x p ( - c ^ )
IMT-double : AI{N) « exp (-e
V
n
(32)
\ ^ \
(33)
(logJV)V
IMT-triple : ,/<»> « exp ( - c ^ - ^ - ^
(34)
DE-formula : AI{N) « exp ( - C p ^ ) •
(35)
3. Analysis of the DE-Formula on Function Spaces
At a research meeting held at the Research Institute for Mathematical Sciences
of Kyoto University in 1985 M.Sugihara presented a detailed theoretical analysis
on the optimality of the DE-formula introducing function spaces for integrands
and his hand-written note on the analysis appeared in [12] in Japanese. Although
he is now preparing a full paper about the details of the analysis, fragrance of
his analysis will be worth while to be given here.
Basically he extended the analysis by Stenger [11] on Hp space to the analysis
on spaces of functions defined not on the unit circle but directly on the real axis
w E (—oo, oo) in the w-plane, where w = u + iv. These spaces are characterized by
the decay of their elements at large |Re w\. First denote the strip domain in the
w-plane
D(d) =A {>f w G ( C |Imw|<|rf}
(36)
sé(D(d)) = {analytic functions on D(d)} .
(37)
and define
He introdeuced a function space
Hdouhk(D(d);A,B)(B<l/d)
= { g G œ?(D(d)) sup {|g(w)| • | exp(,4cosh£w)|} < +oo I
weD(d)
J
(38)
1590
Masatake Mori
with an appropriate norm || g ||double- In short this space is characterized by the
double exponential decay of its elements at large |Re w\. Consider the integral (3)
and an approximation (7) to it in FTdoubie- As to the norm of the error
/•oo
EN=
N
g(u)du - ]T Akg(ak)
• y -°°
(39)
fe=i
Sugihara proved
Theorem 1. In
Hdouhie(D(d);A,B)
inf || EN(a1,...,aN;A1,...,AN)
flfc^eR
||> Cexpf -c-—-)
.
\
log N )
(40)
On the other hand, the following theorem holds for the trapezoidal rule.
Theorem 2. In double (D (d) ; A, B)
|| EN (trapezoidal rule) || < C' exp ( - c ' -—-
) .
(41)
From these two theorems we immediately see that the trapezoidal rule is optimal
i n //double-
Next, in a similar way as Hdoubie> Sugihara introduced another function space
Gingie characterized by the single exponential decay of its elements at large |Re w|,
and showed again the optimality of the trapezoidal rule in ffsingle- However, the
inequality for the error in Hsingie corresponding to (41) is
EN (trapezoidal rule) ||< C'exp j —Vn2dAy N + - j ,
(42)
where A is some constant, so that it is clear that the double exponential transformation asymptotically leads to a more efficient quadrature formula than the
single exponential one. Then it is quite natural to raise a question: is the trapezoidal rule more efficient in a space whose elements decay more rapidly than
those in #doubie? Sugihara answered the question negatively by proving that there
exists no element except zero function in such a space. Consequently we conclude
that the DE-formula is asymptotically optimal.
4. The DE-Formula for Slowly Decaying Oscillatory Integrals
Consider the integral
Jo
./o
f(x)dx
,
. (43)
where
f(x) = fi(x) cosx,
/i(x) = algebraic function .
(44)
In this case f(<j>(w))^(w) does not belong to ifdoubie, so that the DE-formula does
not work well as seen from the analysis in the previous section. Toda and Ono
Double Exponential Formula
1591
[19] applied the DE r formula followed by the Richardson's extrapolation method
successfully to
/»OO
I = lim / f(x) exp(-zx)dx ,
(45)
Afterwards Sugihara showed both theoretically and experimentally [13] that the
Richardson's method is more efficiently applied to
00
,2
= lim / f(x) cxp(-zxl)dx
.
(46)
Recently Ooura and Mori [9] presented an interesting transformation which
gives an efficient formula for integrals such as
•O
/O
I
=
/ /lW
Jo
sul
wxdx
(47)
cos
o)xdx.
(48)
/•oo
J = / /iW
Jo
Consider the variable transformation
Mu
x = Mò(t) =
, rjr . 1 x , M,K = constant .
1 — exp(—K sinh w)
Then (j)(u) satisfies cj) (—oo) = 0 and 0(+oo) = oo. Moreover,
lim ò(u) = u double exponentially
(49)
(50)
M->+00
lim (j)'(u)=0
double exponentially
(51)
hold. If we apply the variable transformation x = M(j)(u) to (47) and compute it
by the trapezoidal rule, we have
h = Mh £
/1(M0(/c/i))sin(coM(/)(/c/i))^/(/c/i) .
(52)
k=—oo
Choose h such that œMh = 2%, then
sin(coM(j)(kh)) ~ sinmMkh = sin2nk = 0 ,
so that we can truncate the sum (52) at some moderate value of fc.
(53)
1592
Masatake Mori
5, Problems Arising in Coding Automatic Integrator
When you write an automatic integrator based on the DE-formula you must be
careful about the loss of significant digits which may occur when computing, say,
(1 -fx)~3/4 in (1) in the close neighborhood of x = —1, and about the overflow.
A device to avoide the loss of significant digits is given in [18].
Recently a useful method to avoide the overflow which may occur when
computing the weights of the DE-formula was presented by Watanabe [21].
Consider again the integral
/ : •L
f(x)dx .
(54)
The weights of the DE-formula obtained by
x = (j)(u) = tanh(— sinh u)
(55)
are
coshkh
, ^ ,, , „
4k = cosh
, 22(f
„ sinhkh)'
• , ,,.» k = °> ±1, ±2,-.- ,
,^
(56)
and a careless coding often gives rise to the overflow when computing the
denominator in (56) because it grows double exponentially as k becomes large.
Watanabe found a recurrence relation
Ak+i =Akxrk
,
(57)
where
rk =
cosh h + sinh h tanh kh
j
(coshs/c + (/)(kh) sinhsfc)
_ox
(58)
and
h
1
sk = n sinh - cosh((fc + -)h) ,
(59)
and showed that the integral (54) can be computed by the following small code:
1=0
DO 10fc= N, 1, - 1
I = (I + f(ak) + f(-ak)) x rk-\
10 CONTINUE
I = y ( I + /(£*>))•
Although the denominator of rk has a double exponential factor sinh s^ its inner
exponential factor has a small coefficient sinh(/z/2), so that the overflow in rk will
not occur untilfcbecomes much larger than suchfcfor which the overflow occurs
in Ak.
Double Exponential Formula
1593
6. Applications of the DE-Formula
The DE-formula is used for multiple integration. See [14] and [1], and the
references therein.
The DE-formula is installed in many computer centers in Japan and is easily
found in subroutine packages in the Japanese market. It is actually used in
various fields of science and technology. In the references one paper is listed
from each of the fields, the boundary element method [2], the suface charge
method [20], filter analysis[4], and molecular chemistry [5]. Very recently also in
the field of statistics it is proved to be quite efficient for numerical evaluation of
risk of improved estimation [15]. For further reference see these papers and the
references therein.
References
1. V.U. Aihie, G.A. Evans: A comparison of the error function and the tanh transformation as progressive rules for double and triple singular integrals. J. Comput. Appi.
Math. 30 (1990) 145-154
2. T. Higashimachi, N. Okamoto, Y. Ezawa, T. Aizawa, A. Ito: Interactive structural
anaysis system using the advanced boundary element method. Boundary Elements
— Proceedings of the Fifth International Conference, Hiroshima, Japan, November
1983, eds. Brebbia, C. A., Futagami, T., Tanaka, M., A Computational Mechanics
Center Publication. Springer, Berlin Heidelberg New York, pp. 847-856
3. M. Iri, S. Moriguti, Y. Takasawa: On a certain quadrature formula. J. Comput. Appi.
Math. 17 (1987) 3-20 — translation of the original paper in Japanese that appeared
in Kokyuroku, RIMS. Kyoto Univ. no. 91 (1970) 82-118
4. T. Kida, T. Kurogochi : A simple numerical derivation of group delay and impulse
response from the prescribed characteristic function of a reactance low-pass filter
(in Japanese). The Transactions of the Institute of Electronics and Communication
Engineers J63 (1980) 421-428
5. T. Momose, T.Shida: Efficient formulas for molecular integrals over the Hiller-SucherFeinberg identity using Cartesian Gaussian functions: Towards the improvement of
spin density calculation. J. Chem. Phys. 87 (1987) 2832-2846
6. M. Mori: An IMT-type double exponential formula for numerical integration. Pubi.
RIMS, Kyoto Univ. 14 (1978) 713-729
7. M. Mori: The double exponential formulas for numerical integration over the half
infinite interval. Numerical Mathematics Singapore 1988, International Series of
Numerical Mathematics, vol. 86. Birkhäuser, Boston 1988, pp. 367-379
8. K. Murota, M. Iri: Parameter tuning and repeated application of the IMT-type
transformtaion in numerical quadrature. Numer. Math. 38 (1982) 327-363
9. T. Ooura, M. Mori: A quadrature formula for oscillatory integrals over the half
infinite interval based on variable transformation (in Japanese). Kokyuroku, RIMS
Kyoto Univ. no. 717 (1990) 68-75
10. M. Sato: Theory of hyperfunctions, I. J. Fac. Sci. Univ. Tokyo 8 (1959) 139-193
11. F. Stenger: Optimal convergence of minimum norm approximation on Hp. Numer.
Math. 29 (1978) 345-362
12. M. Sugihara: On the optimality of the quadrature formula based on the DE-type
variable transformation (in Japanese). Kokyuroku, RIMS Kyoto Univ. no. 585 (1986)
150-175
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Masatake Mori
13. M. Sugihara: Methods of numerical integration of oscillatory functions by the DEformula with the Richardson extrapolation. J. Comput. Appi. Math. 17 (1987) 47-68
14. M. Sugihara: Method of good matrices for multi-dimensional numerical integrations
— An extension of the method of good lattice points. J. Comput. Appi. Math. 17
(1987) 197-213
15. N. Sugiura: Private communication
16. H. Takahasi, M. Mori: Error estimation in the numerical integration of analytic
functions. Rep. Comput. Centre Univ. Tokyo 3 (1970) 41-108
17. H. Takahasi, M. Mori: Quadrature formulas obtained by variable transformation.
Numer. Math. 21 (1973) 206-219
18. H. Takahasi, M. Mori: Double exponential formulas for numerical integration. Pubi.
RIMS Kyoto Univ. 9 (1974) 721-741
19. H. Toda, H. Ono : Some remarks for efficient usage of the double exponential formulas
(in Japanese). Kokyuroku, RIMS, Kyoto Univ. no. 339 (1978) 74-109
20. Y. Uchikawa, T. Ohye, K. Gotoh: Improved surface charge method (in Japanese).
The Transactions of the Institute of Electrical Engineers of Japan A 101 (1981)
263-270
21. T. Watanabe: On the double exponential formula for numerical integration (in
Japanese). Kakuyugokenkyu 63 (1990) no. 5, 397-411