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Electronic Transactions on Numerical Analysis.
Volume 20, pp. 104-118, 2005.
Copyright  2005, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
QUADRATURE OVER THE SPHERE
KENDALL ATKINSON AND ALVISE SOMMARIVA Abstract. Consider integration over the unit sphere in , especially when the integrand has singular behaviour
in a polar region. In an earlier paper [4], a numerical integration method was proposed that uses a transformation
that leads to an integration problem over the unit sphere with an integrand that is much smoother in the polar regions
of the sphere. The transformation uses a grading parameter . The trapezoidal rule is applied to the spherical
coordinates representation of the transformed problem. The method is simple to apply, and it was shown in [4] to
have convergence or better for integer values of . In this paper, we extend those results to non-integral
values of . We also examine superconvergence that was observed when is an odd integer. The overall results
agree with those of [11], although the latter is for a different, but related, class of transformations.
Key words. spherical integration, trapezoidal rule, Euler-MacLaurin expansion
AMS subject classifications. 65D32
1. Introduction. In the earlier paper [4] a quadrature method for the sphere was introduced to deal with an integrand that is singular at either the north or south pole of the sphere.
The present paper addresses some of the conjectures that were left unanswered in that earlier
paper.
The earlier paper studied the more general problem of quadrature over a smooth surface
,
(1.1)
in which
(1.2)
!#" $
&%
is the image of a smooth mapping defined on the unit sphere ')(+*, ,
-
./'1795038;246 :0 7
Using this mapping the quadrature problem reduces to that of integration over ' ,
(1.3)
#<&=
?>
<@" $
%
and that is the case we address here. We assume < is several times continuously differentiable
over the unit sphere ' , with the precise order of differentiability to be specified later. In the
following section we define the numerical method and we give the main results of the paper.
Subsequent sections deal with the proofs of those results.
This problem has also been studied by A. Sidi [11], and some of our tools are closely
related to those used in his paper. In [11] Sidi develops a class of single variable transformations to improve the behaviour of the integrand in (1.3). This class is denoted as the “extended
class ACB ”, or “class AB ” for short, and it is an extension of that developed earlier in [9]. A
B
particular member of this class that is studied in [11] is the D9EGF -transformation, and the numerical examples there are done with this transformation. Some of the tools used in Sidi’s
paper [11] are similar to ones we use, although there are differences as well because our
transformation does not belong to the class he addresses. The overall asymptotic error results
H
Received June 8, 2004. Accepted for publication March 15, 2005. Recommended by F. Stenger.
Depts of Mathematics and Computer Science, University of Iowa
Dept of Pure and Applied Mathematics, University of Padua. Supported by the research project CPDA028291
”Efficient approximation methods for nonlocal discrete transform” of the University of Padua.
104
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QUADRATURE OVER THE SPHERE
105
that we give are, in the end, the same as his, even though the underlying transformations are
different. Our results are not as complete as those of Sidi, due in part to the lack of needed
mathematical tools as compared to those developed in [11] for the class A B transformations
analyzed there.
2. The numerical method. In spherical coordinates this integral (1.3) can be written as
#<&
JI M
L9I
K K <NOQPRD?STD9EGFVUXWYDZEF[STDZEFVUXW9OQPRD U/XD9EGF\U=$]S[$RU
Rather than approximating this integral directly, we begin by introducing a transformation
^
./' 79508;2&6 :0 7 ' . With respect to spherical coordinates on ' ,
(2.1)
^
b .R"_`OP/D?STDZEFVUXW9D9EF[S[D9EGFVU?W9OP/D UR6c
a "d
OQPRD?STDZh EFeU?W9D9EGF[STDZEFfegU?W9OP/D UR
SCW9UR
L
OP/D L U[iMDZEF egU
jlk
In this transformation, monqp is a ‘grading parameter’. The north and south poles of ' remain
fixed, while the region around them is distorted by the mapping.
The integral f<& becomes
#<&=
(2.2)
v w r "|b t
with
(2.3)
X>
^
[{
<sr "b utovRwxr "yb z
t $ %
^
the Jacobian of the mapping ,
v wxr " b t `} ~!
SWZUR\€s~! # SWZU/Q}R
k
k
L
L
DZEF e 42 0 U‚mƒOQPRD L UTiJD9EF UR„
L
‚ DZEF egUTiMOPRD L U „f…†
In spherical coordinates,
(2.4)
I ZD EF L e U‚#m@OP/D L UTiMD9EGF L UR„ LYI
&2 0
#<&= K
L
K <o‡ ZW ˆgWY‰RV$]S[$RU
‚ DZEF egU[iJOQPRD L U „ …†
OQPRD?STD9h EGFfe&UXW9D9EGF[STD9EGFfeU?W9OP/D RU Š‡ W9ˆW3‰
L
D9EGF eUTiMOPRD L U
For ‹Œnqp , let Žl‘‹ , and
S ’z“UQ’T•”/
For a generic function – , introduce the bivariate trapezoidal approximation
8 ˜L 8
I 9L I
L ˜
™
™
K K &– D9EGFVU?W9OP/D UXWYDZEF[SW9OP/D?Sg $]S[$RUN—l ™ š 3K › › ’ š K › › –&D9EGF\U WYOPRDXU W9D9EGF[S ’ W9OQPRD?S ’ in which the superscript notation œ œ means to multiply the first and last terms by L0 before
summing. Apply this to (2.4). Note that the integrand is zero for U“žXW9 and that the
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106
K. ATKINSON AND A. SOMMARIVA
integrand has period Ÿ  in S . Therefore
(2.5)
I YL I
K K g– #DZEFVUXWYOPRDXUXW9D9EGF[SWYOPRDfSX$/ST$U
8˜ ˜ L 8
2&0
L
8
—¡
–g#DZEF\U ™ W9OP/D U ™ W9D9EF[S ’WYOP/D?SX’¢
™š ’ š
j“£
0 0
L
L
D9EF e 2&0 U ‚ m@OP/D L UTiMDZEF U „
<NŠ‡ W9ˆgWY‰
–&D9EGFVUXWYOP/D UXW9D9EF[SCWYOP/D?Sg
L
‚ D9EF eUTiMOP/D L U „ …†
#‡ WZˆgW3‰ as in (2.4).
When Ÿ m is an integer, we were able in [4] to show an accelerated rate of convergence
for this numerical integration of (1.3), as follows.
T HEOREM 2.1. For the grading parameter m in the integral (2.4), assume m¤ndp and Ÿ m
is a positive integer. Introduce ¥
with
¥
§¦ Ÿ Ÿ m]mVW i“pRW¨Ÿ Ÿ mm
even
odd
Assume < is -times differentiable with <ƒ©«ªQ¬­
0 ®'y , the space of Lebesgue integrable
k by (2.5) satisfies
functions on ' . Then the error in approximating (1.3)
(2.6)
5
£
8 l¯+# ª We left some questions unanswered in the earlier paper and two of those are addressed
in this paper.
° First, what happens when Ÿ m is not an integer.
° Second, when Ÿm is an odd integer, what is the actual rate of convergence? We
observed in [4] a much faster rate of convergence in such a case.
The most important tool used in understanding both of these questions is the Euler-MacLaurin
expansion (e.g. see [3, p. 285], [10, Appendix D]) and its generalization in Lyness and
Ninham [6]. A modification of the latter is used in answering the first question given above,
and the regular Euler-MacLaurin expansion is used in exploring the second question. We
¥ present theorems that generalize the above Theorem 2.1, demonstrating ¥them in later sections.
¥
T HEOREM
2.2. Assume the grading parameter m satisfies p²±1mJ±³Ÿ , mlµ
´ p¶«· . Let
¸p¹i)º Ÿm¢» , with º Ÿm¢» denoting the integer part of Ÿm . Assume < is -times differentiable
with all th -derivatives of < belonging to 0 'y . Then
(2.7)
5
k
£
8 q¯ ‚  L e „
After giving a proof in Section 3, we indicate how the theorem may be extended to other
¥
larger non-integral values of Ÿ m . We further note that this theorem corresponds
to Theorem
¥ 4.3 in [11], although the latter ¥ is for a different class of transformations.
T HEOREM 2.3. Assume ms¼p¶«· or m½¾Ÿ ¶«· or m½À¿?¶ · and let ÂÁRm . Assume < is
-times differentiable with all th -order derivatives of < belonging to 0 'y . Then
(2.8)
5
£
8 q¯Ã‚?Ä e „
k
Again, following the proof in Section 4, we indicate how the theorem can be extended to
other cases in which Ÿm is an odd integer. This theorem also agrees with Theorem 4.3 in [11]
for the transformations covered there.
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QUADRATURE OVER THE SPHERE
¥
107
We remark on the differentiability assumptions about < over ' . Suppose < is a function defined on only ' , and suppose all derivatives of < of order Å
, with respect to local
coordinate systems on ' , are continuous. Then it is known that < can be extended to some Æ neighborhood of ' with preservation of the differentiability. In the following theory, without
loss of generality, we assume that the integrand < is defined on an Æ -neighborhood of ' for
some ÆÈÇɝ . Thus we treat <ȇ W9ˆW3‰ as a differentiable function of three variables, not two.
8 into two portions:
As in [4], we decompose the calculation of the error 5
I 9D EGF L e U m@OPRD L UTiMDZEF L U L9£ I
„
42 0 ‚
8 K
L
K <NŠ‡ W9ˆgWY‰¹$/ST$U
£
‚ D9EGF eUTiMOP/D L U „ …†
8˜
L
L ™ L9I
L ™
™
™ ™ ™
5  240 D9EGF e 240 U ‚ m@OQPRD U iMD9EGF U „
L ™
K <Ȋ‡ WZˆ WY‰ \$/S
L
™š
™
‚ D9EGF eU iJOQPRD U „ …†
(2.9)
8˜ 0 L
L
L
240 D9EGF e 240 U ™ ‚ m@OQPRD U ™ iMD9EGF U ™ „
i|
L
™š
‚ D9EGF eU ™ iJOQPRD L U ™ „ …†
0
LYI
˜L 8
™
™
™
5
€ÂÊË K <Ȋ‡ WZˆ W3‰ Ì$]S 
<ȇ ™¢Í ’ W9ˆ ™¢Í ’ W3‰ ™¢Í ’ ÏÎÐ
š’
0
The last portion is the trapezoidal error for a periodic integral over º XWYŸ g» , and it is
straightforward to deal with if < is assumed sufficiently differentiable, obtaining the correct
8 . More precisely, with respect to the integration variable S ,
order of convergence for 5
£
the integrand is a smooth differentiable
periodic function over º ?WYŸ g» . In the remainder of
f<& 5
this paper, we consider only the first portion of the error, that of the trapezoidal rule applied
over NŕU¤Åɏ .
3. Convergence with Ÿ m non-integral. The key tool we use is a modification of a result
of Lyness and Ninham [6]. The proof of the modification (Lyness [7]) is based on recent techniques developed in Monegato and Lyness [8]. We use mostly the notation of [6], specializing
the results in it to our situation. Consider approximating the integral
N
0 ~ Ñ?$RÑ
K x
with ~xÑ? having the form
~xÑ?ғÑ?Ó[9p 5 Ñ?ZÔTŠÑ?
(3.2)
“Ñ?ÓÕ K ŠÑ?`up 5 Ñ? Ô Õ Ñ?
0
with M±1ÖCW ×رÙp . We assume CÑ? is Ú -times continuously differentiable on º XWQpQ»
some Úµn¡p . Consider the error in the trapezoidal rule applied to ,
B
p ˜
”
(3.3)
Û B Ü
~ÞÝ ÜÉß 5 K 0 ~àŠÑ?ƒ$Ñ
’ š K ››
ã
Then
ã
˜ã Õ K © ¬ R ‰¹ 5 Ö 5 ä Û B âá
š K äå ã æ ã Ü Óèç ç 0
ã
˜ã 5 p‘ Õ © ¬ up‘ ‰V 5 × 5 ä (3.4)
iéá
iJêr Ü 2 © Ó¢ç ç 0 ¬ t!iJêr Ü 2 © Ô ç ç 0 ¬ t
æ Ü ç ç
äRå 0
š K
á
á
Ô
0
(3.1)
for
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108
K. ATKINSON AND A. SOMMARIVA
In this we use the Riemann zeta function ‰X ä .
From (2.4), the integrand for our application of (3.4) is the function
(3.5)
9L I
L
L
~xŠÑ?`D9EFŠÑ C3 e &2 0ë ‚ D9EGF ŠÑ C „ K N
< Š‡ W9ˆW3‰o$/SW
¤ÅMÑsÅÃp
where
9 p 5 m ]ì4iMm
Šì e 5 ì4ilp¢ …†
OP/D?STDZEFfî eíÑ C&W9D9EF[S[D9EGFeíŠÑ C 4W9OP/D4ŠÑ C3
‡ WZˆgWY‰R
L
D9EF e@#Ñ CCiMOP/D L Ñ C
ë ìZ
and with < assumed sufficiently smooth on an open neighborhood of the unit sphere.
Let ï be the fractional part of Ÿm ,
ïy¡Ÿm 5 º Ÿ m¢»W
¤±Éïȱ¡p
Rewrite our integrand as
LYI
L
ZD EFŠÑ C ð
L
ð
e
~xŠÑ?lÑXð=Zp 5 Ñ? ð Ý
#
9
D
E
|
F
#
Ñ
C

Y
#
‚
9
D
G
E
F
Š
Ñ
C

3
„
2
&
2
0
ë
K <Ȋ‡ W9ˆgWY‰È$]S
Ñup 5 ?Ñ ß
lÑ ð Õ K ŠÑ?`up 5 Ñ? ð Õ Š Ñ? ¥
0
K
We must show that the function Õ ŠÑ? is -times differentiable with Õ K Ñ? locally integrable
about ѽd ; and similarly with respect to Õ Ñ? about ÑsÂp . We treat separately the cases
0
of p|±ÉmȱqpR¶ · and p¶«·o±+mȱ+Ÿ .
3.1. Case 1: py±Émȱ¡pR¶ · . We have º Ÿ m¢»qŸ and ŸmzqŸ¹iJï ; and then
L9I
ð
L
9
D
G
E
F
Ñ
C

5
~xŠÑ?Ì+ÑXð=9p Ñ? ð Ý
D9EGFÑ C ë ‚DZEF #Ñ CY„ K <ȇ W9ˆW3‰o$/S
Ñy9p 5 Ñ? ß
+Ñ ð Õ K ŠÑ?Ò)Zp 5 Ñ? ð Õ Ñ?
0
where
YL I
L
9D EFŠÑ C ð
Õ K ŠÑ?Ò)Zp 5 Ñ? ð Ý
Z
D
E
F
Š
Ñ

C
9
D
E
F
Š
Ñ
C

< #‡ WZˆgWY‰R$]S
‚
„
ë
K È
Ñup 5 ?Ñ ß
and similarly so for Õ
0
ŠÑ?¶
Note that
D9EFŠÑ C nɏíW
Ñup 5 ?Ñ NÅÉѽÅqp
and then easily
Ý
is analytic on
º XWèp» .
9D EGFÑ C ð
Ñ|9p 5 ?Ñ ß
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109
QUADRATURE OVER THE SPHERE
To obtain an error of size
The error formula becomes
¯ ã ‚Ü 2 Le „
, as asserted in (2.7), we must take Úñ_¿ in (3.4).
ã
˜ã L Õ K © ¬ R ‰V 5 ï 5 ä Û)
ã Ü ð9ç ã ç 0
š K ä å
(3.6)
ã
˜ã L 5 p¢ Õ © ¬ up‘ ‰V 5 ï 5 ä i•¯ ‚ Ü 2 , „
i
äRå 0
Ü ðZç ç
š K
0
L
Ü
Using this to show Û)¡¯d‚ 2 e „ requires:
1. Õ K /=l , Õ Zp¢=¡ ;
0
’
­lò L in neighborhoods of ѕ؝ and ÑJÙp , respectively, and Õ K © ¬ /o
2. Õ K WZÕ
’© ¬ 0
Õ up¢=¡ for ”È)pWYŸ ;
0
3. Õ K ©G,Y¬ WZÕ ©G,Y¬ ­
0 in neighborhoods of Ñ!¡ and юdp , respectively.
0 is immediate.
k
The first condition
We give arguments for only Õ K ìZ , but analogous arguments hold for Õ ŠìZ . To find the
0
derivatives of Õ K ŠÑ? , we must differentiate the product
DZEFyÑ C ð
up 5 Ñ? ð Ý
DZEFyÑ CfCÑ?uóJÑ?
(3.7)
Ñup 5 Ñ? ß
where
L
CÑ? ë ‚ D9EGF Š Ñ C Y„
LYI
óJŠÑ? K <NŠ ‡ ZW ˆgW3‰$]S
(3.8)
We need to consider the behaviour of the derivatives around the endpoints of
first two terms in (3.7),
º XWèp» . Since the
9D EFŠÑ C ð
up 5 Ñ? ð Ý
Ñyup 5 Ñ? ß
are analytic in a neighborhood of ю“ , we need consider only the derivatives of the product
(3.9)
D9EFŠÑ C?ŠÑ?ZóJŠÑ?
Recall
ë ŠìZÒ
u p 5 m ì4iJm
ì e 5 Cì ilp¢ …†
We need the derivatives of
L
ŠÑ?Ò ë ‚DZEF Ñ C3„[ ë ‚ L0 ºp 5 OPRDC#Ÿ;Ñ Cu»Š„
Use the following to do so.
(3.10)
–gÑ? L0 ºôp 5 OP/D4#Ÿ Ñ
ë Š –TÑ?YgW
ë œ õ–|ŠÑ?Y/– œ ŠÑ?
L
L
L
ë © ¬ õ–|ŠÑ?Y&õ– œ ŠÑ?Z i ë œ õ–zÑ?9]– © ¬
L
L
ë ©,9¬ æ Š– œ , iM¿ ë © ¬ æ – œ æ – © ¬ i ë œ æ –
L
L
ë © Ä ¬ æ Š– œ Ä iMö ë ©G,9¬ æ Š– œ æ – © ¬
L
L L
L
iJ¿ ë © ¬ æ rQ– © ¬ t iJÁ ë © ¬ æ – œ æ – ©G,Y¬ i ë
ŠÑ?Ò
 œ ŠÑ?Ò
L
 © ¬ ŠÑ?Ò
 ©G,Y¬ ŠÑ?Ò
 © Ä ¬ ŠÑ?Ò
C®»
ŠÑ?
©G,Y¬
œ æ – ©Ä ¬
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110
K. ATKINSON AND A. SOMMARIVA
These are special cases of the Faa di Bruno [15] formula for the ‹
function:
~
8
˜
ë õ–&ŠÑ?9
(3.11)
with
÷
th
derivative of a composite
™
‹ å
÷ å ÷ 8 å ‚ ~xø ø ë „ õ–&ŠÑ?Z
™
™
™ù
™3ú
0 æQæèæ
8
~ L &– ŠÑ? †
~ –gÑ?
~!–gÑ?
Ý
Ý
€ Ý
p å ß
Ÿ å ß æèæQæ
‹ å ß
a multi-integer satisfying
÷
÷
÷ 8
÷
÷ ` ÷ 0 Wè¶Q¶Q¶QW &÷ W 8 all ’n•
÷ } } ÷ 0 L i æèæQæ i ÷ 8
i•Ÿ i æèæQæ iJ‹
l‹
0
Next, in general for an integer ûzÇɝ ,
ü
– ô© üϬ Ñ?Àý¸þ L0 ϟ;C ü ZD EFyŸ &Ñ?gWÿû
Ï
;
Ÿ
C

O P/D4#Ÿ &Ñ?gW¸û
þ L0
For the function ë
odd
even
ìZ ,
0 ì e 2&0 iJì Üä ìZ#ì e Rì5 ì4
il äèp¢Ü † ê;ê‘ìZVìZ/‹
L ì e 2 L iJì Üä ìZRì
äèÜ ê;ê‘ìZVìZ/‹
L¬
©
ë ìZ
ë œ ìZ
ì e 2&0
ìe2 L
#ì e 5 4ì ilp¢ †
L
Since m|)pƒi L0 ï for the current case, we have that ë © ¬ ŠìZ is singular at ìíl . For a general
m not an integer,
äèÜ ê;ê‘ìZ VìZ /‹!ì e ™
™ ì e 2 ™ i ì Üä ìZ /ì
™
÷
2 W
™
(3.12)
n p
Ã
ë © ¬ ŠìZ
L
L
ì e 5 ì4i“p‘ © ç ,Y¬
Combining these results for derivatives of – and ë , we have
L
L
 œ Ñ?Ò¡¯)rD9EGF&Ñ?9 © e 2&0 ¬ ç 0 ts¡¯)rR#D9EGFy#&Ñ?9 e 2&0 t
¡¯ r D9EGF&Ñ?9 0 4ç ð t
L © e 2 L ¬ ç L Í L © e 2&0 ¬ t½q¯_r/D9EGF&Ñ?3 L e 2 L t
r D9EGF&Ñ?9 ð t
L © e 2 ,Y¬ ç , Í L © e 2 L ¬ ç 0 Í L © e 240 ¬ ç 0 t l¯ r #D9EF|#&Ñ?Y L e 2 , t
r D9EGF&Ñ?9 L
 © ¬ Ñ?Ò¡¯)rD9EGF&Ñ?9
(3.13)
¡¯
 ©G,9¬ Ñ?Ò¡¯
¡¯ r D9EGF&Ñ?9 2&0 ç4ð t
Note that 5 p|± 5 p;i|ïo±É . Thus 4©G,Y¬ŠÑ? is integrable over º XWèp» . Also, C©
L
L
on º ?WQp» with 4© ¬#R=q&© ¬up‘=“ .
What are the derivatives of
óJŠÑ?Ò
9L I
K <NŠ‡ W9ˆW3‰È$/S
L ¬ŠÑ?
is continuous
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111
QUADRATURE OVER THE SPHERE
where
OQPRD?STDZEFfeíÑ C&WYDZh EF[S[D9EFfe@Ñ C4W9OQPRD4Ñ C9
ŠÑ?
L
L
r OPRDfS[D9EGF 0 ç4ð ŠÑ C4W9D9EGF[STD9EGF 0 ç4ð ŠÑ C4W9OP/D4ŠÑ C t
h
ŠÑ?
L
L
L
L
ŠÑ?¡DZEF e@ŠÑ CƒiJOQPRD Ñ C¡D9EGF ç4ð Ñ CiMOPRD #Ñ C
Š‡ WZˆgW3‰
By the chain rule,
YL I
$‡
$ˆ
<
i•< L
i•<
K
$Ñ
0 $RÑ
To obtain the behaviour as a function of Ñ , we use
$R‡
Ê $Ñ Î lyDZEFfe Ñ C mƒh OQPRD4#Ñ C
240
Ë $ˆ Ð
Ñ?
$Ñ
L L
5
5 D9EFŠÑ CXD9EF#Ÿ;Ñ C h mƒD9EF e 2 ŠÑ C
Ÿ Ñ?
Ñ?
OP/D L Ñ C
$]‰
p
5 yDZEFŠÑ C h
i
$RÑ
ŠÑ?
ó œ Ñ?Ò
$/‰
, $RÑ
# ! "
$]S
p
OPRDfS
DZEF[S
L L
m@D9EGF e 2 #Ñ C 5 p
h
Š Ñ?
Ñ?
For our case of ŸmzqŸ¹iJï ,
"
$R‡
m DZEF ð ŠÑ C 5 p
m h OQPRD4Ñ C 5 DZEFyÑ C?DZEFyŸ Ñ C h @
Ê $Ñ Î +D9EGF ð L Š Ñ C @
Ë $Rˆ Ð
ŠÑ?
Ÿ ŠÑ?
Š Ñ?
$Ñ
L
l¯ r DZEF ð Š Ñ C t O ZD PREF[DfS S
OQPRD L Ñ C m@D9EGF ð Ñ C 5 p
p
$/‰
5
h
h
yDZEFŠÑ C
i
$RÑ
ŠÑ?
Ñ?
ŠÑ?
l¯+D9EFŠÑ C3
# # For the second derivative,
"
L
L
L
YL I
Í Ý $‡ ß i•< L Í L Ý $Rˆ ß i•< Í Ý $]‰ ß
®
<
K
, , $RÑ
$RÑ
0 0 $RÑ
$‡ $ˆ
$‡ $]‰
$ˆ $/‰
i•Ÿ< Í L
i•Ÿ< Í ,
i•Ÿ< L Í ,
$Ñ $RÑ
0 $RÑ $Ñ L 0 $RÑ L $RÑ
$ L‡
$L ˆ
$ ‰
i|<
i•<
i•< , L $]S
$Ñ L
$Ñ
0 $RÑ L
For the present case that py±+mo±qp¶«· , we can continue with this to show that
ÍK
L
ó © üϬ Ñ?q¯dr/º D9EGFÑ C®» ð 2 ü ç 0 tÈW
û¹¡XWèpWYŸXW9¿
(3.14)
L
ó © ¬ Ñ?
"
OQPRD?S
D9EGF[S
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112
K. ATKINSON AND A. SOMMARIVA
L
to show the singular nature of óJ© ¬ and ó ©G,Y¬ .
Now to (3.7), we calculate the derivatives of the product
D9EGFÑ CfŠÑ?ZóJŠÑ?
Using Leibniz’s formula,
(3.15)
$ ü
º DZEFŠÑ CfCÑ?uóJÑ?Ï»C
$RÑ ü
%
% % % %
$
%
%
%
š
$
ø $fø ü
˜
$
$
ù
ûå
$ ù
$ †
$
å L å å $RÑ º D9EF|#Ñ Cu» $RÑ º ŠÑ?Ï» $RÑ … º óJÑ?Ï»
†
,
0
where À W L W , ’ nÝ . There are
,
sider û[lŸXW9¿ . 0
° û[lŸ . The corresponding values of
%
L0 û=ilp¢gûiMŸR
$
$
terms in this expansion. Con-
are
Ÿ WYXW9/WèZpWQpRW9/?W¢ZpRW9?WQp¢fW¢XW3Ÿ WYRWè#XWèpWQp‘fWè?W9?WYŸ
° û[“¿
. The corresponding values of
(3.16)
%
are
¿XWYXWYRfWèŸ WQpRW9/Wè#ŸXW9?WQp¢fW¢upW3Ÿ WYRWèZpWèpWQp‘W
9pWYXW3ŸfWè#XW9¿?W9/Wè?WYŸXWQp¢fW¢XWèpW3ŸWè#XWYXW9¿/
It is easily checked that all product terms in (3.15) with û¤§Ÿ are continuous, even though
ó © L ¬ŠÑ? is singular.
™
For ûzd¿ , we need to look only at terms containing ©G,Y¬ŠÑ?Vº §?W9¿?W9R®» or ó © ¬ŠÑ? ,
÷
ٟXW9¿º upRW9?WYŸfW¢XWèpW3ŸW¢?W9XWY¿R®» , in order to check for integrability. It is easily
checked that in these cases the corresponding terms in (3.15) are integrable. Thus Õ K ©G,Y¬ ­
0
in a neighborhood of ѽÝX¶ An exactly analogous proof works in examing the behaviourk of
Õ Ñ? about Ñ _p . This completes the proof of Û_q¯ ‚ Ü 2 L e „ for the case of py±ÉmN±¡p¶«· .
%
%
0
3.2. Case 2:
p¶«·o±+mȱ+Ÿ
. We have
º Ÿ m¢»¡¿
and Ÿmz¡¿[iJï ; and then
LYI
9D EGF#Ñ C ð L
L
5
ð
~xÑ?lÑ ð u p ?Ñ Ý
D9EF #Ñ C ë ‚ DZEF Ñ C „ K <ȇ WZˆgWY‰Ro$]S
Ñup 5 ?Ñ ß
lÑ ð Õ K Ñ?)Zp 5 Ñ? ð Õ Ñ?
0
L9I
L
DZEFyÑ C ð L
9
D
G
E
F
Š
Ñ
C

‚
9
D
G
E
F
Š
Ñ
C

3
„
Õ K ŠÑ?Ò)Zp 5 Ñ? ð Ý
ë
K <Ȋ‡ W9ˆW3‰È$/S
Ñup 5 Ñ? ß
and similarly for Õ ŠÑ?¶
0 of size
To obtain an error
becomes
¯_‚ Ü 2 L e „
ã
, we must take
Ú
ÂÁ
in (3.4). The error formula
ã
˜ã , Õ K © ¬ / V
‰ 5 ï 5 ä
Û_
ã Ü Zð ç ã ç 0
š K Rä å
(3.17)
ã
˜ã , 5 p¢ Õ © ¬ Zp¢ ‰¹ 5 ï 5 ä i
i ¯d‚ Ü 2 Ä „
•
äRå 0
Ü ðZç ç
š K
0
L
Using this to show Û)¡¯d‚ Ü 2 e „ requires:
K
1. Õ #R=lÕÒK œ /“ , Õ Zp¢=+Õ̜ up‘=“ ;
0
0
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113
QUADRATURE OVER THE SPHERE
2. Õ K WZÕ ­xò|, in neighborhoods of Ñ ¡ and юdp , respectively;
0
3. Õ K © Ä ¬ WZÕ © Ä ¬ ­
0 in neighborhoods of Ñ!¡ and юdp , respectively.
0
k is straightforward, because of the presence of D9EGF L ŠÑ
Again, the first condition
mulas for Õ K Ñ? and Õ ŠÑ? .
In analogy with the0 first case, we must consider the derivatives of
L
DZEF Š Ñ C fCÑ?uóJÑ?
L
Note that it contains the higher power D9EF Š Ñ C as compared to the D9EFŠÑ C
C
in the for-
(3.18)
in (3.9).
Proceeding as in the previous case,
(3.19)
™
™
 © ¬ Ñ?Òl¯ r DZEFŠ&Ñ?Y , ç4ð 2 t W
& ð L 2 ü ç L Í K t
ó © ü®¬ ŠÑ?q¯ r º D9EGFÑ C®»
÷
W
of the first case
_pWè¶Q¶Q¶QWZÁ
û¹¡XWèpW3Ÿ W9¿?WZÁ
showing ó
©G,Y¬ŠÑ? and ó © Ä ¬ŠÑ? are singular at Ñ ¡X¶ The second condition stated above, that
­Mò|, in some neighborhoods of Ñ ` and ѲØp , respectively, is satisfied. Simply
0 same approach as in the first case, noting now that in (3.15) the term DZEFŠÑ C is
use the
L
replaced by D9EGF Ñ C . Because of this, all terms arising from (3.16) will be continuous at
ю“ and ю)p .
What remains is to show that Õ K © Ä ¬ ­
0 #XWèp¢ . Returning to (3.15) for the case û!ÂÁ ,
k
the corresponding values of are
Õ K WZÕ
%
Á?W9?W9/Wè¿?WQpRW9RfW¢#¿?W9XWèp¢fWèŸ W3Ÿ W9/WèŸ WQpRWQp‘?W
Ÿ W9?WYŸRWèupRW9¿?W9RfW¢ZpRWYŸ Wèp¢fWèZpWèpWYŸRWèZpW9?W9¿/?W
#XW9Á?W9/W¢XWY¿XWèp¢W¢?WYŸ W3ŸWè?WQpRW9¿RWè#XW9?WZÁ]
%
%
Only when we look at terms containing C© Ä ¬ŠÑ?|º ¨#XWZÁfW9/Ï» or ó © ü®¬3ŠÑ? , û Ø¿?WZÁ½º upRW9XWY¿RW¢XWèpWY¿RW¢?W9XW9Á/®» , is there any need to check for integrability. It is easily checked
that in these cases the corresponding terms in (3.15) are integrable. Thus Õ K © Ä ¬ ­
0 in a
k the
neighborhood of Ñà` . An analogous result holds for Õ about Ñà¾p . This completes
0
proof of Theorem 2.2.
How do we generalize this theorem to larger non-integral values of m ? We would again
look at two cases:
÷
±Émȱ
÷
i L0
÷
i L0 ±+mo±
÷
i“p
™
÷
for some integer . Then we would generalize the formulas for © ¬ ™ and ó ©ôüϬ . The formula
for ó © üϬ generalizes easily; but that for the composite function ƒ© ¬ does not. The latter
and
requires using the Faa di Bruno formula (3.11), and we have not found any general way of
handling this. It is straightforward to do particular cases, however, and we have suitably
generalized (3.13) and (3.19) for mx­qŸXW9¿R and ¿XW9Á/ . With these results in hand, we then
can examine the Leibniz formula (3.15) and show the needed properties for the functions Õ K
and Õ .
0
4. Superconvergence with Ÿm an odd integer. In [4] it was observed experimentally
that with < sufficiently differentiable and m|_p¶«· ,
5
£
'
8 ¡¯¤ ‘
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114
K. ATKINSON AND A. SOMMARIVA
There was no precise estimate of the speed of convergence for the cases my¡ŸX¶ · and myl¿X¶«· ,
although it was clear experimentally that the speed of convergence was very high.
8 when
As noted earlier following (2.9), we are considering the trapezoidal error 5
£
approximating
I
K !ŠU/ !ŠU/@$U
(
(4.1)
L
u p 5 m D9EGF UTiJm
L
!UR=
ZD EF e 2&0 U
L
L
L
#DZEF egUTiM
LYI OPRD UR ,
!ŠU/Ò K <@Š‡ WZˆgW3‰ $]S
(4.2)
(
¥ Here, resorting to the well-known
¥
¥
Euler-MacLaurin expansion
in its standard form (cf. [3],
[6]), we explain the reason for such superconvergence for the cases my_pR¶ ·XWèŸ ¶«· , and ¿X¶«· . Let
+Á/m , and assume < is -times differentiable with all th -order derivatives of < belonging to
Ï'y . Without loss of generality, we can assume that < is defined on some Æ -neighborhood
k ' , call it ' , with < having analogous differentiability properties on ' . Then we show
of
*)
+)
5
£
8 q¯Ã‚?Ä e „
as stated in (2.8) of Theorem 2.3.
Introduce
D9EFfegU
9D EGF eUTiMOPRD L U
OQPRD U
ŠU/ h
L
D9EGF egUTiMOPRD L U
ÕTŠU/ h
,
(4.3)
and then write
Š‡ WZˆgW3‰\­x'
L
(see (2.4)) as
,
Š‡ WZˆgW3‰`ŠÕzŠUR OQPRDSCW9ÕTUR D9EGF[SW ŠUR9
)
Since < is sufficiently differentiable over ' , we can expand it in a Taylor series about
?W9?WQp¢ , corresponding to U¾  , with the series converging in some neighborhood of
?W9?WQp¢ . Using a Taylor series of order Ú 5 p , we can write, roughly speaking,
Í ’ Í ™ ‡ - ˆ ’ ‰ ™ i21 Š‡ WZˆgW3‰
á
- - 0. /
á
with appropriate coefficients - Í ’ Í ™ . The remainder 1 Š‡ W9ˆgWY‰ can be written in a variety
of forms, each depending on Ú -order derivatives of < á . Moreover, all derivatives of 1
of
order ±_Ú equal zero at ‡ WZˆgWY‰Rz¼#XW9?WQp‘ . Using this expansion, we expand !
( UR about
á
(4.4)
˜ 42 0
͒ ͙ K
’ç ç ™
<@Š‡ W9ˆW3‰
á
th
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115
QUADRATURE OVER THE SPHERE
Uol
,
YL I
!ŠU/Ò K
(
- - 0. /
0. /
- ,
0. /
˜ 42 0
Í ’ Í ™ ‡ UXWYSg]ˆ ’ UXWYSgX‰ ™ UXWYSg $]S!i
UR
͒ ͙ K
á
™
ç ’ç
á LYI
˜ 240
™
Í
á
’ Í ™ K ‡ ŠUXW3Sg]ˆ ’ ŠUXW3SgX‰ ŠUXW3SX$/S i
ŠU/
͒ ͙ K
á
™
ç ’ç
ML9I
˜ 240 á Í Í ™
™
’
’ Õ ç ŠU/ UR K OQPRD S[D9EF ’ S[$]S!i
(4.5)
UR
á
͒ ͙ K
á
™
ç ’ç
á
The remainder
UR depends on the Ú th -order derivatives of < and can be written in
an integrated form, á
L9I
ŠU/ K
‡ WZˆgWY‰R $]S
á
á
Thus
ŠU/ is well-defined around UJٝ when the Ú th -order derivatives of < belong to
0 ' Q .á In addition all derivatives of UR of order ±+Ú equal zero at UÈl .
k Denoting by ë the “Beta function” (cf.
á [1, p. 258]), we note that
I L
p
D9EGF S[OQPRD ’ ST$]S½ ë ‚ L0 gilp¢W L0 ”zilp¢9„
K
Ÿ
--
*)
á
-
1
3
-
43
When 3Wϔ are both even, we have
-
YL I
’
K DZEF STOP/D ST$/SslŸ ë ‚ L0 &i“p‘W L0 G”|ilp¢ „
53
3
and this integral equals  in all other cases of 3W” . As consequence,
(4.6)
(!URÒ - Í á ’ ˜ Í ™2&. 0 K - Í ’ Í ™ Õ - ç ’ ŠUR , ™ ŠU/
- ç - ’ Í ’ ç ™/
á
-
JLYI
’
K OPRD STDZEF ST$/S¤i
á
ŠUR
even
This corresponds to Sidi [11, Lemma 4.1 and Theorem 4.2].
Now let mydpR¶ ·X¶ Using Mathematica to simplify the calculations,
76 5 U '
Ÿ
U
L
Õ UR+U , i
, ŠUR_p
76
3
98
iM¯¤U ;
:'
8
5 UR, 5 U i ¿=U iM¯¤U Ÿ
Á
Use this in the expansion of (4.6), noting that giŒ” is always even and therefore
ù
’
L
’
Õ ç UR= Õ ŠUR † © ç ¬
-
-
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116
K. ATKINSON AND A. SOMMARIVA
43
L0 giŒ”/
with
(4.7)
an integer. Then
(!ŠURÒ<; 0 i=; L U , =i ; , U 6 =i ; Ä U '
>?; -@
for suitable
.
Returning to (4.3), we also have
BA ÁU96 Di C9:C U '
¿ U L
· U Ä 5
i
Ÿ
Á
!UR
(4.8)
8
i•¯¤ŠU Combining (4.7) and (4.8), we have
(
!ŠU/ !ŠU/Ò
L
U i
98
iM¯¤U ;
6
LUÄ i
8
'
Ä U iM¯¤U for suitable constants
. A similar Taylor expansion holds for Uo—l . This implies that the
first and third derivatives of !ŠU/ !ŠU/ are zero at Uo¡XW9í¶
We apply the Euler-MacLaurin formula to (4.1). Since < is ö 5 times differentiable with
:
all ö -order derivatives in Ï'y , we can conclude that the order of convergence of the
quadrature rule for m|)p¶«· isk ö .
Use an analogous proof when mŽÀŸX¶ · (and assuming < is pè -times differentiable with
:
all pè -order derivatives in 'y ). Then
k
L
U
pRp4U 5 K
Õ UR=“U i
i
U 0 iM¯¤U 0Y0 ö
Ÿ
> -@
4E
0
(
4E
8
6
, ŠU/_p 5 9U Ÿ 6
5 U
C
8
,U i
9F
7F
K
¿ U 0
5 pp4U i í
i ¯¤ŠU 0Y0 •
p¢Ÿ
pQÁRÁ
:
(!ŠURÒ<; 0 iG; L U 6 i=; , U 8 i=; Ä U F iÉiH; 6 U 0 K iz¯¤ŠU 0Y0 K
·@U Ä
U'
ppèU9I 5 p¢·@U9F
pp¢ U 0
!ŠUR=
iDC
i
i
C7C i•¯¤ŠU 090 Ÿ
p‘Ÿ
pèö
Á
'
J(
¿RŸ ÁR
I
F
K
!ŠUR !ŠURÌ
U Ä i L U i , U i Ä U i•¯¤ŠU 0 0
K
5
8
¯¤ 0 .
¡
This allows us to prove that :
For m!`¿X¶«· (assuming that £ < is Qp Á 5 times differentiable with all pèÁ
in Ï'y )
k
L
U
p U 0Y0 ÁR¿@U 0 , 5
Õ ŠURғU 5
i
i
U 0 ÄÌi•¯¤ŠU 0 ö
p‘Ÿ 
ŸXpèöR
8 9F
, ŠURÒ_p
!UR
6
C
F
8
C
:
6
5 p U i p U 5 p U 090 5 Á/¿ U 0 , i ¿ U 0 Ä i•¯Ã‚U 0 „
Ÿ
¢p Ÿ
;Ÿ ÁR
Á/¿RŸ 
(!ŠURK; 0 iL; L U 8 iL; , U F iL; Ä U 090 i=; 6 U 0 ,
: K ·R· A U 0 L 5 ŸXpQU 0 ,
U ' 5 ¿íU9I
p U 0
C
C i
i
i
Ÿ
4E -order derivatives
Á
Ÿ ÁR
ÁR¿RŸ
Á
=; ' U 0 Ä;iz¯¤ŠU 0 6 A :: ÁRö C Á/U  0 Ä iMŸ U 0 6 iM¯¤U 0 ' i
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QUADRATURE OVER THE SPHERE
J(
!ŠUR !ŠURÌ
9'
0
U Òi
I
K
,U 0 i
LU i
L
ÄU 0 i
6U 0,
Taken together, this implies that the order of convergence is
We would like to generalize this proof to
m| Ü i L0 W
Ü nqp
pèÁ
i
6
117
Ä U 0 ÄÌiM¯¤U 0 .
an integer
but we have been able to do so for only a portion of it. The difficulty can be seen in the form of
the Taylor expansions given above. The various functions are neither even nor odd; but their
lower degree terms have the behaviour needed in order to apply the Euler-MacLaurin error
formula. Nonetheless what we have shown is sufficient for practical purposes, demonstrating
that m of this special form is the preferable choice.
5. Conclusion. Although some of the techniques used in this paper are similar to those
used in Sidi [11],
^ they were obtained independently of that paper. A major difficulty with our
transformation of (2.1) has been the integral term
óJÑ?
YL I
K <o‡ WZˆgWY‰RV$/S
M
of (3.8). We have had to be quite careful in the handling of its derivatives, as we did in
obtaining (3.14) of 3. There is a similar difficulty in Sidi [11, Theorem 4.2], and he has been
able to handle it in a different way, by using the general theory he has developed for the error
analysis of the trapezoidal rule when applied in connection with class A@B transformations
(cf. [11, Theorem 3.1]). See Sidi [14] for an extension of the results of this paper, completed
independently of our present paper. In addition, the papers Sidi [12], [13] also relate to the
numerical method studied in this paper, although again our results are obtained independently
and are somewhat different in approach.
^
In spite of the difficulty in the error analysis of our transformation , we believe it is a
natural way to grade nodes on the sphere and one that needs to be understood more fully.
REFERENCES
[1] M. A BRAMOWITZ AND I. S TEGUN , eds., Handbook of Mathematical Functions, Dover Pub., New York,
1965.
[2] K. ATKINSON , Numerical integration on the sphere, J. Australian Math. Soc., Series B, 23 (1982), pp. 332347.
[3] K. ATKINSON , An Introduction to Numerical Analysis, 2nd edition, John Wiley, New York, 1989.
[4] K. ATKINSON , Quadrature of singular integrands over surfaces, Electron. Trans. Numer. Anal., 17 (2004),
pp. 133-150,
http://etna.mcs.kent.edu/vol.17.2004/pp133-150.dir/pp133-150.pdf
[5] D. E LLIOTT , Sigmoidal transformations and the trapezoidal rule, J. Australian Math. Soc., Series
B(Electronic), 40 (1998), pp. E77–E137.
[6] J. LYNESS AND B. N INHAM , Numerical quadrature and asymptotic expansions, Mathematics of Computation, 21 (1967), pp. 162-178.
[7] J. LYNESS , Private communication, 2005.
[8] J. LYNESS AND G. M ONEGATO , The Euler-Maclaurin expansion and finite-part integrals, Numerische Mathematik, 81 (1998), pp. 273–291.
[9] A. S IDI , A new variable transformation for numerical integration, in Numerical Integration IV, H. Brass and
G. Hämmerlin, eds., Birkhäuser Verlag, Basel, 1993, pp. 359-373.
[10] A. S IDI , Practical Extrapolation Methods, Cambridge University Press, Cambridge, United Kingdom, 2003.
[11] A. S IDI , Application of class
variable transformations to numerical integration over surfaces of spheres,
2003, to appear in Journal of Computational and Applied Mathematics.
[12] A. S IDI , Euler-MacLaurin expansions for integrals with endpoint singularities: A new perspective, Numerische Mathematik 98 (2004), pp. 371-387.
NPO
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118
K. ATKINSON AND A. SOMMARIVA
[13] A. S IDI , Extension of a class of periodizing variable transformations for numerical integration, 2004, to
appear in Mathematics of Computation.
[14] A. S IDI , Analysis of Atkinson’s variable transformation for numerical integration over smooth surfaces in
, Numerische Mathematik, 100 (2005), pp. 519-536.
[15] E. W EISSTEIN , Faá de Bruno’s Formula, in MathWorld - a Wolfram Web Resource,
http://mathworld.wolfram.com/FaadiBrunosFormula.html