MATHEMATICS OF COMPUTATION VOLUME 53, NUMBER 187 JULY 1989, PAGES 279-295
Noninterpolatory Integration Rules for Cauchy Principal Value Integrals By P. Rabinowitz*and Dedicated
to the memory
D. S. Lubinsky** of Peter
Henrici
Abstract. Let w(x) be an admissible weight on [-1,1] and let {p„(x)}g° be its associated sequence of orthonormal polynomials. We study the convergence of noninterpolatory integration rules for approximating Cauchy principal value integrals
(sjiííláx,
-£■ This requires investigation
Ae(-i,i).
of the convergence of the expansion oo
/(/¡A)~ J^(/.Pfc)9*(A).
A€(-1,1),
k=0
in terms of the functions of the second kind {qkW}çf
(f,Pk)-=
fl
ui(x)f(x)pk(x)dx
and
associated with w, where
i"1 Pk(x) qk(X) := -j- w(x)——dx,
fe= 0,l,2,...,A6(-l,l). 1. Introduction. In the third volume of his monumental work, Applied and Computational Complex Analysis, Henrici [8, pp. 139-142] gave an algorithm for the numerical evaluation of Cauchy principal value (CPV) integrals. This algorithm was presented in a more explicit form in a recent paper, by one of the authors [15]. In neither case were convergence questions considered. In this paper, we shall analyze the convergence questions arising from the use of this algorithm. Consider the CPV integral of the form
(1)
/(/; A):= -f w(x)^-dx, 7-1
x -X
-1< X< 1,
where w is an admissible weight function, w € SÍ, that is, w(x) is nonnegative and
integrable in [-1,1] and (2)
m0 := /
w(i w(x)dx > 0.
Received November 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 65D30; Secondary 65D32,
41A55. Key words and phrases. Cauchy principal values, numerical integration, noninterpolatory integration rules, orthogonal polynomials, functions of the second kind. 'Research completed during a visit to N.R.I.M.S. "Part-time at the Department of Mathematics, University of the Witwatersrand, P.O. Wits
2050, Republic of South Africa. ©1989 American 0025-5718/89
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280
P. RABINOWITZ AND D. S. LUBINSKY
For such w, there exist sequences of orthonormal
(3)
polynomials
{Pn(x) := pn(w, x) := knxn H-,
kn > 0},
with respect to the inner product
(4)
(f,g):=l
w(x)f(x)g(x)dx,
satisfying a three-term recurrence relation (5)
xpn(x)
= an+ipn+i(x)
+ ßn+iPn(x)
+ anpn-i(x),
n = 0,1,2,...,
where an:=kn-i/kn,
n > 1;
p_i(x) = 0
and
ßn+i
:= (xpn,pn),
n>0,
—1/2
po(x) = ko = m0
If we define qn(X), the function of the second kind, by
(6)
qn(X):=qn(w,X):=I(pn;X):=-f
7-1
w(x)^¡dx,
-I < X < 1,
x-X
then the qn(X) satisfy the same recurrence relation as the {pn(x)}, namely
(7)
Xqn(X) = an+iqn+i(X)
+ ßn+iqn(X) + anqn-i(X),
n = 0,1,2,...,
with starting values q~i(X) = —1, qo(X) = I(po', A) and ao := m0 ■ If we denote
by (8)
ak:=(f,Pk)
the Fourier coefficient of pk(x) in the formal expansion of f(x), (9)
/(x)~£afcpfc(x), fc=0
then we can write a formal expansion for /(/; X) in terms of the qn(X), oo
(10)
J(/;A)~£>fc9fc(A). fc=0
Hence, an approximation
to /(/; X) will be given by the truncated
sum
N
(11)
SN(f;X):= J2°-kQk(X). k=o
If we now have a sequence of integration rules
(12)
Qm(9) -=^2wimg(xim), t=i
which converges to
(13)
1(g) := j
w(x)g(x)dx
for all g e C[-l, 1] or all g e R[—l, 1], the space of bounded Riemann integrable functions on [—1,1], and if we approximate the Fourier coefficients ak by
(14)
ofcm:= Qm(fpk),
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CAUCHY PRINCIPAL VALUE INTEGRALS
then, in general, we obtain a noninterpolatory
integration rule for I(f;X),
281
namely
N
(15)
Qm(f;X):=J2akrnqk(X). fc=0
The approximations Qm(f; X) can be evaluated in a stable manner using backward recursion by the algorithm given in [15], provided that we have the value of qo(X). We can also express Q%(f;X) in a Lagrangian form that is more useful in the numerical solution of integral equations: m
(16)
Qm(f;X) = ¿2w»m(X)f(xirn),
where the weights N
(17)
■w^m(X):=wim'^2pk(xim)(lk(X),
i = l,2,...,m,
k=0
can also be evaluated in a stable manner by the backward recursion algorithm [15]. As indicated above, this general approach to the numerical evaluation of CPV integrals appears in Henrici [8, pp. 139-142]. However, there is no discussion there of convergence or of the integration rules Qm(g). In fact, it is precisely the freedom in the choice of these rules, subject only to the condition that they converge to
1(g) for all g e C[-l, 1] or all g e R[-l, 1], that affords this method for evaluating CPV integrals considerable interest. Thus, if / is well behaved in most of the interval [-1,1], but is irregular over a small subinterval [a, b] C [—1,1], then we can concentrate most of our integration points x¿m in [a, b]. This was also done by Gerasoulis [7] using a different approach, and the results he achieved were a considerable improvement over those achieved using a conventional spacing of integration points. There have been many approaches to noninterpolatory integration of CPV integrals [4], [14], [17], but these two are the only ones that cater to the situation
indicated
above.
In Section 2, we state and prove Theorems 1 to 5, which deal with convergence of 5tv(/;A) to I(f;X). In Section 3, we state and prove Theorems 6 to 8, which deal with the convergence of Qm(f;X) to I(f;X) as m and N —► oo. It turns out that in the general case we shall be able to prove convergence only for the iterated
limit
(18)
Jim lim Qm(f;X).
N—>oo m—»oo
In fact, we shall show that we cannot in general expect convergence of the double limit. However, in certain cases where we can convert the double limit to a single limit in which m depends on N in some specific manner, we shall again be able to prove convergence. A similar approach was used by Dagnino [3] in studying the convergence of noninterpolatory product integration rules. 2. Convergence Results for Sjv(/; A). Before we can study the convergence of Qm(f; X) to I(f;X), we must establish the convergence of S^(f;X) to I(f;X). To this end, we shall use the methods presented in Natanson [11] and Freud [5] for License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
282
P. RABINOWITZ AND D. S. LUBINSKY
proving convergence of orthonormal
expansions.
Since the proofs in [11] depend on
the Christoffel-Darboux formula na\ (19)
V*« ¡~\n (.a = aw+i-——-, ~ PN+i(x)PN(y) - pN(x)pN+i(y) 2^,Pk(x)pk(y) fc=o x y
we shall first establish an analogous formula for the sum N
(20)
KN(x,X):=J2Pk(x)qk(X). k=0
Throughout, C, Ci, C2,...,
and B, B\, B2,...
denote positive constants indepen-
dent of N, m, x and A. LEMMA 1. Let {pn}o° be a sequence of orthonormal polynomials on [—1,1], with respect to w esrf', and let qn(X) := I(pn', X), n = 1,2,3,..., exist for a given
A€(-1,1).
Then,for N = 1,2,3,...,
(21)
K
(x A) = 0lN+1^PN+1^qN^
~ Pu(x)qN+i(X)} X
+ 1
À
Proof. We have from (5) and (7) that for k = 0,1,2,..., (22)
xpk(x) = afc+ipfc+i(x) + ßk+ipk(x) + cvfcpfc_i(x),
and
(23)
Xqk(X)= ak+iqk+i(X) + ßk+iqkW + ctkQk-iW-
Multiply (22) by qk(X) and multiply (23) by Pfc(x); then subtract the two and sum from k = 0 to N. This yields (x - X)KN(x, X) = aN+i{pN+i(x)qN(X) - a0{po(x)q-i(X) 1 /9
Since p_i(x) = 0, oo
Since
I(P¿;X) = SN(P*N;X), we have
/(/; A) - SN(f; X) = I(rN; X) - SN(rN; A), and it thus remains to show that
lim \I(rN;X)\ = 0.
N—»oo
Now I(rN;X) = /
w(x)
= /
w(x)
rN(x) -rN(X)
x-X rN(x)-rN(X)
dx + rN(X)q0(X)/po
dx + o(l).
x-X
Furthermore, as in [1],
J-l
X-X
J_i
JX-l/N
JX+1
=: Ji + J2 + JzHere -X-l/N
Ji\ < 2EN(f) / J-i
r-Hl^ |x-A| \x-X\
= EN(f)0(logN) = o(l)
as TV-> oo.
Similarly, Jz = o(l) as TV—»oo. Finally,
MI\ix)rjM^Wdz
/ X-l/N
-L
x —X
x+1/N„..,„J(x)-fM w(x)
X-l/N
x-X
dx - If
X+1/N..,^P*N(x)-PnW w(x) x —X Jx-l/N
Since / G DT[—1,1], the first integral on the right-hand side is o(l). second integral, we have from [9] that
|P¿(x)-P¿(A)| x —A
dx. As for the
< max{\P$(t)\: t G [A- 1/TV,A+ 1/TV]}
oo,
since w(x) is uniformly bounded above for A G 3f and TV large enough. This completes our proof. D Remark. Theorem 3 is similar to Theorem 2.2 in [1]. By following the proof of Theorem 3, we can prove a result similar to Theorem 2.1 in [1], namely, that if /
satisfies (28) on J, if w G GSJ, and if for some X e 2, I(f; X) exists, then (32) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
CAUCHY PRINCIPAL VALUE INTEGRALS
287
holds. The proof of Theorem 3 holds in this case too, except that we must show
that
rX+1/N...,.J(*)-fW
I-Á + 1/JS
JN :=
w(x)J-^1—Y^-dx
Jx-l/N
= o(l),
TV-»• oo.
X-X
Since
Jo:= i
7-1
w(x)f{x) ~ {(A) dx = /(/; A) - /(A)J(l; A), x-X
and both /(/; A) and 7(1; A) exist, it follows that Jo exists. Hence J/v = o(l), and the proof is complete. D We now give some additional conditions for (32) to hold, which impose less restrictions on the weight function w G s/, but require more smoothness of /. To this end, we first prove a lemma:
LEMMA2. Let w esrf, and assume that for some X G (—1,1),
(42)
r(A):=/_,
1 \w(x) -w(X)
x —X
dx < oo,
while for some positive e, Bi and B2
(43)
Bi < w(x) < B2
for |x - A| < 2e.
Then there exists a constant Bz > 0 such that n-l
(44)
Tn-i(X):=^2q2k(X)-i+l
= 2i-m/2
J-
E^(f;w)* 0 we can find a positive integer K such that for all large
enough m, N„
(65)
E Qm(fPk)qk(X)< e. k=K
Proof. For any fixed J and all m large enough, Nm
oo
Qmm(/;A)- /(/;A) = Y,Q™(fPk)qk(X) - ¿2(f,Pk)qk(X) k=0
fc=0
K-l
= ¿2{Qm(fPk)-(f,Pk)}qk(X) k=0 oo
Nm
- E(/'Pk)»(A)+ E Qm(fPk)qk(X). k=K
k=K
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CAUCHY PRINCIPAL VALUE INTEGRALS
295
Here, as in the proof of Theorem 6, the first term in this last right-hand side is o(l)
as to —► oo. Further, given e > 0, we can find a K such that the absolute value of the second term in this last right-hand side is bounded above by e. Hence for to large enough, Nm
Qmm (/; A)- /(/; A)- E Qm(fPk)qk(X) < 2e, k=K
which proves the theorem. Department of Applied Mathematics Weizmann Institute of Science
P.O. Box 26 Rehovot 76100, Israel E-mail:
[email protected]
Centre for Advanced Computing
and Decision Support
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ed.), ISNM 45, Birkhäuser
