A new proof of the Euler–Maclaurin expansion for quadrature
over implicitly dened curves
Dr. Muhammed I. Syam∗
Department of Mathematics & Computer Science, Faculty of Science, P.O. Box 17557, United Arab Emirates University, Al-Ain, UAE
Received 23 March 1998; received in revised form 22 July 1998
Abstract
In this paper we describe and justify a method for integrating over implicitly dened curves. This method does not require that the Jacobian be known explicitly. We give a proof of an asymptotic error expansion for this method which is a modication of that of Lyness [4]. c1999 Elsevier Science B.V. All rights reserved.
AMS classication:65
Keywords:Continuation methods; Implicitly dened curve; Modied trapezoidal rule
1. Introduction
The predictor–corrector methods for numerically tracing implicitly dened curves have been devel-oped and investigated in a number of papers and books. For recent surveys, see., e.g. [1, 2]. Siyyam and Syam [5], describe the idea of the predictor–corrector method for tracing an implicitly dened closed curves and they presented a device for reliable stopping. They used the Euler–Predictor and the Gauss–Newton Corrector. They presented the modied trapezoidal rule for approximating the line integrals over implicitly dened closed curve. This rule was developed by Georg [3] to ap-proximate surface integrals for implicitly dened surfaces. Siyyam and Syam [5] proved that the modied trapezoidal rule has an asymptotic error expansion in even powers of 1=m over a scalar and vector eld. But their proof was very long and complicated. In this paper, we present another proof which is shorter and easier than their proof. Moreover, our new proof depends on the idea of the Euler–Maclaurin summation.
∗E-mail: [email protected].
We shall mean that a map is smooth if it has continuous derivatives as the discussion requires. For a smooth map f:Rn→Rn and a smooth curve C with a parametrization, say = (’1; ’2; : : : ; ’n) :
S→Rn; S= [0;1] andC= {(u)|u∈[0;1]}, the line integralR
CfdC can be written as an ordinary integral, i.e.;
Z
C
fdC=
Z 1
0
f((u)): ′(u) du:
For the latter integral the trapezoidal rule takes the form
Z 1
0
f((u)): ′(u) du=1
2 m−1
X
i=0
[f((ui)): ′(ui) +f((ui+1)): ′(ui+1)][ui+1−ui];
over the partition points ui=i=m in [0;1]. Let Xi=(ui) for i= 0 :m, the modied trapezoidal rule takes the form
Z
C
fdC=1 2
m−1
X
i=0
[f(Xi) +f(Xi+1)]·[Xi+1−Xi]; (1.1)
where X0 and Xm are the endpoints of C
Theorem 1.1. The modied trapezoidal rule (1:1) admits an asymptotic error expansion in even
powers of 1=m.
In this paper, we present a new proof for Theorem 1.1.
2. Asymptotic error expansion for the modied trapezoidal rule
In this paper we use the following norm k(1; 2; : : : ; n)k=
q
2
1+22+· · ·+2n. First, we want to prove Theorem 1.1 for a scalar eld f:Rn→R. Letuj=j=m andXj=(uj) forj= 0 :m. Dene
the quadrature
Q(m)f=1 2
m−1
X
j=0
[f(Xi) +f(Xi+1)]kXi+1−Xik: (2.1)
The main result in this paper is that Q(m)f has an h2-expansion. That is, setting h= 1=m, when
f∈C(2p)(C) and (u) is suciently well behaved, we have
Q(m)f=
Z
C
fdc+B2h2+B4h4+· · ·+B2ph2p+(h2p+1); (2.2)
However, the line integral RCfdc can be written as
Z
C
fdc=
Z
S
f((u))J(u) du=
Z 1
0
g(u) du; (2.3)
where J2(u) = (’′
1(u))2+ (’′2(u))2+· · ·+ (’′n(u))2 and
g(u) =f((u))J(u): (2.4)
In Theorem 2.1, we want to study the asymptotic expansion of kXi+1−Xik for any i= 0 :m−1.
Theorem 2.1. Let (u) be C(p)(S); and let =kX
i+1−Xik. Then;
=h(0+h1+· · ·+hp−1p−1) +(hp+1); (2.5)
where 0=J(uj) and s=s;1(uj) where s;1(u) is a function of u having continuous derivatives of
order p.
Proof. It is easy to see that2=kX
i+1−Xik2=Pni=1(’i(uj+1)−’i(uj))2. Expand’i(uj+1) =’i(uj+h) about uj to get
2= n
X
i=1
p−1
X
s=1
’(is)(uj)
hs
s! +’ (p)
i (j)
hp
p!
!2
; (2.6)
where j lies between uj and uj+1. Simplify Eq. (2.6) to obtain
2=h2 J2(uj) + p−1
X
s=1
cs(uj)hs
!
+(hp+2): (2.7)
For suciently small h, we may take the square root of Eq. (2.7) to get
=h(0+h1+· · ·+hp−1p−1) +(hp+1);
where 0=J(uj). Since the derivatives of ’1; ’2; : : : ; ’n of order p are continuous functions of u, it follows that so are the derivatives of cs of order p−1−s. Finally, so long as J(u) is bounded a way from zeros in S, we nd that s also has this order of continuity.
We should note that the proof of Theorem 2.1 is forkXi+1−Xik. The proof is similar forkXi−1−Xik, but
s;1(u) = (−1)ss;−1(u): (2.8)
Moreover, we should note that the functions s; (u) do not depend on m for =−1 and 1. As for
0(u) =J(u), it depends only on ’1; ’2; : : : ; ’n.
Denition 2.2. Let ∈ {−1;1} and B be any interval subset of R. Then dene (m)
j; (B) = 1 or 0 depending on whether or not the interval with endpoints uj and uj+ is a subset of B.
For ∈ {−1;1}, dene the quadratures R(m)g and Q(m)f as
The expression of Theorem 2.1 allows us to establish the following.
Theorem 2.3. Let ∈C(p)(S) and let R(m)
Proof. Using formulas (2.5) and (2.10), we obtain
We should note that g0; (u) coincides with g(u). Also, R
(m)
gs; depend on h via m, so Q
(m)
f is not an h-expansion. If g∈C(p)(S), the following Euler–Maclaurin summation formula is valid,
R(m)g=
and Bj(R;g) is independent of m. Substitute formula (2.14) in (2.12) to establish the following:
Q(m)f = coecients in the Euler–Maclaurin expansion.
Theorem 2.4. The coecients in the Euler–Maclaurin expansion (2:14) satisfy
Bj(R;g) = (−1)jBj(R−;g): (2.18)
Proof. One can establish this result by direct evaluation of the coecients in terms of Bernoulli function. But our approach is based on the fact that a symmetric rule has an m2-expansion. It is easy to see that Rg=1
Hence, we establish the relation (2.18).
From Eqs. (2.8) and (2.13), we see that
gs; (u) = (−1)sgs;−(u): (2.19)
Dene the quadrature
Thus,
Hence, quadrature (2.21) is the modied trapezoidal rule. In Theorem 2.5, we establish the asymp-totic error expansion of quadrature (2.21).
Theorem 2.5. Let and f() be C(p)(S). The quadrature (2:21) has an h2-error expansion.
Proof. From Eqs. (2.17), (2.18) and (2.19) we obtain
e coecient vanishes when s is odd.
Theorem 2.5 gives us the asymptotic error expansion for the modied trapezoidal rule for a scalar eld. To prove Theorem 1.1, let f= (f1; f2; : : : ; fn) :Rn→Rn be a given function and let C1 be a
while the modied trapezoidal rule is given by
1
which follows from the approximation that
Z 1
0
fi((u))′i(u) du≈ 1
References
[1] E.L. Allgower, K. Georg, Numerical Continuation Methods, Springer, New York, 1990.
[2] E.L. Allgower, K. Georg, Continuation and path following, Acta Numerica, Cambridge Univ. press, Cambridge, 1993, pp. 1– 64.
[3] K. Georg, Approximation of integrals for boundary element methods, SIAM J. Sci. Statist. Comput. 12 (1991) 443 – 453.
[4] J.N. Lyness, Quadrature over curved surfaces by Extrapolation, SIAM J. Numer. Anal. 15 (1993).
