- Email: [email protected]
Numerical evaluation of Cauchy principal value integrals based on local spline approximation operators
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
ELSEVIER
Journal of Computational and Applied Mathematics 76 (1996) 231-238
Numerical evaluation of Cauchy principal value integrals based on local spline approximation operators 1 C. Dagnino *, P. Lamberti Dipartimento di Matematica, Universit°li Studi di Torino, Via Carlo Alberto 10, 1-10123 Torino, Italy Received 3 January 1996; revised 23 July 1996
Abstract
In this paper a local spline approximation method, defined for any function f C L1[-1, 1], is applied to evaluate Cauchy principal value integrals and convergence results with an error bound are established. Some comparisons with other methods and numerical examples are given. Keywords: Cauchy singular integrals; Approximating splines; B-splines AMS classification." 65D30; 65D32
1. Introduction We consider the evaluation of the Cauchy principal value integral (CPV), J(Kf;2)=
K(x)
dx
-1
<2<
1,
(1)
1
where K and f are such that J ( K f ; 2) exists. Recently, several authors [1-4, 6, 7, 9-12] have studied quadratures for (1) defined for any f E C'-1 [_ 1, 1], s ~> 1 and based on the approximation of f by splines o f the form Qf = ~
2,fNi(x),
(1')
where {Ni} is a sequence of B-splines and {2i} is a sequence of linear functionals involving point evaluation of the function f [8]. In this paper we propose the numerical evaluation of (1) by rules once more based on schemes of the form (1'), but defined for a larger class of functions f and * Corresponding author. E-mail: [email protected]. l Work sponsored by "Consiglio Nazionale delle Ricerche" of Italy. 0377-0427/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PH S0377-0427(96)00105-7
C. Dagnino, P. LambertilJournal o f Computational and Applied Mathematics 76 (1996) 231-238
232
involving some modified moments o f f . In Section 2 we present the approximation spline operator, in Section 3 we define the corresponding quadratures for (1) and we prove some convergence results. Finally, in Section 4 we give several numerical applications, where we compare our new quadratures with the old ones [1-4, 6, 7, 9-12].
2. On the approximation spline operator Let Y,: Yo - - 1 < YI < Y2 < "'" < Yn-1 < Y , -- +1 be a partition o f I - [ - 1 , 1] and let h i = y~+l - Y i , i = 0, 1,..., n - 1. We call l"Iu ~-(Xi}iNo a nondecreasing sequence obtained from {yi}in 0 by repeating Yi exactly d; times and so giving N = ~"=-11 di + 1. We say that the sequence o f partitions { Yn) is locally uniform if, for some constant A ~> 1, hi~h/<<. A for all 1 ~< i ~< n - 1 and j = i 4- 1. If k is an integer and k > d i , i = 1 , 2 , . . . , n - 1, we denote by Sk, nN the class of all polynomial spline functions of order k so defined as Sk,11N = {g/g[(yl,yi+l) E [~k, i ----0, 1 , . . . , n -- 1 and 9(J~(y +) = g(J)(YZ),
j = 0, 1 , . . . , k -
d~ - 1, i = 1 , 2 , . . . , n -
1},
where Pk is the set of polynomials o f order k, i.e. of degree < k. Let HNo = "trX;S;=l-klN+k-1be the extended partition associated to HN where x~ < x;+k, i = 1 - k , . . . , N - 1 and - 1 - Xl-k - XZ-k ~ "'" ~ X-1 ~ XO, XN ~ XN+I ~ "'" ~ XN+k--1 ~ 1. We call a sequence o f spline spaces locally uniform if they are based on a sequence of locally uniform partitions. The normalized B-splines {Nik(x)}i=~_ N-~ k of order k [5, 13], associated to FINe form a basis for Sk, l-IN•
For each j = 1 , 2 , . . . , k let /3j be the classical Gauss-Legendre orthogonal polynomial of order j (degree j - 1) [14]. We recall that /3/ has ( j - 1) distinct zeros in ( - 1 , 1): ~ j l , ~ j z , . . . , ~ j j - 1 and P j ( Y ) : q j ( Y - ~yl ) ( Y - ~jz) " " ( Y - ~jj-l) with qj = ( 2 j - 2 ) ! / [ 2 J - ~ ( j - 1)!2]. Moreover, let hj = 2 / ( 2 j - 1) be the normalization constant, i.e. f_l I ~ j ( y ) ~ ( y ) d y 6ju Kroneker's delta. Now, given at < bi such that [ai, bi] C [xi, xi+k], we consider the polynomial pij(x)=
~j \
bi-ai
that we can write as follows p i f i x ) = trij(x - Zijl )(x - zij2 ) " " (x - zijj-1 )
with
and zip=
÷
r=1,2,
...
,j-1.
: fish/, with
C. Daonino, P. LambertilJournal of Computational and Applied Mathematics 76 (1996) 231-238 For any f c L I [ - 1 ,
20f =
ft
1] we can define
/bi_ai I Pj(Y)f ~ Y
bi+ai)dy, 2
+
j = l,2,
""
.,k"
From [8] we remark ~i#Pij : (~j#, j, I~ = 1,2,..., k with support Thus, the following spline operator: N--1
233
(2)
[ai, bi]
and det(~ijx #-1 )j,/~=0k~ 0.
k
Qf(x) = ~
~ ~ij i~;fN,.k(x)
(3)
i=l--k j = l
with j-1 ~ i j = Gij
Z(--|) v
symmv(zi;1,... ,zijj-1 ) s y m m j _ v _ l ( X i + l , .
. . ,Xi+k-1
)
k-1 (j--v--l)
v=O
(Y)
maps Ll[--1, 1] into Sk,n~ and it is local. Moreover, from Theorem 3.3 of [8] it follows that Q is exact for Pk. Generally (3) is not a projector, i.e. it does not reproduce splines. However, if [ai, bi] C [x~,,x~,+l] C [xi,xi+k], i = 1 - k , . . . , N 1, then we can verify Q is a projector of L l [ - 1 , 1] into &,nx. Since the modified moments (2) are required to construct Q f , we assume they can be evaluated accurately, either analytically or numerically.
3. Quadrature rules for CPV integrals (1) based on the local splines (3) We consider the following integration rules for (1): N-1
J ( K f ; 2) _~ J ( K Q f ; 2) = ~
k
~
#oJ(KNik; 2),
(4)
i=l--k j = l
where
Pij = gij " 2gjf . We remark that if K and f E L l [ - 1 , 1] A DT(Na(2)) then (1) exists and the quadratures (4) are defined. We denote by DT(I) a subclass of continuous functions, the functions of Dini type on the interval I, and by Na(2) the interval [2 - 6, 2 + 6], where 6 > 0 is chosen so that Na(2) C ( - 1, 1). Now we will study the convergence of J ( K Q f ; 2) to d ( K f ; 2). For fixed - 1 ~< t ~< 1, let m be such that Xm <<.t < Xm+l. NOW let us introduce the following parameters: 2 m ~---
max
m+l--k <~i<~m+k-- I
Am k--r = "
AN =
min
(xi+l - xi),
rn+l--k+r<<.v<~m
(xv+k-r -- xv),
max ( x i + 1 - - x i ). 1-k~i~N+k--I
c. Dagnino, P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231~38
234
We recall the modulus o f continuity o f a function g E C(I), where I is an arbitrary interval, is given by ~(g; A ; I ) = ~max Ig(x + h) - g(x)l. ,x+hEl O<.h<~A
Moreover, we denote by Im the smallest interval containing [xm,Xm+,] and [ai, bi], with m such that Xm < Xm+,. For any s such that 1 ~< s ~< k we define
SDr(f -Qf)(t), DrQf(t),
Er, s( t )
O<.r
We assume henceforth that AN --~ 0 as N ~ oo. In our discussion we need the following lemmas. We remark L e m m a 1 is a special case o f Theorem 6.1 o f [8]. L e m m a 1. Let 1 <<.s <<.k. I f f --A--r--,
max
t C [Xm,Xm+1]
then f o r 0 <<.r < k
E Cs-l[lm], s
IEr,s(t)l <~ RmA m
1
--
~o(D - f ;Am;lm)
(5)
with R ?n
(s
--
1)!
-
~
~(2pm)J-l'
,= hj
-
where Xi+ k -- X i Pm =
max m+,-k~i~m
hi
- - gli
and Fk~--
(k-r-l)!
[r/2]
"
L e m m a 2. I f pro is uniformly bounded f o r all m and N, then f o r all f E C [ - 1 , 1] (6)
[[f - Qf[[oo ~ O as N ~ oo. Furthermore, if-Am/Am, k-1 is uniformly bounded f o r all N, then max
[DQf(t)[ <<.C,A,,'~(f;Am;Im).
(7)
Xm ~ l ~ Xm+ ]
Proof. From L e m m a 1 with s = 1, r = 0 we have k
max Xm ~t
[ f ( t ) - Qf(t)[ <~ k 2 ~
j=l
~(2p.,)J-'o~(f;-A,.;I.,)
hj
<. C2o (f ;-dN;Z), where C2 is a constant independent on N.
(8)
C Dagnino. P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231-238
235
Therefore (8) implies (6), since f E C [ - 1 , 1] and AN ~ 0 as N ~ c~. If we now take r - - 1, we get from Lemma 1 max
[DQf(t)l <<.k2Fkl .-A"
Xm <~t
/]m,k-1
~
qj ~2,~ -1 co(f;A,,;I,,), ~-~, vm,~J-1A m
(9)
"~.= h j
which implies (7) holds because of the hypothesis on Pm and "Am/Am,k_l. [] From Lemma 2 we can deduce that (i) If [ai, bi] ~- [xv,,xv,+l] C [xi,xi+k] with vi such that Xvi+l - - X v. =
'
max (xi+2+1 --
0~<2~k--1
Xi+.~),~
(10)
then Pm ~ k. Therefore (6) holds. (ii) If the spline spaces {Q f } are locally uniform with constant A for all m and N, then for any
[ai, bi] - [Xv~,X~,+l] C [xi,Xi+k]
(10t)
k-1 A~, and Am~Am.k-1 it results that Pm < 2 ~j=0 <<.Ak_t. Therefore (6) and (7) hold. Lemma 3. Let the spline spaces {Q f } be locally uniform and [ai, bi] be defined by (10') for any i. Then c o ( Q f ; 3N; I ) :
O(co(f; AN; I)).
( 1 1)
Proof. From the above remarks on the operator Q we can prove Q satisfies the hypothesis of Theorem 4 of [10], then from such a theorem we can conclude (1 1) holds. [] Theorem 4. Let {Q f } be locally uniform and [ai, bi],i = 1 - k,... , N - 1 be defined by (10'). Let K E L l [ - 1 , 1] 71DT(N~(2)) and f E C [ - 1 , 1] N DT(N~(2)) for some 2 E ( - 1 , 1) and some 6 such that N ~ ( 2 ) C ( - 1 , 1). I f AN ---+0 as N --~ c~, then
J ( K Q f ; 2) ~ J ( K f ; 2).
(12)
Proof. From the hypothesis it follows that Lemmas 2 and 3 hold. Therefore, the proof of (12) is similar to that of Theorem 8 of [9] and we do not report it here. [] Now we shall consider the quadrature error
E ( K f ; 2) ----J ( K f ; 2) - J ( K Q f ; 2)
(13)
for which in the following theorem we shall derive a bound for sufficiently smooth functions f . Theorem 5. Let 2 <. s <~ k. Let the hypothesis o f Theorem 1 be verified. Then for all f E CS-l[-1, 1] --s-2
s 1
--
E(Kf;)~) = O(d u ~o(D - f ; A u ; I ) ) .
(14)
236
C. Dagnino, P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231-238
Proof. With the help o f the mean-value theorem and Lemma 1 where we put successively r = 0
and r = 1 the thesis follows.
[]
4. C o m p a r i s o n o f C P V quadrature rules and numerical applications
Numerical quadratures for (1) have several interesting applications, particularly they m a y be usefully applied to solve Cauchy singular integral equations [10]. Recently [ 1 4 , 6, 7, 9-12] q.i. spline local methods for (1) defined for any f E C ( I ) have been studied and tested. In Section 3 of this paper we have proposed quadratures (4) for (1) defined for a wide class of functions f . Moreover, we have studied their convergence and derived an error bound. Now we present some numerical applications, obtained when rules (4) and other old ones [1, 9] of the form (1') with the same initial partition Yn, are applied to (1) for several functions f . For our examples we have chosen the following points as Yn partition: U: Uniform, i.e. Y n = { Y i = - - I + ~;i=0,1,...,n}; P: Perfect, i.e. Y , = { y i = c o s ( - u )n - - i Tr, i = 0 , 1 , . . . , n }. Moreover, we assume N = n. We denote by: - E the truncation error (13) of quadratures (4); - E d the truncation error o f the rules proposed in [1]; - E e the truncation error o f the formulas presented in [9]. The Tables 1-4 present the absolute errors for several 2, n and k. The exact test integrals values have been reported in [1], therefore we do not write them here. Table 1 Absolute errors for K(x) = 1, f ( x ) = ,/1 - x 2, k = 4, Yn = U n = 15
n = 31
n = 63
)~
[El
IEal
IERI
IEI
IEal
IERI
IEI
IEal
IERI
0.3 0.4 0.5 0.6 0.7 0.8
9.5(-4) 1.3(-3) 1.9(-3) 2.7(-3) 4.3(-3) 8.0(-3)
5.4(-3) 9.7(-3) 1.7(-2) 3.1(-2) 2.2(-2) 4.2(-2)
6.7(-3) 9.6(-3) 1.3(-2) 1.8(-2) 2.5(-2) 3.2(-2)
2.8(-4) 4.2(-4) 5.9(-4) 8.4(-4) 1.2(-3) 2.1(-3)
1.0(-3) 1.6(-3) 2.4(-3) 3.9(-3) 7.4(-3) 2.1(-2)
2.4(-3) 3.5(-3) 4.8(-3) 6.8(-3) 9.8(-3) 1.5(-2)
9.7(-5) 1.4(-4) 1.9(-4) 2.8(-4) 4.1(-4) 6.8(-4)
2.6(-4) 3.9(-4) 5.7(-4) 8.7(-4) 1.4(-3) 2.9(-3)
8.6(-4) 1.2(-3) 1.7(-3) 2.4(-3) 3.5(-3) 5.6(-3)
Table 2 Absolute errors for K(x) = 1, f ( x ) = (y2 _ x2)-1/2, y = 1.1, k = 4, Yn= U n = 13
n = 17
n = 63
2
]EI
IEal
[ER]
]EI
lEa[
]ER[
[EI
lEa[
[ERI
0.25 0.99
5.8(-4) 3.7(-2)
1.2(-2) 2.2(-2)
1.3(-2) 2.3(-1)
2.1(-4) 1.6(-2)
5.8(-3) 3.9(-3)
9.0(-3) 1.7(-1)
2.1(-6) 5.0(-4)
1.2(-4) 3.4(-3)
9.3(-4) 3.2(-2)
C. Dagnino, P. LambertilJournal of Computational and Applied Mathematics 76 (1996) 231-238 Table 3 Absolute errors for K(x) = (1 - x2) -1/2,
f(x)
n=5
=
237
(X2 -]- y2)--1, y = 5, k = 4, Y, = U
n = 13
n=23
A
IEI
[Edl
IERb
IEI
IEdl
IERI
IEI
lEd[
IERI
0.25 0.99
9.5(-8) 4.0(-6)
7.5(-6) 4.1(-6)
4.8(-5) 2.9(-4)
9.7(-10) 1.2(-7)
8.3(-8) 2.6(-7)
4.5(-6) 7.1(-5)
1.0(-9) 1.5(-8)
7.5(-9) 3.8(-8)
1.2(-6) 2.9(-5)
Table 4 Absolute errors for K(x) = 1, f ( x ) = ex, Yn = P 2 -----0.1
k
2 = 0.5
2 = 0.9
n
Igl
IEal
IERI
IEI
IEal
IERI
[El
IUI
IERI
10 20 42
1.5(-3) 3.1(-4) 4.4(-5)
5.7(-3) 5.1(-4) 1.5(-5)
1.2(-2) 3.3(-3) 8.3(-4)
1.4(-3) 5.0(-5) 1.6(-5)
2.6(-3) 3.0(-4) 4.7(-6)
1.2(-2) 3.3(-3) 6.0(-4)
1.2(-3) 1.5(-4) 3.2(-5)
5.5(-3) 6.7(-5) 1.3(-4)
4.5(-2) 8.1(-3) 7.1(-4)
10 20 42
1.8(-4) 4.8(-6) 7.4(-8)
1.3(-4) 3.6(-5) 2.8(-6)
1.2(-2) 4.0(-3) 1.0(-3)
1.9(-4) 1.8(-5) 1.1(-6)
2.4(-3) 1.6(-4) 8.5(-6)
1.7(-2) 4.6(-3) 5.7(-4)
6.1(-4) 3.9(-5) 2.0(-6)
1.9(-3) 2.0(-4) 1.1(-5)
5.7(-2) 1.4(-2) 2.0(-3)
5. Final remarks
In this paper we have presented a method for generating quadrature rules for (1), based on approximating splines and defined for a wide class of functions f . We have obtained some convergence results and derived an error bound when f E C'-1[ - 1, 1], s/> 1. We remark that an interesting open question concerning these formulas could be to prove convergence results for a larger class of integrand functions f [12]. Now we are investigating this problem and we propose to report some new results in a forthcoming paper.
References [1] C. Dagnino, V. Demichelis and E. Santi, Numerical integration based on quasi-interpolating splines, Computing 50 (1993) 149-163. [2] C. Dagnino, V. Demichelis and E. Santi, An algorithm for numerical integration based on quasi-interpolating splines, Numer. Algorithms 5 (1993) 443-452. [3] C. Dagnino and P. Rabinowitz, Product integration of singular integrands based on quasi-interpolating splines, Comput. Math. to appear, special issue dedicated to 100 birthday of Cornelius Lanczos. [4] C. Dagnino, V. Demichelis and E. Santi, Local spline approximation methods for singular product integration, Approx. Theory Appl. 12 (1996) 1-14. [5] C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences, Vol. 27 (Springer, Berlin, 1978). [6] V. Demichelis, The use of modified quasi-interpolating splines for the solution of Prandtl equation, J. Comput. Appl. Math. 59 (1996) 329-338. [7] V. Demichelis, Quasi-interpolatory splines based on Schoenberg'points, Math. Comput. 65 n.215 (1996) 1235-1247. [8] T. Lyche and L.L. Schumaker, Local spline approximation methods, J. Approx. Theory 15 (1975) 294-325.
238
C. Dagnino, P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231-238
[9] P. Rabinowitz, Numerical integration based on approximating splines, J. Comput. Appl. Math. 33 (1990) 73-83. [10] P. Rabinowitz, Application of approximating splines for the solution of Cauchy singular integral equations, Appl. Numer. Math. 15 (1994) 285-297. [11] P. Rabinowitz and E. Santi, On the uniform convergence of Cauchy principal value of quasi-interpolating splines, Bit 35 (1995) 277-290. [12] P. Rabinowitz, Numerical evaluation of Cauchy principal value integrals with singular integrands, Math. Comput. 55 (1990) 265-276. [13] L.L. Schumaker, Spline Functions (Wiley, New York, 1981). [14] G. Szeg6, Orthogonal Polynomials, AMS Colloq. Publications, Vol. XXIII, 1939.
ELSEVIER
Journal of Computational and Applied Mathematics 76 (1996) 231-238
Numerical evaluation of Cauchy principal value integrals based on local spline approximation operators 1 C. Dagnino *, P. Lamberti Dipartimento di Matematica, Universit°li Studi di Torino, Via Carlo Alberto 10, 1-10123 Torino, Italy Received 3 January 1996; revised 23 July 1996
Abstract
In this paper a local spline approximation method, defined for any function f C L1[-1, 1], is applied to evaluate Cauchy principal value integrals and convergence results with an error bound are established. Some comparisons with other methods and numerical examples are given. Keywords: Cauchy singular integrals; Approximating splines; B-splines AMS classification." 65D30; 65D32
1. Introduction We consider the evaluation of the Cauchy principal value integral (CPV), J(Kf;2)=
K(x)
dx
-1
<2<
1,
(1)
1
where K and f are such that J ( K f ; 2) exists. Recently, several authors [1-4, 6, 7, 9-12] have studied quadratures for (1) defined for any f E C'-1 [_ 1, 1], s ~> 1 and based on the approximation of f by splines o f the form Qf = ~
2,fNi(x),
(1')
where {Ni} is a sequence of B-splines and {2i} is a sequence of linear functionals involving point evaluation of the function f [8]. In this paper we propose the numerical evaluation of (1) by rules once more based on schemes of the form (1'), but defined for a larger class of functions f and * Corresponding author. E-mail: [email protected]. l Work sponsored by "Consiglio Nazionale delle Ricerche" of Italy. 0377-0427/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PH S0377-0427(96)00105-7
C. Dagnino, P. LambertilJournal o f Computational and Applied Mathematics 76 (1996) 231-238
232
involving some modified moments o f f . In Section 2 we present the approximation spline operator, in Section 3 we define the corresponding quadratures for (1) and we prove some convergence results. Finally, in Section 4 we give several numerical applications, where we compare our new quadratures with the old ones [1-4, 6, 7, 9-12].
2. On the approximation spline operator Let Y,: Yo - - 1 < YI < Y2 < "'" < Yn-1 < Y , -- +1 be a partition o f I - [ - 1 , 1] and let h i = y~+l - Y i , i = 0, 1,..., n - 1. We call l"Iu ~-(Xi}iNo a nondecreasing sequence obtained from {yi}in 0 by repeating Yi exactly d; times and so giving N = ~"=-11 di + 1. We say that the sequence o f partitions { Yn) is locally uniform if, for some constant A ~> 1, hi~h/<<. A for all 1 ~< i ~< n - 1 and j = i 4- 1. If k is an integer and k > d i , i = 1 , 2 , . . . , n - 1, we denote by Sk, nN the class of all polynomial spline functions of order k so defined as Sk,11N = {g/g[(yl,yi+l) E [~k, i ----0, 1 , . . . , n -- 1 and 9(J~(y +) = g(J)(YZ),
j = 0, 1 , . . . , k -
d~ - 1, i = 1 , 2 , . . . , n -
1},
where Pk is the set of polynomials o f order k, i.e. of degree < k. Let HNo = "trX;S;=l-klN+k-1be the extended partition associated to HN where x~ < x;+k, i = 1 - k , . . . , N - 1 and - 1 - Xl-k - XZ-k ~ "'" ~ X-1 ~ XO, XN ~ XN+I ~ "'" ~ XN+k--1 ~ 1. We call a sequence o f spline spaces locally uniform if they are based on a sequence of locally uniform partitions. The normalized B-splines {Nik(x)}i=~_ N-~ k of order k [5, 13], associated to FINe form a basis for Sk, l-IN•
For each j = 1 , 2 , . . . , k let /3j be the classical Gauss-Legendre orthogonal polynomial of order j (degree j - 1) [14]. We recall that /3/ has ( j - 1) distinct zeros in ( - 1 , 1): ~ j l , ~ j z , . . . , ~ j j - 1 and P j ( Y ) : q j ( Y - ~yl ) ( Y - ~jz) " " ( Y - ~jj-l) with qj = ( 2 j - 2 ) ! / [ 2 J - ~ ( j - 1)!2]. Moreover, let hj = 2 / ( 2 j - 1) be the normalization constant, i.e. f_l I ~ j ( y ) ~ ( y ) d y 6ju Kroneker's delta. Now, given at < bi such that [ai, bi] C [xi, xi+k], we consider the polynomial pij(x)=
~j \
bi-ai
that we can write as follows p i f i x ) = trij(x - Zijl )(x - zij2 ) " " (x - zijj-1 )
with
and zip=
÷
r=1,2,
...
,j-1.
: fish/, with
C. Daonino, P. LambertilJournal of Computational and Applied Mathematics 76 (1996) 231-238 For any f c L I [ - 1 ,
20f =
ft
1] we can define
/bi_ai I Pj(Y)f ~ Y
bi+ai)dy, 2
+
j = l,2,
""
.,k"
From [8] we remark ~i#Pij : (~j#, j, I~ = 1,2,..., k with support Thus, the following spline operator: N--1
233
(2)
[ai, bi]
and det(~ijx #-1 )j,/~=0k~ 0.
k
Qf(x) = ~
~ ~ij i~;fN,.k(x)
(3)
i=l--k j = l
with j-1 ~ i j = Gij
Z(--|) v
symmv(zi;1,... ,zijj-1 ) s y m m j _ v _ l ( X i + l , .
. . ,Xi+k-1
)
k-1 (j--v--l)
v=O
(Y)
maps Ll[--1, 1] into Sk,n~ and it is local. Moreover, from Theorem 3.3 of [8] it follows that Q is exact for Pk. Generally (3) is not a projector, i.e. it does not reproduce splines. However, if [ai, bi] C [x~,,x~,+l] C [xi,xi+k], i = 1 - k , . . . , N 1, then we can verify Q is a projector of L l [ - 1 , 1] into &,nx. Since the modified moments (2) are required to construct Q f , we assume they can be evaluated accurately, either analytically or numerically.
3. Quadrature rules for CPV integrals (1) based on the local splines (3) We consider the following integration rules for (1): N-1
J ( K f ; 2) _~ J ( K Q f ; 2) = ~
k
~
#oJ(KNik; 2),
(4)
i=l--k j = l
where
Pij = gij " 2gjf . We remark that if K and f E L l [ - 1 , 1] A DT(Na(2)) then (1) exists and the quadratures (4) are defined. We denote by DT(I) a subclass of continuous functions, the functions of Dini type on the interval I, and by Na(2) the interval [2 - 6, 2 + 6], where 6 > 0 is chosen so that Na(2) C ( - 1, 1). Now we will study the convergence of J ( K Q f ; 2) to d ( K f ; 2). For fixed - 1 ~< t ~< 1, let m be such that Xm <<.t < Xm+l. NOW let us introduce the following parameters: 2 m ~---
max
m+l--k <~i<~m+k-- I
Am k--r = "
AN =
min
(xi+l - xi),
rn+l--k+r<<.v<~m
(xv+k-r -- xv),
max ( x i + 1 - - x i ). 1-k~i~N+k--I
c. Dagnino, P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231~38
234
We recall the modulus o f continuity o f a function g E C(I), where I is an arbitrary interval, is given by ~(g; A ; I ) = ~max Ig(x + h) - g(x)l. ,x+hEl O<.h<~A
Moreover, we denote by Im the smallest interval containing [xm,Xm+,] and [ai, bi], with m such that Xm < Xm+,. For any s such that 1 ~< s ~< k we define
SDr(f -Qf)(t), DrQf(t),
Er, s( t )
O<.r
We assume henceforth that AN --~ 0 as N ~ oo. In our discussion we need the following lemmas. We remark L e m m a 1 is a special case o f Theorem 6.1 o f [8]. L e m m a 1. Let 1 <<.s <<.k. I f f --A--r--,
max
t C [Xm,Xm+1]
then f o r 0 <<.r < k
E Cs-l[lm], s
IEr,s(t)l <~ RmA m
1
--
~o(D - f ;Am;lm)
(5)
with R ?n
(s
--
1)!
-
~
~(2pm)J-l'
,= hj
-
where Xi+ k -- X i Pm =
max m+,-k~i~m
hi
- - gli
and Fk~--
(k-r-l)!
[r/2]
"
L e m m a 2. I f pro is uniformly bounded f o r all m and N, then f o r all f E C [ - 1 , 1] (6)
[[f - Qf[[oo ~ O as N ~ oo. Furthermore, if-Am/Am, k-1 is uniformly bounded f o r all N, then max
[DQf(t)[ <<.C,A,,'~(f;Am;Im).
(7)
Xm ~ l ~ Xm+ ]
Proof. From L e m m a 1 with s = 1, r = 0 we have k
max Xm ~t
[ f ( t ) - Qf(t)[ <~ k 2 ~
j=l
~(2p.,)J-'o~(f;-A,.;I.,)
hj
<. C2o (f ;-dN;Z), where C2 is a constant independent on N.
(8)
C Dagnino. P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231-238
235
Therefore (8) implies (6), since f E C [ - 1 , 1] and AN ~ 0 as N ~ c~. If we now take r - - 1, we get from Lemma 1 max
[DQf(t)l <<.k2Fkl .-A"
Xm <~t
/]m,k-1
~
qj ~2,~ -1 co(f;A,,;I,,), ~-~, vm,~J-1A m
(9)
"~.= h j
which implies (7) holds because of the hypothesis on Pm and "Am/Am,k_l. [] From Lemma 2 we can deduce that (i) If [ai, bi] ~- [xv,,xv,+l] C [xi,xi+k] with vi such that Xvi+l - - X v. =
'
max (xi+2+1 --
0~<2~k--1
Xi+.~),~
(10)
then Pm ~ k. Therefore (6) holds. (ii) If the spline spaces {Q f } are locally uniform with constant A for all m and N, then for any
[ai, bi] - [Xv~,X~,+l] C [xi,Xi+k]
(10t)
k-1 A~, and Am~Am.k-1 it results that Pm < 2 ~j=0 <<.Ak_t. Therefore (6) and (7) hold. Lemma 3. Let the spline spaces {Q f } be locally uniform and [ai, bi] be defined by (10') for any i. Then c o ( Q f ; 3N; I ) :
O(co(f; AN; I)).
( 1 1)
Proof. From the above remarks on the operator Q we can prove Q satisfies the hypothesis of Theorem 4 of [10], then from such a theorem we can conclude (1 1) holds. [] Theorem 4. Let {Q f } be locally uniform and [ai, bi],i = 1 - k,... , N - 1 be defined by (10'). Let K E L l [ - 1 , 1] 71DT(N~(2)) and f E C [ - 1 , 1] N DT(N~(2)) for some 2 E ( - 1 , 1) and some 6 such that N ~ ( 2 ) C ( - 1 , 1). I f AN ---+0 as N --~ c~, then
J ( K Q f ; 2) ~ J ( K f ; 2).
(12)
Proof. From the hypothesis it follows that Lemmas 2 and 3 hold. Therefore, the proof of (12) is similar to that of Theorem 8 of [9] and we do not report it here. [] Now we shall consider the quadrature error
E ( K f ; 2) ----J ( K f ; 2) - J ( K Q f ; 2)
(13)
for which in the following theorem we shall derive a bound for sufficiently smooth functions f . Theorem 5. Let 2 <. s <~ k. Let the hypothesis o f Theorem 1 be verified. Then for all f E CS-l[-1, 1] --s-2
s 1
--
E(Kf;)~) = O(d u ~o(D - f ; A u ; I ) ) .
(14)
236
C. Dagnino, P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231-238
Proof. With the help o f the mean-value theorem and Lemma 1 where we put successively r = 0
and r = 1 the thesis follows.
[]
4. C o m p a r i s o n o f C P V quadrature rules and numerical applications
Numerical quadratures for (1) have several interesting applications, particularly they m a y be usefully applied to solve Cauchy singular integral equations [10]. Recently [ 1 4 , 6, 7, 9-12] q.i. spline local methods for (1) defined for any f E C ( I ) have been studied and tested. In Section 3 of this paper we have proposed quadratures (4) for (1) defined for a wide class of functions f . Moreover, we have studied their convergence and derived an error bound. Now we present some numerical applications, obtained when rules (4) and other old ones [1, 9] of the form (1') with the same initial partition Yn, are applied to (1) for several functions f . For our examples we have chosen the following points as Yn partition: U: Uniform, i.e. Y n = { Y i = - - I + ~;i=0,1,...,n}; P: Perfect, i.e. Y , = { y i = c o s ( - u )n - - i Tr, i = 0 , 1 , . . . , n }. Moreover, we assume N = n. We denote by: - E the truncation error (13) of quadratures (4); - E d the truncation error o f the rules proposed in [1]; - E e the truncation error o f the formulas presented in [9]. The Tables 1-4 present the absolute errors for several 2, n and k. The exact test integrals values have been reported in [1], therefore we do not write them here. Table 1 Absolute errors for K(x) = 1, f ( x ) = ,/1 - x 2, k = 4, Yn = U n = 15
n = 31
n = 63
)~
[El
IEal
IERI
IEI
IEal
IERI
IEI
IEal
IERI
0.3 0.4 0.5 0.6 0.7 0.8
9.5(-4) 1.3(-3) 1.9(-3) 2.7(-3) 4.3(-3) 8.0(-3)
5.4(-3) 9.7(-3) 1.7(-2) 3.1(-2) 2.2(-2) 4.2(-2)
6.7(-3) 9.6(-3) 1.3(-2) 1.8(-2) 2.5(-2) 3.2(-2)
2.8(-4) 4.2(-4) 5.9(-4) 8.4(-4) 1.2(-3) 2.1(-3)
1.0(-3) 1.6(-3) 2.4(-3) 3.9(-3) 7.4(-3) 2.1(-2)
2.4(-3) 3.5(-3) 4.8(-3) 6.8(-3) 9.8(-3) 1.5(-2)
9.7(-5) 1.4(-4) 1.9(-4) 2.8(-4) 4.1(-4) 6.8(-4)
2.6(-4) 3.9(-4) 5.7(-4) 8.7(-4) 1.4(-3) 2.9(-3)
8.6(-4) 1.2(-3) 1.7(-3) 2.4(-3) 3.5(-3) 5.6(-3)
Table 2 Absolute errors for K(x) = 1, f ( x ) = (y2 _ x2)-1/2, y = 1.1, k = 4, Yn= U n = 13
n = 17
n = 63
2
]EI
IEal
[ER]
]EI
lEa[
]ER[
[EI
lEa[
[ERI
0.25 0.99
5.8(-4) 3.7(-2)
1.2(-2) 2.2(-2)
1.3(-2) 2.3(-1)
2.1(-4) 1.6(-2)
5.8(-3) 3.9(-3)
9.0(-3) 1.7(-1)
2.1(-6) 5.0(-4)
1.2(-4) 3.4(-3)
9.3(-4) 3.2(-2)
C. Dagnino, P. LambertilJournal of Computational and Applied Mathematics 76 (1996) 231-238 Table 3 Absolute errors for K(x) = (1 - x2) -1/2,
f(x)
n=5
=
237
(X2 -]- y2)--1, y = 5, k = 4, Y, = U
n = 13
n=23
A
IEI
[Edl
IERb
IEI
IEdl
IERI
IEI
lEd[
IERI
0.25 0.99
9.5(-8) 4.0(-6)
7.5(-6) 4.1(-6)
4.8(-5) 2.9(-4)
9.7(-10) 1.2(-7)
8.3(-8) 2.6(-7)
4.5(-6) 7.1(-5)
1.0(-9) 1.5(-8)
7.5(-9) 3.8(-8)
1.2(-6) 2.9(-5)
Table 4 Absolute errors for K(x) = 1, f ( x ) = ex, Yn = P 2 -----0.1
k
2 = 0.5
2 = 0.9
n
Igl
IEal
IERI
IEI
IEal
IERI
[El
IUI
IERI
10 20 42
1.5(-3) 3.1(-4) 4.4(-5)
5.7(-3) 5.1(-4) 1.5(-5)
1.2(-2) 3.3(-3) 8.3(-4)
1.4(-3) 5.0(-5) 1.6(-5)
2.6(-3) 3.0(-4) 4.7(-6)
1.2(-2) 3.3(-3) 6.0(-4)
1.2(-3) 1.5(-4) 3.2(-5)
5.5(-3) 6.7(-5) 1.3(-4)
4.5(-2) 8.1(-3) 7.1(-4)
10 20 42
1.8(-4) 4.8(-6) 7.4(-8)
1.3(-4) 3.6(-5) 2.8(-6)
1.2(-2) 4.0(-3) 1.0(-3)
1.9(-4) 1.8(-5) 1.1(-6)
2.4(-3) 1.6(-4) 8.5(-6)
1.7(-2) 4.6(-3) 5.7(-4)
6.1(-4) 3.9(-5) 2.0(-6)
1.9(-3) 2.0(-4) 1.1(-5)
5.7(-2) 1.4(-2) 2.0(-3)
5. Final remarks
In this paper we have presented a method for generating quadrature rules for (1), based on approximating splines and defined for a wide class of functions f . We have obtained some convergence results and derived an error bound when f E C'-1[ - 1, 1], s/> 1. We remark that an interesting open question concerning these formulas could be to prove convergence results for a larger class of integrand functions f [12]. Now we are investigating this problem and we propose to report some new results in a forthcoming paper.
References [1] C. Dagnino, V. Demichelis and E. Santi, Numerical integration based on quasi-interpolating splines, Computing 50 (1993) 149-163. [2] C. Dagnino, V. Demichelis and E. Santi, An algorithm for numerical integration based on quasi-interpolating splines, Numer. Algorithms 5 (1993) 443-452. [3] C. Dagnino and P. Rabinowitz, Product integration of singular integrands based on quasi-interpolating splines, Comput. Math. to appear, special issue dedicated to 100 birthday of Cornelius Lanczos. [4] C. Dagnino, V. Demichelis and E. Santi, Local spline approximation methods for singular product integration, Approx. Theory Appl. 12 (1996) 1-14. [5] C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences, Vol. 27 (Springer, Berlin, 1978). [6] V. Demichelis, The use of modified quasi-interpolating splines for the solution of Prandtl equation, J. Comput. Appl. Math. 59 (1996) 329-338. [7] V. Demichelis, Quasi-interpolatory splines based on Schoenberg'points, Math. Comput. 65 n.215 (1996) 1235-1247. [8] T. Lyche and L.L. Schumaker, Local spline approximation methods, J. Approx. Theory 15 (1975) 294-325.
238
C. Dagnino, P. Lamberti/Journal of Computational and Applied Mathematics 76 (1996) 231-238
[9] P. Rabinowitz, Numerical integration based on approximating splines, J. Comput. Appl. Math. 33 (1990) 73-83. [10] P. Rabinowitz, Application of approximating splines for the solution of Cauchy singular integral equations, Appl. Numer. Math. 15 (1994) 285-297. [11] P. Rabinowitz and E. Santi, On the uniform convergence of Cauchy principal value of quasi-interpolating splines, Bit 35 (1995) 277-290. [12] P. Rabinowitz, Numerical evaluation of Cauchy principal value integrals with singular integrands, Math. Comput. 55 (1990) 265-276. [13] L.L. Schumaker, Spline Functions (Wiley, New York, 1981). [14] G. Szeg6, Orthogonal Polynomials, AMS Colloq. Publications, Vol. XXIII, 1939.