AMS Subj. Classiﬁcation: 65D07, 65D30.We gave a short review of several results which are related to the role of splines
(cardinal, centered or interpolating) in numerical integration. Results deal with the problem
of approximate computation of the integrals with spline as a weight function, but also with
the problem of approximate computation of the integrals without weight function. Besides, we
presented an algorithm for calculation of the coeﬃcients of the polynomials which correspond
to the cardinal B-spline of arbitrary order and described ﬁve methods for calculation of the
moments in the case when cardinal B-spline of order m,m ∈ N, is a weight function

Udovičić, Zlatko

Bulgarian Digital Mathematics Library at IMI-BAS

M a t h e m a t i c aB a l k a n i c a—————————New Series Vol. 24, 2010, Fasc.3-4Splines in Numerical IntegrationZlatko Udovicˇic´We gave a short review of several results which are related to the role of splines(cardinal, centered or interpolating) in numerical integration. Results deal with the problemof approximate computation of the integrals with spline as a weight function, but also withthe problem of approximate computation of the integrals without weight function. Besides, wepresented an algorithm for calculation of the coefficients of the polynomials which correspondto the cardinal B-spline of arbitrary order and described five methods for calculation of themoments in the case when cardinal B-spline of order m,m ∈ N, is a weight function.AMS Subj. Classification: 65D07, 65D30.Key Words: cardinal B-spline, coefficients, moments, rectangular rule, interpolating quadraticspline, hat function, cubic B-spline.1. IntroductionIn the approximation theory, cardinal B-splines have a very importantplace (in different methods for solving initial and boundary value problems,spline interpolation, multiresolution approximation...). The main topic of theauthor’s PhD thesis is the use of the cardinal B-splines in numerical integration.The obtained results can be grouped in two ways, depending on the type ofintegral under consideration or the order of the spline applied.The first grouping refers to the results where the problem of approximatecomputation of the integral(1.1)∫ m0ϕm(x)f(x)dx(integrals with cardinal B-spline of order m as a weight function) is consideredand those where the classical problem of numerical analysis, i.e. the problem ofapproximate computation of the integral(1.2)∫ baf(x)dxis considered.352 Zlatko Udovicˇic´In organizing this paper (as well as in the PhD thesis) we opt for thesecond possibility. In the next section, we deal with splines of arbitrary order,while the final one presents the results related to the splines of a low order.Let us recall here the definition of the cardinal B-spline and list its basicproperties.Definition 1. Cardinal B-spline of the first order, denoted by ϕ1(·), is thecharacteristic function of the interval [0, 1), i.e.ϕ1(x) ={1, x ∈ [0, 1)0, otherwise.Cardinal B-spline of order m,m ∈ N, denoted by ϕm(·), is defined as a convolu-tionϕm(x) = (ϕm−1 ∗ ϕ1) (x) =∫Rϕm−1(x− t)ϕ1(t)dt=∫ 10ϕm−1(x− t)dt.Theorem 1. Cardinal B-spline of order m,m ∈ N, has the following properties(1) suppϕm(·) = [0,m];(2) ϕm(·) ∈ Cm−2[0,m];(3) at each interval [k, k + 1], 0 ≤ k ≤ m − 1, cardinal B-spline of order mis a polynomial of degree equal to m− 1;(4)(1.3) (∀t ∈ [0,m])ϕm(t) = tm− 1ϕm−1(t) +m− tm− 1ϕm−1(t− 1),m ≥ 2;(5)(1.4) (∀t ∈ [0,m])ϕ′m(t) = ϕm−1(t)− ϕm−1(t− 1),m ≥ 2;(6) Cardinal B-spline is symmetric at the interval [0,m], i.e.(∀t ∈ [0,m])ϕm (t) = ϕm (m− t);(7)(1.5) (∀a ∈ R)∑i∈Zϕm(i− a) = 1;(8) For any m time differentiable function g(·) holds∫Rϕm(x)g(m)(x)dx =m∑k=0(−1)m−k(mk)g(k);Splines in Numerical Integration 353(9) The cardinal B-spline is a solution of the so-called dilatation equation(∀t ∈ R)ϕm(t) = 12m−1m∑k=0(mk)ϕm(2t− k).The proof of the theorem, and more details about cardinal B-splines onecan find in [1] or [2].2. Splines of arbitrary order2.1. Calculation of the coefficients of the cardinal B-spline. The firstresult presented in this paper is an algorithm for calculation the coefficients ofthe polynomials which correspond to the cardinal B-spline of order m. By usingequalities (1.3) and (1.4) it is easy to obtain differential equation(m− x)ϕ′m(x) + (m− 1)ϕm(x) = mϕm−1(x).Evidently, the solution of this equation is a cardinal B-spline of orderm. Furthermore, let x ∈ [k, k + 1] and let ϕm(x) =m−1∑i=0a(m,k)i xi. By using theprevious differential equation and the symmetry of the cardinal B-spline one canprove that the following reccurence relations hold:a(m,0)m−1 =1m− 1a(m−1,0)m−2 ,a(m,0)i = 0,m− 2 ≥ i ≥ 0,a(m,k)m−1 =1(m− 1)(m− k)[ma(m−1,k)m−2 − k(m− 1)a(m,k−1)m−1−a(m,k−1)m−2],a(m,k)i =mi+ 1−m[(i+ 1)a(m,k)i+1 − a(m−1,k)i],m− 2 ≥ i ≥ 0,a(m,m−k−1)m−1 = (−1)m−1a(m,k)m−1 ,a(m,m−k−1)j =(−1)jj!m−j−1∑i=0(i+ 1)(i + 2) . . . (i+ j) · a(m,k)i+j mi,0 ≤ j ≤ m− 2.Those relations enable us to construct an algorithm for calculating thecoefficients of the cardinal B-spline of order m.354 Zlatko Udovicˇic´2.2. Calculation of the moments of the cardinal B-spline. Calculatingmoments of the given weight function has essential significance in constructionof the orthogonal polynomials and quadrature rules, as well as in other fieldsof the approximation theory. On the other hand, a very frequent weight func-tion is exactly the cardinal B-spline (finite elements method, multiresolutionapproximation,...). In the paragraph below, five methods (without proofs) forcalculation of momentsMn,m =∫ m0ϕm(t)tndt, n ∈ N0.are provided.(1)Mn,m =12m(2n − 1)m∑k=1n−1∑l=0(mk)(nl)kn−lMl,m;(2)Mn,m =n∑k=0(nk)Mk,m−1 − 1m− 1n−1∑k=0(nk)Mk+1,m−1= 1 +n∑k=1[(nk)− 1m− 1(nk − 1)]Mk,m−1;(3)Mn,m =1n+ 1n∑k=0(n+ 1k)Mk,m−1;(4)Mn,m =m∑k=0(−1)m−k(mk)km+n(m+ n)(m+ n− 1) . . . (n+ 1) .The last method is, of course, the most important result of this para-graph. This method generalizes equality (1.5) and we will formulate it as atheorem.Theorem 2. For any a ∈ R and every m ∈ N,m ≥ 2, the following equalityholdsMn,m =∑i∈Zϕm(i− a)(i− a)n, 0 ≤ n ≤ m− 1.In the case when a is an integer and m is odd, one can easily check thatthe previous equality also holds for n = m. Those results were recently published[8].Splines in Numerical Integration 3552.3. One point quadrature rule with cardinal B-spline. In this paragraphwe are presenting probably the most important result of our research which waspublished in [7]. The basic formula is a classical one point quadrature rule(2.6)∫ baf(x)dx ≈ (b− a)f(X),with choosen point X = (1− λ)a+ λb, for some λ ∈ [0, 1].Let x1 < x2 < . . . < xp−1 be an arbitrary points from the interval (0, 1)and let the partition0, x1, x2, . . . , xp−1,1, x1 + 1, x2 + 1, . . . , xp−1 + 1,...............m− 1, x1 +m− 1, x2 +m− 1, . . . , xp−1 +m− 1,m,of the interval [0,m] be given. We call this kind of mesh “quaziuniform”. Fur-thermore, let the points Xk+ i, 0 ≤ i ≤ m−1, where Xk = (1−λk)xk+λkxk+1,for some λk ∈ [0, 1], 0 ≤ k ≤ p − 1, at each interval of the given partition, bechosen (naturally, x0 = 0 and xp = 1).At each interval of partition, we will use the formula (2.6) with chosenpoints Xk + i, 0 ≤ i ≤ m − 1, 0 ≤ k ≤ p − 1, to approximately compute theintegral (1.1). After summation, we will obtain(2.7)∫ m0ϕm(x)f(x)dx ≈p−1∑j=0(xj+1 − xj)m−1∑i=0ϕm(Xj + i)f(Xj + i).Theorem 3. If f(x) = xn, 0 ≤ n ≤ m − 1, then the quadrature rule (2.7) isexact, for each m ∈ N.In accordance with the previous theorem formula (2.7) has, conditionallyspeaking, algebraic degree of exactness equal to m − 1. In some special casesthis degree of exactness can be improved up to m.2.4. Numerical integration using cardinal B-splines. One of the classicalproblems in the mathematical analysis is, of course, the calculation of the inte-gral (1.2). Since this integral cannot, in general, be calculated exactly, there isa large number of methods for its approximate computation. We are presentinghere a new method which is obtained by using cardinal B-splines. The main ideais to approximate the function f(·) by its projection on the cardinal B-splinespacef̂(x) =∑k∈Zckϕ(j,k)m (x),356 Zlatko Udovicˇic´where ϕ(j,k)m (x) = 2j2ϕm(2jx − k). The coefficients of the approximation aresolution of the corresponding system of linear equations. This approach leavesm − 2 coefficients free, so additional conditions can be chosen depending onthe concrete situation. One possibility is to choose some integer nodes to havemultiplicity two. Finally, one obtains∫ m0f(x)dx ≈∫ m0f̂(x)dx = 2−j2(−1∑k=−m+1ck(1− µ−k0,m)+(2j−1)m∑k=0ck +2jm−1∑k=(2j−1)m+1ckµ2jm−k0,m ,whereµxn,m =∫ m0ϕm(t)tndtare the so-called shortened spline moments. Those moments, for example, canbe calculated recursivelyµxn,1 =∫ x0ϕ1(t)tndt =0, x < 0xn+1n+1 , x ∈ [0, 1]1n+1 x > 1,µxn,m =∫ x0ϕm(t)tndt=xn+1ϕm(x)n+ 1− 1n+ 1∫ x0ϕm−1(t)tn+1dt+1n+ 1∫ x−10ϕm−1(t)(t+ 1)n+1dt=1n+ 1(xn+1ϕm(x)− µxn+1,m−1 +n+1∑k=0(n+ 1k)µx−1k,m−1).This result was inspired by the paper [5].3. Splines of given (low) order3.1. Some modifications of the trapezoidal rule. This section begins withthe result recently published in [6]. The main idea is based on the fact that theconstruction of the quadratic interpolating spline leaves one parameter free. Byan appropriate choice of the free parameter we obtain reproduction of some wellknown quadrature rules, but also a wide class of new quadrature rules.Splines in Numerical Integration 357Except the reproduction of the Simpsons and Ermits quadrature rule, weobtain the following formulas:∫ baf(x)dx ≈ h2(f0 + 22m−1∑k=1fk + f2m)− h36m∑k=1f ′′2k−1;≈ h2(f0 + f1)− h3f ′′(X)12+QS[f, x1, b, 2m];≈ h2(f0 + 22m∑k=1fk + f2m+1)− h36(f ′′(X)2+m∑k=1f ′′2k);≈ h2(f0 + f1)− h212(f ′1 − f ′0) +QS [f, x1, b, 2m];≈ QT [f, x0, x1]− h312[f ′′(X) + f ′′′(X)(x0 +h2−X)]+QS[f, x1, b, 2m].In the second and the third formulas X is any point from the interval [x0, x1]such that f ′′(X) = λf ′′0 + (1 − λ)f ′′1 , where λ ∈ [0, 1], while in the last formulaX is any point from the interval [x0, x1]. Furthermore, QT and QS denote aclassical trapezoidal and the Simpsons rule respectively. The accuracy of theobtained formulas is O(h4).3.2. A certain class of quadratures with specific weight function. Thelast result was inspired by the papers [3] and [4], and it is about the so-calledquadrature formulas of “practical type”. Hence, we are considering the quadra-ture formula ∫ m0ϕm(x)f(x)dx ≈5∑i=1Aif(xi).This formula is of “practical type” if the following conditions hold:• coefficients Ai, 1 ≤ i ≤ 5, are symmetric, i.e. (A1 = A5 and A2 = A4);• nodes xi, 1 ≤ i ≤ 5, are symmetric, rational numbers from the interval[0,m].We proved that in the case when the weight function is cardinal B-splineof order two (hat function), i.e. in the case when the weight function is acardinal B-spline of order four (cubic B-spline), the maximal algebraic degreeof exactness of such type of quadratures is equal to five.358 Zlatko Udovicˇic´References[1] C . C hu i , An Introduction to Wavelets, Academic Press, inc., Boston SanDiego New York London Sydney Tokyo Toronto, 1992.[2] C . C h u i , Wavelets: A mathematical tool for signal analysis, Society forIndustrial and Applied Mathematics, Philadelphia, 1997.[3] E . C o n s t a n t i n e s c u , A certain class of quadratures, Acta Univ. Apu-lensis Math. Inform., 7, 2004, 119-116.[4] E . C o n s t a n t i n e s c u , A certain class of quadratures, Gen. Math., 10,No. 1-2, 2002, 75-84.[5] H . H a s h i s h , S . H . B e h i r y , N . A . E l - S h amy , Numerical Inte-gration Using Wavelets, Appl. Math. Comput., 2009[6] Z . Ud o v i cˇ i c´ , Some modifications of the trapezoidal rule, Sarajevo J.Math., 2(15), 2006, 237-245.[7] Z . Ud o v i cˇ i c´ ,One point quadrature rule with cardinal B-spline, J. Math.Anal. Appl., 360, 2009, 432-438.[8] Z . Ud o v i cˇ i c´ , Calculation of the moments of the cardinal B-spline, Sara-jevo J. Math., accepted.[9] Z . U d o v i cˇ i c´ , A certain class of quadratures with hat function as a weightfunction, Acta Univ. Apulensis Math. Inform., accepted.Faculty of Sciences,Department of MathematicsZmaja od Bosne 3571000 SarajevoBosnia and HerzegovinaE-Mail: zzlatko@pmf.unsa.ba

Splines in Numerical Integration

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