SAKAI, Manabu

SHIMAUCHI, Hiroyuki

島内, 博行

酒井, 宦

English

Kagoshima University Repository

Rep. Fac. Sci. Kagoshima Univ. (Math., Phys. & Chem.),No. 27, 27-33, 1994Improvement of Numerical Integration Formulasby Iterated Cubic SplinesManabu SAKAIO and Hiroyuki SHIMAUCHI2(Received September 2, 1994)AbstractWe improve numerical integration with a fixed number of evaluation points basedon iterated cubic splines. Some numerical examples are given to illustrate usefulnessof our methods.1. Introduction and Description of Methods●                                                                      ●Iterated cubic splines are useful for order-preserving approximation to a given function.There is computational evidence that they give better results than a single spline ( [1], [3]).Recently we have considered an application of the iterated cubic splines to evaluation ofintegrals on subintervals [xjf xj+i] or the ratios to the whole interval [0, 1] required instatistics:f(x)dxorIf(x)dx/If(x)dxforsomeorallj(0<;xiJXjJo≦サーl)whereletn≧1,xj-jh(-j/n)([5]).Usethenotation:fj-f(xj),smj-sm(xj),s㌫J-S㌫Uy)andfj+i/2-f((xj+xj+i)/2).Then,theiteratedcubicsplinessm(m≧0)arerecursivelydefinedasfollows.First,letsobetheusualcubicsplineinterpolantof/ontheuniformpartitionof[0,1]withknotsxjfi.e.,(2)S｡j-fj(0≦j≦0subjecttoAk<?｡,0-vkS｡,n-owherefromnowon,k∈to,i,…,n-liisfixedandA(V)istheforward(backward)differenceoperator.Next,letsm(m≧1)bethecubicsplineinterpolantofs㌫-ionthesameuniformpartition,Le.,Department of Mathematics, Faculty of Science, Kagoshima University, 1-21-35 Korimoto,Kagoshima 890, JapanDepartment of Child Education, Faculty of Home Economics, Nakamura Gakuen University,5-7-1 Befu, Jonan-ku Fukuoka 814-01, Japan28(3)Manabu SAKAI and Hiroyuki SHIMAUCHIsinj S㌫-lj (0≦j≦n) subjectto Aks㌫0-17*5㌫*-O.In computation of the iterated cubic splines sm (m≧0), it is convenient to rewrite the endcondition Ak s㌫0-0 as follows:(4) ck S㌫o+5ん,i-Lk(di, d2,..., dk-i)where for ck, Lk and computational comment, see [3] or [5]. The end condition A sん,-oused in the numerical examples is equivalent to(5)　　(71/265) sん,｡+S㌫ 1- (92017rfi-24637d2+6567d3-1715rf4+419d5-87d6+ 13d7-rf8) /57240where dj is the right hand side of the consistency relation for the cubic spline:(6)　　　　　　(sm.J+l+4s㌫j+S^,;-1)/6- ¥sm,j+i Snトd/(2h) (-rfy).For a periodic function メ the end conditions (2)-(3) are to be replaced by(7)　　　　　　　　　　　　　鑑)- (r) (r¥≦r≦2,m≧0).The following asymptotic error estimates for the m-th derivative f{m) by the m-th iteratedcubic spline sm are based on results in [3] (nonperiodic) and [6] (periodic) where CHO, 1]denotes the set of periodic functions in C9(-∞, ∞) with period one.Lemma ([31-[　For l≦m≦9 andf∈Cp [0, 1] (periodic case) and l≦m≦k≦n-1, k≦9 and f∈CIO[0, 1] (nonperiodic case), then(8)srn,j-fj-f(m)-mh6芸f(-+4)-Ji扇 j　　トcmhsf/m+4)+O(hL) (o≦j≦n)where ci-l/25920, L-10-m or L-k+l-mfor♪eriodic case or nonperiodic case. Note thatthe hs term is absorbed into the order term for m≧2.By making use of the iterated cubic splines sm as apporximation to the derivatives in theasymptotic error estimate in Simpsons rule:(9)　/ /(*)dx-(A/6)(/,+4/W+/-)-X (-Dkh2k+2 CkAf/2k+1)+O(hn),Improvement of Numerical Integration Formulas by Iterated Cubic Splines　　　　　29we can get an improvement of the rule based on the iterated splines:ml(10)　　SmAh) - (A/6) (/i+4/,+1/2+/i+i) + ∑ (-1)k h2k+2百As2k+u (0≦m≦3)k=lwhere (Cu C2, C3)-(1/2880, 1/96768, 1/3686400) and (G, C2, C3)-(1/2880, 1/96768,67/1059200). For the errors in Smj(h), we obtainTheorem 1. Let k∈ (2m+3, 2m+4,..., n-¥) and k≦9 be fixed. (This restriction on k,defined in (2) and (3), is necessary in the nonperiodic case only.) Iff∈Ci-[0, 1] or ∈CIO[0,1],thenf(x)dx-SmJ(h) -O(h2m+5) (o≦m≦3).Similarly as in Simpson's rule, the asymptotic relation is known for the midpoint one:(ll)fxt+1f(x)dx-h/fi+1/2- Z (-1)k+1 h2k Dk Af/2k-サ+O(h9)from which we get the following formulas:●m(12)　　　　　　MmJ(h) -hfj+U2+ ∑ (-Dk+1 h2kDkAs2k-U (0≦m≦3)k=lwhere (A, D2, D3) - (1/24, 7/5760, 31/967680) and (Du D2, D3)- (1/24, 7/5760, 17/64512).For the errors in MmJ(h), we obtainTheorem 2. Let k∈ {2m+l, 2m+2,…, n-1) and k≦7 be fixed. (This restriction on k,defined in (2) and (3), is necessary in the nonperiodic case only.) Iff∈Cf[0, 1] or ∈C8[0, 1],thenf(x)dx-MmJ(h) -O(h2m+3) (o≦m≦3).Proof. Under the condition on /c, m and / in Theorem 2, we have only to check(13)h6180"　l扇蝣f/-+6)]+O(hL) (o<j<n)where L-8-m or L-k+¥-m for periodic case or nonperiodic case.30 Manabu SAKAI and Hiroyuki SHIMAUCHIWe can also consider improvement of the product trapezoidal rule when w(x) -xα(α>-1) or ln(x) for which the following asymptotic error formula is obtained:(14)P+1w(x)f(x)dx-h{po(j)fj+qo(j)fj+i)Jxi3-∑h2m{pm(j)ffm-v+qm(j)mP)}+0(h9+minW-α)orO(h9In(/?)).m=¥Here, first coefficients qm (-qm(j)) (0≦m≦3) are determined by substitution of (x-xj)2m(1≦m≦3), (x-xj)7 into (14) without order terms, and next coefficients ♪ (-pm(j)) aresuccessively done by substituting (x-x})2m l (1≦m≦3) as follows:1517/wo= -4c7+14c6-35c4+42c2, 34hql=4c7-Uc6+35c4-25c2204hq2= -2c7+7c6-9Ci+4c2, 12240hq3-12c7-25c6+20c4-7c217hpo=4c7-Uce+35c4-42c2+17cof 34hp1-4c7-Uc6+35c4-59c2+34c1204/^2= -2c7+7c6-26c4+34c3-13c2,12240/#3= 17c7-59c6+102c5-65c4+10c2where cm(-cm(j)) - / dm w(xj+hd)dd are successively determined as follows:for w(x)-xα, h(l+α)co-xj+¥　Xj α, h(m+l+α)cm-x^1a-mjhcn_1 (m≧1);for w{x) -¥n{x)f hco-xj+i ln(.r;+i) -Xj ¥n(xj) -h,h(m+l)cm-xj+i ln(xj+i) -mjhcm-i+mh/(m+1) (m≧1).For α-0, (Po, tu tt, *s)- (flo, -Qu -Q2, -qz)- (1/2, -1/12, 1/720, -1/30240), i.e., (14)reduces to a special case of the well-known Euler-Machaurin summation formula. Use sm asan approximation of f(m) to give the following integration formulas:m(16)　TmAh) -h{p｡(j)fi+qo(j)fj+i} + ∑ h2k{pic(j)s2k-ij+qk(j)s2k-ij+i} (0≦m≦3)k=lwith (pk(j), qk(j))-{pk{j), qk{))) (k-l, 2) and (p3(j), q3(j))-(p3(j), Qzi)))+(pi(j),tfi(/'))/180. As in the proof of Theorem 2, we haveTheorem 3. Under the same assumption on k, m and f in Theorem 2,:x'+1w(x)f(x)dx-TmJ(h)-o(h2m+3+min(0-x)α,)OrOU2w+3|ln(/0|)(0≦m≦3)Improvement of Numerical Integration Formulas by Iterated Cubic Splines　　　　　31for w(x)-xα (α>-1) or ¥n(x).Similarly we improve the trapezoidal rule for integrals of the form /3:i+lw(x)f(x) dx withan oscillatory weight w(x)-cos(kx) or sin(kx) for a relatively large value of k. Forcos(kx) or sin(kx), cm(-cm(j)) -17Or6m w(xj+h6)dd are determined byhkco(j) -sin(kxj+1) -sin(/ex,), h2 k2 Ci(j) -hk sm(kxj+i) +cos(kxj+i) -cos(fcr,),h k2cm(j)-hksin(kxj+i)+mcos(kxj+i)-m(m-1)cm-2(j) ¥m≧2)hkco(j) -cos(kxj) -cos(fcr;+i), h2 k2 Ci(j) - -hk cos(kxj+i) -sin(kxj+i) +sin(kxd18h k2cm(j)--hkcos(kxj+i)-msin(kxj+i)+m(m-1)cm-2(j) (m≧2).2. Numerical ExamplesFirst, we consider an application of the above stated numerical formulas Smj(h) andMm>j(h) by taking two functions fix) -exp(5x) and sin(47T,r). In Tables 1-4, are given theobserved maximum absolute errors in the formulas on subintervals [xj, xj+i] (0 ≦j≦n-1) and the observed orders of convergence from the numerical results with n-32, 64.Figures in parentheses behind the observed orders of convergence are the theoretical onespredicted in Theorems where a-b means ax 10 ¥　For reference, the absolute errors in thecomposite formulas on the whole interval [0, 1] using Sm,,(l/64) (or.Mm,y(l/64)), 0≦m≦3can be improved with a fixed umber of the evaluation points as 3.81-7 -* 6.90-ll -* 2.85-13→ 1.15-14 (or 7.50-3 - 1.33-6 - 1.76-9 - 1.92-12) where the orders of convergence in thecomposite rules using Smj(h) (or MmJ(h)) are approximately equal to the theoretical ones,2m+5 (or 2m+3), respectively. Next, we consider an application of the product trapezoidalrule when w(x)-l/､斥¥n(x) and 1 to the same functions exp(5x) and sin(4?r.r). We useT3j(h/2) as the unknown exact values to bound the errors in TmJ(h) (0<m<2). In Table4, note that the theoretical rates of convergence are different from the others since themaximum absolute errors occurred near at x-0. Finally we improve the product trapezoid-al rule for evaluating two integrals [2J:(19)　X(20)flJoU-2)-1(1-x)~1M(1+x)-3/4 dx (- -1.944905429166746...)exp(ux)cos(kx)dx (-exp(u)(ucosk+ksink)-u)(u2+k2) v (ォ-1, 5)32 Manabu SAKAI and Hiroyuki SHIMAUCHIThe integral (19) whose weight is singular at both end-points is subdivided at ∫-0 producingtwo integrals with a single end-point singularity. For (19), the absolute observed errors inn-¥the composite trapezoidal rule　∑ Tm>j(h) with n-32, i.e., 33 function evaluation points on;-0[-1, 1] were significantly improved as 1.82-5 -> 1.43-7 -> 4.40-9 -+ 1.28-10 for O<m<3.For (20) with 17 evaluation points, the observed absolute errors in evaluation ofJ 0exp (〝∬)×cos(kx)dx with k-l, 10, 102, 103, 104 were given in Table 5. Our methods with a fixednumber of evaluation points would be useful when finer meshes are not acceptable.Table 1. Comparison of the maximum absolute errors in the numerical evaluation of/.xj+lf(x)dx using i-proved Simpson's rules Sm>j(h) (0<;≦ 1-1)fix)　　　　　　nonperiodic exp (5x)　　　　　　　　　　periodic sin i47tx)n＼m16　　　2.63-5　　5.34-8　　2.10-8　　1.81-8　　7.57-6　　1.89-7　　5.44-8　　5.83-932　　　　8.88-7　　6.19-10　　2.45-11　1.64-11　2.52-　　　1.26-9　　1.07-10　　2.73-1264　　　2.88-8　　5.21-12　　2.77-14　　9.66-15　　8.02-9　　9.40-12　　2.09-13　1.31-15orders　　4.9(5)　6.9(7)　9.8(9) 10.7(ll)　5.0(5)　7.1(7)　9.0(9) ll.0(ll)Table2.Comparisonofthemaximumabsoluteerrorsinthenumericalevaluationof/.xj+lf(x)dxusingi-provedmidpointrulseMmJ(h)(0<jxj≦-1)fix)　　　　　　nonperiodic exp (5x)　　　　　　　　　　periodic sin (47rx)n＼m o 116　　　3.23-2　　9.02-532　　　　4.37-3　　3.09-664　　　　5.67-4　　1.01-7orders　　2.9 (3)　4.9 (5)2　　　　　　3　　　　　　0　　　　　1　　　　　2　　　　　　31.98-6　　8.45-8　　1.47-3　　9.97-　　　3.90-6　　4.56-71.65-8　　8.61-11　1.97-4　　9.10-　　　3.03-8　　8.68-101.34-10　1.56-13　　2.50-5　　2.83-8　　2.36-10　1.68-126.9(7)　9.1 (9)　3.0(3)　5.3(5)　7.0(7)　9.0(9)Table 3. Comparison of the maximum absolute errors in the numerical evaluation ofw(x)sin(4:7zx)dx using improved product trapezoidal rules Tmj(h) (0≦j≦n-1)w(x)　　　　　xト咋　　　　　　　　InU)n＼m16　　　6.5ト　　1.03-4　　3.56-6　　5.00-2　　9.09-6　　2.88-8　　6.47-2　1-02-4　　3.81-632　　　8.63-3　　3.53-6　　3.06-8　　6.25-4　　2.91-7　　2.25-9　　8.73-3　　3.52-6　　3.ll-864　　　1.54-3　　3.31-7　　4.68-10　1.28-4　　4.56-8　　5.39-ll 1.13-3　1.15-7　　2.51-10orders　2.5(2.5) 3.4(4.5) 6.0(6.5) 2.3(2.7) 2.7(4.7) 5.4(6.7) 2.9(3)　4.9(5)　7.0(7)Improvement of Numerical Integration Formulas by Iterated Cubic Splines　　　　　33Table 4. Comparison of the maximum absolute errors in the numerical evaluation ofw(x)exp(5.r)dx using improved product trapezoidal rules Tmj(h) (0≦j≦n-1w(x)　　　　　　咋　　　　　　　　InU)n＼m16　　　9.55-3　　7.99-5　1.32-7　　7.15-3　　7.08-5　1.0ト8　　2.92-3　　3.68-5　　7.23-632　　　1.97-　　2.53-6　1.69-9　　8.98-　　2.24-6　　8.00-10　3-92-　　1.06-6　　5.68-864　　　5.52-4　1.29-7　　2.92-ll 1.60-4　　7.ll-8　　9.54-12　4.99-5　　3.25-8　　4.44-10orders 1.8(2.5) 4.3(4.5) 5.9(6.5) 2.5(2.7) 4.7(4.7) 6.5(6.7) 3.0(3)　5.0(5)　7.0(7)Table 5. Comparison of the maximum absolute errors in the numerical evaluation ofn-¥exp (ux) cos (kx) dx using composite improved product trapezoidal rules ∑ TmAh);-o〝　　　　　　　　　　　　　　　　1　　　　　　　　　　　　　　　　　　　　　　　　　　5机m3.84-4　　2.73-8　　3.50-11　2.03-14　1.55-1　　2.49-4　　8.77-6　　3.45-810　　　2.33-4　　9.0ト　　　2.41-11　1.40-14　1.18-　　　1.60-4　　6.63-6　　1.13-7102　　　1.88-3　　6.45-8　　1.63-10　　9.00-14　　2.70-1　　2.94-　　1.47-5　　1.07-7103　　　9.30-8　　3.10-11　8.61-15　　4.88-18　1.61-4　　4-36-7　　9.05-　　　4.19-10104　　　3.59-8　　4.6-15　　3.04-15　1.42-18　　7.12-6　　3.75-10　　3.78-10　4.06-12References[ 1 ] J. Ahlberg, E. Nilson and J. Walsh, Theory of Splines and Their Applications, Academic Press, NewYork, 1967.[2] G. Evans, Practical Numerical Integration, John Wiley & Sons, New York, 1993.[ 3 ] M. Sakai and R. Usmani, On consistency relations for cubic splines-on-splines and their asymptoticerror estimates, J. Approx. Theory, 55(1985), 195-200.[ 4 ] M. Sakai and R. Usmani, On spline-on-spline numerical integration formula, J. Approx. Theory, 59(1989), 350-355.[ 5 ] M. Sakai and R. Usmani, Numerical integration formulas based on iterated cubic splines, Computing,52(1994), 309-314.[ 6 ] M. Shelly and G. Baker, On order-preserving approximation to successive derivatives of periodicfunctions by iterated splines, SIAM J. Numer. Anal., 25(1988), 1442-1452.

スプライン関数による数値積分の改良

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スプライン関数による数値積分の改良

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