SAKAI, Manabu

USMANI, Riaz A.

酒井, 宦

English

Kagoshima University Repository

Rep. Fac. Sci. Kagoshima Univ. (Math., Phys. & Chem.),No. 26, 1-9, 1993.Polynomial Approximations Based onIterated Cubic Splines and their ApplicationsManabu SAKAI and Riaz A. USMANI(Received July 6, 1993)AbstractWe consider an application of iterated cubic splines to Hermite intepolation whichis of much use for development of numerical integration formulas of singular integrals.Some numerical examples are given to illustrate the usefulness of our methods.●1. Introduction and description of the methodsIterated splines are of much use for order-preserving approximations to a given function.●                                                                                   ●There is computational evidence that these give better results than a single spline (【11, 【4】,[6]). For n>l and a sufficiently smooth function /defined on [0, 1], we consider an applicationof the iterated cubic splines to the Hermite interpolant p2m+i of the function /at two points x%and xi+i (0≦i≦n-1), i.e.,(1)　　　　主監+iU)-/ォ*'(*,) (j-i, t+l;O≦k≦m-1)where xj-jh(-j/ny nh-l).Note that polynomial p2m+i of degree 2m+l is given in ([1], [2]):(2)　　　p2m+1(x) -fl Tm,｡(6)+fi+x Tm,｡a-O) +h{f; TmA(d) -fM 7^(1-0)} +-+h>l{fim)Tm,m(d)+(-1)mfifi) Tm,m(l-d)} (x-xi+dh, 0≦0≦1)with(3)mk¥TmA6)-dk-∑(-1)∫(m-k+]¥(2m-k+l¥nm+s+1¥j)¥m-i)V;-0Department of Mathematics, Faculty of Science, Kagoshima University, 1-21-35 Korimoto,Kagoshima 890, JapanDepartment of Applied Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2Manabu SAKAI and Riaz A. USMANIThe iterated cubic spline sm　¥m≧0) can be recursively defined as follows. Let so and sm(m≧1) be the usual cubic spline interpolants of / and s㌫_i, respectively, i.e.,(4)　　　s｡j(-s｡(xj))-fjf smj(-sm(xj)) -s完,∫ (0≦j≦n)subject to(5)　　　　Ahs左,0-V"s完,乃-0 (m≧0)where O≦k≦n and A (V) is the forward (backward) difference operator. For the periodicfunction /, end conditions (5) are to be replaced by(6)　　　　鑑0-5鑑　(0≦r≦2, m≧0).Here we have to notice that the coefficient matrices for determining the iterated cubic splinesKm≧0) are exactly the same, i.e., sm (m≧1) are easily obtained with little additional effort(【1], page 14). For practical treatment of end conditions (5) (in order to get a tridiagonalsystem for determining the iterated cubic spline sm), see 【41. For example, A9 s完｡-O (whichwill be used in numerical examples) can be equivalently rewritten as follows:(7)　　　　㌫ o+ (265/71) 5ffl>i- (92017^1-24637rf｡+6567rf3- 1715d4+419^5-87^6+13^-^)/15336where dj is the right hand side of the consistency relation for the cubic spline sm:(8)(s完,j+l+4s'mj+S㌫;-1)/6-¥Sm,j+l-smJ-i)/2h.Togetanideaofhowthem-thiteratedcubicsplinesmapproximatesthem-thderivative/,wegivethefollowingtwolemmasbasedonresultsin[4](non-periodiccase)and[6](periodiccase)wheretheterm0(hp)(p>0)denotesaquantitywhosesizeisproportionaltoh♪orpossiblysmallerandC}[0,1]isthesetoffunctionsinCq{-…,…)whichareperiodicwithperiodone.Lemma 1. For l≦m≦9 andf∈C|[0, 1] (periodic case) and l≦m≦k≦9 andf∈C9[0, 1] (non-periodic case), the iterated cubic spline sm (m≧0) can be uniquely and recursively determinedfor sufficiently small h under (4) and (5) (or (6)), and-f.bn)-mnf(m+i)1f(m+6)¥A-f)(ljL)mll80/;1512/;¥^u^n>Polynomial Approximations Based on Iterated Cubic Splines and their Applicationswith L-9-m in the periodic case and L-min(9-my k+l-m) in the non periodic case, wherefor m+4>9 and m+6>9, the terms fj{m+*] and fim+6) are absorbed in the order term 0(hL),respectively.Proof. The non periodic cases rn-l, 2 were covered in [4] while the periodic cases for all mwere done in 【61. The similar technique in 【41 gives the desired relations for ∽≧3. Just use ∽+4 and m+6 in ､place of m in the above lemma to obtain(i) h4sm+u-h4f/m+4) +0(hL) (m>4)(9)(ii) hesm+6J-h6fr+a +O(hL) (m<2).Combining the above asymptotic relations (9) with Lemma 1, we haveLemma 2. For m>l, under (4) and (5) {or (6))fr=smAm (窓180 W+4J~ 1512Sm+6,j㌫sm+6,j蝣+0(hL) (o≦j≦n).On making use of the above Lemma 2, we can obtain useful approximations to thederivatives /(γ) (1≦γ≦3):(10)(i) for k≧3,//-su+O(A3)(ii) for k≧5, fi-su+蓋s5J+O(h5), f;'-s2J+O(h4)(iii) for k≧7, fi-su+蓋*U-T㌫s7J+O(h7)f;'-s2j+^s6J+o(h6), fr-s3j+^s7J+o(h5).Hence we have the following theorem where p2m+x of degree 2m+l is the polynomial definedby (2) with the derivatives fir) (j-i, i+1, 1≦r≦m) approximated by using (10) (i)-(iii) for∽-1, 2 and 3, respectively.Theorem 1. For k≧2m+l (1≦m≦3) (The restriction on k, defined in (4), is not necessary inthe periodic case) and f∈cno, 1] or f∈C9[O, l], thenp2m+l(x)-f(x)-O(h2m+2) (o≦x≦1)Next, for an approximation of the derivative function /', we can use Lemma 2 again toobtain more accurate approximations to fj　(1≦r≦4) at node points:(ll)Manabu SAKAI and Riaz A. USMANI(i)　for k≧4, f-sx.+O{h*), fi -s2j+O(h3)(ii) for /c>6, fj'-su+蓋s5,,+O(/z6), fr-s2J+芸s7J+O(hs)//3>-ssj+O(A4)(iii) for k≧8, fj -Si,;+蓋S5,ノー了㌫s7J+O(h8)h6fi'-s2,∫+怠se,r面ssj+O(h7)D　　　　　　　　　　　　　　　　45Hence as an approximation of the first derivative, we have a polynomial否2m+i of degree2m+l which is the Hermite interpolant of/'given by (2) with the derivatives/ (j-i, i+1,l<r<m+l) approximated by use of the right hand sides of ll (i)-(iii) without the orderterms.Theorem 2. For k≧2m+2 (1≦m≦3) (This restriction on k, defined in (4), is not necessary inthe periodic case) and f∈C?[0, 1] or f∈C9[0, 1],Q2m+l(x)-f′(x)-O(h2m+2) (o≦x≦1).For an approximation of the second derivative /", there exists a polynomial r2m+i ofdegree 2m+1 which is the Hermite interplant of /" given by (2) with the derivatives fi(r)(j-i, i+1, 2≦r≦m+2) approximated by using Lemma 2:12)(i)fork>5,f,,-s2J+O(h*¥fr-s3J+0(h3)(ii)fork>7,f;'-s2J+-^se,,+O(h6),f/3)-s3j+-^s7j+O(h5)//4)-54J+O(A4)h4h6(iii)for/c>9,fir-s2j+-^s6j-^QSsj+O(hs),h4∫一′/r-s3J+品S7J一盲面S9,i+Oihl), fjU)=stj+妄ssj+0mh4　　　h6//5)-s5,;+^s9J+ OU5).Theorem 3. For k≧2m+3 (1≦m≦3) {This restriction on k, defined in (4), is not necessary inthe periodic case) and f∈cm l] or f∈C9[0, 1],r2m+i(x)-f"(x)-O(h2m+2) (o≦x≦1).Polynomial Approximations Based on Iterated Cubic Splines and their ApplicationsFinally we note that polynomials p2m+i, fcm+i and f2m+i of degree 2m+l are all m-timescontinuously differentiable functions on [0, 1] and order-preserving, i.e., O (/z2m+2)-approxima-tions to /(㍗) (0≦∬≦2), respectively.2. Numerical integration formulas for singular integralsWe consider an application of the above stated polynomial approximation to numericalintegration formulas for singular integrals of the form:●(13)　　f w(x)f(x)dx - " a linear co-bination offt (0≦i≦n-1)for some or all j (0≦j≦n-1) where w{x)-x- (o>-1) or log(x). Now use of the abovestated polynomial approximation p2m+i to / leads to following numerical integration formulaslmj(h) (1≦m≦3):w(x)p2m+l(x)dxl -i: hi{αj'7/"+ (-D'#'>/S>with(15)αjg-hfw(xj+dh)Tmi(d)ddandJ｡酢-h¥w(xi+dh)T-i{l-6)dd.J｡FortheerrorinIm>j(h),bymeansofTheorems1-3,itisstraightforwardtoshowthefollowingtheorem.●Theorem4.Fork≧2m+l(1≦m≦3)(Thisrestrictiononkisnotnecessaryintheperiodiccase),J*xj+irxj+iw(x)f(x)dx-I-J(h)+O(h2m+2)¥w(x)¥dx(0<;xjJxi≦M-1)Corollary.UnderthesamerestrictiononmandkinTheorem4,w-1w(x)f(x)dx- ∑ Imj(h)+O(h2m+2);-0while the error in compound Simpson rule is 0(h ln h) ([21).For the calculation of the weights α,(～ and B i} (o≦i≦m), define the following auxiliaryintegrals:Manabu SAKAI and Riaz A. USMANI才>-h[1diw(xi+hd)d6 (0≦i≦2-+l)●which can be evaluated by means of the following recurrence formulas:(i)forw(x)-xα(α>-1),(1+α¥(0)-^(1+)Cj-Xj+iα-->(Hxjα)(i+l+α¥(i)-(1+)Cj-Xj+iα)-､IJC(/-I)(1≦i≦2m+l)17(ii)forw(x)-¥0g(x),cr-xj+1log(xj+1)-Xj¥0g(Xj)-h(i+l)c}n-xj+ilog(x,+1)-h-ijcj'-v+ih/d+l)(1≦i≦2m+l).Then,theweightsintheintegrationformulasImj(h)(1≦m≦3)canberepresentedintermsofc(0≦i≦2m+l):2m+1                            2m+l(18)　　　　α(i)-　∑　um(ir)cji} and i酢)-　∑ ;u(*サcj'>.r=m+l                           r=m+lHere values of fim(i,r) and人,Ur) (1≦m≦3; i<m+l≦r≦2m+1) are given in Table 1.Table 1. Values offjtm andスm (1≦m≦3).γ＼　0　1　2.～…｢lii (i, r)　　　　　　　h (i, r)coo<r>coii~二4　　　　5　　　　3　　　　415　　　-6　　　10　　-15o-1　　　　　　　-2LO CD COTable 1 (continued)(JLz (i, r)Hr＼　O i-H CM COJ～ 4　　　　5　　　　6-35　　　84　　-70-20　　　45　　-36-10　　　20　　-15-4　　　　　　　　-4t>サ　O O t*　r*2　1*3 (i, r)4　　　　535　　-8415　　-395　　-141　　-3t^　O O　^H i-HI ItD o　^　co en7　3　1Polynomial Approximations Based on Iterated Cubic Splines and their Applications3. Numerical ExamplesFirst we consider an application of the above stated methods by taking some examplesf(x)-l/(l+25x2) (~-1≦x≦1) (or l/(l+100(x-1/2)2): o≦x≦1) and sin(47rx) (0≦x≦1).In Table 2-4, we give the observed maximum absolute errors of the function, first and second●derivative values at mid-points where a-b-ax 10 . Methods I-III mean the ones by use ofp2m+i, Q2m+i and hm+i (1≦m≦3), respectively. Rates are the observed ones obtained from thenumerical results with ^-64 and 128 while the figures in parentheses are the predicted ones●given in Theorems 1-3.Table2The observed maximum absolute errors of the function values at mid-points.l/(l+25x2) : -1≦x≦1n＼Method163264128rateⅠ　　　　　　ⅠⅠ3.79-2　　　5.67-2.47-4　　　2.02-44.02-5　　　1.37-62.38-6　　　2.57-84.1(4)　　　5.7(6)sin(47rx) : 0≦x≦1Ill II III3.94-2　　　1.06-3　　　5.41-5　　　3.17-61.55-4　　　6.31-5　　　1.ll-6　　　2.99-83.09-7　　　3.89-6　　　1.83-8　　　1.15-101.76-9　　　2.42-7　　　2.90-10　　4.47-137.5(8)　　　4.0(4)　　　6.0(6)　　　8.0(8)Tabe3The observed maximum absolute errors of the丘rst derivatives at mid-points.l/(l+25x2) : -1≦x≦1n＼Method3264128rateⅠ　　　　　　Ⅰ11.04-1　　　7.27-25.69-3　　　1.26-33.12-4　　　1.97-54.2(4)　　　6.0(6)sin(47r.r) : 0≦x≦1Ill II III5.56-2　　　2.45-3　　　8.75-5　　　3.14-53.00-4　　　1.52-　　　　1.37-6　　　8.31-91.47-6　　　9.53-6　　　2.14-8　　　3.24-ll7.7(8)　　　4.0(4)　　　6.0(6)　　　8.0(8)Tabe4The observed maximum absolute errors of the second derivatives at mid-points.l/(l+25x2) : -1≦x≦　　　　　　　　　sin(4tzx) - 0≦x≦1n＼Method64128rateII III II III5.78-1　　　8.29-2　　　4.71-2　　　3.22-3　　　6.05-6　　　1.03-73.69-2　　　3.41-4　　　1.80-4　　　2.01-4　　　9.26-8　　　4.00-104.0(4)　　　6.6(6)　　　8.0(8)　　　4.0(4)　　　6.0(6)　　　8.0(8)Manabu SAKAI and Riaz A. USMANINext we consider an application of the numerical integration formulas Imj (h ) (1≦m≦3)for the weight function w{x) -¥/､斥or log(x). In the following Tables 5-6, we give theobserved absolute errors of the integration formulas on subintervals and the whole interval[0, 1] in the case when f(x) -:exp(5x). The rates are the ones from the numerical resultswith n-32 and 64 while figures in parentheses are the theoretical ones given in Theorem 4.Note that the observed maximum absolute errors of the integration formulas on subintervalsoccurred at node points bounded away from x-0. In comparison with the proposed methodsin [3】, the errors in the cases when (w,f) - (1/諺, exp(x)) and (log(x), exp(x)) are 9.32-7 and 8.89-7 (〟-400), while the errors of our methods (1≦∽≦3) are 5.94-8, 2.93-ll,3.15-14 and 2.76-8, 1.36-ll, 1.47-14 (w-16), respectively. Taking into account ofthese results, our methods would be of much use in the case when a finer mesh is notacceptable.Table5The observed maximum absolute errors of the integration formulas on subintervals.w(x)-¥ ¥fxn＼m1.03-4　　　1.08-63.55-6　　　1.08-81.16-7　　　8.93-ll4.9(5)　　　6.9(7)w(x) -¥0g(x)1　　　　　　　2　　　　　　　32.90-7　　　9.92-6　　　1.30-7　　　1.10-73.02-10　　3.16-7　　　9.85-10　　3.27-ll2.50-13　　9.91-9　　　7.66-12　　1.60-1410.2(9)　　5.0(5)　　　7.0(7)　　11.0(9)Tabe6The observed absolute errors of the integration formulas on the whole interval [0, lj.w(x) - ¥l¥fx w(x) -¥0g(x)n＼m4.38-4　　　5.23-62.81-5　　　　3.77-81.77-　　　　1.37-94.0(4)　　　6.0(6)5.04-7　　　9.85-5　　　1.14-6　　　1.12-76.55-10　　6.25-6　　　1.95-8　　　1.14-101.65-12　　3.93-　　　　3.03-10　　3.43-138.6(8)　　　4.0(4)　　　6.0(6)　　　8.4(8)References1. J. Ahlberg, E. Nilson and J. Walsh, Theory of Splines and Their Applications, Academic Press, NewYork, 1967.2. K. Atkinson, An Introduction to Numerical Analysis, John Wiley & Sons, New York, 1978.3. J. Bohman and Carl-Erik Froberg, On numerical computation of singular integrals, BIT 24 (1984),113-116.Polynomial Approximations Based on Iterated Cubic Splines and their ApplicationsM. Sakai and R. Usmani, On consistency relations for cubic splines-on-splines and asymptotic errorestimates, J. Approx. Theory 45 (1985), 195-200.5. M. Sakai and R. Usmani, On spline-on-spline numerical integration formula, J. Approx. Theory 59(1989), 350-355.6. M. Shelley 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多項式近似とその応用

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スプライン関数による多項式近似とその応用

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