Let s be a cubic spline, with equally spaced knots on [a,b], interpolating a given function y at the knots. The parameters which determine s are used to construct a piecewise defined polynomial P of degree four. It is shown that P can be used to give better orders of approximation to y and its derivatives than those obtained from s. It is also shown that the known superconvergence properties of the derivatives of s, at specific points [a,b], are all special cases of the main result contained in the present paper

Behforooz, GH

Papamichael, N

English

Brunel University Research Archive

             TR/83                   JUNE 1978 IMPROVED ORDERS OF APPROXIMATION DERIVED FROM INTERPOLATORY CUBIC SPLINES                    BY G.H. BEHFOROOZ and N.PAPAMICHAEL                                     W9260375                  ABSTRACT  Let s be a cubic spline, with equally spaced knots on [a,b], interpolating a given function y at the knots. The parameters which determine s are used to construct a piecewise defined polynomial P of degree four. It is shown that P can be used to give better orders of approximation to y and its derivatives than those obtained from s. It is also shown that the known superconvergence properties of the derivatives of s, at specific points [a,b], are all special cases of the main result contained in the present paper.  1 1. IntroductionLet s be a cubic spline on [a,b] with equally spaced knots xi = a + ih ;    i=0,1,...., k , (1.1) where h=(b-a)/k. Then sєC2[a,b] and in each of the intervals [x.i-1 , xi ] ; i =1,2, ,.,k, s is a cubic polynomial. Given the set of values yi ;i=0,l,...,k, where yi= y(xi.) ; y є c n  [a,b], n ≥ 4  consider the problem of constructing an interpolatory s such that s(xi) = yi ;    i =0, 1,..., k. (1.2) If the values mi = s(1)(xi) ; i = 0,1,... , k are known then, by use of Hermite's two point interpolation formula, s is given by ,mhx(xi)x(xmh)xxx)(xyhh}x){2(x)x-(xyhh})x{2(xx)(xs(x)i221-i1i21i2ii3i21i1i31i2i−−−−−++−++−−=−−−−−      xε[xi-1,xi] ; i= 1, 2, …., k.          (1.3)   Equivalently, if the values Mi = s(2) (xi) ; i = 0,1,.. . ,k, are known, s can be obtained in [xi-1 ,xi] by integrating { }i1i1ii)2( M)xx(M)xx(h1)x(s −− −+−= twice with respect to x and using the interpolation conditions s(xi-1) =yi-1 , s(xi) =y for the determination of the two constants of integration. To determine either of the k + 1 parameters mi or Mi the consistency relations { };)yyh3m4m 1i1i1i1i −++− −=+   i= 1, 2, ……,k-1,  (1.4) or  { ;yy2yh6MM4M 1ii1i21ii1i +−+− +−=++ }  i= 1, 2, …….., k-1,   (1.5) are used, these being direct consequences of the continuity constrains on s. Since either of (1.4) or (1.5) provide only k-1 linear equations, 2 it follows that the interpolation conditions (1.2) are not sufficient to determine s uniquely. Two additional linearly independent conditions are always needed for this purpose. These are usually taken to be end conditions, i.e. conditions imposed on s, s(1) or s(2) near the two endpoints a and b. As might be expected the choice of end conditions plays a critical role on the quality of the spline approximation. It is well-known that if y∈C4[a,b] and s satisfies appropriate end conditions then || s ( r ) -y( r ) | |= 0(h4-r) ;    r =0,1,2,3, (1.6) where ||·|| denotes the uniform norm on [a,b]. Furthermore, with r=0, (1.4) gives the best order of approximation to y which can be achieved by an interpolatory cubic spline s. It is also known that if yєC5[a,b] then, for a variety of end conditions, mi = y i ( 1 )  + 0(h4) ;      i = 0,l,...,k, (1.7) where mi = s(1) (xi) and yi (1) =y(1) (xi). More generally, if (1.7) holds then improved orders of approximation to the first three derivatives of y are obtained at certain specific points of [a»b] as follows,  s(r)(xi-1+αrh)=y(r)(xi-1+αrh) + 0(h5-r): r=l,2,3,; i=l,2,..,k, (1.8) where the admissible values of a are respectively α1 =1/2, α2 = (3±/3)/6 and α3 = l/2. (1.9) The class of end conditions for which (1.7) holds includes the conditions mo=yo(1) ,                 mk = yk(1)       ,                                           (1.10) ∆mo= ∆yo(1),           ∇mk = ∇yk(1) ,                                            (1.11)   ∆3Mo=∇3Mk= 0, (1.12)   ∆4Mo= ∇4Mk= 0, (1.13)                                ∆s(a+0.5h) = ∆y(a +0.5h) , ∇s(b-0.5h) = ∇y(b-0.5h)  ,           (1.14) and, for periodic y, the conditions    mo = mk ' Mo = mk , (1.15) of the periodic spline. (In the above A and V are respectively the usual forward and backward difference operators and, as before mi = s(1) (xi.) , Mi =s(2) (xi)).   3 For splines with end conditions (1.10) the result (1.7) has been established by Birkhoff and De Boor C4] , Hall [6] and Kershaw [7]. For periodic splines, (1.7) has been established by Kershaw [7] and, under the assumption that yєC6[a,b], by Albasiny and Hoskins [2]. More recently Lucas [8] has established (1.7) and also (1.8) -(1.9) for cubic splines satisfying any of the end conditions (1.10)- (1.14) as well as some other end conditions which are listed in [8]. (We note that if yєC6[a,b] then these results of Lucas can also be derived from the results of Daniel and Swartz [5].) Lucas has also shown that when y€C5Ca,b] and s satisfies end conditions such that (1.7) holds then, 0(h ) estimates of the derivatives yi(r) = y(r) ( xi );r = 2,3,4, at the knots can be obtained in terms of linear functionals of Mi= s (2) (xi). For i - 1,2,..., k - 1, these estimates are, )18.1(),h(0}MM2M{h1y)17.1(),h(0}MM{h21y)16.1(),h(0}MM10M{121y1ii1i2)4(i21i1i)3(i31ii1i)2(i++−=+−=+++=+−−++− In the present paper we take yєC5[a,b] and consider interpolatory cubic splines for which (1.7) holds. We show that, for such splines, formulae of the form (1.16)-(1.18) can be used to produce 0(h5-r ) approximations to y(r) ; r = 1,2,3,4, not only at the knots but at any point xє[a,b], For this we construct a piecewise defined polynomial P of degree 4, whose coefficients are given in terms of yi and mi= s(1) (xi), i.e. in terms of the parameters which determine s. We prove that ||y(r)- P(r)||=0(h5-r) ; r=0,1,2,3,4,                                     (1.19) and show that the results (1.8)-(1.9) .and (1.16)-(1.18) are all special cases of (1.19). More generally, our result shows that P can be used with very little additional computational effort to compute, at any point xє[a,b] , more accurate approximations to y and its derivatives than those obtained from s. 2. Improved Orders of ApproximationGiven the values yi = y(xi) ; i=0,l,...,k, where x. are the equally spaced points (1.1), let Hi. ; i=l,2,..,,k-l, denote the quartic Hermite 4 polynomials which are such that Hi (xj ) = yj        ; j =i-l, i, i +1, and Hi(1)(xj) = yj(1)  ; j =i-l,i, where yj(1) = y(1) (xj). Then, Hi (x) =θi (x)yi-1 + ni(x)yi+ vi (x)yi+1               + φi(x)y(1)i-1+ψi(x)yi(l) ;         i-1,2,…, k -1 ,                       (2.1)  where ⎪⎪⎪⎭⎪⎪⎪⎬⎫−−−−=ψ−−−−=φ−−−=ν−−−=η−+−−−=θ+−+−−+−+.h/)xx)(xx()xx()x(,h2/)xx()xx)(xx()x(,h4/)xx()xx()x(,h/)xx()xx()x(,h4/)}xx(5h7){xx()xx()x(31ii21ii31i2i1ii42i21ii421i21ii4i1i2ii    (2.2) Also, if y є C5 [ a,b], then ;]x,x[x),xx()xx()xx(!5)(y)x(y)x(H 1i1i1i2i21ii)5(i +−+− ε−−−ξ=−     (2.3) where ξi. is some point in [xi-1 , xi+1 ] ; see e.g. Ralston [9 :p65]. It follows that |Hi(r)(x)-y(r)(x)|≤Krh5- r)5(y ; r = 0,l,2,3,4, x є[xi-1 ,xi+1] ; i= 1, 2, …….., k-1,      (2.4) where, from (2.3), K0 = 1/240 , (2.5)  and, by Taylor series expansions, K1 =99/60, K2 = 484/60, K3= 1093/60 and K4 =1128/60. (In (2.4) and throughout this paper || • || denotes the uniform norm on [a,b].) We point out that the error bounds given by (2.4) and (2.6), for r = 1,2,3,4, are by no means optimal. Sharper values for the Kr(2.6). 5 can be found by use of Peano’s theorem. However,since our purpose is to establish asymptotic orders of convergence, we prefer the simplicity of Taylor's theorem. Let s be an interpolatory cubic spline which agrees with a function y at the equally spaced knots (1.1), and denote by p. the quartic polynomials  Pi(x) =θ i(x)yi-1+ ni (x)yi+vi(x)yi + 1 + φi (x)mi-1+ψi(x)mi; i=1, 2, ……., k-1 ,          (2.7) where m. = s(1) (xi) and θi ,ni ,vi ,φi and ψi are given by (2.2). Definition 1.   The piecewise defined polynomial ⎢⎢⎢⎣⎡ε−=εε=−−−,]x,x[x,)x(p,1k.....,,.........3,2i;]x,x[x,)x(p,]x,x[x,)x(p)x(Pk1k1ki1ii101(2.8) where the p. are given by (2.7), will be called the piecewise Hermite quartic induced by s. It follows at once from the definition that PεC1[a,b]. The main result of this paper is contained in the following theorem. Theorem 10 Let s be an interpolatory cubic spline which agrees with the function y є C5[a,b] at the equally spaced knots (1.1) and satisfies end conditions such that )1(ii ym −  ≤ Ah4 ; i =0,1,.., k , (2.9)  where A is a constant. Let P be the piecewise Hermite quartic induced by s. Then there exist constants C ; r =0,1,2,3,4 such that    r5r)r()r( hCyP −≤−; r =0,1,2,3,4.              (2.10) Proof From (2.1) and (2.7), { } )1(jjkj0)r(i)r(i)r(i)r( i ymmax)x()x()x(H)x(P −ψ+φ≤− ≤≤ ; r-0,1,2,3,4,         (2.11)                xε[xi-1,xi+1] ; i=1, 2, …….., k-1 , where, by determining the maximum values of  |φ(r)(x)  and  |ψi(r)(x)| , |φ(r)(x)| + |ψi(r)(x)|≤Lrh1-r; r=0,1,2,3,4, xε[xi-1,xi+1],         (2.12) with L0= 41/8, L1 =5, L2 = 21, L3=41 and L4=36         (2.13)           6 Hence, from (2.11), (2.12) and (2.9), |p(r)(x) -Hi(r)(x)| ≤LrAh5-r ; r =0,1,2,3,4, x є[xi-1 ] xi+1, ];i = l,2,...,k-l. (2.14) Finally, from (2.4), (2.14) and the definition of P, the result (2.10) follows with Cr=Kr|| y(5)||+ LrA ; r= 0,1,2,3,4.     (2.15) The piecewise Hermite quar.tic P, of Definition 1, is defined completely by the parameters which determine the cubic spline s. Thus, under the conditions of Theorem 1, P can be used with very little additional computational effort to produce, at any point Xε[a,b], more accurate approximations to y and its derivatives than those obtained from s. Theorem 2 below shows that the results (1.8) and (1.16)-(1.18), derived by Lucas [8] are all special cases of the result (2.10) of Theorem 1 Theorem 2. Let s be an interpolatory cubic spline with equally spaced knots (1.1) matching the values y. ;i=0,l,...,k at the knots. Let P be the piecewise Hermite quartic induced by s. Then, PP(r)(x  +α h)=si-1 r (r)(x . +a  h) ;r=0,l,2,3; i-1,2,…………k,    (2.16) i-1 rfor all choices of the yi, if and only if   α0 = 0,1, αx =0,1/2,1, α2 = (3 ±3)/6 , α3 = 1/2.               (2.17) Also ⎢⎢⎢⎣⎡=++−=++=++=−−+−,Ki;12/)M13M2M(,1K.,,.........2,1i;12/)MM10M(0i;12/)MM2M13()x(Pk1k2k1ii1i210i)2(     (2.18) ⎢⎢⎢⎣⎡=+−−=−=−+−=−−−+,Ki;12/)M3M4M(,1K.,,.........2,1i;h2/)MM(0i;h2/)MM4M3()x(Pk1k2k1i1i210i)3(       (2.19) ⎢⎢⎢⎣⎡ε+−−=ε+−ε+−=−−−−+−,]x,x[x,/)MM2M(,1K.,,.........3,2,1i,]x,x[x,h/)MM2M(,]x,x[x,h/)MM2M()x(Pk1kk1k2ki1i21ii1i102210i)4(                              (2.20) where Mi =s(2)(xi).    Proof At any point xє[xi-1,xj];i=l,2,... ,k-l, s is given by (1.3) in terms of yi-1 , yi , mi-1 and mi , whilst P is given, by (2.7)-(2.8), in terms of these four parameters and also yi+1 . The consistency relation (1.4) shows that yi+1 cannot, in general, be expressed in terms of yi-1 , yi ,mi-1 and mi only. It follows, from (2.7), that a necessary condition for (2.16) to hold, for all choices of the y., is that the points xi-l+αrh; r=0,1,2,3; i = l,2,...,k-l are respectively real zeros in [xi-1 , xi ] of the polynomials  )r(iv  (x); r =0,1,2,3; i=l,2,...,k - 1. It can be easily shown that such zeros occur only at the points defined by (2.21) and (2.17). (2.21) The results                                    P(xi) = yi= s(xi)      ; i=0,l,2,..,k, and PP(l)(x )  = mi i       = s(l)(xi)  ; i=0,l,2,...,k, follow at once from the definition of P. The other results contained in (2.16) can be established easily from (1.3), (2.7),(2.8) and the consistency relation (1.4). Thus,  .)h5.0x(s)mm(41)yy(h23m43yh43yh23)yh43mh1m(m43yh43yh23)yh3m(41)h5.0x(P,1k,......,2,1i);h5.0x(s)mm(41)yy(h23)h5.0x(P1k)1(k1k1kk1kk1kkk1k1kk1k2k2k1k)1(ik1i)1(i1i1ii1i)1(i+=+−−=++−+−−=++−+=+−=+=+−−=+−−−−−−−−−−−−−−−−and hence, PP(1)(x  +0.5h) =si-1 (l)(x  +0.5h);  i=1, 2, ….,k. i-1Similarly, it can be shown that, PP(2) (x α h)=si-1+ 2 (2)(x α h) ;   i=1, 2, …,k,i-1+ 2 8 where α2 = (3 ± 3 ) /6, and PP(3)(x  +0.5h) = si-1 (3)(x  +0.5h); i=l,2,...,k. i-1This completes the proof of the first part of the theorem. For the proofs of (2.18) -(2.20) the following cubic spline identity is needed, ; l-k0,1,...,i  ;  )y-(yh1M6h-M 3h =m i1i1iii =+ ++ (2.22) see e.g. Ahlberg, Nilson and Walsh [1]. Using (2.22) and the consistency relation (1.5), the polynomials p. in (2.7) can be written in the form Pi(x) = {θi(x)-vi(x) - h1 (Φi(x)+ψi(x)) }yi-1                            + {η i(x)+2vi(x)+ h1 (Φi(x)+ψi(x))}yi                            + 6h{hvi (x) – 2Φi (x) +ψi (x)}Mi-1                            + 6h{4hvi (x) - Φi (x) +2ψi (x)}Mi                            + 6h2  vi(x)Mi+1 ; i = 1,2,..., k-1. (2.23) The results (2.18)-(2.20) then follow from the derivatives of (2.23), by direct substitution. Numerical Results and DiscussionLet s be the cubic spline with knots     xi = 0.05i ;   i=0, 1, ……20,      (3.1) which interpolates the function     y(x) =exp(x) , at the knots and satisfies the end conditions (1.12), i.e.                            Δ 3M== ∇3Mo0= 0,  Mi.= s(2)(xi.). In Table 1 we list values of   e(r)(x) =|s(r)(x) -exp(x)| ; r-0,1,2,3,           (3.2) and  E(r)(x) =|P(r)(x) -exp(x)| ; r-0,1,2,3,4,         (3.3) computed at various points of [0,1] by constructing this s and the corresponding Hermite quartic P induced by s. The results illustrate the improvement in accuracy obtained when P is used 9 instead of s, and confirm some of the theoretical results contained in Theorem 2. We note that the spline s considered above is an E(3) cubic spline. By the definition of Behforooz and Papamichael [3], an E(α) cubic spline is an interpolatory cubic spline with equally spaced knots (1.1) and end conditions (2 -α)∆3M + (9-3α)∆2Mo = 0., (2 -α) ∇3Mk + (9-3α)∇2Mk=0 , For any a in the domain defined by α < 11/3 and α > 19/5, an E (α) cubic spline exists uniquely and, for yεC5[a,b], it satisfies (1.6). However, the approximation of the first derivative at the knots is, in general, 0(h3) and only the value a -3 yields an E(α) spline for which (1.7) holds; see [3]. We also note that, for y∈C5[a,b], the interpolatory cubic spline with equally spaced knots (1.1) and end conditions   , M)2(00 yM = k = Yk(2) gives mi =yi(1) + 0(h3) ;   i=0,l,...,k. However, it has been shown by Kershaw [7] that if h < 1 and k is sufficiently large then there exists an integer p, where - logh/log(2+ 3 ) ≤ p < 1 -logh/log(2+ 3 ), such that     mi-yi(1) = 0(h4) ;    p ≤ i ≤ k-p    (3.4) By direct application of the technique of [7], results similar to (3.4) can also be established for other interpolatory cubic splines for which (1.7) does not hold. This occurs when the matrix of the linear system which determines the parameters mi of the spline can be reduced to the tri-diagonal form ⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡a114·1·1·411a with a = 2 or 4. In particular results similar to (3.4) hold for the E(α) cubic splines which correspond respectively to the values of α =0,1/4, 1/2, 2,7/2 and 4. It follows, from the proof of Theorem 1, that if P is the 10 piecewise Hermite quantic induced by any of these splines then the result P(r)(x) =y(r)(x) +0(h5 - r) ; r = 0,l,2,3,4, does not hold in the full range [a,b] but only in intervals bounded away from the two end points. To illustrate this we construct the piecewise Hermite quartic induced by the E(2) cubic spline s interpolating the function y(x) =exp(x) at the equally spaced knots (3.1). In Table 2 we present the values of the errors (3.2) and (3.3) computed, by using these P and s, at various points of [ 0,1 ]. We are grateful to Dr.J.A.Gregory for his helpful comments. 11 TABLE (1)   r  x 0 1 2 3 4 e(r)(x) 0.13x10-7 0.14x10-5 0.28x10-4 0.15x10-1 _ E(r) (x)0.0375    0.33x10-8 0.28x10-6 0.24x10-6 0.10x10-2 0.56x10-1e(r) (x) 0.11x10-7 0.12x10-5 0.30x10-4 0.16x10-1 - E(r)(x) 0.2375    0.46x10-9 0.24x10-7 0.29x10-5 0.16x10-3 0,16x10-1e(r)(x, 0.14x10-7 0.14x10- 1 5      0.40x10-4 0.18x10-1 - E(r)(x) 0.3625   0.16x10-9 0.44x10 0.14x10 0,72x10 0.55x10 e(r) (x)     0.25x10-7 0.22x10-7 0.16x10-3   0.16x10-3 - E(r)(x), 0.4250   0.65x10-9  0.22x10-7    0.41x10-5  0.16x10-3 0.39x10-1e(r) (x) 0.16x10-7 0.17x10- 5 0.43x10-4 0.23x10-1 - E(r) (x)0.5875     0 .67x10 - 9     0.36x10-7 0.41x10-5  0.24x10-3 0.23x10-1e(r)(X) 0.12xl0-13 0.88x10-7 0.46X10-3 0.55X10-1 — E(r)(x) 0.8000   0. 12xl0- 1 3  0.88x10-7 0.36x10-7 0.49x10-3 0.19x10-2e(r)(x) 0.15xl0-7 0.18x10-6 0.66x10-6 0.29x10-1 - E(r) (x)0.9625   0.76x10-8 0.66x10-6 0.94x10-6 0.23x10- 2 0.13x10012 TABLE (2)   r   x 0 1 2 3 4 e(r)(x) 0.91xl0-7 0.76x10-5 0.55x10-4 0.28x10-1 - E(r)(x) 0.0375 0.91x10-7 0.76x10-5 0.55x10-4 0.28x10-1 0.10x10-1e(r)(x) 0.11x10-7 0.12x10-5 0.30x10-4 0.16x10- 1 - E(r,(x) 0.2375 0.94x10-9 0.64xl0-7 0.32xl0-5 0.31x10-3 0.11xl0-1e(r)(x) 0.14xlo-7 0.14x10-5 0.40X10-4 0.13x10-1 - E(r)(x) 0.3625 0.16xl0-9 0.44x10-7 0.15x10-5 0.73x10-3 0.55X10-1e(r)(x) 0.25x10-7 0.22x10-7 0.16x10-5 0.16x10-3 - E(r)(x) 0.4250 0.65x10-9 0.22x10-7 0.41x10-5 0.16x1O-3 0.39x10-1e(r)(x) 0.16x10-7 0.17x10-5 0.43x10-4 0.23X10-1 - E(r)(x)0.5875 0.69xl0-9  0.36x10-7 0.42x10-5  0.24x10-3 0.23x10-1e(r) (x) 0.12x10-13 0.38x10-6 0.48x10-3 -0.56x10-1 - E(r)(x) 0.8000 0.12x10-13 0.38x10-6 0.l0x10-4 0.12x10-2 0.49X10-1e(r)(x) 0.22x10-6 0.18x10-4 0.12x10-3 0.69X10-1 - E(r)(x) 0.96 25 0.22xl0-6 0.18x10-4 0.12x1o-3 0.69X10-1 0.26x10-113 REFERENCES 1. J.H. Ahlberg, E.N.Nilson and J.L. Walsh, "The theory of splines and their applications", London: Academic Press 1967. 2. E.L. Albasiny and W.D. Hoskins, "Explicit error bounds for periodic splines of odd order on a uniform mesh". J,Inst.Maths Applies 12 (1973), 303-312. 3. G.H. Behforooz and N. Papamichael, "End conditions for cubic spline interpolation", Technical Report TR/80, Dept.of Maths, Brunei University, 1978. A. G. Birkhoff and C. De Boor, "Error bounds for spline interpolation", J.Math. Me ch .13 (1964), 827-835. 5. J.W. Daniel and B.K. Swartz, "Extrapolated collocation for two-point boundary value problems using cubic splines", J.Inst.Maths Applies 16 (1975), 161-174. 6. C.A. Hall, "On error bounds for spline interpolation", J.Approx. Theory 1 (1968), 209-218. 7. D. Kershaw, "The orders of approximation of the first derivative of                                   cubic splines at the knots", Math.Comp.26 (1972), 191-198. 8. T.R. Lucas, "Error bounds for interpolating cubic splines under various end conditions", SIAM J.Numer.Anal. 11 (1974), 569-584. 9. A. Ralston, "A first course in numerical analysis", New York: McGraw-Hill, 1965.         NOT TO BE REMOVED  FROM THE LIBRARY XB 2356816 X  

A first course in numerical analysis&quot;,

End conditions for cubic spline interpolation&quot;,

Error bounds for interpolating cubic splines under various end conditions&quot;,

Explicit error bounds for periodic splines of odd order on a uniform mesh&quot;.

Extrapolated collocation for two-point boundary value problems using cubic splines&quot;,

On error bounds for spline interpolation&quot;,

The orders of approximation of the first derivative of cubic splines at the knots&quot;,

The theory of splines and their applications&quot;,

Improved orders of approximation derived from interpolatory cubic splines

Improved orders of approximation derived from interpolatory cubic splines

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