Advances in Numerical Analysis Volume 204, Article ID 35394, 8 pages ttp://dx.doi.org/0.55/204/35394 Researc Article Error Analysis for a Noisy Lacunary Cubic Spline Interpolation and a Simple Noisy Cubic Spline Quasi Interpolation Feng-Gong Lang and Xiao-Ping Xu Scool of Matematical Sciences, Ocean University of Cina, Qingdao, Sandong 26600, Cina Correspondence sould be addressed to Feng-Gong Lang; fenggonglang@sina.com Received 8 May 204; Revised 30 August 204; Accepted 30 August 204; Publised 0 September 204 Academic Editor: Weizu Bao Copyrigt 204 F.-G. Lang and X.-P. Xu. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. We mainly present te error analysis for two new cubic spline based metods; one is a lacunary interpolation metod and te oter is a very simple quasi interpolation metod. Te new metods are able to reconstruct a function and its first two derivatives from noisy function data. Te explicit error bounds for te metods are given and proved. Numerical tests and comparisons are performed. Numerical results verify te efficiency of our metods.. Introduction Cubic spline, as te most commonly used spline in practice, is a fundamental approximation tool [ ]. Nowadays, it as been widely used in many fields suc as numerical analysis, computer aided geometric design, matematical modeling, and engineering problems. Essentially, cubic spline is a twice differentiable piecewise cubic polynomial defined over a partitioned interval. Matematically, cubic spline interpolation is often introduced as follows. Let yx be a function defined over [a, b], let y j =yx j, j=0,,...,n, be a set of given function data at te nodes a=x 0 <x < <x n <x n =b, 2 and let y 0 =y x 0, y n =y x n 3 be two boundary derivatives. Ten, tere exists a unique cubic spline sx satisfying s x 0 =y 0, s x n =y n, sx j =y j, j=0,,...,n. 4 However, we often meet two troubles in te practical applications of cubic spline interpolation. Te first trouble is tat we cannot obtain te precise function values in. Tey generally involve some unavoidable measurement noise. Te second trouble is tat it often lacks te boundary derivatives in 3. To deal wit te troubles, in tis paper, we give two new effective cubic spline based metods for reconstructing yx, y x,andy x from te given noisy data y j =yx j +ε j, j=0,,...,n, 5 were ε j is te measurement noise. Te first one is a noisy lacunary interpolation metod Metod I and te second one is a very simple noisy quasi interpolation metod Metod II. Te error bounds of te metods, wic ave not been studied before and are important and useful for te users of cubic spline, are mainly studied in tis paper. We organize te remainder of tis paper as follows. In Section 2, we present some useful preliminaries; in Section 3, we give te new metods; in Section 4, wepresentteteoretical results of te errors; in Section 5, weperformsome numerical tests to verify te error analysis; finally, we conclude tis paper in Section 6.
2 Advances in Numerical Analysis 2. Preliminaries 2.. Cubic B-Splines. We assume tat te nodes in 2 are equidistant because tis case is very common in practice. Te nodes produce a uniform partition Δ for [a, b] wit mes size = b a/n.tedimensionoftecubicsplinespaceoverδ is n+3. Te corresponding cubic B-splines are given below [4 8]. For i=2,3,...,n 2,let x x i 2 3, if x [x i 2,x i ] B i x= x x i 2 3 4x x i 3, if x [x i,x i ] 6 3 x i+2 x 3 4x i+ x 3, if x [x i,x i+ ] x i+2 x 3, if x [x i+,x i+2 ] 0, else. 6 Te oter six B-splines B x, B 0 x, B x, B n x, B n x, and B n+ x are generated by te translation, were B x = 3 6 3 x x, if x [x0,x ] 0, else, B 0 x = 6 3 x 2 x 3 4x x 3, if x [x 0,x ] x 2 x 3, if x [x,x 2 ] 0, else, x x 0 + 3 4x x 0 3, if x [x 0,x ] B x = x 3 x 3 4x 2 x 3, if x [x,x 2 ] 6 3 x 3 x 3, if x [x 2,x 3 ] 0, else, x x n 3 3, if x [x n 3,x n 2 ] B n x= x x n 3 3 4x x n 2 3, if x [x n 2,x n ] 6 3 x n x+ 3 4x n x 3, if x [x n,x n ] 0, else, x x n 2 3, if x [x n 2,x n ] B n x= 6 3 x x n 2 3 4x x n 3, if x [x n,x n ] 0, else, B n+ x = 6 3 x x 3 n, if x [xn,x n ] 0, else. Tey are linearly independent, nonnegative, and locally supported. Moreover, 7 B k i x =B k i+ x+, k=0,,2; i=2,3,...,n 3. 8 Te values of B i x, B i x, andb i x attenodesarelisted in Table. 2.2. Approximate Boundary Derivatives. Using two-point numerical differentiation formula, we ave y 0 = y y 0 = y y 0 =y 0 + 2 y ξ + ε ε 0 y n = y n y n = y n y n + ε ε 0 =y n 2 y ξ 2 + ε n ε n =y 0 +ε 0, + ε n ε n =y n +ε n. Similar results can be obtained by using tree-point and fivepoint numerical differentiation formulae. See Tables 2 and 3, were ξ x 0,x, ξ 2 x n,x n, ξ 3 x 0,x 2, ξ 4 x n 2,x n, ξ 5 x 0,x 4,andξ 6 x n 4,x n and ε 0 and ε n represent te computational truncated errors to y x 0 and y x n. Tey arise from te used numerical differentiation formulae and te above-mentioned measurement noise. 3. Two New Metods 3.. Metod I. We study te following noisy lacunary cubic spline interpolation NLCSI problem.we ope to find a cubic spline sx satisfying 9 sx j = y j, j=0,,2,...,n. 0 To make te NLCSI problem uniquely solvable, it requires using two approximate boundary derivatives in Section 2.2. Obviously, tere also exists a unique noisy lacunary cubic spline sx satisfying s x 0 = y 0, s x n = y n, sx j = y j, j=0,,2,...,n. Let sx = n+ i= c ib i x be te cubic spline determined by, were te unknown coefficients c i i =,0,...,n+ can be obtained by solving te linear system 0 d d d 0 c 0 6 y 0 c 6 y. =. c n 6 y n c n 6 y n c c n+ 2 y 0 2 y n 2 followed from Table. Furtermore,wecanuse s x = n+ i= c ib i x and s x = n+ i= c ib i x to approximate y x and y x,respectively.
Advances in Numerical Analysis 3 Table : Te values of B k i nodes. B i x 6 B i x 2 xi =, 0,..., n + ; k = 0,, 2 at te x i x i x i+ Else 4 6 6 0 2 B i x 2 2 2 2 0 Table 2: Te approximate boundary derivatives and teir errors I. y 0 ε 0 y n ε n Two-point results y y 0 2 y ξ + ε ε 0 y n y n 2 y ξ 2 + ε n ε n Tree-point results 3 y 0 +4 y y 2 2 2 3 y ξ 3 + 3ε 0 +4ε ε 2 2 y n 2 4 y n +3 y n 2 2 3 y ξ 4 + ε n 2 4ε n +3ε n 2 Table 3: Te approximate boundary derivatives and teir errors II. Five-point results 0 0 Proof. Addcolumnonetocolumntreeandalsoaddcolumn n+3to column n+, and we get a strictly diagonally dominant matrix A = 0 0 4 2 d d d 2 4 0 0 n+3 n+3 Obviously, A is invertible and A =AP,were We ave P= 0 0 d 0 0 n+3 n+3. 4. 5 A =PA, A, P =2. 6 Hence, we ave A = PA P A 2. y 0 ε 0 y n ε n 25 y 0 +48 y 36 y 2 +6 y 3 3 y 4 2 4 5 y5 ξ 5 + 25ε 0 + 48ε 36ε 2 + 6ε 3 3ε 4 2 3 y n 4 6 y n 3 +36 y n 2 48 y n +25 y n 2 4 5 y5 ξ 6 + 3ε n 4 6ε n 3 + 36ε n 2 48ε n + 25ε n 2 Let sx = n+ i= c ib i x be te cubic spline determined by 4, C=c,c 0,...,c n,c n+ T, Y=2y 0,6y 0,...,6y n,2y n T, andtenweaveac = Y.Let 7 3.2. Metod II. By using te given function data, we can directly get a cubic spline n+ s x = y i B i x, 3 i= were y =yx 0 +ε and y n+ =yx n ++ε n+.we can also use sx, s x,and s x to approximate yx, y x, and y x,respectively. Te metod is very simple and effective metod for noisy data because it avoids using approximate boundary derivativesandalsoavoidssolvingtelinearsystem2. 4. Main Results 4.. Error Analysis for Metod I. We denote 2by A C = Y. Lemma. A is invertible and A 2. and we ave ε = max ε 0, ε n }, ε=max 0 i n ε i }, 8 Y Y max 2ε,6ε}; 9 see Table 4 for te results, were M 5 = y 5 x = max a x b y 5 x and M 4, M 3,andM 2 are defined similarly. Lemma 2. Consider C C 2 Y Y. Proof. Consider C C = A Y Y A Y Y 2 Y Y. Lemma 3. Consider n+ i= B ix, n+ i= B i x 3/2, and n+ i= B i x 4/2.
4 Advances in Numerical Analysis Table 4: Te bounds of ε and Y Y. Two-point results Tree-point results Five-point results ε Y Y 2 M 2+ 2ε max M 2 2 +4ε,6ε} 3 M 3 2 + 4ε 2 3 M 3 3 +8ε 5 M 5 4 + 32ε 3 2 5 M 5 5 + 64ε 3 Proof. Because of property 8, we only need to ceck tem over a typical subinterval [x j,x j+ ]. By differentiating 6, for a general i,weave x x i 2 2, if x [x i 2,x i ] B i x= x x i 2 2 4x x i 2, if x [x i,x i ] 2 3 x i+2 x 2 +4x i+ x 2, if x [x i,x i+ ] x i+2 x 2, if x [x i+,x i+2 ] 0, else, B i x = 3 x x i 2, if x [x i 2,x i ] 3x x i 2 +4x i, if x [x i,x i ] x i+2 4x i+ +3x, if x [x i,x i+ ] x i+2 x, if x [x i+,x i+2 ] 0, else. 20 All of tem are locally supported over four adjacent subintervals. i By te nonnegativity and partition of unity of cubic B-splines, for x [x j,x j+ ],weave n+ i= n+ B i x = B i x = i= ii For x [x j,x j+ ],weave j+2 B i x. 2 i=j = 3 [x j+ x+ 3x+x j+2 4x j+ + 3x x j +4x j +x x j] 4 6x x j, x [x j,x j + 3 ] = 2, x [x j + 3 3,x j + 2 3 ] 6x x j 2, x [x j + 2 3,x j+] 4 2. 23 Lemma 4. Let sx and sx be te cubic spline interpolants of yx determined by 4 and,respectively.ten we ave s x s x 2 Y Y, s x s x 3 Y Y, s x s x 8 2 Y Y. Proof. First of all, for k=0,,2,weave 24 n+ B i x i= = B j x + B j x + B j+ x + B j+2 x = 2 3 [x x j+ 2 + 4x x j+ 2 x x j+2 2 + x x j 2 4x x j 2 +x x j 2 ] = 2 3 [ 4x x j 2 +4x x j +2 2 ] 3 2. iii For x [x j,x j+ ],weave n+ B i x i= = B j x + B j x + B j+ x + B j+2 x 22 n+ sk x s k x = c i c i B k i x i= C C n+ i= Bk i x. AndtenusingLemmas2 and 3, we get tese results. 25 Teorem 5. Let sx be te noisy lacunary cubic spline interpolant of yx determined by.ten one as s x yx 5 384 M 4 4 +2 Y Y, s x y x 24 M 4 3 + 3 Y Y, s x y x 3 8 M 4 2 + 8 2 Y Y. 26
Advances in Numerical Analysis 5 Proof. Tese results follow from te traditional cubic spline interpolation error teory [, 3, 7], Lemma 4,andtefollowing triangle inequality sk x y k x sk x s k x + sk x y k x, k=0,,2. 27 4.2.ErrorAnalysisforMetodII. From 3 and Table, for j=0,,...,n,weave s x j = y j +4 y j + y j+, 28 6 s x j = y j+ y j, 29 2 s x j = y j 2 y j + y j+ 2. 30 It is a surprise to find tat s x j and s x j are te same as te well-known central numerical differentiation formulae. Lemma 6. Let sx be te noisy cubic spline quasi interpolant of yx determined by 3.Ten for j=0,,...,n,one as s x j yx j 6 M 2 2 +ε, 3 s x j y x j 6 M 3 2 + ε, 32 s x j y x j 2 M 4 2 + 4ε 2. 33 Proof. By 30, we ave s x j y x j y j 2 y j + y j+ = 2 y x j y j 2y j +y j+ = 2 y x j + ε j 2ε j +ε j+ 2 2 M 4 2 + 4ε 2. 34 Te proofs of 3 and32 are similar, wic are omitted. Teorem 7. Let sx betenoisycubicsplinequasiinterpolant of yx determined by 3.Ten weave s x yx 384 M 4 4 + 24 M 3 3 + 6 M 2 2 + 5 4 ε, 35 s x y x 5 48 M 4 3 + 6 M 3 2 + 3 ε, 36 s x y x 5 24 M 4 2 + 4 ε. 37 2 Proof. We first prove 37. sx is a cubic spline; ence s x is a piecewise continuous linear function over [a, b] wit respect to te partition Δ. LetLx be te piecewise linear interpolant to y x wit respect to Δ.Forj =,2,...,n,let s j x and L jx be te restriction of s x and Lx over [x j,x j ].Tenweave s j x = s x j x j x L j x =y x j x j x + s x j x x j, +y x j x x j. For j=,2,...,n,by38and33, we get s j x L j x = s x j y x j x j x Hence, for all x [a,b],weave 38 + s x j y x j x x j 2 M 4 2 + 4 2 ε. 39 s x Lx 2 M 4 2 + 4 ε. 40 2 Moreover, by te piecewise linear polynomial interpolation teory [, 3, 7], for all x [a,b],weave y x Lx 8 M 4 2. 4 Ten 37followsimmediatelyfrom40and4. Next, we prove 36. For j =,2,...,n,let s j x be te restriction of s x over [x j,x j ].Tenforx [x j,x j ],by 32 and37, we ave x s j s j x y x = t y tdt+ s j x j y x j, x [x j,x j + x j 2 ] x s j t y tdt+ s j x j y x j, x [x j + x j 2,x j]
6 Advances in Numerical Analysis x j +/2 x j x j x j +/2 s j t y t dt + s j x j y x j, x [x j,x j + 2 ] s j t y t dt + s j x j y x j, x [x j + 2,x j] 5 48 M 4 3 + 6 M 3 2 + 3 ε. 42 Finally, we prove 35. For every subinterval [x j,x j ], j =,2,...,n,wegive α j x = 2x x j 3 3x x j 2 + 3 3, betecubichermiteinterpolantofyx over [x j,x j ];ten for x [x j,x j ],byusing3, 32, 44, 45, and 46, we ave y x H j x 384 M 4 4, α j2 x = 2x x j 3 +3x x j 2 β j x = x x j 2 x x j 2, β j2 x = x x jx x j 2 3, 2. 43 s j x H j x = sx j yx j α j x + sx j yx j α j2 x + s x j y x j β j x + s x j y x j β j2 x 47 Tey are very useful in cubic Hermite interpolation, and we also ave α j x 0, α j2 x 0, β j x 0, β j2 x 0, α j x +α j2 x =, β j x β j2 x = x j xx x j 4. 44 Let s j x = sx [xj,x j ] be te restriction of sx over [x j,x j ];tenitcanalsobewrittenas 6 M 2 2 +ε+ 4 6 M 3 2 + ε = 24 M 3 3 + 6 M 2 2 + 5 4 ε. Bytetriangleinequality,weget35. 5. Numerical Tests and Discussions 5.. Numerical Tests. In tis section, we perform numerical tests by Matlab. Te following examples 5 f x =, x [, ], +x2 f 2 x =e x, x [, ], 48 Let s j x = sx j α j x + sx j α j2 x + s x j β j x + s x j β j2 x. H j x =yx j α j x +yx j α j2 x +y x j β j x +y x j β j2 x 45 46 are considered. In every numerical test, te mes size and te measurement noise bound ε are bot given. Because te measurement noises ε i i = 0,,...,n are random, we let ε i = ε r i, were r i i = 0,,...,n are random numbers and satisfy r i. In Tables 5 and 6, Metods I-, I-2, and I-3 represent Metod I wit two-point, tree-point, and five-point approximate boundary derivatives, respectively. CSM represents te cubic spline metod in []. E 0, E,andE 2 are te maximum absolute error of te function, te first order derivative, and te second order derivative, respectively.
Advances in Numerical Analysis 7 Table 5: Numerical results of f x. ε=0 4 = 0.2 ε=0 4 = 0. ε=0 3 = 0.2 ε=0 3 = 0. Metod I- I-2 I-3 II CSM [] E 0 7.8 0 3.6 0 3 5.9 0 4 6.0 0 2 4.4 0 3 E 2.4 0 4.7 0 2.3 0 2.4 0 2. 0 2 E 2 4.6 0.75 0.4 0.38 0.45 E 0 2.0 0 3.3 0 4.2 0 4.6 0 2 3.4 0 4 E.2 0 4.7 0 3 3.0 0 3 3.8 0 2 3. 0 3 E 2 4.29 0.5 0.4 0.0 0.5 E 0 8.0 0 3.8 0 3.3 0 3 6.0 0 2 4.5 0 3 E 2.4 0 4.7 0 2.8 0 2.5 0 2.3 0 2 E 2 4.7 0.75 0.43 0.39 0.45 E 0 2.2 0 3.0 0 3 9. 0 4.6 0 2 9.4 0 4 E.2 0.6 0 2.0 0 2 4.0 0 2. 0 2 E 2 4.29 0.65 0.42 0.29 0.44 Table 6: Numerical results of f 2 x. ε=0 5 = 0.2 ε=0 5 = 0. ε=0 4 = 0.2 ε=0 4 = 0. Metod I- I-2 I-3 II CSM [] E 0 8.0 0 3 9.8 0 4.7 0 5.3 0 2 8.2 0 5 E 6.8 0 2 8.3 0 3 2.3 0 4.4 0 2. 0 4 E 2.7 0.3 6.8 0 3 7.3 0 3 7.3 0 3 E 0 2.0 0 3.3 0 4.0 0 5 3.9 0 3.2 0 5 E 3.5 0 2 2.2 0 3 2.7 0 4 4. 0 3. 0 4 E 2.2 7.5 0 2 8.3 0 3 3.6 0 3 8.4 0 3 E 0 8.0 0 3 9.8 0 4.0 0 4.3 0 2.2 0 4 E 6.8 0 2 8.3 0 3.4 0 3.4 0 2 4.8 0 4 E 2.7 0.3 0.09 0.00 0.02 E 0 2.0 0 3.5 0 4 7.3 0 5 3.9 0 3 8.5 0 5 E 3.5 0 2 2.6 0 3 2.9 0 3 4.0 0 3.0 0 3 E 2.22 0.08 0.08 0.03 0.09 5.2. Discussions. Generally, te maximum absolute errors E 0, E,andE 2 vary if one of ε and does. If is fixed and ε decreases, ten te maximum absolute errors E 0, E,andE 2 will decrease. But if ε is fixed wile decreases, te errors will not decrease necessarily; tey maybe increase sometimes. See teteoreticalresultsinteorem5 and Teorem 7 and te numerical results in Tables 5 and 6. Wen and ε arebotfixedinaspecifictest,itiseasyto find tat E 0 and E of Metod I-2 and Metod I-3 are better tan tose of Metod I- and Metod II, wile E 2 of Metod I-3 and Metod II are better tan Metod I- and Metod I-2. See Tables 5 and 6. It is very reasonable to compare our metods wit te cubic spline metod CSM in [] becauseourmetodsare also based on cubic spline. From Tables 5 and 6,wefindtat te errors of Metod I-3 are overall better tan CSM in []. At te same time, E 2 of Metod II are better tan CSM in []. In summary, wen approximating a function, we advise using Metod I-3, Metod I-2, and CSM []; wen approximating its first order derivative, we advise using Metod I-3 and CSM []; wen approximating its second order derivative, we advise using Metod I-3, Metod II, and CSM []. Wen ε = 0, Metod I- and Metod II are O 2 metods, Metod I-2 is an O 3 metod, and Metod I- 3 is an O 4 metod. Te cubic spline metod CSM [] isalsoano 4 metod,wiletemetodin[2] is an O 2 metod; te metod in [3] isano 2.5 log metod conditionally, only if te sape parameter c = O terein. Obviously, te approximation orders of Metod I-2 andmetodi-3areigertantemetodsin[2, 3], te approximation orders of Metod I- and Metod II equal tat of te metod in [2], and te approximation order of Metod I-3 equals tat of CSM []. Undoubtedly, our metods are full of approximation ability. Furtermore, [2, 3] ave not studied first order and second order derivative approximations. At te same time, our metods are more suitable for noisy data tan te metods in [2, 3]. Hence, Metod I-2, Metod I-3, and CSM [] are more preferable tan oters. 6. Conclusions Te explicit error bounds for a noisy lacunary cubic spline interpolation and a simple noisy cubic spline quasi interpolation are well studied in tis paper; see Teorems 5 and 7.
8 Advances in Numerical Analysis Tese new results are very useful in numerical approximation and related practical fields. Moreover, tese results are also verified by some numerical examples. In a word, bot teoretical analysis and numerical tests sow tat our metods are well beaved. We end te paper wit te following remarks. i Te main contributions of te paper include i studying two new metods to approximate a function and its first order and second order derivatives from te given noisy data and ii analyzing te explicit error bounds for te metods. ii Te main advantages of our new metods include te following: i tey are very simple; ii tey are not only applicable to noisy data but also applicable to exact data; iii Metod I-2 and Metod I-3 ave better performance in function approximation and first order derivative approximation tan oter metods; Metod I-3 and Metod II ave better performance in second order derivative approximation tan oter metods. [] P.Sablonnière, Univariate spline quasi-interpolants and applications to numerical analysis, Rendiconti del Seminario Matematico UniversitàePolitecnicodiTorino,vol.63,no.3,pp.2 222, 2005. [2] Z. M. Wu and R. Scaback, Sape preserving properties and convergence of univariate multiquadric quasi-interpolation, Acta Matematicae Applicatae Sinica,vol.0,no.4,pp.44 446, 994. [3] L. Ling, A univariate quasi-multiquadric interpolation wit better smootness, Computers & Matematics wit Applications,vol.48,no.5-6,pp.897 92,2004. Conflict of Interests Te autors declare tat tere is no conflict of interests regarding te publication of tis paper. Acknowledgments Te autors appreciate te reviewers and editors for teir careful reading, valuable suggestions, and timely review and reply. References [] E.Süli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, UK, 2003. [2] C. A. Hall, Optimal error bounds for cubic spline interpolation, Approximation Teory, vol.6,no.2,pp.05 22, 976. [3] A. Quarteroni, R. Sacco, and F. Saleri, Numerical Matematics, Springer, Berlin, Germany, 2nd edition, 2007. [4] I. J. Scoenberg, Contribution to te problem of approximation of equidistant data by analytic functions, Quarterly of Applied Matematics,vol.4,pp.45 99and2 4,946. [5] C. de Boor, A Practical Guide to Splines, Springer, New York, NY, USA, 978. [6] F. Lang and X. Xu, A new cubic B-spline metod for linear fift order boundary value problems, Applied Matematics and Computing, vol. 36, no. -2, pp. 0 6, 20. [7] R. H. Wang, Numerical Approximation, Higer Education Press, Beijing, Cina, 999. [8] L. L. Scumaker, Spline Functions: Basic Teory, Cambridge University Press, Cambridge, UK, 3rd edition, 2007. [9] C. Zu and W. Kang, Numerical solution of Burgers-Fiser equation by cubic B-Spline quasi-interpolation, Applied Matematics and Computation,vol.26,no.9,pp.2679 2686,200. [0] Z. W. Jiang and R. H. Wang, An improved numerical solution of Burgers equation by cubic B-spline quasi-interpolation, Journal of Information and Computational Science, vol.7,no.5,pp. 03 02, 200.
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