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1 A 851 WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER A CLASS OF LOCAL EXPLICIT MANY-KNOT SPLINE INTERPOLATION SCHEME--ETC(U) JUL 81 D X ai DAAG29-80-C-0041 UNCLASSIFIED MRC-TSR-2238 NL :hlllllllli mmmllr

2 t MRC Technical Summary Report # 2238 M A CLASS OF LOCAL EXPLICIT MANY-KNOT SPLINE INTERPOLATION SCHEMES * D. X.Q Mathematics Research Center University of Wisconsin- Madisonl 510 Walnut Street Madison. Wisconsin July (Received May 26, 1981) S EB IP Approved for public relese Distributien unlimited I North Sponsored by U. S. Army Research office P. 0. BOX Research Triangle Park Carolina * ~088039

3 Acceon 7Far UNIVERSITY VVIC OF T". WISCONSIN-MADISON cd rts ; MATHEMATICS RESEARCH CENTER 'Can' A CLASS OF LOCAL EXPLICIT MANY-KNOT SPLINE cty--t INTERPOLATION SCHEMES ' b '..... doa5ts a D. X. A, L/ Technical Summary Report #2238 July 1981 ABSTRACT The purpose of this paper is to present a new local explicit method for an approximation of real-valued functions defined on intervals. The operators of the form Qf } x f q.,k are studied under a uniform mesh, where {q i,k} comes from a linear combination of B-splines. This paper contains the definition of (q comments on its existence, proof of reproduction of the operator Q for appropriate classes of polynomials, and a note about some applications. AMS (MOS) Subject Classification: 41A15 Key Words: Many-knot spline function, local, explicit, spline interpolation. Work Unit Number 3 - Numerical Analysis and Computer Science Department of Mathematics, Jilin University, Changchun, China. Sponsored by the United States Army under Contract No. DAAG29-80-C i,

4 SIGNIFICANCE AND EXPLANATION The variation diminishing method established by Schoenberg and the quasiinterpolant method developed by de Boor and Fix take the form Qf = A f N where (N i is a sequence of B-splines and (AI is a i i,k i,k i sequence of linear functionals. This form is convenient in practices. We would like to keep this form but replace B-spline Nik with another function q*,k" i.e. we consider a different operator Qf [ x Alf q,k where qi,k has small support, satisfies qik = 8, and Af - f(xo). Thus, the operator Q becomes interpolant, and Qf is in a class of the so-called many-knot" splines. The paper proves that Q reproduces appropriate classes of polynomials. This operator can be used to fit curves or surfaces. I The responsibility for the wording and views expressed in this descriptive summary lies with MRC, and not with the author of this report. S',ai

5 A CLASS OF LOCAL EXPLICIT KANY-KNOT SPLINE INTERPOLATION SCHEMES "). X. Qi* As is well known, it is very important to study both theory and application of local spline approximation, such as the variation diminishing method established by Schoenberg, the quasi-interpolant method developed by de Boor and Fix and so on. Those authors studied operators of the form Qf If N ik, where (Ni} is a sequence of B-splines and {Xi} is i i ik i a sequence of linear functionals (see (11, [21, [31, [4]). The purpose of this paper is to present a new method, to get an approximation of real-valued functions defined on intervals. In this method, I use {q i,k to substitute for (N i, k mentioned above as a basic function. The functions qi,k possess the following characteristics: (i) small support (it makes operators of the form Qf " i i f i,k local); (ii) qi,k( J ) - dij" Here I would only like to discuss how to construct the basic functions {q I under Ai flxi. 4i,k i i Let A be a uniform mesh: a - x 0, b - xn, xi - x + ih (i- 0,1,...,N), and ~p additional nodes x_ 1, x_ 2,.., and xn+1, xn+ 2, **. Let (Ak) denote the set of spline functions whose knots are {xi, xi + }. Then Qf 6 Sp (A,k). This paper contains the following three parts: i) definition of a certain basis (q i,k of p (,k) and comments on its existence, (ii) proof that Q reproduces approrpriate classes of polynomials, and (iii) a note about some applications. Department of Mathematics, Jilin University, Changchun, China. Sponsored by the United States Army under Contract No. DAAG29-80-C ;- H I -I!!

6 4 1. Construction of { ie and let I..(k-2)... k-2). Then the functions Mk('-),. i are of order S-splines k on the knot sequence Z + k/2, hence independent over the points 1/2 by the Schoenberg-Whitney Theorem [6] since Mk(i - /2) I 0 for i I. Consequently, the functions Mk('-J/2), j e I are independent over I. In particular, there exists exactly one choice of SY : (Y so that JeI " [ jqk Mk ((-j/2) 0.1) satisfies qk(1) 8, 0 all i e 1 (1.2) Note that Y- Y by uniqueness and symmetry (which can be used to simplify -j Ti the calculation of Y) and that I " k~i)u - J/2) iei iex jei M (i - j ) - jei Y ( M k(i - J/2) Y 1 iei ke since M k.(i- J/2)- K k(i - J/2) - 1, all j e I. iel I Now we define (1.3) q 1 (,k *) :- qk(- The following are the table of coefficients Y and drawings of qk when k = 2,3, * -* --

7 I-!,k Yo0 Y 1 Y q 2 3 q Figure 1 D. X. Qi (1975) has already constructed a class of many-knot spline.'4 interpolating functions for solving curve fitting problems ([2], [51). The main difference between the previous study and the present one is in their basic function. Pk that appeared in [2] and (5] is not the same as qko - 2. The interpolation scheme leaves Pk fixed In this section I want to prove that Q reproduces certain polynomials. I will use the symbols: -3- * '1._

8 ~i1j 2'' V V V (V 1,...,V) 1 2 V e f1,2,...,k}, V V (i $ j), gymo) -, (Ii) : sm - k-1-2- h-3 2.., + k-i -2)I. k) U 0 The letters Pk denote the set or linear space of all polynomials of order k, i.e., of degree < k. Lemma (simple consequence of M~arsden's identity for a uniform partition [41).-i - [ t")n(x-i), x e [a,b) i (2.1) -I ~P, 0,1,.ook-1 Theorem 1 Q1 "k Proof It is enough to prove X" [ (i q,k (x), x e ls,b] i i -' 0,1,...,k-1. (2.2) Now we use induction as follows. Evidently (2.2) holds for 0-0. Let us assume (2.2) holds throughout i - 0,I,...,m-1. We will prove it holds for P m. Notice (1.1) q- W Y M(jX + -i and by lemma i I

9 Therefore P Y (x + 1 F)1j 1 (x (2.3) Since Y 1 jei p Wx - Y ()x iv) j2 = x' + P~()x IA P (2.4) v Y-i))~ By induction hypothesis and (2.3), (2.4), - in r-v x P () - I(')P (O)x C (M) q - I (Mp 0 (i)uivqj (X) i iv ~ VVi - I(r) - (')()i (M) ) m -V Then, fromt (2.3) and q (V) - 6i k-i k-i% P (0) ~ q (0)= (J), ji i,k O0 2~j ~ k 2' However

10 i _m) = 1 - symm(i s-y-,.., k-3 + k-1) -)- - I k in 2' 2'' 2 k-i1 ~ 1 in Smv(- 2 -, 2 -V -V k 1 -()" k V k-1 F -I k-v k- 1-V in 2 2 "m-v V 2 = i m. The last identity is gotten by using a well known fact about elementary symmetric function. From Theorem 1, we can get a result about approximation order. Theorem 2 If f e ck+ [a,b], then Rk :- f - Qf -I (s), I= max ls(a)1. k+1-9 a+(k-1) h~xb-(k-1)h I (x)i - s 0,1,...,k 3. Applications in CAGD - I Ai By convention, let {P i denote a set of ordered points in R n. We hope to get a curve through {P }. It is known that people in Computer Aided ii Geometric (CAGD) like and are used to the parametric form. So the curve, as may be imagined, can be represented as follows: Qk(t) = qk (t-j)p (3.1) We can get with ease from this representation and (1.1) in case of k 3,4: Q;(: '- i (P:J+,, -P.), 1 Q (:J)-i (1PJ+1-P I-1) 1PJ+2 PJ-2) Q;(J) - 3(PP 2P- + P2 2(P J+2"2P j +P J - 2 etc. 4 ~ J+ - JP P 2-2 ( V..II ' j.,, : >

11 It is simple and useful in CAGD that the interpolating curve is represented by a matrix. (i) Firstly, we consider a quadratic many-knot spline. Let te, 1 -. We can find 2 (q 3 (t+1), g 3 (t), q 3 (t-1), q 3 (t-2) (t 2t,1) ) (3.2) and with the help of symmetry : (t 2,t,1)M 3 2 T1 3(t) = (tq (-t) (Pi PiP t [0,t, Pi+,Pi,P, t e [, 1] * (ii) Secondly we consider a cubic many-knot spline. Let t e [0, Then 3 2 /7 _ (q 4 (t+2),q 4 (t+1),.**q 4 (t-3)) (t (3.3) : :(t3,t2,t,1)m and with the help of symmetry (t3t2tlm 4 (P i-2, 1+ 3 )T t e [0, 11 1 [ (lt) 3, ( t) 2, t, I)M4-41 (Pi3- P12 V-Pi2 T, m I I! l T -7-

12 I~~~ C.. i - n.. I Mi +3 Figure 2 As the parameter t increases from 0 to 1, the segment on the manyknot interpolating spline curve will be traversed from Pi to Pi 11 (see Figire 2). If we want to get many-knot spline surfaces when the points {P 1 are given (i = 0,1,...,Np j - 0,1,o...M), we could represent the surface as follows: [+. (u'w) ~~Qk qk(u-v)q k(u-w)pv'p0<v;..m, - 0 4V 4N, 0O4w 4 1 and this satisfies Qk"J - ij The representation by matrix for k 3 is: 2T 2 T1 (1) Q 3 (u,w) - (u,u,i)m PM (w,w,1), 0 u, w ( P P P i-1gj-1 i1,j i,j j+2 P i+2,j-1... P i+ 2, + ) - (, )V i t - (II) Q 3 (u,w) ((1-u)2,1_U,')MPMT(w 2 3 w,)t 0 2 )i-1 j+2 VU V-i+2, U, J-1 II '" i I a II i +I I i l -

13 (III) Q(u,v) =(U 2 ul)mpm ((1) 2, W 1) 0 (u ~ i w 1 P (p 1i+2 j-1 VU =1-1 1iJ+2 (]:V) Q (u,w) (1-U) 2 1)T PM T 2-U 2 -W ) TV I UW 3 lii33 3* 2(v 2W w 1 p~ ili J-1 P PVtd V-i+2, li-j+2 Their figures are shown in Figure 3. PP *1 j-l~j-l Figure 3 in the case of k =4 the representation and figures can be given as follows: 3 2 ( 4 uw) = ( U,AI& 1 T3 P&M (w 4 2 T1, W,1),0 < Ui~w <u 2u P (p 1i+i3 j+3 V,U 9=i-2, PU-J

14 Q (u,vw) (1-U),(1-u),1-u, 1)m P1 4 (W,v t,1)t, 4 u 4 1, 0 W v 4 -P -(P 1i2 J+3 V, J V-j+3, U*J T 3 2 T i Q(u, u,u,u,l)m 4 PM 4(1-v) O(-'a),-)) 1-,) 0 Cu 2' 2 w i+3 J-2 *1 (PV (I )3 2 T 3 2 T I Q 4 (u,w) (-u),(1-u),t-u,1)m 4 PM 4 (1-),(1-W),1-w,l), U, W 4 p - p 1-2 J-2 p PV, 11 V-ij+3, IimJ+3 P - i+3,j- /P /+,+ pi- 2,j- 2 i-2,j+3 Figure 4 Acwledement I would like to express my sincere appreciation to Professor Carl de Roar for his valuable suggestions.

15 K ,,,a L i! : Jm e - -o N, m a p REFERENCES [I] C. de Boor and G. J. Fix, Spline approximation by quasi-interpolants, J. Approx. Theory, 8 (1973), [2) Y. S. Li and D. X. Qi, The Methods of Spline Function, Academic Press, Peking, China, [31 T. Lyche and L. L. Schumaker, Local spline approximation methods, J. Approx. Theory, 15 (1975), [4] M. Marsden, An identity for spline functions with application to variation diminishing spline approximations, J. Approx. Theory, 3 (1970), [5] D. X. Qi, On cardinal many-knot 6 -spline interpolation (I), Acta. -I Scientiarum Natur. Universitatis Jilinensis, 3 (1975), [6) I. J. Schoenberg and A. Whitney, On Polya frequency functions III, Trans. Amer. Math. Soc., 74 (1953), DXQ/Jvs. -11-

16 SECURITY CLASSIFICATION OF THIS PAGC (0hNen Dwta '.etered) REPORT DOCUMENTATION PAGE r '^ IP.' T, 1os I. REPORT NUMO".R G. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMDER # TITLE an0br1te).. TYPE OF REPORT & PERIOD COVERED AoaSummary Report - no specific A Class of Local Explicit many-knot Spline reporting period Interpolation Schemes# S. PERFORMING ORG. REPORT NUMBER V. AUTHOR(&)... I 6. CONTRACT OR GRANT NUMBER(s) D.(// DAAG980CArd~41 9. PERFORN TION NA(ANO ADOR I0. PROGRAM ELEMENT. PROJECT. TASK MateOmaics OReearchATO en-te,.. of. AREA & WORK UNIT NUMBERS Mathematics Research Center, University of Work Unit Number Walnut Street Wisconsin Numerical Analysis & Computer Madison, Wisconsin ne ft. CONTROLLING OFFICE NAME AND ADDRESS REPORT DATE U. S. Army Research Office /// July 1981 P. 0. Box j S..lES Research Triangle Park, North Carolina MONITORING XGENCY NAME & ADORESS(Il ditlerenlt from Cutrolling Office,) IS. SECURITY CLASS. (of t Is report) UNCLASSIFIED "/T-t ier. DECLASSI FICATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) * Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abstract entered Inr Bock 20, it different tcm Report) 1. SUPPLEMENTARY NOTES I. KEY WORDS (Continue on reore aide itn ecoesary and Identify by block n.umber) Many-knot spline function, local, explicit, spline interpolation. I 11 * ( *.-/,. t 9 ' -. STRACT (Continue an reverse aide it necoesoy and identify by block number) The purpose of this paper is to present a new local explicit method for an approximation of real-valued functions defined on intervals. Trhe operators of the form Qf A fq are studied under a uniform mesh, where comes.fro from a linear combination of B-splines. This paper contains the definition of q 1 ]), comments on its existence, proof of reproduction of the operator Q for appropriate classes of polynomials, and a note about some applications. DD IjANf EDITION OF I NOV GS IS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (141kon Data Enteo, i