A rational cubic spline based on function values

A rational cubic spline based on function values

PII: Comput. & Graphics, Vol. 22, No. 4, pp. 479±486, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0097-8493(98)00...

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PII:

Comput. & Graphics, Vol. 22, No. 4, pp. 479±486, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0097-8493(98)00046-6 0097-8493/98 $19.00 + 0.00

Technical Section

A RATIONAL CUBIC SPLINE BASED ON FUNCTION VALUES QI DUAN1,2, K. DJIDJELI2, W. G. PRICE2 and E. H. TWIZELL3{ Department of Mathematics and Physics, Shandong University of Technology, Jinan, 250061, People's Republic of China 2 Department of Ship Science, University of Southampton, Southampton, SO17 1BJ, England 3 Department of Mathematics and Statistics, Brunel University, Uxbridge UB8 3PH, Middlesex, England

1

AbstractÐA rational spline based on function values only is constructed, which is of the same order as a cubic spline. The rational spline may be used to control the position and shape of a curve or surface. The approximation properties of this spline are studied. # 1998 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION

2. INTERPOLATION

Spline interpolation is a useful and powerful tool for designing a curve or surface, such as the outer shape of a ship, car or aeroplane. Many authors have studied many kinds of spline for curve and surface control [1±4] and in recent years the rational spline, especially the rational cubic spline with quadratic and cubic denominators, and its application to shape control, have received attention in the literature [5±8]. In [9], a rational cubic spline with linear denominator was constructed and analysed and was used to control the shape and second derivative of the interpolatory functions; some of its properties were studied in [10]. For this kind of interpolation (including those used in papers [5±8]), function values and derivatives at the knots are used. Unfortunately, in some manufacturing processes the derivatives are dicult to obtain and, because of this, a rational cubic spline is developed in the present paper which is based on the use of function values only. In general, this spline is the C1-continuous interpolant but C2-continuity may be obtained by controlling the parameters (ai and bi) of the spline. This spline is constructed in Section 2. In Section 3 the constrained curve-interpolation problem is considered and necessary and sucient conditions are obtained for a C1-continuous interpolant to be above or below a straight line and/or a quadric curve in an individual knot interval. In Section 4, the method to control the second-order derivative of the interpolatory function P(t) to be in some interval [N, M] is obtained, and in Section 5 the approximation property of interpolatory function is studied. { Author for correspondence. 479

n

Let fi$R , i = 0, 1, . . . n, n + 1 be a given set of data points, where t0
pi …t† ; qi …t†

i ˆ 0; 1; . . . n ÿ 1;

…1†

is considered, where pi …t† ˆ …1ÿy†3 ai fi ‡y…1ÿy†2 Vi ‡y2 …1ÿy†Wi ‡y3 bi fi‡1 ; qi …t† ˆ …1 ÿ y†ai ‡ ybi ; y ˆ …t ÿ ti †=hi ; hi ˆ ti‡1 ÿ ti ; and Vi ˆ …ai ‡ bi †fi ‡ ai fi‡1 ; Wi ˆ …ai ‡ 2bi †fi‡1 ÿ hi bi Di‡1 ; with ai>0, bi>0, and Di=(fi+1ÿfi)/hi. The rational cubic interpolatory function based on function values in [t0, tn] de®ned by (1) is denoted by P(t). It is easy to show that the interpolatory function P(t) which satis®es: p(ti) = fi, p(ti + 1) = fi + 1, p'(ti + 1) = Di + 1, exists and is unique for the given data. In fact, let P…t† ˆ

a…1 ÿ y†3 ai ‡ by…1 ÿ y†2 ‡ cta2 …1 ÿ y† ‡ dy3 bi …1 ÿ y†ai ‡ ybi

then P(t) satis®es p…ti † ˆ fi ; p…ti‡1 † ˆ fi‡1 ; p0 …ti † ˆ Di ; p0 …ti‡1 † ˆ Di‡1 ; if and only if a = fi, b = Vi, c = Wi and d = fi + 1. Obviously, the value of P(t) is related to ai, bi, but, when baii is ®xed, the value of P(t) is unrelated to the values of ai, bi, and is just related to the value of baii .

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Fig. 1. The C1-continuous, rational cubic spline [Equation (2)] with ai=1 and bi=1, and the original function f(t) = cos10(t).

If the knots are equally spaced, Equation (1) becomes   …1ÿy†2 …ai ‡bi y†fi ‡y ai ÿai y‡3bi yÿ2bi y2 fi‡1 ÿ y2 …1 ÿ y†bi fi‡2 …1 ÿ y†ai ‡ ybi

P…t† ˆ

t 2 ‰t1 , ti‡1 Š:

; …2†

Furthermore, if ai=bi,

ÿ y …1 ÿ y†fi‡2 ; t 2 ‰ti ; ti‡1 Š:

  aiÿ1 b hi 1 ‡ …Di ÿ Diÿ1 †‡hiÿ1 i …Di‡1 ÿ Di † ˆ 0; biÿ1 ai i ˆ1; 2;    ; n ÿ 1;

…3†

It is of interest that, for suitably selected parameters ai, bi, the piecewise rational cubic interpolatory function P(t) can be C2-continuous in [t0, tn]. In fact, let P00 …ti ‡† ˆ P00 …ti ÿ†; i ˆ 1; 2;    ; n ÿ 1;

…4†

may be obtained. Again, when the knots are equally spaced, (4) becomes   aiÿ1 b 1‡ …fi‡1 ÿ2fi ‡fiÿ1 †‡ i …fi‡2 ÿ2fi‡1 ‡fi † ˆ 0; biÿ1 ai i ˆ1; 2;    ; nÿ1:

  P…t† ˆ…1 ÿ y†2 …1 ‡ y†fi ‡ a 1 ‡ 2y ÿ 2y2 fi‡1 2

Fig. 3. The C1-continuous, rational cubic spline [Equation (2)] with ai=1 and bi=100, and the original function f(t) = cos10(t).

…5†

Some examples of the rational cubic spline interpolation are given in Figs 1±5. Figures 1±3 show the C1-continuous, rational cubic spline [Equation (2)] using di€erent values of ai and bi. It can be seen from Fig. 3 that the error between the C1-continuous, rational cubic spline (ai=1, bi=100), and the original function f(t) = cos10(t) is slightly larger than those in Fig. 1 (ai=1, bi=1) and Fig. 2

then the equations concerning the parameters ai and bi

Fig. 2. The C1-continuous, rational cubic spline [Equation (2)] with ai=10 and bi=1, and the original function f(t) = cos10(t).

Fig. 4. The C1-continuous, rational cubic spline [Equation (1)] with ai=10 and bi=1, and the original curves (f(t)2ÿt = 0, f(t)2+4f(t) + _t + 2 = 0).

A rational cubic spline based on function values

481

Table 2. Values of ci for bi=1 with various values of ai i

ai

1 2 3 4 5 6 7 8 9 10 11 12

1.0 1.1 1.2 1.5 2.0 3.0 5.0 10.0 20.0 50.0 100.0 1000.0

bi

ci

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.301805 0.297633 0.293937 0.285055 0.275151 0.264602 0.256627 0.252003 0.250556 0.250095 0.250024 0.250000

P…ti † ˆ fi ; P0 …ti † ˆ Di ; i ˆ 0; 1;    ; n; 2

Fig. 5. The C -continuous, rational cubic spline [Equations (1) and (4)], and the original function f(t) = cos10(t).

(ai=10, bi=1). This is expected from Tables 1 and 2 (see Section 5, error estimation). Figure 4 shows a C1-continuous, rational cubic spline [Equation (1)] with ai=10 and bi=1, using a di€erent set of interpolation points (f(t)2ÿt = 0, f(t)2+4f(t) + t + 2 = 0). Finally, Fig. 5 shows a C2-continuous, rational cubic spline [(1), (4)], together with the original function f(t) = cos10(t). The results obtained by the C2-continuous, rational cubic spline are very similar to the original function. However, it is found that the interpolation points have to be carefully chosen.

Given a function g(x) and a data set {(ti, fi):i = 0, 1,. . ., n, n + 1} with or fi Rg…ti †; i ˆ 0; 1;    ; n;

let P(t) be a rational cubic interpolatory function de®ned by (1) which satis®es the following conditions: Table 1. Values of ci for ai=1 with various values of bi i

ai

1 2 3 4 5 6 7 8 9 10 11 12

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

bi 1.0 1.1 1.2 1.5 2.0 3.0 5.0 10.0 20.0 50.0 100.0 1000.0

fi rg…ti †; i ˆ 0; 1; . . . ; n: From (1) it is known that qi(t)r 0 for t $ [ti, ti + 1], so P…t† ˆ

pi …t† rg…t† qi …t†

is equivalent to pi …t† ÿ qi …t†g…t†r0: Let

3. CONSTRAINED INTERPOLATION

fi rg…ti †

if P(t) rg(t)(or P(t) R g(t)) for all t $ [t0, tn], then P(t) is called a constrained interpolation above (or below) g(t). Within this context, the following two cases are considered. 3.0.1. Case 1. Let g(t) be the piecewise linear function on the partition D:t0
ci 0.301805 0.306085 0.310069 0.320499 0.334042 0.352379 0.372697 0.393554 0.406906 0.416295 0.419715 0.422937

Ui …t† ˆ pi …t† ÿ qi …t†g…t†; then Ui …t† ˆ …1ÿy†3 ai fi ‡y…1ÿy†2 Vi ‡y2 …1ÿy†Wi ‡y3 bi fi‡1 ÿ ÿ  ÿ …1ÿy†ai ‡ ybi …1ÿy†gi ‡ygi‡1 r0; …6† where gi, gi+1 represent g(ti), g(ti+1) respectively. Since ÿ ÿ  …1 ÿ y†ai ‡ ybi …1 ÿ y†gi ‡ gi‡1 y ˆ …1 ÿ y†2 ai gi ‡ y…1 ÿ y†…ai gi‡1 ‡ bi gi † ‡ y2 bi gi‡1 ˆ …1 ÿ y†3 ai gi ‡ y…1 ÿ y†2 …ai gi‡1 ‡ bi gi ‡ ai gi † ‡ y2 …1 ÿ y†…ai gi‡1 ‡ bi gi ‡ bi gi‡1 † ‡ y3 bi gi‡1 ; (6) becomes Ui …t† ˆ …1 ÿ y†3 ai …fi ÿ gi † ‡ y…1 ÿ y†2 Ai ‡ y2 …1 ÿ y†Bi ‡ y3 bi …fi‡1 ÿ gi‡1 †r0; where

…7†

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Ai ˆ ai …fi‡1 ‡ fi ÿ gi‡1 ÿ gi † ‡ bi …fi ÿ gi †;

where a ˆ bi …fi‡1 ÿ gi‡1 †; b ˆ Bi ; g ˆ Ai ; d ˆ ai …fi ÿ gi †:

Bi ˆ bi …2fi‡1 ÿ gi‡1 ÿ gi ÿ hi Di‡1 † ‡ ai …fi‡1 ÿ gi‡1 †: …8† Since fiÿgir0, fi+1ÿgi+1r0, then the following theorem follows 3.1. Theorem 3.1 Given {(ti, fi, gi), i = 0, 1, . . . , n, n + 1} with firgi, i = 0, 1, . . . , n, the sucient condition for the rational cubic spline P(t) to lie above the straight line g(t) in [ti, ti+1] is that the parameters ai, bi satisfy Bir0. For the equally spaced partition, Theorem 3.1 has the following 3.1.1. Corollary 3.1. For the equally spaced partition, the sucient condition for the rational cubic spline P(t) to lie above the straight line g(t) in [ti, ti+1] is that the parameters ai, bi satisfy the condition bi …3fi‡1 ÿ gi‡1 ÿ gi ÿ fi‡2 † ‡ ai …fi‡1 ÿ gi‡1 †r0: For a given data set {(ti, fi), i = 0, 1, . . . , n, n + 1} the corresponding numbers Bi de®ned by (8) are called criterion numbers for the rational cubic interpolant above the straight line in the subinterval [ti, ti + 1]. In the same way as above, the sucient condition for the rational cubic spline P(t) to lie below a piecewise line or between two given piecewise lines can be obtained as follows. 3.2. Theorem 3.2 Given {(ti, fi, gi* ), i = 0, 1, . . . , n, n + 1} with firgi* , i = 0, 1, . . . , n, the sucient condition for the rational cubic spline P(t) to lie below the straight line g*(t) in [ti, ti + 1] is that the parameters ai, bi satisfy ÿ  ÿ  bi 2fi‡1 ÿ gi‡1 ÿ gi ÿ hi Di‡1 ‡ ai fi‡1 ÿ gi‡1 R0:

Obviously, a r 0, d r 0, so the following theorem is obtained 3.4. Theorem 3.4 Given {(ti, fi, gi), i = 0, 1,. . ., n, n + 1} and firgi, i = 0, 1, . . . , n, the rational cubic spline P(t) lies above the straight line g(t) in [ti, ti + 1] if and only if the positive parameters ai, and bi satisfy either (a) Bir0, or (b) 4bi(fi+1ÿgi+1)Ai3+4ai(fiÿgi)B3i +27a2i b2i (fi+1ÿgi+1)2(fiÿgi)2ÿ18aibi(fi+1ÿgi+1) (fiÿgi)AiBiÿA2i B2ir0. 3.4.1. Case 2. Let g(t) be a quadratic function, and let firg(ti). In the same way as in case 1, where t $ [ti, ti + 1], since ÿ  gi …t† ˆ …1 ÿ y†2 gi ‡ y…1 ÿ y† 2gi ‡ g0i hi ‡ y2 gi‡1 ; where gi ˆ g…ti †; gi‡1 ˆ g…ti‡1 †; g0i ˆ g0 …ti †; it follows that P…t† ˆ

pi …t† rg…t† qi …t†

is equivalent to Ui …t† ˆ …1 ÿ y†3 ai …fi ÿ gi † ‡ y…1 ÿ y†2 Ci ‡ y2 …1 ÿ y†Di ‡ y3 bi …fi‡1 ÿ gi‡1 †e0;

…9†

where

ÿ  Ci ˆ ai fi‡1 ‡ fi ÿ 2gi ÿ hi g0i ‡ bi …fi ÿ gi †; …10†

3.3. Theorem 3.3 Given {(ti, fi, gi, gi*), i = 0, 1, . . ., n, n + 1} with girfirgi*, i = 0, 1, . . . , n, the sucient condition for the rational cubic spline P(t) to lie below the straight line g*(t) and above the straight line g(t) in [ti, ti + 1] is that parameters ai, bi satisfy the following inequality system b …2fi‡1 ÿgi‡1 ÿgi ÿhi Di‡1 †‡ai …fi‡1 ÿgi‡1 †r0; ÿi  ÿ  bi ÿ2fi‡1 ‡gi‡1 ‡gi ‡hi Di‡1 ‡ai ÿfi‡1 ‡gi‡1 r0: Using the method in [11, p. 343, Proposition 2], a necessary and sucient condition for this interpolation process may be obtained. Let y = s/(s + 1); it is easy to see that (7) is equivalent to U …s† ˆ as3 ‡ bs2 ‡ gs ‡ dr0; for all sr0;

Di ˆ ai …fi‡1 ÿ gi‡1 † ‡ bi …2fi‡1 ÿ 2gi ÿ hi Di‡1 ÿ hi g0i †: …11† For a given data set {(ti, fi), i = 0, 1, . . . , n, n + 1}, and a given quadratic (maybe piecewise) function g(t), the corresponding Ci, Di de®ned in (10) and (11) are called the criterion numbers for the rational cubic interpolant above the quadratic curve in the subinterval [ti,ti + 1]. In the same way as in case 1, the following theorem is obtained 3.5. Theorem 3.5 Let {(ti, fi), i = 0, 1, . . ., n, n + 1} be a given data set, and g(t) be a given quadratic equation satisfying firgi, i = 0, 1, . . . , n, then the sucient

A rational cubic spline based on function values

condition for the rational cubic spline P(t) to lie above the quadratic curve g(t) is that the parameters ai, bi are such that Cir0, Dir0. 3.5.1. Corollary 3.2. For the equally spaced partition, the sucient condition for the rational cubic spline P(t) to lie above the quadratic curve g(t) in [ti, ti + 1] is that the parameters ai, bi satisfy  ai …fi‡1 ‡ fi ÿ 2gi ÿ hi g0i † ‡ bi …fi ÿ gi †r0; ÿ  ai …fi‡1 ÿ gi‡1 † ‡ bi 3fi‡1 ÿ 2gi ÿ fi‡2 ÿ hi g0i r0:

 ÿ 3 ÿ1 : P00 …t† ˆ h2i …1 ÿ y†ai ‡ ybi (



where a ˆ bi …fi‡1 ÿ gi‡1 †;

4. CONSTRAINT ON THE SECOND-ORDER DERIVATIVE OF THE INTERPOLANT

The second-order derivative of an interpolant has been used in estimating the strain energy and, consequently, smoothness of the interpolant. Smaller energy generally implies smoother shape. However, it is possible that the overall energy of an interpolant is small while great enough to generate an abnormal shape at some points or even some small intervals. A better way would be to control the second-order derivative directly. An e€ective method can be developed for a rational cubic interpolant with linear denominator to restrict its second-order derivative in a desired interval [N, M]. When t $ [ti, ti + 1], from (1) it is easy to show that

…1 ÿ y†3 ai fi ‡ y…1 ÿ y†2 Vi ‡ y2 …1 ÿ y†Wi 3

‡ y bi fi‡1



) :

Let P0(t)R M, then ÿ 3 Q…y† ˆMh2i …1 ÿ y†ai ‡ ybi ‡ ( ÿ 2 ÿ ÿ …1 ÿ y†ai ‡ ybi 6…1 ÿ y†ai fi ‡ …6y ÿ 4†Vi ‡ …2 ÿ 6y†Wi ‡ 6ybi fi‡1 ÿ  ‡ 2…bi ÿ ai † …1 ÿ y†ai ‡ ybi :    ÿ3…1 ÿ y†2 ai fi ‡ 1 ÿ 4y ‡ 3y2 Vi

(a) Cir0, Dir0, or (b) 4bi(fi+1ÿgi+1)C3i +4ai(fiÿgi)D3i +27a2i b2i (f 2i+1ÿgi+1)2(fiÿgi)2ÿ18aibi(fi+1ÿgi+1) (fiÿgi)C2i D2ir0. Necessary and sucient conditions for constrained interpolation curves to lie above a quadratic curve are given in Theorem 3.5 and Theorem 3.6. In the same way, conditions for constrained interpolation which lies below a quadratic curve or between two given quadratic curves can also be given.



   2 2 ‡ 2 y ÿ 3y Wi ‡ 3y bi fi‡1 ‡ 2…bi ÿ ai †2

U …s† ˆ as3 ‡ bs2 ‡ gs ‡ dr0;

3.6. Theorem 3.6 Let {(ti, fi), i = 0, 1,. . ., n, n + 1} be a given data set, and let g(t) be a given quadratic function satisfying firgi, i=0, 1,. . ., n. The necessary and sucient conditions for the rational cubic spline P(t) to lie above the quadratic curve g(t) in [ti, ti + 1] are that the parameters ai, bi satisfy either

ÿ 2 ÿ …1 ÿ y†ai ‡ ybi 6…1 ÿ y†ai fi

‡ …6y ÿ 4†Vi ‡ …2 ÿ 6y†Wi ‡ 6ybi fi‡1 ÿ  ÿ 2…bi ÿ ai † …1 ÿ y†ai ‡ ybi  ÿ3…1 ÿ y†2 ai fi ‡ …1 ÿ 4y ‡ 3y2 †Vi

Furthermore, let y = s/(1 + s), then (9) may be written as

b ˆ Di ; g ˆ Ci ; d ˆ ai …fi ÿ gi †:

483



   ‡ 2y ÿ 3y2 Wi ‡ 3y2 bi fi‡1  ÿ 2…bi ÿ ai †2 …1 ÿ y†3 ai fi ‡ y…1 ÿ y†2 Vi 2

3

‡ y …1 ÿ y†Wi ‡ y bi fi‡1



) r0:

Since

 2 Q0 …y† ˆ …1 ÿ y†aa ‡ ybi h i 3…bi ÿ ai †Mh2i ‡ 6…ai fi ÿ Vi ‡ Wi ÿ bi fi‡1 † ;

so Q(y) is monotone in [0, 1] and Q…0† ˆ2a2i bi …fi ÿ fi‡1 ‡ hi Di‡1 † ‡ a3i Mh2i ;   Q…1† ˆb3i Mh2i ÿ 2fi ‡ 2fi‡1 ÿ 2hi Di‡1 ‡ 2ai b2i …fi‡1 ÿ fi ÿ hi Di‡1 †: Therefore, the following theorem is obtained. 4.1. Theorem 4.1 For the rational cubic interpolant function P(t) de®ned by (1), the second-order derivative P0(t) is less than or equal to a given real number M in [ti, ti + 1] if and only if the positive parameters ai, bi

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satisfy the following inequality system 2bi …fi ÿ fi‡1 ‡ hi Di‡1 † ‡

ai Mh2i r0;

…12†

  bi Mh2i ÿ 2fi ‡ 2fi‡1 ÿ 2hi Di‡1 ‡ 2ai …fi‡1 ÿ fi ÿ hi Di‡1 †r0:

…13†

Similarly, for the second-order derivatives of the interpolant function P(t) to be greater than or equal to a given real number N or to be bounded in the given interval [N,M], the following theorems hold. 4.2. Theorem 4.2 For the rational cubic interpolant function P(t) de®ned by (1), the second-order derivative P0(t) is greater than or equal to a given real number N in [ti, ti + 1] if and only if the positive parameters ai, bi satisfy the following inequality system 2bi …fi ÿ fi‡1 ‡ hi Di‡1 † ‡ ai Nh2i R0;

…14†

  bi Nh2i ÿ 2fi ‡ 2fi‡1 ÿ 2hi Di‡1 ‡ 2ai …fi‡1 ÿ fi ÿ hi Di‡1 †R0:

…15†

4.4. Theorem 4.4 For the given data {fi, i = 1, 2,. . .} and the given real number M, there must exist parameters ai>0, bi>0 for the interpolation function P(t) de®ned by (1) such that the second-order derivative of P(t) remains less than or equal to M in [ti, ti + 1] except the following cases: 1) a < 0 and M R 0; or 2) a = 0 and M < 0; or 3) a>0 and Mhi2Ra. In a similar way, for Theorem 4.2, there is the following existence theorem. 4.5. Theorem 4.5 For the given data {fi, i = 1, 2,. . .} and the given real number N, there must exist parameters ai>0, bi>0 for the interpolation function P(t) de®ned by (1) such that the second-order derivative of P(t) remains greater than or equal to N in [ti, ti + 1] except in the following cases: 1) a>0 and N r 0; or 2) a = 0 and N>0; or 3) a < 0 and Nhi2ra. For Theorem 4.3, assume M>0 and N < 0, then the inequalities (16)±(19) become

4.3. Theorem 4.3 For the rational cubic interpolant function P(t) de®ned by (1), the second-order derivative P0(t) in [ti, ti + 1] is bounded in a given interval [N, M] if and only if the positive parameters ai, bi satisfy the following inequality system (16)±(19) 2bi …fi ÿ fi‡1 ‡ hi Di‡1 † ‡ ai Mh2i r0;

‡ 2ai …fi‡1 ÿ fi ÿ hi Di‡1 †r0;

…17†

ÿ2bi …fi ÿ fi‡1 ‡ hi Di‡1 † ‡ ai Nh2i r0;

…18†

  ÿbi Nh2i ÿ 2fi ‡ 2fi‡1 ÿ 2hi Di‡1 …19†

As far as the existence condition for the positive parameters ai, bi in Theorem 4.1 is concerned, by setting li=bi/ai, where li>0, and letting a ˆ 2fi ÿ 2fi‡1 ‡ 2hi Di‡1 ; the conditions (12) and (13) become ali ‡ Mh2i r0; 

 Mh2i ÿ a li ÿ ar0:

The following theorem follows.

…20†

  Mh2i ÿ a li ÿ ar0;

…21†

ÿali ÿ Nh2i r0;

…22†

 a ÿ Nh2i li ‡ ar0:

…23†



…16†

  bi Mh2i ÿ 2fi ‡ 2fi‡1 ÿ 2hi Di‡1

ÿ 2ai …fi‡1 ÿ fi ÿ hi Di‡1 †r0:

ali ‡ Mh2i r0;

4.6. Theorem 4.6 For the given data {ti, fi, i = 1, 2, . . . } and the given real numbers M>0 and N < 0, there must exist parameters ai>0, bi>0 for the interpolation function P(t) de®ned by (1) such that the secondorder derivative of P(t) in [ti, ti + 1] remains on [N, M] if the given data satisfy 1) a = 0; or 2) a>0, a < Mhi2 and a2R(a ÿ Mhi2)Nhi2; or 3) a < 0, a>Nhi2 and a2R(a ÿ Nhi2)Mhi2. 4.7. Proof (a) If a = 0, it is obvious that for any li$R+, (20)± (23) hold. (b) If a>0, from Theorem 4.4, when a < Mhi2, let any lirl*0 (20) and (21) l0 ˆ Mha2 ÿa, then for i ÿNh2i  hold. Let l1 ˆ a ; from Theorem 4.5, for any li0.

A rational cubic spline based on function values

485

All of these show that when t*
The proof is complete.

q…t†r0; 5. ERROR ESTIMATION

For the error estimation of the piecewise rational cubic interpolatory function (1), it is necessary only to deal with the case when the knots are equally spaced. Without loss of generality, it is necessary to consider just the subinterval [ti, ti + 1]. When f(t) $ C2[a, b] and P(t) is the rational cubic interpolatory function of f(t) in [ti, ti + 1], using the Peano±Kernel Theorem gives Z ti‡2   R‰ f Š ˆ f…t† ÿ P…t† ˆ f …2† …t†Rt …t ÿ t†‡ dt; …24† ti

where

8 1 > > > …t ÿ t† ÿ > …1 ÿ y†a > i ‡ bi y > h > > 2 > > y…a ÿ a y ‡ 3b y i i i ÿ 2bi y † > > i > > > > …ti‡1 ÿ t† ÿ y2 …1 ÿ y†bi …ti‡2 ÿ t† ; > > > > ti < t < t; > > > > > > >   > > > 1 >   < ÿ …1 ÿ y†a ‡ b y y ai ÿ ai y i Rt …t ÿ t†‡ ˆ i > > 2 > ‡3b …ti‡1 ÿ t† y ÿ 2b y > i i > >  > > > 2 > > ÿy …1 ÿ y†b …t ÿ t† ; > i i‡2 > > > > t < t < ti‡1 ; > > > > > > > > > 1 > > > y2 …1 ÿ y†bi …ti‡2 ÿt†; > > …1 ÿ y†a ‡ b y i > i : ti‡1 < t < ti‡2 ; 8 < p…t†; ti < t < t; ˆ q…t†; t < t < ti‡1 ; : r…t†; ti‡1 < t < ti‡2 : Now the properties of the kernel function Rt[(t ÿ t)+] of the variable t in [ti, ti + 2] are considered. It is easy to see that r(t) r 0 for all t $ [ti + 1, ti + 2], and q(t) can be written as   bi y3 ‡ …ai ÿ 2bi †y2 ÿ ai y …ti‡1 ÿ t† ‡y2 …1 ÿ y†bi hi …1 ÿ y†ai ‡ bi y

q…t† ˆ Since q…ti‡1 † ˆ

y2 …1 ÿ y†bi hi r0; …1 ÿ y†ai ‡ bi y

q…ti † ˆ ÿyhi R0; and it can be proven that the zero point t* of q(t) satis®es ti < t < t ˆ ti‡1 ‡

y2 …1 ÿ y†bi hi ti‡1 : bi y ‡ …ai ÿ 2bi †y2 ÿ ai y 3

and when t < t < t*, q…t†R0: Finally, consider p(t); since p(t) = (t ÿ t) + q(t), and p…ti † ˆ …t ÿ ti † ‡ q…ti † ˆ 0; p…t† ˆ q…t† < 0; then for all t $ [ti, t], p…t†R0: Thus, j R‰ f Š jR k f …2† k Z ‡ ˆ

ti‡1

t

"Z

t ti

Z j p…t† j dt ‡ Z

q…t†dt ‡

ti‡2

ti‡1

t

t

j q…t† j dt

# r…t†dt

h2i w…ai ; bi ; y† k f …2† k ; v…ai ; bi ; y† 2

where

  w…ai ; bi ; y† ˆy ai bi ÿ 3b2i y4   ‡ 7b2i ÿ 5ai bi ‡ a2i y3   ÿ 4b2i ÿ 7ai bi ‡ 3a2i y2    ‡ 3 a2i ÿ ai bi y ÿ a2i ;  ÿ  v…ai ; bi ; y† ˆ …1ÿy†ai ‡bi y bi y2 ‡…ai ÿ2bi †yÿai :

For the given ai and bi, let ci ˆ max

0
w…ai ; bi ; y† : v…ai ; b; y†

…25†

This leads to the theorem

5.1. Theorem 5.1 If f(t) $ C2[a, b], and D:t0
h2i k f …2† …t† k ci ; 2

…26†

where hi=ti + 1ÿti, and the optimal error constant ci does not depend on the subinterval [ti, ti + 1]; it just depends on the parameters ai, bi, as shown in (25). Some values ci for di€erent ai, bi are given in Table 1 and Table 2.

486

Q. Duan et al.

From the tables, it seems that ci is bounded for any other ai, bir0. In fact, the following theorem can be proved. 5.2. Theorem 5.2 For any ai>0, bi>0, ci is bounded, and 1 3y3 ÿ 7y2 ‡ 4y Rci R max ˆ 0:42330428    : 0RyR1 4 2ÿy …27† 5.2.1. Proof. De®ne W…y; bi ; ai † ˆ

w…ai ; bi ; y† : v…ai ; bi ; y†

Let bi=liai, then li>0, and h   l2i ÿ3y5 ‡ 7y4 ÿ 4y3   ‡li y5 ÿ 5y4 ‡ 7y3 ÿ 3y2 i ‡y4 ÿ 3y3 ‡ 3y2 ÿ y W…y; bi ; ai † ˆ U…li ; y† ˆ : ‰l2i y2 …y ÿ 2† ‡ li y…y ÿ 1† …3 ÿ y† ÿ …1 ÿ y†2 Š Since y2 …yÿ1†3 h i 2 2 2li y …yÿ3†‡2li y…3yÿ4†ÿ…yÿ1†…yÿ3† U0 …li † ˆ h i2 ; l2i y2 …yÿ2†‡li y…yÿ1†…3ÿy†ÿ…1ÿy†2 it can be shown that, when li>0, for any ®xed y $ (0, 1), then U'(li)>0, so U(li, y) is a strictly increasing function of li. Since lim U…li ; y† ˆ

y4 ÿ 3y3 ‡ 3y2 ÿ y ; ÿ…1 ÿ y†2

lim U…li ; y† ˆ

ÿ3y3 ‡ 7y2 ÿ 4y ; yÿ2

li 40

li 41

it follows that y4 ÿ 3y3 ‡ 3y2 ÿ y 1 ˆ 0
ÿ3y3 ‡ 7y2 ÿ 4y ˆ 0:42330248    : 0
Thus, the proof is complete.

6. CONCLUDING REMARKS

. The interpolation given in this paper is C1-continuous, but a C2-continuous interpolation can be constructed by choosing the parameters ai, bi that satisfy the relation (4) or (5). . Theorem 4.6 gives not only the existence condition for the second-order derivative P0(t) of the interpolant function P(t) de®ned by (1) to be bounded in a given interval [N, M], but also a method on how to ®nd the parameters ai and bi. . Section 5 shows that this interpolation is very stable, no matter how the parameters ai, bi are chosen. The error coecient is bounded, and lies in the interval [0.25, 0.42330428].

AcknowledgementsÐThe support of the Royal Society of London is gratefully acknowledged.

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