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The poisedness of interpolation problem for splines
Applied Numerical Mathematics 54 (2005) 95–103 www.elsevier.com/locate/apnum
The poisedness of interpolation problem for splines ✩ Ren-Hong Wang a , Jing-Xin Wang a,b,∗ a Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China b Mathematics College, Liaoning Normal University, Dalian 116029, People’s Republic of China
Available online 25 November 2004
Abstract We study the poisedness of interpolation problem for univariate spline spaces and the location of interpolation points relative to spline knots, and obtain two new complete characterizations of poisedness condition. They are equivalent to the Schoenberg–Whitney theorem, but they are very valid for poisedness test of sample points for a spline space. We present the concepts of local poised set, minimal poised set and perfect local poised set for the configuration of interpolation points with respect to spline space and obtain complete characterizations of them. 2004 Published by Elsevier B.V. on behalf of IMACS. Keywords: Splines; Schoenberg–Whitney theorem; Poisedness of interpolation problem; Perfect local poised set; Minimal poised set
1. Introduction It is well known that splines are a useful tool of computational mathematics and computer aided geometrical design. Location of sample points in domain of a spline space that guarantees poisedness of the interpolation problem from a spline space is a very important problem in theory and application. An univariate polynomial of degree n can be determined completely by values of the polynomial at any n + 1 sample points in A = {t1 , t2 , . . . , tn , tn+1 } which is a set of n + 1 distinct points in real field R. Whereas a spline is piecewise polynomial with some analysis, so location of sample points relative to spline knots is the critical factor for guaranteeing poisedness of the interpolation problem from a spline ✩
The work is supported by NNSF of China (No. 69973010, No.10271022, No.60373093, No. 10171042).
* Corresponding author.
E-mail addresses: [email protected] (R.-H. Wang), [email protected] (J.-X. Wang). 0168-9274/$30.00 2004 Published by Elsevier B.V. on behalf of IMACS. doi:10.1016/j.apnum.2004.08.001
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space. Schoenberg–Whitney theorem (see [3,5]) has provided a good characterization for it, but it does not express the laws in which the poised interpolation sets is made up. In this paper, we solve the problem completely for the univariate spline case. In what follows, we consider univariate splines, the piecewise polynomials of degree n and n − 1 times continuous differentiable on the domain of a spline. By Sn [x0 , x1 , . . . , xN +1 ] we denote the spline space on interval [x0 , xN +1 ], with degree n, and with spline knots x1 , . . . , xN . Where, −∞ x0 < x1 < · · · < xN < xN +1 +∞. If x0 = −∞, then take [x0 , b] = (−∞, b]. In the same way, if xN +1 = +∞, take [a, xN +1 ] = [a, +∞). When x0 and/or xN +1 are real numbers, they are also called spline knots. It is simply for dealing with the spline spaces over bounded interval and unbounded interval uniformly. The difference between the two cases is that x0 , xN +1 may be taken as spline knots when they are not ±∞. That explains the reason for using interval signal [· · ·] in Sn [x0 , x1 , . . . , xN +1 ]. In addition, when there is no possible confusion we shall simply write Sn in place of Sn [x0 , x1 , . . . , xN +1 ]. Definition 1.1. A set A = {t1 , . . . , tm } with m distinct points in R is said to be poised with respect to Sn , if for any given set of data y1 , . . . , ym ∈ R, there exists one and only one spline s ∈ Sn such that s(ti ) = yi , i = 1, . . . , m. If A is poised with respect to Sn , A is also said to be poised set of Sn . It has been known that univariate spline s ∈ Sn [x0 , x1 , . . . , xN , xN +1 ] has following representation: s(x) = pn (x) +
N
cj (x − xj )n+
(−∞ < x < +∞),
(1.1)
j =1
where pn (x) is a univariate polynomial of degree n, u+ = u for u 0 and u+ = 0 for u < 0. {1, x, . . . , x n , (x − 1)n+ , . . . , (x − xN )n+ } is a basis of Sn [x0 , x1 , . . . , xN , xN +1 ]. Obviously, if A is poised with respect to Sn [x0 , x1 , . . . , xN +1 ], then m = N + n + 1. We always assume that xi < xj , ti < tj provided i < j . And for any i ∈ {1, 2, . . . , N + 1}, x0 ti xN +1 . By (1.1) we know that if the representation of spline s has been determined over [xi , xi+1 ], then one and only one condition corresponding to the interval (xi+1 , xi+2 ] (for example, one interpolation point) be needed to determine the representation of s over (xi+1 , xi+2 ]. In the same reason, a condition corresponding to [xi−1 , xi ) can help us to determine the representation of s on [xi−1 , xi ). For the statement above we can refer to [5]. The poisedness of the interpolation problem has been characterized by Schoenberg and Whitney in 1953 [3], that is, the famous Shoenberg–Whitney theorem. Theorem 1.1 (Schoenberg and Whitney [3]). Let A = {ti | i = 1, 2, . . . , N + n + 1} be a set consisting of N + n + 1 real numbers, x0 < t1 < · · · < tN +n+1 xN +1 . Then A is a poised set with respect to spline space Sn [x0 , x1 , . . . , xN +1 ] if and only if ti < xi < ti+n+1
(i = 1, . . . , N ).
(1.2)
The Schoenberg–Whitney theorem essentially says that if a set of sample points is poised with respect to Sn [x0 , x1 , . . . , xN , xN +1 ], then every support of B-spline contains at least one point in its interior.
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However, there is no definite result on the configuration of the interpolation points relative to the spline knots. In this paper, we introduce the concepts of local poised sets, the minimal poised sets and perfect local poised sets (in Section 2), give some characterizations of poisedness conditions that are different from Schoenberg–Whitney condition, and solve the configuration problem of sample points with respect to spline spaces (in Section 3). Section 4 will be concerned with the construction of perfect local poised set with respect to spline space Sn . In Section 5 we conclude the paper with a statement about the significance of the results in theory and application. 2. Some concepts We introduce several concepts in this section. Sn [x0 , x1 , . . . , xN , xN +1 ] and A are the same as the above statement. Definition 2.1. (1) For any i, j , 0 i < j N + 1, [xi , xj ] is said to be a connect interval of Sn [x0 , x1 , . . . , xN , xN +1 ]. (2) For connect interval [xi , xj ] of Sn = Sn [x0 , x1 , . . . , xN , xN +1 ], if E = [xi , xj ] ∩ A is a poised interpolation set of Sn [xi , xi+1 , . . . , xj ], then E is said to be a local poised set of Sn [x0 , x1 , . . . , xN , xN +1 ], and [xi , xj ] is a local poised interval of A. The local poised interval of A is also said to be interval determined by a set of sample points E. (3) If [xi , xj ] ∩ A is a local poised set of Sn [x0 , x1 , . . . , xN , xN +1 ], and for any connect interval [xs , xt ] [xi , xj ] of Sn [x0 , x1 , . . . , xN , xN +1 ], [xs , xt ] ∩ A will not become local poised set of Sn [x0 , x1 , . . . , xN , xN +1 ], then [xi , xj ] ∩ A is said to be a minimal local poised set of Sn , and [xi , xj ] is a minimal local poised interval of A. (4) Let E be a local poised set of Sn , and for any set of sample points B which contains E as a proper subset, B is not local poised set of Sn provided B does not contain any other minimal local poised set of Sn which is not contained in E, then E is said to be a maximal local poised set of Sn . (5) If E is a maximal local poised set of Sn which contains one and only one minimal local poised set, then E is said to be a perfect local poised set of Sn . In this case, the interval [xi , xj ] determined by E is said to be a perfect local poised interval of A. It is obvious that when [xi , xi+k ] is a local poised interval of A, there are exactly n + k + 1 points in [xi , xi+k ] ∩ A. It is easy to know that any local poised set has to contain a minimal local poised set. 3. Poisedness conditions of sample points from spline spaces Theorem 3.1. If A = {t1 , t2 , . . . , tN +n+1 } is a set contained N + n + 1 real numbers, and x0 t1 < · · · < tN +n+1 xN +1 , then A is a poised set of Sn [x0 , x1 , . . . , xN , xN +1 ] if and only if (1) for any k, i, 0 i < i + k N + 1, there are at most n + k points in [xi , xi+k ] ∩ A; (2) for any j = 1, . . . , N + 1, there are at least j points in [x0 , xj ) ∩ A.
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Proof. Necessity. If A is a poised set of Sn , then it follows from Theorem 1.1 that ti < xi < ti+n+1
(i = 1, . . . , N ).
(3.1)
So [xi , xi+k ] contains at most n + k points ti+1 , . . . , ti+k+n . On the other hand, tj < xj (j = 1, 2, . . . , N ) and tN +1 < xN +1 , so [x0 , xj ) ∩ A contains at least j points. Sufficiency. Assume that (1) for any k, i, 0 i < i + k N + 1, there are at most n + k points in [xi , xi+k ] ∩ A, and (2) for any j = 1, . . . , N + 1. Then there are at least j points in [x0 , xj ) ∩ A. For any i ∈ {1, 2, . . . , N + 1}, ti < xi . If this is not so, then there is a j , such that xj tj . Thus there are only j − 1 points in [x0 , xj ) ∩ A, and this contradict (2). At the same time, for any i ∈ {1, 2, . . . , N }, xi < ti+n+1 . Otherwise, there should be j such that tj +n+1 xj . Thus there are at least j + n points in [x0 , xj ) ∩ A, and this contradict (1). The proof has been completed. 2 The following result can be obtained in terms of the above theorem immediately. Corollary 3.2. A, Sn are the same as the above statement. Connect interval [xi , xj ] is a local poised interval of A if and only if (1) for any k, s, i s < s + k j , there are at most n + k points in [xs , xs+k ] ∩ A; (2) for u = i + 1, . . . , j , there are at least u − i points in [xi , xu ) ∩ A. This result is valid for judging a set of sample points to be a poised set. Following lemmas can be proved in terms of the above theorem and corollary. Lemma 3.3. If connect interval [xi , xj ] is a local poised interval of A, and connect interval [xu , xv ] ⊂ [xi , xj ] (i < u < v < j ) is also a local poised interval of A, then [xi , xv ] and [xu , xj ] are all local poised intervals of A. Proof. Since [xi , xj ] and [xu , xv ] are all local poised intervals of A, there are j − i + n points in [xi , xj ] ∩ A, and v − u + n points in [xu , xv ] ∩ A (see Fig. 1). Assume that there are n1 points in [xi , xu ) ∩ A, and n2 points in (xv , xj ] ∩ A (see Fig. 1). From Corollary 3.2(2) we know that there are at least u − i points in [xi , xu ) ∩ A. So n1 u − i. From Corollary 3.2(1) we know there are at least points in [xi , xv ] ∩ A. So there are at least (j − i + n) − (v − i + n) = j − v points in (xv , xj ] ∩ A, That is, n2 j − v. In addition, j − i + n = n1 + (v − u + n) + n2 (u − i) + (v − u + n) + (j − v) = j − i + n. {n1 points} xi
{v − u + n points} xu
{n2 points} xv
Fig. 1. k points means k points of A.
xj
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Therefore n1 = u − i, n2 = j − v, and there are v − i + n points in [xi , xv ] ∩ A and j − u + n points in [xu , xj ] ∩ A. Now observe interval [xi , xv ]. For any natural number s, k, i s < s + k v, since [xs , xs+k ] ⊂ [xi , xj ], there are at most k + n points in [xs , xs+k ] ∩ A. But for any m, i < m v, there are at least m − i points in [xi , xm ) ∩ A. By Corollary 3.2 we know that [xi , xv ] is a local poised interval of A. If v < m j , then there are at least m−i points in [xi , xm )∩A. But there exist u−i points [xi , xu )∩A, so there are at least (m − i) − (u − i) = m − u points in [xu , xm ) ∩ A. By the Corollary 3.2 we know that [xu , xj ] is also a local poised interval of A. The proof has been completed. 2 The following Lemma 3.4 can be easily obtained in terms of the above theorem. Lemma 3.4. If A is a poised set of Sn , A = {t1 , . . . , ti1 , . . . , ti2 , . . . , tN +n+1 }, and A1 = {tj | j = i1 + 1, . . . , i2 } is also a local poised set of Sn , then A0 = {tj | j = 1, 2, . . . , i1 , . . . , i2 } is a local poised set of Sn . Lemma 3.5. Let A be a poised set of Sn , A1 ⊂ A, A2 ⊂ A, and A1 , A2 be two distinct local poised sets of Sn . If A1 ∩ A2 = ∅, then A1 ∩ A2 is also a local poised set of Sn . Proof. Let x0 < t1 < · · · < xk1 ti1 xk1 −1 ti2 xk2 < · · · < ti3 xk3 , A1 = {t1 , . . . , ti2 }, A2 = {ti1 , . . . , ti3 }, and [x0 , xk2 ] and [xk1 , xk3 ] are poised intervals determined by A1 and A2 , respectively (see Fig. 2). Consider [xk1 , xk2 ] ∩ A. Since [xk1 , xk2 ] ⊂ [xk1 , xk3 ], the conditions (1) and (2) of Corollary 3.2 have been satisfied; hence it is sufficient to show that there are k2 − k1 + n points in [xk1 , xk2 ] ∩ A. In fact, there are k2 + n points in [x0 , xk2 ] ∩ A, and k2 − k1 + n points in [xk1 , xk3 ] ∩ A. Suppose there are j points in [xk1 , xk2 ] ∩ A, then (k2 + n) + (k3 − K1 + n) − j = k3 + n, that is, j = k2 − k1 + n. This completes the proof. 2 Theorem 3.6. A = {ti | i = 1, 2, . . . , N + n + 1} is a poised set of spline space Sn = Sn [x0 , x1 , . . . , xN +1 ] if and only if A is a union set of finite perfect local poised sets of Sn that any two of them are disjoint. And any two neighboring perfect poised intervals of A is segregated by n − 1 spline knots. Proof. Firstly to prove the sufficiency. There is no harm in supposing that A is a union set of A1 and A2 , which are two disjoint perfect local poised sets of Sn , and there are n − 1 spline knots that segregate the two perfect local poised intervals determined by A1 and A2 , respectively. {t1 , . . . , ti1 −1 } x0
{ti1 , . . . , ti2 } xk1
{ti2 +1 , . . . , ti3 } xk2
Fig. 2.
xk3
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Let A1 = {t1 , . . . , ti1 } contains i1 distinct points and [x0 , xk1 ] be the poised interval determined by it, then k1 = i1 − n. Suppose that A2 = {ti1 +1 , ti1 +2 , . . . , ti2 } contains i2 − i1 distinct points, and that [xk2 , xk3 ] is the poised interval determined by it, then k2 = i1 , t2 − n = k3 = N + 1. By Corollary 3.2, for any k, i, if 0 i < i + k k1 , there are at most n + k points in [xi , xi+k ] ∩ A1 . For any k, i, if k2 i < i + k k3 , then there are at most n + k points in [xi , xi+k ] ∩ A2 . But (xk1 , xk2 ) ∩ A = ∅, and there exist n − 1 spline knots between xk1 and xk2 . (1) For any k, i, when 0 i < i + k k3 , there are at most n + k points in [xi , xi+k ] ∩ A. But for any j = 1, . . . , k1 , there are at least j points in [x0 , xj ) ∩ A. Note that there are i1 points in A1 , n − 1 spline knots in (xk1 , xk2 ), and k2 = i1 , so for u ∈ {1, . . . , k2 }, there are at least u points in [x0 , xu ) ∩ A. (2) In terms of Corollary 3.2 we know that for any u ∈ {k2 + 1, . . . , k3 }, there are at least u − k2 points in [xk2 , xu ) ∩ A. So there are at least u − k2 + i1 = u − k2 + k2 = u points in [x0 , xu ) ∩ A. That is, for any j = 1, . . . , k3 , there are at least j points in [x0 , xj ) ∩ A. By (1), (2) and Theorem 3.1 we know that A is a local poised set of Sn . So the sufficiency is true. Now let us prove the necessity. Suppose A is a poised set of Sn [x0 , x1 , . . . , xN , xN +1 ]. Obviously, {t1 } is not a local poised set of Sn , but there must exist a natural number k such that {t1 , . . . , tk } is a local poised set of Sn . Let k1 be the smallest number of such kind of k. The set {t1 , . . . , tk1 } is a local poised set of Sn , so there exists a natural number k ∈ {1, . . . , k1 } such that {tk , tk+1 , . . . , tk1 } is a local poised set of Sn . By k0 we denote the largest of the k’s. Then B1 = {tk0 , tk0 +1 , . . . , tk1 } is a minimal local poised set of Sn . In fact, if tk0 ti < tj tk1 , then E = {ti , . . . , tj } is a subset of B1 . Suppose E is also a local poised set of Sn . If j < k1 , then by Lemma 3.3 we know that {tk0 , tk0 +1 , . . . , tj } is a local poised set; therefore {t1 , tk0 +1 , . . . , tj } is a local poised set. This contradicts the choice of k1 . So tj = tk1 . If k0 < i, then by Lemma 3.3 we know that {ti , . . . , tk1 } is a local poised set of Sn . This contradicts with the choice of k0 . Therefore tk0 = ti , thus B1 = E. It follows that B1 is a minimal local poised set of Sn . We use A1 to denote the perfect local poised set of Sn which contains B1 . And let ti1 = max A1 . If A1 = A, then the conclusion is true. Otherwise, A1 ∪ {ti1 +1 } is not local poised set of Sn , thereby there exist k ∈ {i1 + 1, i1 + 2, . . . , N + n + 1} such that A1 ∪ {ti1 +1 , tk } is a local poised set of Sn . By k2 we denote the smallest of the k’s. Here, A1 ∪ {ti1 +1 , . . . , tk2 } contains two different minimal local poised sets B1 and B2 . B1 ∩ B2 = ∅. If this is not so, by Lemma 3.5 we know that B1 ∩ B2 is also a local poised set. From B1 ∩ B2 B1 , B2 , we know this contradicts the fact that B1 and B2 are all minimal local poised sets. Use the same method we can prove that A1 ∩ B2 = ∅. By A2 we denote perfect local poised set of Sn containing B2 . It follows from Lemma 3.5 that A1 ∩ A2 = ∅. Suppose the local poised intervals of A1 and A2 are [x0 , xs1 ] and [xs2 , xs3 ], respectively, it can be proved that s2 − s1 = n. In fact, by Lemma 3.4, we know that A1 ∪ A2 is a poised set of Sn , therefore [x0 , xs3 ] contains s3 + n points of A, [x0 , xs1 ] contains s1 + n points of A, and [xs2 , xs3 ] contains s3 − s2 + n points of A. It follows
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that s1 + n + s3 − s2 + n = s3 + n, namely, s2 − s1 = n. This means there are exactly n − 1 spline knots that separate the interval [x0 , xs1 ] from [xs2 , xs3 ]. Since A contains finite points, the process can be repeated finite times, therefore we can decompose A into union of finite perfect local poised sets of Sn , any two of them are not intersect, and any two neighboring perfect local poised intervals are separated by n − 1 spline knots. This completes the proof. 2 4. Perfect local poised sets Definition 4.1. Let A = {ti | i = 1, 2, . . . , N + n + 1} be a poised set of spline space Sn = Sn [x0 , x1 , . . . , xN +1 ], k is a natural number. (1) Let Er = (xj , xj +k ] ∩ A contain exactly k points. If [xu , xu+n ] ⊂ [xj , xj +k ], (xu , xu+n ) ∩ A = ∅; but if m < k, the number of points of (xj , xj +m ] ∩ A does not reach m, then Er is said to be a right trivial extension set of sample points in A. And interval (xj , xj +k ] is a right trivial extension interval. (2) Let El = [xi−k , xi ) ∩ A contain exactly k points. If [xu−n , xu ] ⊂ [xi−k , xi ], (xu−n , xu ) ∩ A = ∅; but if m < k, the number of points of [xi−m , xi ) ∩ A does not reach m, then El is said to be a left trivial extension set of sample points in A. And interval [xi−k , xi ) is a left trivial extension interval. Both the left trivial extension sets and right trivial extension sets are called trivial extension sets. Both the left trivial extension intervals and right trivial extension intervals are called trivial extension intervals. Definition 4.2. If B is a minimal local poised set of Sn and B contains no any trivial extension set, then B is said to be a minimal poised set of Sn , or briefly, a minimal poised set. Theorem 4.1. Let B1 be a minimal local poised set of Sn , [xi , xj ] be a connect interval determined by B1 , and A1 be a set of sample points which contains B1 , then A1 is a perfect local poised set of Sn if and only if A1 consists of finite left, right trivial extension sets and B1 . Proof. The sufficiency is obvious, it is sufficient to prove the necessity. If A1 is a perfect local poised set of Sn which contains B1 , then there should exist k0 i, k1 j such that [xk0 , xk1 ] is the connect interval determined by A1 . We might as well suppose k0 < i < j < k1 , by Lemma 3.5, we know that [xi , xk1 ] is a poised interval of A. But there are j − i + n points in [xi , xj ] ∩ A so there are exactly k1 points in (xj , xk1 ] ∩ A. Since there must exist a natural number s such that A ∩ [xi , xs ] is local poised set of Sn (because [xi , xk1 ] is such one), we can take the smallest of such a kind of s and denote it by s1 , then 1 s1 k1 . If s1 = k1 , then we have achieved the conclusion. Otherwise, since [xi , xs1 ] and [xi , xk1 ] are poised intervals, in the same way we can find that there exists s2 such that 1 s2 n and there are exactly s2 points in (xs1 , xs1 +s2 ] ∩ A. After finite steps, we then extend it to [xi , xk1 ]. The process of left extending is the same as the above. After several steps, the interval [xi , xj ] is extended trivially to the interval [xk0 , xk1 ] and B1 is extended trivially to A1 . This completes the proof. 2 It follows from the above discussion that minimal poised sets are very important for understanding the configuration of sample points with respect to Sn .
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Since every spline knot is the common end point of two conjoint spline intervals, so in the process of studying the configurations, it is advisable to choose one form from the following three representations of intervals. (1) The first interval is closed one and others are intervals which are open on the left and closed on the right: [· , ·](· , ·] · · · (· , ·](· , ·]. (2) The last interval is closed one and others are intervals which are closed on the left and open on the right: [· , ·)[· , ·) · · · [· , ·)[· , ·]. (3) A closed interval is in-between, and the intervals on the left are closed on the left and open on the right. the intervals on the right are open on the left and closed on the right: [· , ·)[· , ·) · · · [· , ·)[· , ·](· , ·] · · · (· , ·](· , ·]. The configurations of minimal poised sets of spline spaces with degree 1–3 are listed in Table 1, where the numeral in a interval means the number of sample points which fall into the interval. The configurations of general minimal local poised sets of Sn can be obtained by adding finite trivial extension sets into the above minimal poised sets. The following are several examples of configurations of minimal local poised sets of splines with degree 3. Example 1. [2](2](1](0](0](3](2] 7 intervals, 10 knots. Example 2. [2)[1)[1)[2)[2] 5 intervals, 8 knots. Example 3. [2)[2)[0)[2](1](1](0](2](2] 9 intervals, 12 knots. Where the underlined parts with are all trivial extension sets. 5. Conclusions The significance of the results in this paper are as follows: First, for univariate spline space Sn [x0 , x1 , . . . , xN +1 ] and the sample set A which contains N + n + 1 distinct points, Theorems 3.1 and 3.6 provide two methods to judge that A is a poised set of Sn . And Table 1 Configurations of minimal poised sets of spline spaces with degrees 1–3 Degree of spline
Number of intervals
Configuration
1
1
[2]
2
1 2
[3] [2](2]
1 2 3
[4] [3](2]; [2](3] [2](2](2]; [3](0)(3]
3
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at the same time, it is easy to know how many perfect local poised sets make A, the poised set of Sn (namely, how many minimal poised sets are there in A). Second, for given sample points, in terms of Theorem 3.6 we can group it suitably by inserting spline knots to compose the minimal local poised sets and perfect local poised sets so as to obtain the interpolation spline which we want. The open question is if the similar results about interpolation set of spline spaces can be generalized in multivariate spline cases. In fact, the interpolation problem of multivariate spline is more complex and more difficult then that of univariate spline cases, even if for bivariate splines. Some discussions and results can be found in [1,4,2]. We will present some of our works on it in other papers.
References [1] O. Davydov, On almost interpolation, J. Approx. Theory 91 (1997) 398–418. [2] O. Davydov, M. Sommer, H. Strauss, Interpolation by bivariate linear splines, J. Comput. Appl. Math. 119 (2000) 115–131. [3] I.J. Schoenberg, A. Whitney, On Po’lya frequency function, III: The positivity of translation determinants with an application to the interpolation problem by spline curves, Trans. Amer. Math. Soc. 74 (1953) 246–259. [4] M. Sommer, H. Strauss, Interpolatio by uni- and multivariate generalized splines, J. Approx. Theory 83 (1995) 423–447. [5] R.-H. Wang, Numerical Approximation, Higher Education Press, Beijing, 1999.
The poisedness of interpolation problem for splines ✩ Ren-Hong Wang a , Jing-Xin Wang a,b,∗ a Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China b Mathematics College, Liaoning Normal University, Dalian 116029, People’s Republic of China
Available online 25 November 2004
Abstract We study the poisedness of interpolation problem for univariate spline spaces and the location of interpolation points relative to spline knots, and obtain two new complete characterizations of poisedness condition. They are equivalent to the Schoenberg–Whitney theorem, but they are very valid for poisedness test of sample points for a spline space. We present the concepts of local poised set, minimal poised set and perfect local poised set for the configuration of interpolation points with respect to spline space and obtain complete characterizations of them. 2004 Published by Elsevier B.V. on behalf of IMACS. Keywords: Splines; Schoenberg–Whitney theorem; Poisedness of interpolation problem; Perfect local poised set; Minimal poised set
1. Introduction It is well known that splines are a useful tool of computational mathematics and computer aided geometrical design. Location of sample points in domain of a spline space that guarantees poisedness of the interpolation problem from a spline space is a very important problem in theory and application. An univariate polynomial of degree n can be determined completely by values of the polynomial at any n + 1 sample points in A = {t1 , t2 , . . . , tn , tn+1 } which is a set of n + 1 distinct points in real field R. Whereas a spline is piecewise polynomial with some analysis, so location of sample points relative to spline knots is the critical factor for guaranteeing poisedness of the interpolation problem from a spline ✩
The work is supported by NNSF of China (No. 69973010, No.10271022, No.60373093, No. 10171042).
* Corresponding author.
E-mail addresses: [email protected] (R.-H. Wang), [email protected] (J.-X. Wang). 0168-9274/$30.00 2004 Published by Elsevier B.V. on behalf of IMACS. doi:10.1016/j.apnum.2004.08.001
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space. Schoenberg–Whitney theorem (see [3,5]) has provided a good characterization for it, but it does not express the laws in which the poised interpolation sets is made up. In this paper, we solve the problem completely for the univariate spline case. In what follows, we consider univariate splines, the piecewise polynomials of degree n and n − 1 times continuous differentiable on the domain of a spline. By Sn [x0 , x1 , . . . , xN +1 ] we denote the spline space on interval [x0 , xN +1 ], with degree n, and with spline knots x1 , . . . , xN . Where, −∞ x0 < x1 < · · · < xN < xN +1 +∞. If x0 = −∞, then take [x0 , b] = (−∞, b]. In the same way, if xN +1 = +∞, take [a, xN +1 ] = [a, +∞). When x0 and/or xN +1 are real numbers, they are also called spline knots. It is simply for dealing with the spline spaces over bounded interval and unbounded interval uniformly. The difference between the two cases is that x0 , xN +1 may be taken as spline knots when they are not ±∞. That explains the reason for using interval signal [· · ·] in Sn [x0 , x1 , . . . , xN +1 ]. In addition, when there is no possible confusion we shall simply write Sn in place of Sn [x0 , x1 , . . . , xN +1 ]. Definition 1.1. A set A = {t1 , . . . , tm } with m distinct points in R is said to be poised with respect to Sn , if for any given set of data y1 , . . . , ym ∈ R, there exists one and only one spline s ∈ Sn such that s(ti ) = yi , i = 1, . . . , m. If A is poised with respect to Sn , A is also said to be poised set of Sn . It has been known that univariate spline s ∈ Sn [x0 , x1 , . . . , xN , xN +1 ] has following representation: s(x) = pn (x) +
N
cj (x − xj )n+
(−∞ < x < +∞),
(1.1)
j =1
where pn (x) is a univariate polynomial of degree n, u+ = u for u 0 and u+ = 0 for u < 0. {1, x, . . . , x n , (x − 1)n+ , . . . , (x − xN )n+ } is a basis of Sn [x0 , x1 , . . . , xN , xN +1 ]. Obviously, if A is poised with respect to Sn [x0 , x1 , . . . , xN +1 ], then m = N + n + 1. We always assume that xi < xj , ti < tj provided i < j . And for any i ∈ {1, 2, . . . , N + 1}, x0 ti xN +1 . By (1.1) we know that if the representation of spline s has been determined over [xi , xi+1 ], then one and only one condition corresponding to the interval (xi+1 , xi+2 ] (for example, one interpolation point) be needed to determine the representation of s over (xi+1 , xi+2 ]. In the same reason, a condition corresponding to [xi−1 , xi ) can help us to determine the representation of s on [xi−1 , xi ). For the statement above we can refer to [5]. The poisedness of the interpolation problem has been characterized by Schoenberg and Whitney in 1953 [3], that is, the famous Shoenberg–Whitney theorem. Theorem 1.1 (Schoenberg and Whitney [3]). Let A = {ti | i = 1, 2, . . . , N + n + 1} be a set consisting of N + n + 1 real numbers, x0 < t1 < · · · < tN +n+1 xN +1 . Then A is a poised set with respect to spline space Sn [x0 , x1 , . . . , xN +1 ] if and only if ti < xi < ti+n+1
(i = 1, . . . , N ).
(1.2)
The Schoenberg–Whitney theorem essentially says that if a set of sample points is poised with respect to Sn [x0 , x1 , . . . , xN , xN +1 ], then every support of B-spline contains at least one point in its interior.
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However, there is no definite result on the configuration of the interpolation points relative to the spline knots. In this paper, we introduce the concepts of local poised sets, the minimal poised sets and perfect local poised sets (in Section 2), give some characterizations of poisedness conditions that are different from Schoenberg–Whitney condition, and solve the configuration problem of sample points with respect to spline spaces (in Section 3). Section 4 will be concerned with the construction of perfect local poised set with respect to spline space Sn . In Section 5 we conclude the paper with a statement about the significance of the results in theory and application. 2. Some concepts We introduce several concepts in this section. Sn [x0 , x1 , . . . , xN , xN +1 ] and A are the same as the above statement. Definition 2.1. (1) For any i, j , 0 i < j N + 1, [xi , xj ] is said to be a connect interval of Sn [x0 , x1 , . . . , xN , xN +1 ]. (2) For connect interval [xi , xj ] of Sn = Sn [x0 , x1 , . . . , xN , xN +1 ], if E = [xi , xj ] ∩ A is a poised interpolation set of Sn [xi , xi+1 , . . . , xj ], then E is said to be a local poised set of Sn [x0 , x1 , . . . , xN , xN +1 ], and [xi , xj ] is a local poised interval of A. The local poised interval of A is also said to be interval determined by a set of sample points E. (3) If [xi , xj ] ∩ A is a local poised set of Sn [x0 , x1 , . . . , xN , xN +1 ], and for any connect interval [xs , xt ] [xi , xj ] of Sn [x0 , x1 , . . . , xN , xN +1 ], [xs , xt ] ∩ A will not become local poised set of Sn [x0 , x1 , . . . , xN , xN +1 ], then [xi , xj ] ∩ A is said to be a minimal local poised set of Sn , and [xi , xj ] is a minimal local poised interval of A. (4) Let E be a local poised set of Sn , and for any set of sample points B which contains E as a proper subset, B is not local poised set of Sn provided B does not contain any other minimal local poised set of Sn which is not contained in E, then E is said to be a maximal local poised set of Sn . (5) If E is a maximal local poised set of Sn which contains one and only one minimal local poised set, then E is said to be a perfect local poised set of Sn . In this case, the interval [xi , xj ] determined by E is said to be a perfect local poised interval of A. It is obvious that when [xi , xi+k ] is a local poised interval of A, there are exactly n + k + 1 points in [xi , xi+k ] ∩ A. It is easy to know that any local poised set has to contain a minimal local poised set. 3. Poisedness conditions of sample points from spline spaces Theorem 3.1. If A = {t1 , t2 , . . . , tN +n+1 } is a set contained N + n + 1 real numbers, and x0 t1 < · · · < tN +n+1 xN +1 , then A is a poised set of Sn [x0 , x1 , . . . , xN , xN +1 ] if and only if (1) for any k, i, 0 i < i + k N + 1, there are at most n + k points in [xi , xi+k ] ∩ A; (2) for any j = 1, . . . , N + 1, there are at least j points in [x0 , xj ) ∩ A.
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Proof. Necessity. If A is a poised set of Sn , then it follows from Theorem 1.1 that ti < xi < ti+n+1
(i = 1, . . . , N ).
(3.1)
So [xi , xi+k ] contains at most n + k points ti+1 , . . . , ti+k+n . On the other hand, tj < xj (j = 1, 2, . . . , N ) and tN +1 < xN +1 , so [x0 , xj ) ∩ A contains at least j points. Sufficiency. Assume that (1) for any k, i, 0 i < i + k N + 1, there are at most n + k points in [xi , xi+k ] ∩ A, and (2) for any j = 1, . . . , N + 1. Then there are at least j points in [x0 , xj ) ∩ A. For any i ∈ {1, 2, . . . , N + 1}, ti < xi . If this is not so, then there is a j , such that xj tj . Thus there are only j − 1 points in [x0 , xj ) ∩ A, and this contradict (2). At the same time, for any i ∈ {1, 2, . . . , N }, xi < ti+n+1 . Otherwise, there should be j such that tj +n+1 xj . Thus there are at least j + n points in [x0 , xj ) ∩ A, and this contradict (1). The proof has been completed. 2 The following result can be obtained in terms of the above theorem immediately. Corollary 3.2. A, Sn are the same as the above statement. Connect interval [xi , xj ] is a local poised interval of A if and only if (1) for any k, s, i s < s + k j , there are at most n + k points in [xs , xs+k ] ∩ A; (2) for u = i + 1, . . . , j , there are at least u − i points in [xi , xu ) ∩ A. This result is valid for judging a set of sample points to be a poised set. Following lemmas can be proved in terms of the above theorem and corollary. Lemma 3.3. If connect interval [xi , xj ] is a local poised interval of A, and connect interval [xu , xv ] ⊂ [xi , xj ] (i < u < v < j ) is also a local poised interval of A, then [xi , xv ] and [xu , xj ] are all local poised intervals of A. Proof. Since [xi , xj ] and [xu , xv ] are all local poised intervals of A, there are j − i + n points in [xi , xj ] ∩ A, and v − u + n points in [xu , xv ] ∩ A (see Fig. 1). Assume that there are n1 points in [xi , xu ) ∩ A, and n2 points in (xv , xj ] ∩ A (see Fig. 1). From Corollary 3.2(2) we know that there are at least u − i points in [xi , xu ) ∩ A. So n1 u − i. From Corollary 3.2(1) we know there are at least points in [xi , xv ] ∩ A. So there are at least (j − i + n) − (v − i + n) = j − v points in (xv , xj ] ∩ A, That is, n2 j − v. In addition, j − i + n = n1 + (v − u + n) + n2 (u − i) + (v − u + n) + (j − v) = j − i + n. {n1 points} xi
{v − u + n points} xu
{n2 points} xv
Fig. 1. k points means k points of A.
xj
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Therefore n1 = u − i, n2 = j − v, and there are v − i + n points in [xi , xv ] ∩ A and j − u + n points in [xu , xj ] ∩ A. Now observe interval [xi , xv ]. For any natural number s, k, i s < s + k v, since [xs , xs+k ] ⊂ [xi , xj ], there are at most k + n points in [xs , xs+k ] ∩ A. But for any m, i < m v, there are at least m − i points in [xi , xm ) ∩ A. By Corollary 3.2 we know that [xi , xv ] is a local poised interval of A. If v < m j , then there are at least m−i points in [xi , xm )∩A. But there exist u−i points [xi , xu )∩A, so there are at least (m − i) − (u − i) = m − u points in [xu , xm ) ∩ A. By the Corollary 3.2 we know that [xu , xj ] is also a local poised interval of A. The proof has been completed. 2 The following Lemma 3.4 can be easily obtained in terms of the above theorem. Lemma 3.4. If A is a poised set of Sn , A = {t1 , . . . , ti1 , . . . , ti2 , . . . , tN +n+1 }, and A1 = {tj | j = i1 + 1, . . . , i2 } is also a local poised set of Sn , then A0 = {tj | j = 1, 2, . . . , i1 , . . . , i2 } is a local poised set of Sn . Lemma 3.5. Let A be a poised set of Sn , A1 ⊂ A, A2 ⊂ A, and A1 , A2 be two distinct local poised sets of Sn . If A1 ∩ A2 = ∅, then A1 ∩ A2 is also a local poised set of Sn . Proof. Let x0 < t1 < · · · < xk1 ti1 xk1 −1 ti2 xk2 < · · · < ti3 xk3 , A1 = {t1 , . . . , ti2 }, A2 = {ti1 , . . . , ti3 }, and [x0 , xk2 ] and [xk1 , xk3 ] are poised intervals determined by A1 and A2 , respectively (see Fig. 2). Consider [xk1 , xk2 ] ∩ A. Since [xk1 , xk2 ] ⊂ [xk1 , xk3 ], the conditions (1) and (2) of Corollary 3.2 have been satisfied; hence it is sufficient to show that there are k2 − k1 + n points in [xk1 , xk2 ] ∩ A. In fact, there are k2 + n points in [x0 , xk2 ] ∩ A, and k2 − k1 + n points in [xk1 , xk3 ] ∩ A. Suppose there are j points in [xk1 , xk2 ] ∩ A, then (k2 + n) + (k3 − K1 + n) − j = k3 + n, that is, j = k2 − k1 + n. This completes the proof. 2 Theorem 3.6. A = {ti | i = 1, 2, . . . , N + n + 1} is a poised set of spline space Sn = Sn [x0 , x1 , . . . , xN +1 ] if and only if A is a union set of finite perfect local poised sets of Sn that any two of them are disjoint. And any two neighboring perfect poised intervals of A is segregated by n − 1 spline knots. Proof. Firstly to prove the sufficiency. There is no harm in supposing that A is a union set of A1 and A2 , which are two disjoint perfect local poised sets of Sn , and there are n − 1 spline knots that segregate the two perfect local poised intervals determined by A1 and A2 , respectively. {t1 , . . . , ti1 −1 } x0
{ti1 , . . . , ti2 } xk1
{ti2 +1 , . . . , ti3 } xk2
Fig. 2.
xk3
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Let A1 = {t1 , . . . , ti1 } contains i1 distinct points and [x0 , xk1 ] be the poised interval determined by it, then k1 = i1 − n. Suppose that A2 = {ti1 +1 , ti1 +2 , . . . , ti2 } contains i2 − i1 distinct points, and that [xk2 , xk3 ] is the poised interval determined by it, then k2 = i1 , t2 − n = k3 = N + 1. By Corollary 3.2, for any k, i, if 0 i < i + k k1 , there are at most n + k points in [xi , xi+k ] ∩ A1 . For any k, i, if k2 i < i + k k3 , then there are at most n + k points in [xi , xi+k ] ∩ A2 . But (xk1 , xk2 ) ∩ A = ∅, and there exist n − 1 spline knots between xk1 and xk2 . (1) For any k, i, when 0 i < i + k k3 , there are at most n + k points in [xi , xi+k ] ∩ A. But for any j = 1, . . . , k1 , there are at least j points in [x0 , xj ) ∩ A. Note that there are i1 points in A1 , n − 1 spline knots in (xk1 , xk2 ), and k2 = i1 , so for u ∈ {1, . . . , k2 }, there are at least u points in [x0 , xu ) ∩ A. (2) In terms of Corollary 3.2 we know that for any u ∈ {k2 + 1, . . . , k3 }, there are at least u − k2 points in [xk2 , xu ) ∩ A. So there are at least u − k2 + i1 = u − k2 + k2 = u points in [x0 , xu ) ∩ A. That is, for any j = 1, . . . , k3 , there are at least j points in [x0 , xj ) ∩ A. By (1), (2) and Theorem 3.1 we know that A is a local poised set of Sn . So the sufficiency is true. Now let us prove the necessity. Suppose A is a poised set of Sn [x0 , x1 , . . . , xN , xN +1 ]. Obviously, {t1 } is not a local poised set of Sn , but there must exist a natural number k such that {t1 , . . . , tk } is a local poised set of Sn . Let k1 be the smallest number of such kind of k. The set {t1 , . . . , tk1 } is a local poised set of Sn , so there exists a natural number k ∈ {1, . . . , k1 } such that {tk , tk+1 , . . . , tk1 } is a local poised set of Sn . By k0 we denote the largest of the k’s. Then B1 = {tk0 , tk0 +1 , . . . , tk1 } is a minimal local poised set of Sn . In fact, if tk0 ti < tj tk1 , then E = {ti , . . . , tj } is a subset of B1 . Suppose E is also a local poised set of Sn . If j < k1 , then by Lemma 3.3 we know that {tk0 , tk0 +1 , . . . , tj } is a local poised set; therefore {t1 , tk0 +1 , . . . , tj } is a local poised set. This contradicts the choice of k1 . So tj = tk1 . If k0 < i, then by Lemma 3.3 we know that {ti , . . . , tk1 } is a local poised set of Sn . This contradicts with the choice of k0 . Therefore tk0 = ti , thus B1 = E. It follows that B1 is a minimal local poised set of Sn . We use A1 to denote the perfect local poised set of Sn which contains B1 . And let ti1 = max A1 . If A1 = A, then the conclusion is true. Otherwise, A1 ∪ {ti1 +1 } is not local poised set of Sn , thereby there exist k ∈ {i1 + 1, i1 + 2, . . . , N + n + 1} such that A1 ∪ {ti1 +1 , tk } is a local poised set of Sn . By k2 we denote the smallest of the k’s. Here, A1 ∪ {ti1 +1 , . . . , tk2 } contains two different minimal local poised sets B1 and B2 . B1 ∩ B2 = ∅. If this is not so, by Lemma 3.5 we know that B1 ∩ B2 is also a local poised set. From B1 ∩ B2 B1 , B2 , we know this contradicts the fact that B1 and B2 are all minimal local poised sets. Use the same method we can prove that A1 ∩ B2 = ∅. By A2 we denote perfect local poised set of Sn containing B2 . It follows from Lemma 3.5 that A1 ∩ A2 = ∅. Suppose the local poised intervals of A1 and A2 are [x0 , xs1 ] and [xs2 , xs3 ], respectively, it can be proved that s2 − s1 = n. In fact, by Lemma 3.4, we know that A1 ∪ A2 is a poised set of Sn , therefore [x0 , xs3 ] contains s3 + n points of A, [x0 , xs1 ] contains s1 + n points of A, and [xs2 , xs3 ] contains s3 − s2 + n points of A. It follows
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that s1 + n + s3 − s2 + n = s3 + n, namely, s2 − s1 = n. This means there are exactly n − 1 spline knots that separate the interval [x0 , xs1 ] from [xs2 , xs3 ]. Since A contains finite points, the process can be repeated finite times, therefore we can decompose A into union of finite perfect local poised sets of Sn , any two of them are not intersect, and any two neighboring perfect local poised intervals are separated by n − 1 spline knots. This completes the proof. 2 4. Perfect local poised sets Definition 4.1. Let A = {ti | i = 1, 2, . . . , N + n + 1} be a poised set of spline space Sn = Sn [x0 , x1 , . . . , xN +1 ], k is a natural number. (1) Let Er = (xj , xj +k ] ∩ A contain exactly k points. If [xu , xu+n ] ⊂ [xj , xj +k ], (xu , xu+n ) ∩ A = ∅; but if m < k, the number of points of (xj , xj +m ] ∩ A does not reach m, then Er is said to be a right trivial extension set of sample points in A. And interval (xj , xj +k ] is a right trivial extension interval. (2) Let El = [xi−k , xi ) ∩ A contain exactly k points. If [xu−n , xu ] ⊂ [xi−k , xi ], (xu−n , xu ) ∩ A = ∅; but if m < k, the number of points of [xi−m , xi ) ∩ A does not reach m, then El is said to be a left trivial extension set of sample points in A. And interval [xi−k , xi ) is a left trivial extension interval. Both the left trivial extension sets and right trivial extension sets are called trivial extension sets. Both the left trivial extension intervals and right trivial extension intervals are called trivial extension intervals. Definition 4.2. If B is a minimal local poised set of Sn and B contains no any trivial extension set, then B is said to be a minimal poised set of Sn , or briefly, a minimal poised set. Theorem 4.1. Let B1 be a minimal local poised set of Sn , [xi , xj ] be a connect interval determined by B1 , and A1 be a set of sample points which contains B1 , then A1 is a perfect local poised set of Sn if and only if A1 consists of finite left, right trivial extension sets and B1 . Proof. The sufficiency is obvious, it is sufficient to prove the necessity. If A1 is a perfect local poised set of Sn which contains B1 , then there should exist k0 i, k1 j such that [xk0 , xk1 ] is the connect interval determined by A1 . We might as well suppose k0 < i < j < k1 , by Lemma 3.5, we know that [xi , xk1 ] is a poised interval of A. But there are j − i + n points in [xi , xj ] ∩ A so there are exactly k1 points in (xj , xk1 ] ∩ A. Since there must exist a natural number s such that A ∩ [xi , xs ] is local poised set of Sn (because [xi , xk1 ] is such one), we can take the smallest of such a kind of s and denote it by s1 , then 1 s1 k1 . If s1 = k1 , then we have achieved the conclusion. Otherwise, since [xi , xs1 ] and [xi , xk1 ] are poised intervals, in the same way we can find that there exists s2 such that 1 s2 n and there are exactly s2 points in (xs1 , xs1 +s2 ] ∩ A. After finite steps, we then extend it to [xi , xk1 ]. The process of left extending is the same as the above. After several steps, the interval [xi , xj ] is extended trivially to the interval [xk0 , xk1 ] and B1 is extended trivially to A1 . This completes the proof. 2 It follows from the above discussion that minimal poised sets are very important for understanding the configuration of sample points with respect to Sn .
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Since every spline knot is the common end point of two conjoint spline intervals, so in the process of studying the configurations, it is advisable to choose one form from the following three representations of intervals. (1) The first interval is closed one and others are intervals which are open on the left and closed on the right: [· , ·](· , ·] · · · (· , ·](· , ·]. (2) The last interval is closed one and others are intervals which are closed on the left and open on the right: [· , ·)[· , ·) · · · [· , ·)[· , ·]. (3) A closed interval is in-between, and the intervals on the left are closed on the left and open on the right. the intervals on the right are open on the left and closed on the right: [· , ·)[· , ·) · · · [· , ·)[· , ·](· , ·] · · · (· , ·](· , ·]. The configurations of minimal poised sets of spline spaces with degree 1–3 are listed in Table 1, where the numeral in a interval means the number of sample points which fall into the interval. The configurations of general minimal local poised sets of Sn can be obtained by adding finite trivial extension sets into the above minimal poised sets. The following are several examples of configurations of minimal local poised sets of splines with degree 3. Example 1. [2](2](1](0](0](3](2] 7 intervals, 10 knots. Example 2. [2)[1)[1)[2)[2] 5 intervals, 8 knots. Example 3. [2)[2)[0)[2](1](1](0](2](2] 9 intervals, 12 knots. Where the underlined parts with are all trivial extension sets. 5. Conclusions The significance of the results in this paper are as follows: First, for univariate spline space Sn [x0 , x1 , . . . , xN +1 ] and the sample set A which contains N + n + 1 distinct points, Theorems 3.1 and 3.6 provide two methods to judge that A is a poised set of Sn . And Table 1 Configurations of minimal poised sets of spline spaces with degrees 1–3 Degree of spline
Number of intervals
Configuration
1
1
[2]
2
1 2
[3] [2](2]
1 2 3
[4] [3](2]; [2](3] [2](2](2]; [3](0)(3]
3
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at the same time, it is easy to know how many perfect local poised sets make A, the poised set of Sn (namely, how many minimal poised sets are there in A). Second, for given sample points, in terms of Theorem 3.6 we can group it suitably by inserting spline knots to compose the minimal local poised sets and perfect local poised sets so as to obtain the interpolation spline which we want. The open question is if the similar results about interpolation set of spline spaces can be generalized in multivariate spline cases. In fact, the interpolation problem of multivariate spline is more complex and more difficult then that of univariate spline cases, even if for bivariate splines. Some discussions and results can be found in [1,4,2]. We will present some of our works on it in other papers.
References [1] O. Davydov, On almost interpolation, J. Approx. Theory 91 (1997) 398–418. [2] O. Davydov, M. Sommer, H. Strauss, Interpolation by bivariate linear splines, J. Comput. Appl. Math. 119 (2000) 115–131. [3] I.J. Schoenberg, A. Whitney, On Po’lya frequency function, III: The positivity of translation determinants with an application to the interpolation problem by spline curves, Trans. Amer. Math. Soc. 74 (1953) 246–259. [4] M. Sommer, H. Strauss, Interpolatio by uni- and multivariate generalized splines, J. Approx. Theory 83 (1995) 423–447. [5] R.-H. Wang, Numerical Approximation, Higher Education Press, Beijing, 1999.