Unisolvency for multivariate polynomial interpolation in Coatmèlec configurations of nodes

Unisolvency for multivariate polynomial interpolation in Coatmèlec configurations of nodes

Applied Mathematics and Computation 217 (2011) 7427–7431 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homep...

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Applied Mathematics and Computation 217 (2011) 7427–7431

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Unisolvency for multivariate polynomial interpolation in Coatmèlec configurations of nodes Miguel Ángel García-March a, Fernando Giménez b, Francisco R. Villatoro c, Jezabel Pérez b,⇑, Pedro Fernández de Córdoba b a

Department of Physics, Colorado School of Mines, Golden, Colorado, USA Instituto Universitario de Matemática Pura y Aplicada – IUMPA, Universidad Politécnica de Valencia, Valencia, Spain c Departamento de Lenguajes y Ciencias de la Computación E.T.S.I. Industriales, Universidad de Málaga, Málaga, Spain b

a r t i c l e

i n f o

Keywords: Multivariate interpolation Properly posed set of nodes Geometric characterization Coatmèlec lattices

a b s t r a c t A new and straightforward proof of the unisolvability of the problem of multivariate polynomial interpolation based on Coatmèlec configurations of nodes, a class of properly posed set of nodes defined by hyperplanes, is presented. The proof generalizes a previous one for the bivariate case and is based on a recursive reduction of the problem to simpler ones following the so-called Radon–Bézout process. Ó 2011 Published by Elsevier Inc.

1. Introduction The problem of polynomial interpolation of one-dimensional data has a widely known solution. However, despite its apparent simplicity, multivariate polynomial interpolation remains a topic of current research [1–3]. The existence and uniqueness of the interpolation polynomial strongly depends on the geometrical distribution of the interpolation points. The distribution of points for which the interpolation problem is unisolvable is referred to as properly posed set of nodes (PPSN). The mathematical characterization of the most general PPSN is not currently known. The configurations of nodes based on algebraic varieties, such as those of Bos [4] and Liang et al. [5,6], are very general but non-constructive. In a computational setting, configurations based on hyperplanes, such as those of Coatmèlec [7] and Chung and Yao [8], are preferred. Surprisingly, the configuration of nodes introduced by Coatmèlec [7] in the plane has received several names: DH-set [2], straight line type node configuration [5], PPSN with node configuration A [9], straight line type node configuration A [10], PPSN by the recursive construction theorem using lines [11], and PPSN by line-superposition process [12]. In this paper, a new proof of the unisolvability of the interpolation problem for Coatmèlec configuration of nodes in arbitrary dimensions is presented. The proof is based on a Bézout–Radon process [13,14]. Chui and Lai [9] present a proof for the bivariate case only, state the result in arbitrary dimension, but did not prove it because of complications in their notation. Multidimensional interpolation is the basis to develop different numerical methods. The results of this paper permit to design, for example, generalized finite difference methods in irregular meshes based on Coatmèlec configuration of nodes in two [15] or more dimensions. The contents of this paper are as follows. The definitions and notation required to set our main theorem are presented in the next section. The proof of this theorem is detailed in Section 3. Finally, in the last section, the main conclusions are summarized.

⇑ Corresponding author. E-mail address: [email protected] (J. Pérez). 0096-3003/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.amc.2011.02.034

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2. Presentation of the problem Let Pm ðRk Þ be the vector space of multivariate polynomials of degree not greater than m with k variables. Let w ¼ ðx1 ; . . . ; xk Þ> 2 Rk , where

>

j

j

j

denotes transpose, N0 ¼ N [ f0g, j ¼ ðj1 ; . . . ; jk Þ> 2 C :¼ Nk0 , jjj = j1 +    + jk, wj ¼ x11 x22    xkk , k

and Cm :¼ {j 2 C : jjj 6 m}. The set of multivariate monomials fwj gj2Cm is a basis of Pm ðR Þ, i.e., every polynomial pm(w) P may be written uniquely as j2Cm aj wj , with aj 2 R. Hence, the vector space Pm ðRk Þ has dimension N ¼ C kkþm , where C kn is   n the binomial coefficient . k Pm k1 k1 s s s Let C :¼ {j 2 Cm : jjj = s}, s = 0, 1, . . ., m. Note that Cm ¼ [m s¼0 C , the cardinal #C ¼ C k1þs , and #Cm ¼ s¼0 C k1þs ¼ N. The set of sth degree monomials may be represented as a column vector of length #Cs given by > s 1 1 s1 s > wðsÞ :¼ ðxs1 ; xs1 i1 6 i2 6    6 is 6 k. Note that 1 x2 ; . . . ; xi1 xi2    xis ; . . . ; xk1 xk ; xk Þ , for all i ¼ ði1 ; . . . ; is Þ 2 N0 , and 1 6 wð0Þ ¼ ð1Þ 2 R, wð1Þ ¼ w 2 Rk , and each component of the vector w(s) corresponds to a unique monomial wj with j 2 Cs. Using P P j this notation, every polynomial pm ðwÞ 2 Pm ðRk Þ may be written as m s¼0 j2Cs aj w . k Here on, a node refers to a point in R and a configuration of nodes (CN) is a set of pairwise distinct nodes X m ¼ fwi gNi¼1 where wi  ðxð1;iÞ ; xð2;iÞ ; . . . ; xðk;iÞ Þ> 2 Rk . The Lagrange interpolation problem may be stated as follows: Given a CN Xm and an arbitrary set of real numbers ffi 2 RgNi¼1 , find a polynomial pm ðwÞ 2 Pm ðRk Þ such that

pm ðwi Þ :¼

X

aj wji ¼ fi ;

i ¼ 1; 2; . . . ; N:

ð1Þ

j2Cm

This problem is properly posed with respect to Xm if it has a unique solution (unisolvability) for every set ffi gNi¼1 . Compared with the one-dimensional case where the solvability is always assured, the solvability of multivariate interpolation depends strongly on the geometrical distribution of the nodes. A CN Xm is said to be a properly posed set of nodes (PPSN) if the Lagrange interpolation problem is properly posed with respect to Xm. Eq. (1) is a system of N linear equations with a multivariate Vandermonde matrix Vm, i.e., ðV m Þij ¼ wji , where j 2 Cm, wi 2 Xm, and 1 6 i 6 N. Note that this matrix looks a little bit bizarre since rows and columns are indexed by different structural entities. A graded lexicographical order in the set of multiindices Cm may be introduced to enhance the notation (see Ref. [16]) but this is not required in this paper. The following theorem summarizes some previously known results. Theorem 1. Let X m ¼ fwi gN i¼1 be a CN in k dimensions and Vm the corresponding multivariate Vandermonde matrix, then the following expressions are equivalent: (i) Xm is a PPSN in Rk . (ii) Vm is a nonsingular matrix, i.e., det(Vm) – 0. (iii) rank(Vm) = N. Let X m  X ðm;kÞ ¼ fwi gNi¼1  Rk be a CN with N ¼ C kmþk nodes in k dimensions. Let us define by induction on k the following CNs, first introduced by Coatmèlec [7,9]. S Definition 2. A CN X m  X ðm;kÞ  Rk is Coatmèlec in k dimensions if X ðm;kÞ ¼ m X with #X ðp;k1Þ ¼ C k1 pþk1 and there Sp¼0 ðp;k1Þ c , for 0 6 p 6 m  1, with each X(p,k1) exists m + 1 hyperplanes c0,c1,. . .,cm such that X(m,k1)  cm and X ðp;k1Þ  cp n m q¼pþ1 q being Coatmèlec in (k  1) dimensions by identifying each hyperplane cp with Rk1 . Note that, in one dimension, every CN X m  X ðm;1Þ  R is Coatmèlec because all its nodes are pairwise distinct, i.e., wi – wj, if i – j. Note also that, in Definition 2, only one node belongs to the hyperplane cm. The main result of this paper is a proof of the following theorem. Theorem 3. Every Coatmèlec CN Xm in k dimensions is a properly posed set of nodes in Rk . 3. Proof of the main theorem Our proof makes use of the following lemmas. k b Lemma 4. Let us take the CN Xm where the nodes fwi gN i¼1 are represented as column vectors in R , and the CN X m whose nodes are b m be the ^ i ¼ w0 þ Hwi ; i ¼ 1; . . . ; N, where w0 is an arbitrary vector and H is a non-singular matrix of dimension k. Let Vm and V w b b Vandermonde matrices associated to the CNs Xm and X m , respectively. If rank(Vm) = N, then rankð V m Þ ¼ N.

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Proof of Lemma 4. For every set of real numbers ffi 2 RgNi¼1 , there exists one and only one interpolating polynomial such ^ m ðw ^ i Þ ¼ fi , given by p ^m ð^ that p xÞ ¼ pm ðH1 ðx  w0 ÞÞ where pm(x) is the unique interpolating polynomial for Xm given by Theb orem 1. Therefore, rankð V m Þ ¼ N. Lemma 5. Let f^ xi : i ¼ 1; . . . ; kg be an orthonormal basis of Rk , and n1 an arbitrary vector. There always exists an orthogonal ^ 1 ¼ n1 =kn1 k. matrix H, representing a rotation in Rk , which transform the vector ^ x1 onto H^ x1 ¼ n ^1 ¼ ^ x1 , then H = I, the identity matrix. Otherwise, let us apply the procedure of Gram–Schmidt ortProof of Lemma 5. If n honormalization to vectors f^ x1 ; n1 g, yielding

^1 ¼ ^x1 ; q

q2 q ^2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ 2 ; q2  q2 kq2 k

^ 1 Þq ^1 ; q2 ¼ n1  ðn1  q

where the dot is the ordinary Euclidean dot product. An arbitrary vector q can be written as q = q\ + qVert, where ^ 1 Þq ^1 þ ðq  q ^ 2 Þq ^1 ¼ QQ > q, where Q ¼ ½q ^1 ; q ^2  is the rectangular matrix whose columns are the vectors q ^i ; note that qk ¼ ðq  q > Q Q is the identity matrix of dimension 2. Taking the vector q\ = q  qk as the rotation axis for the rotation matrix H results in Hq = q\ + Hqk = (I  QQ>)q + QRQ>q, where R is the standard two-dimensional rotation matrix





cos h  sin h sin h

cos h

 ;

^1 ; cos h ¼ ^x1  n

sin h ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 1 Þ2 : 1  ð^x1  n

>

^1 . Hence, H = I  QQ + QRQ> is a rotation matrix (HH> = H>H = I and det(H) = 1) such that H^ x1 ¼ n Proof of Theorem 3. Let us use the induction principle over m and k. Let us first consider m = 0 and any k 2 N. Clearly X0 = w1 and rank(V0) = 1 = N. We consider next k = 1 and m – 0. The corresponding CN is Coatmèlec in one dimension and the coefficient matrix is a (one-dimensional) Vandermonde matrix with maximal rank C 1mþ1 ¼ m þ 1 ¼ N, since the nodes are pairwise distinct. By the induction hypothesis, let us assume that the theorem holds for either m  1 or k  1, and let us prove that it holds for m and k. Here on, let us take n = m + k. Since Xm is a Coatmèlec CN in k dimensions, the following conditions are fulfilled

n o X ðm;k1Þ ¼ w1 ; w2 ; . . . ; wC k1  cm ; n1 n o X ðm1;k1Þ ¼ wC k1 þ1 ; . . . ; wC k1 þC k1  cm1 n cm ; n1 n2 n n1 o X ðm2;k1Þ ¼ wC k1 þC k1 þ1 ; . . . ; wC k1 þCk1 þC k1  cm2 n cm1 [ cm ; n1

X ð0;k1Þ

n2

n1

n2

n3

.. . ¼ fwN g  c0 n c1 [    [ cm ;

where

X m ¼ X ðm;k1Þ [ X ðm1;k1Þ [    [ X ð0;k1Þ : The multivariate Vandermonde matrix associated to the Lagrange interpolation problem in the CN Xm may be written as

0

1

B wð1Þ B 1 B B ð2Þ B V m ¼ B w1 B B .. B . @ ðmÞ w1

1 ð1Þ

w2

ð2Þ

w2 .. .

ðmÞ

w2



1

1

ð1Þ wCk C C n C ð2Þ C    wCk C: n C C .. C . C A ðmÞ    wC k



n

^ ¼ w0 þ Hw to all the nodes of the CN, where H is the orthogonal matrix given in Let us apply the affine transformation w Lemma 5, that transforms the xk coordinate axis in Rk into the normal vector to the hyperplane cm, and w0 is the distance between the intersection point of the (new) rotated xk axis and the hyperplane cm. ^ 1; w ^ 2; . . . ; w ^ C k1 g, hence The application of the affine transformation nullifies the kth coordinates of the vectors fw n1 j b m , where ð V b mÞ ¼ w ^ i ¼ ð^ ^ w xð1;iÞ ; ^ xð2;iÞ ; . . . ; ^ xðk1;iÞ ; 0Þ> . Let V , be the coefficient matrix of the transformed linear system of ij i b equations. From Lemma 4, rankðV m Þ ¼ rankð V m Þ. b m may be sorted by renaming the nodes w ^ i to w ~ i , in order to group all its zero The rows and columns of the matrix V e m has the following structure elements into its left-bottom part. This process preserves the rank. The resulting matrix V



A

B

0

A0 D

 ;

ð2Þ

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k1 where A is the C n1  C k1 n1 matrix given by

0

1

1

B ð1Þ ~ Bw B 1 B B ð2Þ B ~ A ¼ B w1 B B . B . B . @ ~ 1ðmÞ w B is the

k1 C n1



0

~ ð1Þ w 2 ~ ð2Þ w 2 .. . ~ 2ðmÞ w

C kn1

D is the



0

B B B B D¼B B B @ A0 is the

C kn1

n1

1



~ ð1Þk1 w C þ2



n1

~ ð2Þk1 w C þ2 n1

.. . ~ ðmÞ w C k1 þ2

n1

C kn1

C ~ ð1Þk1 C  w C n1 C C C ~ ð2Þk1 C  w C n1 C; C .. .. C C . . C A ~ ðmÞ  w k1 C

matrix

1

B ð1Þ ~ k1 Bw þ1 B C n1 B B ð2Þ B ~ k1 þ1 B ¼ B wC n1 B B . B .. B @ ~ ðmÞ w C k1 þ1 C kn1

1

1



n1

1

1

C ~ ð1Þk C w Cn C C C ~ ð2Þk C  w C n C; C .. .. C C . . C A ~ ðmÞ  w k C n

diagonal matrix

1

^xðk;C k1 þ1Þ

0



0

0

^xðk;C k1 þ2Þ



0

.. .

.. .

..

.. .

0

0

   ^xðk;C k Þ

C kn1

matrix given by

n1



0

n1

1

B ð1Þ ~ Bw þ1 B Ck1 B n1 0 A ¼B .. B . B @ ~ ðmÞ w k1 C n1 þ1

n

1



~ ð1Þk1 w þ2 C



n1

.. . ~ ðmÞ w C k1 þ2 n1

.

C C C C C; C C A

1

1

C ~ ð1Þk C w Cn C C ; .. .. C C . . C A ~ ðmÞ  w Ck n

k k1 k and finally 0, cf. Eq. (2), represents the null matrix of dimensions C kn1  C k1 n1 . We recall that C n ¼ C n1 þ C n1 . ~ The square matrix A is a multivariate Vandermonde matrix in (k  1) variables and the C k1 nodes f w g i are a Coatmèlec n1 CN in (k  1) dimensions. Therefore, by the induction hypothesis, rankðAÞ ¼ C k1 n1 . k ^ The diagonal matrix D is nonsingular, i.e., ^ xk;i –0, for i ¼ C k1 n1 þ 1; . . . ; C n , because if there existed at least an i with xk;i ¼ 0, k1 then there would be at least C n1 þ 1 different nodes lying in the hyperplane cm, but this is not possible because Xm is a Coatmèlec CN. Hence, rank(A0 D) = rank(A0 ). Moreover, the matrix A0 is also a multivariate Vandermonde matrix corresponding to the C kn1 nodes that do not belong to the hyperplane cm. Since the Coatmèlec property of a CN does not k ~ i g, i ¼ C k1 change under either rotation or translation of all the nodes, the CN fw n1 þ 1; . . . ; C n , is also a Coatmèlec CN. The induction hypothesis yields that the rank of matrix A0 is C kn1 . e m is rankðAÞ þ rankðA0 Þ ¼ C k1 þ C k ¼ C k , and the theorem is proved. Finally, the rank of the C kn  C kn matrix V n1 n1 n

4. Conclusions The unisolvency of the problem of multivariate polynomial interpolation in a Coatmèlec CN, a kind of properly posed set of nodes defined by hyperplanes, has been shown through a new and straightforward proof. This proof uses elementary techniques from linear algebra. This fact permits the understanding of the topic by nonexperts and opens the possibility of it being incorporated in numerical analysis textbooks. The geometrical condition characterizing Coatmèlec CNs is one of the most general conditions currently available for the characterization of properly posed set of nodes defined by hperplanes, which is easier and more efficient to be checked by an automatic computational software than the widely known geometrical characterization of Chung and Yao [8]. Therefore, Coatmèlec CNs are useful in mesh generation for the numerical solution of partial differential equations in irregular domains, such as generalized finite difference methods.

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Acknowledgements The authors thank to Drs. Mariano Gasca and Juan I. Ramos for pointing us some references and for their useful comments which have greatly improved the presentation. The authors also thank a reviewer for pointing out a mistake in the original Proof of Lemma 5. The research reported in this paper was partially supported by Project MTM2010-19969 from the Ministerio de Ciencia e Innovación of Spain and Grant PAID-06-09-2734 from the Universidad Politécnica de Valencia. M.A.G.M. acknowledges support from the Spanish Ministry of Science and Education (MEC), Fulbright Commission, and FECYT. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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