Explicit solution of the polynomial least-squares approximation problem on Chebyshev extrema nodes

Transcription

1 Explicit solution of the polynoial least-squares approxiation proble on Chebyshev extrea nodes Alfredo Eisinberg, Giuseppe Fedele Dipartiento di Elettronica Inforatica e Sisteistica, Università degli Studi della Calabria, 87036, Rende (Cs), Italy Abstract In this paper we propose an explicit solution to the polynoial least squares approxiation proble on Chebyshev extrea nodes. We also show that the inverse of the noral atrix on this set of nodes can be represented as the su of two syetric atrices: a full rank atrix which adits a Cholesky factorization and a -rank atrix. Finally we discuss the nuerical properties of the proposed forulas. Key words: Polynoial approxiation, Cholesky factorization, Cobinatorial identities 1 Introduction The polynoial least squares proble (PLSP) has so any applications [1]. The proble is forulated as follows: given a set of points Θ = {(x i,f i ), i = 1,,...,n}, find a polynoial p(x) of degree less than or equal to 1 with coefficients c 1,c,...,c such that the least squares criterion n ɛ(c 1,c,...,c ) = (p(x i ) f i ) i=1 Corresponding author. Tel.: ; fax: E-ail address: fedele@si.deis.unical.it (G. Fedele) Preprint subitted to Elsevier Science 5 March 007

2 is iniized. In general, is uch saller than n. The proble can be reforulated as follows in c V c f, (1) V is a Vanderonde atrix of order n, depending on the observation points x i V (i,j) = x j 1 i, i = 1,,...,n, j = 1,,...,, () and f R n and c R are vectors containing experiental data f i and coefficients c i, respectively. Finding the solution of proble (1) is equivalent to copute the best approxiate solution [1] of the over-deterined linear syste V c = f. This is given by c = (V T V ) 1 V T f = V f, V is the Moore-Penrose pseudo-inverse of V, that is well-defined if x i x j. The nuerical solution of PLSP is usually ill-conditioned. The ost reliable algoriths either use orthogonal transforations, for exaple the well-known QR ethod, or copute the Cholesky factorization of the noral atrix B = V T V. The choice between the two ethods is still an open proble. Golub and Van Loan [8] reported soe guidelines for the above choice, based on the value of the ratio n/ and on that of the residual ρ = in c V c f, but they recognized that both ethods can give inaccurate solutions when they are applied to probles with large value of κ (V ). Rectangular Vanderonde atrices on not structured nodes have been considered in [4], fast algoriths for the Cholesky factorization of the noral atrix B = V T V and for the QR factorization of V have been given. Here we give explicit solution to the proble (1) for Chebyshev extrea [] (also called Gauss-Lobatto [7] or Clenshaw-Curtis) nodes x k = cos ( ) k 1 n 1 π, k = 1,,...,n. (3) First we present soe properties of the noral atrix defined on these nodes and show that it can be represented as a su of two syetric atrix: the first adits a Cholesky factorization, the second is a -rank atrix. Moreover we propose an explicit forula for the Moore-Penrose pseudoinverse of V. This work generalizes soe results given in [6] when n =, that corresponds to the polynoial interpolation proble. As in [6] we obtain the results using cobinatorial identities and other arguents fro nuber theory. Fro the practical point of view the proposed forulas give an algorith that is fast and accurate. The effect of finite precision arithetic is investigated by perforing several nuerical experients.

3 The structure of the noral atrix In this section we present soe properties of the noral atrix B = V T V. Such a atrix has a very siple structure; in particular its entries exhibit a chessboard pattern, and they are rational nubers. Proposition 1 B(i, j) = 1 + ˆB(i, j), i,j = 1,,...,, B(i 1, j 1) = 1 + ˆB(i 1, j 1), i,j = 1,,..., (4) ˆB(i, j) = ˆB(i 1, j 1) = ( ) n 1 i+j i+j i+j 1, i,j = 1,,...,, (5) ( ) n 1 i+j 4 i+j 4 i+j, i,j = 1,,...,. Proof. It is easy to see [9] that B(i,j) = 0 when i + j is odd. If i + j is even, by Lea 1 (see Appendix), then eq. (4) follows. Note that B = ˆB + u 1 u T 1 + u u T, (6) u 1 (i) = Mod[i, ], i = 1,,...,, (7) u (i) = Mod[i + 1, ], i = 1,,...,, therefore atrix B is the su of a full-rank atrix ˆB and a -rank atrix. The following proposition gives the Cholesky factorization of the atrix ˆB. Proposition ˆB = ˆR T ˆL ˆR, (8) ˆR(i, j) = ( ) i j j 1 j i, i = 1,,...,, j = i,i + 1,...,, ˆR(i 1, j 1) = ( ) i j j j i, i = 1,,...,, j = i,i + 1,..., (9) 3

4 and 1, i = 1, ˆL(i,i) = (n 1) 1, i =, 3,...,. i 3 (10) Proof. By (9) and (10), eq. (8) can be rewritten as: ˆB(i, j) = 3 i j (n 1) k 1 ( i 1 i k ˆB(i 1, j 1) = 4 i j (n 1) ( i i i j (n 1) k j 1, (11) j k j j 1 ( i i k By using Lea and standard arguents we can get the (8). Further result is an explicit expression for the inverse of ˆR. Proposition 3 ˆR 1 (1, 1) = 1, (1) j j k. ˆR 1 (1, j 1) = ( 1) j+1 1, j =, 3,..., j 3, ˆR 1 (i 1, j 1) = ( 1) i+j 1 j 1 j i i 1 ( ) i+j 3 i 3, i =, 3,...,, j = i,i + 1,...,, ˆR 1 (i, j) = ( 1) i+j 1 j 1 j i i 1 ( ) i+j i, i = 1,,...,, j = i,i + 1,...,. (13) Proof. It ust be shown that ˆR 1 ˆR = δi,j, i,j = 1,,...,. (14) For brevity, we consider only the even rows, then eq. (14) becoes ( ˆR 1 ˆR)i,j = ( 1) i j i (i 1) ( j i + k ( 1) k (k 1) k=i i j 1. (15) j k 4

5 By using forula (3) in [5] we have ( ˆR 1 ˆR)i,j = δ i,j. Siilarly we can obtain the case for odd rows. Now the explicit expression of inverse noral atrix is given. Theore 1 and M (i, j) = ( 1) i+j i+j (n 1)(n+ 1 Mod[,]) i,j = 1,,...,, M (i 1, j 1) = ( 1) i+j i,j = 1,,...,. B 1 = M 1 M (16) M 1 = ˆB 1, (17) ( +i Mod[,] i 1 i+j 4 (n 1)(n+ +Mod[,]) Proof. By Sherann-Morrison s forula [8], we have +j Mod[,], j 1 ( +i 4+Mod[,] i +j 4+Mod[,], j (18) M 1 (u 1 u T 1 + u u T ) [ I + M 1 (u 1 u T 1 + u u T ) ] 1 M1 = M, (19) then ˆBM B = u 1 u T 1 + u u T. (0) For brevity we consider the case = q. and ( ˆBM B) i,j = ( 1)q n+ 1 ( ˆBM B) i 1,j 1 = ( 1)q n+ q ( 1) k ( ) k 1 q+k 1 k 1 + ( 1) q n 1 q (n+ 1) j 1 ( 1) k ( q+k 1 k 1 q ( 1) k ( ) k q+k k + ( 1) q n 1 q (n+ ) j ( 1) k ( q+k k (1) k+j k+j 1, () k+j 4 k+j. 5

6 Since [10] ( ) q q + k 1 ( 1) k k 1 = ( 1) q q, (3) k 1 using Lea 3, the (16) follows. Proposition 4 [6] V T = DU T H, (4) 1, i = 1, D(i,i) = (5) 1, i =, 3,...,, i U(1, j 1) = ( ) j 3 j 1, j = 1,,...,, U(i 1, j 1) = ( ) j j i, i =, 3,...,, j = i,i + 1,...,, (6) and U(i, j) = ( ) j 1 j i, i = 1,,...,, j = i,i + 1,..., [ ] (i 1)(j 1) H(i,j) = ( 1) i+1 cos π, i = 1,,...,, j = 1,,...,n. (7) n 1 By ultiplying Q 1 = M 1 DU T and Q = M DU T, we obtain the final result: The Moore-Penrose atrix pseudo-inverse of V is V = (Q 1 Q )H, (8) Q 1 (i,i) = i 1 n 1, i = 1,,...,, i+j i Q 1 (i, j) = ( 1) j 1 n 1 j i ( i+j j i 1), i = 1,,...,, j = i + 1,i +,...,, ( i+j i 1 j Q 1 (i 1, j 1) = ( 1) i+j 3 n 1 j i j i 1), i = 1,,...,, j = i + 1,i +,...,, (9) 6

7 Q (i, j) = i = 1,,..., +i Mod[,] ( ( 1) +i Mod[,] (n 1)(n+ 1 Mod[,]) i i 1, j = 1,,...,, ), ( +i 4+Mod[,] Q (i 1, j 1) = (n 1)(n+ +Mod[,]) i 1 i i = 1,,...,, j =, 3,...,, ( 1) +i+mod[,] ), (30) ( 1) +i+mod[,] ( +i 4+Mod[,] Q (i 1, 1) = (n 1)(n+ +Mod[,]) i i i = 1,,...,. ), 3 Nuerical properties and experients In this section we present soe properties of the proposed forulas and the results of nuerical experients carried out in order to deonstrate the perforance of the new algorith. First note that the atrix V can be rewritten as V = 1 n 1 S( ˆQ 1 ˆQ )H (31) S(i, i) = 1 n+ 1 Mod[,], i = 1,,...,, (3) 1 S(i 1, i 1) =, i = 1,,..., n+ +Mod[,], ˆQ 1 = (n 1)S 1 Q 1 (33) and ˆQ = (n 1)S 1 Q. (34) 7

8 Moreover the atrix Q 1 = (n 1)Q 1 can be constructed using the following properties: Q 1 (1, 1) = 1, Q 1 (i,i) = Q 1 (i 1,i 1), i =, 3,...,, Q 1 (1, j 1) = ( 1) j+1, j =, 3,...,, Q 1 (i,j) = Q 1 (i 1,j 1) Q 1 (i,j ), i =, 3,...,, j = i +,i + 3,...,, i + j even. (35) It is iportant to underline that both the entries of ˆQ1 and ˆQ are integer nubers which can be stored without rounding. To construct ˆQ, let c 1 and c be defined as c 1 = ( 1) Mod[ + 1, ], c = ( 1) + Mod[, ] and consider the vector v defined as follows: v(1) = ( 1) +1+ Mod[,], v(i) = ( 1 + c 1 i 1) v(i 1), i = 1,,...,, then v(i + 1) = ( ) 1 c i v(i), i = 1,,...,, i + 1, ˆQ (i,j) = v(i), i = 1,,...,, j =, 3,...,, i + j even, ˆQ (i 1, 1) = v(i 1), i = 1,,...,. Note that the product b = H f can be constructed efficiently by considering the syetric properties of the atrix H. Define two vectors f 1 and f as: f 1 = { } n f(i), i = 1,,...,, 8

9 then b(i) = n j=1 and if n is odd then { } n f = f(n + 1 i), i = 1,,...,, H(i,j)(f 1 (j) + ( 1) i+1 f (j)), i = 1,,...,, ( ) n + 1 b(i 1) = b(i 1) + ( 1) i+1 f, i = 1,,...,. Here we report soe nuerical experients aied at investigating the effectiveness of the factorization (31). The proposed ethod costs n O() flops. We copare our algorith (EF) with the proposed one in [3] (CB) it costs 10n flops. These algoriths have been ipleented in Matheatica [13], which allows arbitrary precision nubers. For soe values of and n, we have generated one thousand vectors f, with entries uniforly distributed in [ 1, 1], and have coputed the exact solution of the proble (1) using extended precision of 56 digits. For each algorith we have coputed the axiu coponent wise relative errors E EF = ax 1 i ĉ EF i c i c i (36) E CB = ax 1 i ĉ CB i c i c i (37) ĉ EF i and ĉ CB i are the approxiate solutions (coputed in achine precision) of EF and CB algorith respectively. The ean and the axiu of E EF and E CB are reported in Tables 1 and. Tables 1 and report also the fraction of trials in which the proposed algorith gives equal or ore accurate result than the CB algorith. 4 Conclusions In this paper we have considered the proble of finding an explicit factorization of both the noral atrix, defined on Chebyshev extrea nodes, and its inverse. Such a factorization allows us to design an efficient algorith to solve least-square probles on these nodes. Fro the practical point of view the proposed forulas give an algorith that is fast and accurate as it is confired by several nuerical experients. 9

10 CB EF s.r. ax ean ax ean EF vs CB Table 1 LMS. n = 100. Maxiu and ean value of E CB and E EF. Success rate of EF algorith over 1000 runs. CB EF s.r. ax ean ax ean EF vs CB Table LMS. n = Maxiu and ean value of E CB and E EF. Success rate of EF algorith over 1000 runs. Appendix Lea 1 [ n γ = cos ( )] q k 1 n 1 π = q ( ) q (n 1) (A-1) q 10

11 Proof. By using the identity [9]: (cos x) q = 1 q ( ) q + 1 q 1 q q s=0 ( ) q cos [(q s)x], s (A-) we get: γ = 1 q ( ) q n + q n q 1 1 q s=0 ( ) [ q cos (q s) k 1 ] s n 1 π (A-3) Since [ n cos (q s) k 1 ] n 1 π = 1, (A-4) and q 1 1 q s=0 ( ) q Γ(q + 1/) = 1 s πγ(q + 1) (A-5) the (A-1) follows. Lea [1] in(i,j) ( i 1 i k j 1 = 1 j k ( ) i + j. (A-6) i + j 1 Proof. By the syetric property of binoial coefficients we get: in(i,j) ( i 1 i k j 1 = j k in(i,j) ( i 1 j 1. i 1 + k j 1 + k Supposing i = in(i,j) and taking into account the identity [10] we have: k ( ( ) r s r + s =, + k n + k r + n 11

12 0 i ( i 1 i k j 1 + j k i ( i 1 i k j 1 = j k ( ) i + j. i + j 1 Lea 3 ( q i + k q + k 1 ( 1) k = ( 1) q i 1. (A-7) i + k 1 k 1 Proof. Let f(q,k,i) be the quantity defined as follows: f(q,k,i) = ( 1) k ( i + k i + k 1 Such a quantity satisfies the recursive property: q + k 1. (A-8) k 1 f(q,k,i + 1) = ( 4 ) f(q,k,i), i = 1,,...,q 1. (A-9) i + k By suing each ter in (A-9) for k fro 1 to q, we get: S(q,i + 1) = 4S(q,i) q ( 1) k i + k ( i + k i + k 1 q + k 1, (A-10) k 1 S(q,i) = q ( 1 ( ) k i+k q+k 1 i+k 1 k 1. The suation in (A-10) can be written as: ( ) i i q i H [{i + 1/, 1 q, 1 + q}, {i +, 3/}, 1], i = 1,,...,q. (A-11) The hypergeoetric function involved, satisfies Saalschüts s conditions [11], then (A-11) has the following closed-expression: 1

13 0, i = 1,,...,q 1, ( 1) q+1 1 q q+1( q q ), i = q. (A-1) Since S(q, 1) = ( 1) q then (A-7) follows. References [1] A. Björck, Nuerical Methods for Least Squares Probles, SIAM, [] L. Brutan, Lebesgue functions for polynoial interpolation - a survey, Annals of Nuer. Math., (1997), [3] S.D. Conte, C. De Boor, Eleentary Nuerical Analysis, McGraw-Hill, Second. ed., 197. [4] C.J. Deeure, Fast QR factorization of Vanderonde atrices, Linear Algebra Appl., 1 (1989), [5] A. Eisinberg, G. Franzé, N. Salerno, Rectangular Vanderonde on Chebyshev nodes, Linear Algebra Appl., 338 (001), [6] A. Eisinberg, G. Fedele, Vanderonde systes on Gauss-Lobatto Chebyshev nodes, Applied Math. Cop., 170 (005), [7] W. Gautschi, Orthogonal Polynoials. Coputation and Approxiation, Oxford Science Publ., 004. [8] G.H. Golub, A. van Loan, Matrix Coputations, Johns Hopkins U. Press, [9] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, third ed., Acadeic Press, New York, [10] D.E. Knuth, The Art of Coputer Prograing, vol. 1, second ed., Addison- Wesley, Reading, MA, [11] Y.L. Luke, The special functions and their approxiations, vol. 1, Acadeic Press, [1] P. Pugliese, On the noral atrix of the polynoial LS proble over the Chebyshev points, Linear Algebra Appl., 78 (004), [13] S. Wolfra, Matheatica: a Syste for Doing Matheatics by Coputers, Second. ed., Addison-Wesley,