We present fast evaluating and differentiating algorithms for the Hermite interpolating polynomials with the knots of multiplicity 2, which are generated dynamically in a field K = (K,+,⋅) by the recurrent formula of the formX^i = αx^i-1+β (i=1,2,..,n-1; x^0=γ).As in the case of Lagrange-Newton interpolating algorithms, the running time of these algorithms is C(n) + O(n) base operations from the field K, where C(n) = O(nlog n) denotes the time needed to compute the wrapped convolution in Kn

Kapusta, Joanna

Smarzewski, Ryszard

English

University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

 Annales UMCS Informatica AI 6 (2007) 95-102 Annales UMCS Informatica Lublin-Polonia  Sectio AI http://www.annales.umcs.lublin.pl/  Fast algorithms for polynomial evaluation and differentiation  at special knots  Joanna Kapusta*, Ryszard Smarzewski**  Department of Mathematics and Computer Sciences, The John Paul II Catholic University  of Lublin, Konstantynów 1H, 20-708 Lublin, Poland  Abstract We present fast evaluating and differentiating algorithms for the Hermite interpolating polynomials with the knots of multiplicity 2, which are generated dynamically in a field ? ?, ,K K? ? ?  by the recurrent formula of the form  ? ?1 01,2,.., 1;i ix x i n x? ? ??? ? ? ? ? . As in the case of Lagrange-Newton interpolating algorithms, the running time of these algorithms is ? ? ? ?C n O n?  base operations from the field K, where ? ? ? ?logC n O n n?  denotes the time needed to compute the wrapped convolution in Kn.   1. Introduction and preliminaries Let ? ?, ,K K? ? ?  be a field and let ( 0,1,.., 1)ix i n? ?  be n  pairwise distinct points in the field K. Additionally, let the values   ? ?, 0,1,..., 1i iy z K i n? ? ?  be given. Then there exists [1] a unique polynomial  ? ? ? ? ? ?1 10 0n ni i i ii ip x y c x z d x? ?? ?? ?? ?  (1) in the space 2 1[ ]nK x?  of all polynomials of degree less than 2n, which is determined by the following Hermite interpolating conditions   ? ?i ip x y?  and ? ?' i ip x z? ? ?0,1,..., 1i n? ? . (2) Moreover, the basic polynomials ci(x) and di(x) are given by the following formulae                                                  *E-mail address: jkapusta@kul.lublin.pl **E-mail address: rsmax@kul.lublin.pl Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSJoanna Kapusta, Ryszard Smarzewski 96 ? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ?' 221 ,0,1,2,..., 1i i i i i i ii i ic x x x l x l x l xd x x x l x i n? ?? ? ? ?? ?? ? ? ?  (3) with  ? ? 10,nkik k i i kx xl xx x?? ??? ??   In numerical computations [1,2], the Newton – Hermite interpolating formula   ? ? ? ? ? ?11 22 2 10 0knk k k vkp x f f x x x x???? ?? ?? ? ? ? ?? ??  (4) is preferred, where the generalized divided differences fi (i = 0,1,…,2n – 1) are computed by the usual recurrent formulae, which requires O(n2) base operations from the field K. A faster algorithm is announced in Section 3 without proofs.  2. Fast evaluation and differentiation of polynomials Let xi (i = 0,1,…,n – 1) be n  pairwise distinct points from the field K generated dynamically by the following recurrent formula   ? ???? ????? ? 01 ;1,..,2,1 xnixx ii , (5) where ? ? 0, ?, ?  are fixed in the field K. In this case, it is possible to present a fast algorithm for computation of values  ? ? ? ?   0,1,..., 1i iy p x i n? ? ?  and derivatives  ? ? ? ?'     0,1,..., 1i iz p x i n? ? ?  of Hermite interpolating polynomial  ? ? ? ? ? ?11 20 0knk k k vkp x g h x x x x????? ?? ?? ? ?? ??  (6) where coefficients gk and hk (k = 0,1,…,n – 1) are given. Of course, these coefficients are generalized divided differences  ? ?? ? ? ?0 0 1 10 0 1 1, ,..., , , ,, ,..., , , , 0,1,.., 1k k k kk k k k kg p x x x x xh p x x x x x x k n? ?? ??? ? ?  of the polynomial p(x). Indeed, one can substitute xi into formula (6) and differentiate (6) at x = xi to get   ? ? ? ? ? ?1 112 20 00 0k ki ii k i v k i k i vk ky g x x h x x x x? ?? ??? ?? ?? ? ? ? ?? ??  (7)  Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSFast algorithms for polynomial evaluation and differentiation … 97 and   ? ? ? ? ? ?? ? ? ? ? ?1 1 1120 00 0 01 11 10 00 022 .k i ii ii k i v i i v ik vvk ki kk k i k i v ik vvz h x x g x x x xg h x x x x x x?? ? ???? ??? ? ??? ?? ? ??? ?? ?? ?? ??? ? ? ? ?? ?? ?? ?? ? ? ? ?? ?? ? ? ?? ?? ? ? ?? ? ? (8) Next we rewrite formula (5) in the form  ? ?1 2 ... 1i i iix ? ? ? ? ?? ?? ? ? ? ? . (9) By inserting xi into formulae (7) and (8) we obtain   2 2110 0i ik i k ii k k k i ki k i kk kq p q py g h r tp p?? ?? ?? ?? ? ? ?? ?? ? ? ?? ? ? ?? ?  and  ? ?2 110 021 1 111 10 0 0121 1 1  2 ,i ik ii k i ii k iki i i kk ik k k i ki k i k i kkq pz h q pp r tq pg h r tp r t r r t? ??? ? ? ?? ??? ? ?? ?? ? ? ?? ?? ? ? ? ? ?? ? ?? ?? ?? ?? ?? ?? ?? ? ?? ?? ? ? ?? ? ? ?? ?? ? ? where  ,1e ?? ?? ? ?     ,jjr ??      ? ?1 1jjt e ? ?? ? , (10)  10jjq??????? , ? ?1 101jjp e??????? ??   ? ?0,1,..., 1j n? ? . In these formulae, it is assumed that products are equal to 1 and sums are equal to 0  whenever their upper indices are smaller than the lower ones. As in the Lagrange-Newton interpolating algorithm [3], a computation of these formulae uses only the following six different vector operations in Kn:  – coordinatewise vector addition, subtraction, multiplication, division and multiplication by scalars, – wrapped convolution defined as  ? ?0 1 1, ,.., na b c c c ?? ? ,  where  ? ?0 1 1, ,..., na a a a ?? , ? ?0 1 1, ,..., nb b b b ??     Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSJoanna Kapusta, Ryszard Smarzewski 98and  ? ?00,1,..., 1ii k i kkc a b i n??? ? ?? .  Indeed, if we denote   ? ? 10nig g ?? , ? ? 10nih f ?? , ? ? 102 ni i id p q w ?? ,   ? ? 10niy y ?? , ? ? 10niz z ?? , ? ? 10nir r ?? , ? ? 10nit t ?? , (11)  ? ? 10nip p ?? , ? ? 10niq q ?? , ? ? 10niw w ?? ,   ? ? 101 nj ?? , ? ? 12 0niv q ?? , ? ? 12 0niu p ??   then one can get  ? ? ? ?w j r j t? ?  and   ? ? ? ? ? ? ? ?* * * *y g v j u h r v t u u? ? ? ? ? ?? ?   ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ?* 2* * * * * 2* * * * * .z h v j u w g v u h r v t ug v r w u h v t w u u d? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ?? ? Now we present the algorithm to compute the required values and derivatives. It uses two classes KType and KTypeVector, which should make it possible to perform operations in K and Kn.  Algorithm 1. Polynomial evaluation and differentiation at knots  ? ?1 01,2,..., 1;i ix ax b i n x c?? ? ? ? ?  Input: A vectors ? ?0 1 1, ,..., ng g g g ?? , ? ?0 1 1, ,..., nnh h h h ?? ??  and three scalars 0? ? , ?  and ? in a field K. Output: nzy ??, . 1. Set /(1 )e c b a? ? ? , 0 1p ? , 0 1q ? , 0 1r ?  and 0 0v ? . 2. For k from 1 to n – 1 do the following: 2.1. arr kk ?? ?1 , ? ? ert kk ??? 1 , 2.2. 1 1k k kp p r? ?? ? , 1 1k k kq q t? ?? ? . 3. Set ? ?1 1 1n nt r a e? ?? ? ? ? . 4. Compute ? ?w j r j t? ? , 2p p p? ? , v q q? ? , gv g v? ? , hrv hv r? ? , tu t u? , ju j u? , 2d p q w? ? ? ? . 5. Compute ? ? .y gv ju hrv tu u? ? ? ? ?  Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSFast algorithms for polynomial evaluation and differentiation … 99 6. Compute  ? ? ?? ? ? ? ? ?? ??22 .z hv ju w gv u hrv tugv r w u hv tu w u? ? ? ? ? ? ? ? ?? ? ? ? ? ?  7. Return y, z.  Since the wrapped convolutions   ? ?,conv p q p q? ? , with  ? ?0 1 1, ,..., np p p p ??  and ? ?0 1 1, ,..., nq q q q ?? , can be computed by an algorithm having a running time of O(nlogn) base field operations in K [4], it follows that computation of values and derivates requires only O(nlogn).  3. Fast computation of generalized divided differences For the completeness, we include also a very short description of the inverse algorithm to Algorithm 1. Since the generalized divided differences f2k+1 (k = 0,1,…,n – 1) are coefficients at x2k+1 [1] in Hermite polynomials of degree 2k + 1 determined by the interpolating conditions  ? ?i is x y?  and ? ?' i is x z?  ? ?0,1,...,i k? , it follows that   ? ?? ?? ?10,2 12 200, 0,2.kj jkj jk k kjj j vj jy x xzfx x x x?? ??? ? ? ??? ???? ? ? ?? ??? ?? ?? ?? ?? ?? ?? ?? ??? ? ? (12) On the other hand, the generalized divided differences f2k (k = 0,1,…,n – 1) are coefficients at x2k [1] of  Hermite polynomials of degree 2k determined by the interpolating conditions  ? ?i iw x y? , ? ?' i iw x z?   ? ?0,1,..., 1i k? ?  and ? ?k kw x y? . Hence we get ? ?? ? ? ?? ?? ? ? ?? ? ? ? ? ?11 10, 0,2 1 100, 0, 0, 0,11 1200, 0, 0.k kj jkj jk jk k k kjj j j jj j j jkj kk k kjj j kj jx x x xf yx x x x x x x xz yx x x x x x? ?? ? ? ?? ? ? ?? ? ? ? ? ? ? ?? ? ?? ? ? ? ??? ?? ? ? ?? ??? ? ? ? ? ? ? ??? ??? ? ? ? ?? ?? ?? ?? ?? ?? ? ?? ?? ?? ? ? ?? ?? ?? ?? ? ?? ?? ? ? ? ?? ? ? ? (13) Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSJoanna Kapusta, Ryszard Smarzewski 100These formulae can be used to derive a O(nlogn) – algorithm to compute generalized divided differences in the case when the interpolating knots are generated by formula (9). Now we present the algorithm to compute the required generalized divided differences in the Hermite interpolating formula (2). It uses two classes KType and KTypeVector, which make it possible to perform operations in K and Kn nK . First we perform initial computation preparing to get generalized divided differences.  Algorithm 2. Computation of generalized divided differences with respect to knots  ? ?1 01,2,..., 1;i ix ax b i n x c?? ? ? ? ?  of multiplicity 2. Input: A vectors ? ?0 1 1, ,..., ny y y y ?? , ? ?0 1 1, ,..., nnz z z z ?? ??  and three scalars 0? ? , ? and ? in a field K. Output: , ng h?? . 1. Set ? ?/ 1e c b a? ? ? , 0 1p ? , 0 1q ? , 0 1r ?  and 0 0v ? . 2. For k from 1 to n – 1 do the following: 2.1. 1k kr r a?? ? , ? ?1 1k kt t e?? ? ? , 1 1k k kq q t? ?? ? . 2.2. 1 1k k kp p r? ?? ?  1 11k k kv v t? ?? ? . 3. Set ? ?1 1 1n nt r a e? ?? ? ? ? . 4. Compute ? ?w j r j t? ? , 2p p p? ? , 2q q q? ? ,  2r r r? ? , ? ?2 2d y p q? ? , 2 2u p q? , s u t? ,  2 2rq r q? ? , yw y w? ? . 5. Compute ? ? ? ?? ? ? ?? ?2 2 2 2 2.g yw q u y rq u z q u p? ? ? ? ? ? ?  6. Compute  ? ? ? ? ? ?? ?? ? ? ? ? ?? ?2 2 2 2 22 2 2 2 2 .h yw rq s t y r p u py r rq u t z rq s p d? ? ? ? ? ? ? ?? ? ? ? ? ? ?  7. Return g, h.  The details of the proofs connected with this algorithm will  be presented elsewhere.  It should be noticed that Algorithms 1 and 2 give a fast way to pass between the representations of a polynomial p(x) in K2n–1[x] with respect to the Lagrange-Hermite base   0 0 1 1 1 1, , , ,.., ,n nc d c d c d? ?  and the Newton base   Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSFast algorithms for polynomial evaluation and differentiation … 101  ? ? ? ? ? ? ? ?? ? ? ? ? ?2 2 20 0 0 2 12 2 20 2 11, , ,..., ... ,... .n nn nx x x x x x x x x xx x x x x x? ?? ?? ? ? ? ?? ? ?  3. A complexity of algorithm In this section we consider complexity of computation of Algorithm 1. The running time of this algorithm is O(nlogn) base operations from the field K, while the running time of classical algorithm is O(n2). For example, if we choose n of order 214, then we save about 98 percent of base operations. More precisely, if n = 214 and if we use Algorithm 1, then the number of base operations for computing values and derivatives is approximately equal to 7.62?106, whereas the number of base operations in the classical algorithm is 5.37?108. Hence the number of saved base operations for Algorithm 1 equals %42.1 .  -20%0%20%40%60%80%100%16 32 64 12825651210242048409681921638432768 Fig. 1. The numbers of saved operations in percentage terms  The graph in Figure 1 shows the percentage of saved operations by Algorithm 1 in comparison with the classical O(n2) – algorithm for different values of n. The similar complexity results are also true for Algorithm 2.  4. Conclusions In this paper, we have presented fast evaluating and differentiating algorithm for Hermite interpolating polynomial with n knots of multiplicity 2. These knots are generated dynamically in a field Kby the recurrent formula of the form    ? ?1 01,2,.., 1;i ix x i n x? ? ??? ? ? ? ? . Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSJoanna Kapusta, Ryszard Smarzewski 102The algorithm computes values and derivatives with the running time of C(n) + O(n) base operations from the field K, where C(n) = O(nlogn) denotes the complexity of computation of the wrapped convolution in Kn. On the other hand, the classical algorithm requires O(n2) of base operations in K. Numerical experiments show that this algorithm can be useful in practice, whenever n is sufficiently large. For example, such a situation occurs during the computation of shares and recovering keys in secret sharing schemes of Shamir type [5,6]. The similar results hold also for the inverse algorithm, which computes generalized divided differences.  References [1] Ralston A., A first Course in Numerical Analysis. McGraw-Hill, New York, (1965). [2] de Boor C., A Practical Guide to Splines. Springer-Verlag, New York, (2001). [3] Smarzewski R., Kapusta J., Fast Lagrange-Newton transformations. Journal of Complexity, doi:10.1016/j.jco.2006.12.004, in press. [4] Aho A.V., Hopcroft J.E., Ullman J.D., The Design and Analysis of Computer Algorithms. Addison-Wesley, London, (1974). [5] Kapusta J., Smarzewski R., Fast interpolating algorithms in cryptography. Annales UMCS Informatica AI, 5 (2006) 37. [6] Smarzewski R., Kapusta J., Algorithms for multi-secret hierarchical sharing schemes of Shamir type. Annales UMCS Informatica AI, 3 (2005) 65.   Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 20:55:32UMCSPowered by TCPDF (www.tcpdf.org)

Fast algorithms for polynomial evaluation and differentiation at special knots

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Fast algorithms for polynomial evaluation and differentiation at special knots

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