The formulae and algorithms for Lagrange-power basis transformation and Lagrange–Newton transformation

Applied Mathematics and Computation 218 (2012) 5861–5866

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

The formulae and algorithms for Lagrange-power basis transformation and Lagrange–Newton transformation Dian-jun Lu a,⇑, Keh-Shin Lii b, Yu Wang a a b

Department of Mathematics and Information Science, Qinghai Normal University, Xining 810008, PR China Department of Statistics, University of California, Riverside, CA 92521, USA

a r t i c l e

i n f o

Keywords: Lagrange-power basis transformation Lagrange–Newton transformation Inverse transformation Lagrange basis Newton basis Power basis Vandermonde matrix

a b s t r a c t The interpolation polynomials based on Lagrange, Newton and power basis play important roles in applied mathematics, computing method and many other emerging applications. In this paper, we present some coordinate transformation formulae and algorithms as demonstrated below. Firstly, we put forward the formulae of the Lagrange-power basis transformation and its inverse transformation, and as a byproduct, we provide a new method to arrive at the inversion of the Vandermonde matrix. Secondly, we give the formulae of Lagrange–Newton transformation and its inverse transformation. Moreover, we construct related algorithms of Lagrange-power basis transformation, Lagrange–Newton transformation and their inverse transformations. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Since Shamir [1] presented the idea of threshold secret sharing in 1979, many cryptographic schemes [2,3] which employed power basis and Lagrange interpolation polynomial were advanced. Later many theoretical and applied researches relate to Lagrange basis, Newton basis and power basis were carried out [4–7]. In 1998, based on duality, Lodha and Goldman [8] researched into lattices and algorithms for bivariate Bernstein, Lagrange, Newton and other related polynomial bases. In 2007, Smarzewski and Kupusta [9] presented a fast Lagrange–Newton transformation formula which based on the wrapped convolution, Horner transformation, iterative product and coordinate vector operations. In this paper, based on Smarzewski’s work, we furthermore research the transformation issue among Lagrange basis, power basis, and Newton basis. And we present explicit vector formulae for the Lagrange-power basis transformation and its inverse transformation. In general, for a given power basis polynomial p(x) = a0 + a1  x +    + an1  xn1 and bi = p(xi) (i = 0, 1, 2, . . . , n  1), if we want calculate the value of the vector a = (a0, a1, . . . , an1) from vectors b = (b0, b1, . . . , bn1) and x = (x0, x1, . . . , xn1), we need traditionally to invert the Vandermonde matrix. Concerning the inversion of the Vandermonde matrix, there are many methods [10–13]. However, here we employ the Lagrange’s special structure to construct the inverse transformation formulae without employing the inversion of the Vandermonde matrix directly, and as a byproduct, we provide a new method to arrive at the inversion of the Vandermonde matrix such as those described in Jog’s [10], and we present it as a corollary of Theorem 2. These formulae are related to the pairwise distinct knots xi = li (i = 0, 1, 2, . . . , n  1) and interpolating knots xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c). We also include the Lagrange–Newton transformation formula and its inverse transformation formula with respect to the pairwise distinct knots xi = xi1  a + b. In addition, we design algorithms related to Lagrange-power basis transformation, Lagrange–Newton transformation and their inverse transformations. Based on these, Newton-power basis transformation and its inverse transformation can be obtained. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (D.-j. Lu). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.003

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P Let a = (a0, a1, . . . , an1), b = (b0, b1, . . . , bn1), and ci ¼ ik¼0 ak  bik , then the wrapped convolution is defined as Pn1 c = (c0, c1, . . . , cn1) = a  b. Let ci ¼ k¼0 ak  bnk1 ði ¼ 0; 1; 2; . . . ; n  1Þ, then the wholly-wrapped convolution is defined as c = (c0, c1, . . . , cn1) = a 0 b. These notations will be used later. P Pn1 Pn1 i Let K be a field, we define the expression pðxÞ ¼ n1 i¼0 bi  li ðxÞ; pðxÞ ¼ i¼0 ai  x and pðxÞ ¼ i¼0 ci  mi ðxÞ as the Lagrange interpolation polynomial, power basis polynomial, and the Newton interpolation polynomial respectively, where Q Pi Qi bi ¼ pðxi Þ; li ðxÞ ¼ n1 and m0(x) = 1, k–i;k¼0 ðx  xk Þ=ðxi  xk Þ; ci ¼ p½x0 ; x1 ; . . . ; xi ; p½x0 ; x1 ; . . . ; xi  ¼ j¼0 pðxj Þ= k¼0;k–j ðxj  xk Þ mi(x) = (x  x0)  (x  x1)      (x  xi1 ), (i = 1, 2, . . . , n  1). According to above expression, we call the computational formula for vector b from given vectors a and x = (x0, x1, . . . , xn1) as the Lagrange-power basis transformation formula which is denoted as b = Lp(a, x). Conversely, the computational formula of vector a from given vectors b and x is defined as the inverse Lagrange-power basis transformation formula which is denoted as a ¼ L1 p ðb; xÞ. Similarly, we denote the computational formula for vector c from given vectors b and x as the Lagrange–Newton transformation formula, which is denoted by c = LN(b, x), and we denote the computation of vector b from given vectors c and x as the inverse Lagrange–Newton transformation formula, which is denoted by b ¼ L1 N ðc; xÞ. Lemma 1. Let K be a field and the pairwise distinct knots x0, x1, . . . , xn1 which are generated by the recursive formulae Q Q Q xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K, then xi ¼ ai  c þ b  Si ; j1 ðx  xk Þ ¼ dj  jk¼1 Sk  j1 ak ; k¼0 j k¼0 Qn1 Q P nj1 i1 m nj1 j nj1 ðx  x Þ ¼ d  ð a Þ  S and S = a  S + 1, where S ¼ a and d = ( a  1)  c + b. i+1 i j i k k¼jþ1 k m¼0 k¼1 Proof. By recursive formulae xi = xi1  a + b = (xi2  a + b)  a + b =    = ai  c + b  (ai1 + ai2 +    + 1) = ai  c + b  Si. Qj1

Qj1 



Q

aj  ða  1Þ  Skj  c þ b  aj  Skj ¼

Qn1







Q

ak  ðajk  1Þ  c þ b  ak  Sjk ¼ j1 ðajk  1Þ  c þ b  Sjk  ak ¼ j1 ða  1Þ  ðajk1 þ k¼0 k¼0   Q Q Q Qj1 j1 j1 k ajk2 þ    þ 1Þ  c þ b  Sjk   a ¼ k¼0 ðða  1Þ  c þ bÞ  Sjk  ak ¼ k¼0 ½d  Sjk   ak ¼ dj  j1 S  a : k¼0 jk k¼0 Also

And

Qn1

k¼jþ1 ðxk

Since Qnj1 k¼1

Sk ; we

k¼0

ðxj  xk Þ ¼

 xj Þ ¼

Qn1  k¼jþ1

k¼0 k







P Q Pjk1 t t  Pj2 ¼ j1 S ¼ j1 t¼0 t¼0 t¼0 k¼0 jk k¼0 Q Q Q k have j1 ðx  xk Þ ¼ dj  jk¼1 Sk  j1 and k¼0 j k¼0 Qj1

Since Siþ1 ¼

a

k¼jþ1 ½

at     

a

a

Pi

Pi Pi1 m m m m¼0 a ¼ 1 þ m¼1 a ¼ 1 þ a  m¼0 a ¼

Qn1

aj  Skj  d ¼ dnj1  ðaj Þnj1 

Pn1

 Qj1 Pk t t ¼ Qj S t¼0 a ¼ k¼0 t¼0 a k¼1 k

k¼jþ1 Skj :

P0

k¼jþ1 ðxk

nj1

 xj Þ ¼ d

j nj1

 ða Þ

a  Si þ 1, we have Si+1 = a  Si + 1.



Qnj1 k¼1

and

Qn1

k¼jþ1 Skj

¼

Sk :

h

2. Lagrange-power basis transformation formulae 2.1. Lagrange-power basis transformation formulae with xi = li

Theorem 1. Let K be a field with a primitive root l from unity of degree n. According to the Lagrange interpolation polynomial P Pn1 i pðxÞ ¼ n1 i¼0 bi  li ðxÞ and the power basis polynomial pðxÞ ¼ i¼0 ai  x , if the pairwise distinct knots x0, x1, . . . , xn1 are generated i by xi = l (i = 0, 1, . . . , n  1), then the Lagrange-power basis transformation formula b = Lp (a, x) and its inverse transformation P P ij ij formula a ¼ L1 can be expressed as explicit vector formulae bi ¼ n1 and ai ¼ 1n  n1 j¼0 aj  l j¼0 bj  l p ðb; xÞ ði ¼ 0; 1; 2; . . . ; n  1Þ. 2 3 1 1 ... 1 6 P x1 . . . xn1 7 j 6 x0 7: Proof. Since bi ¼ pðxi Þ ¼ n1 j¼0 aj  xi , then [b0, b1, . . . , bn1] = [a0, a1, . . . , an1]  An, where An ¼ 4 . . . ... ... ... 5 n1 n1 n1 x0 x1 . . . xn1 Pn1 i ij Notice that xi = l (i = 0, 1, . . . , n  1), hence bi ¼ j¼0 aj  l . 2 3 1 1 ... 1 n1 6 7 l ... l 1 7: On the other hand, since xi = li (i = 0, 1, . . . , n  1), then An is changed to Fourier matrix: F ¼ 6 4... 5 ... ... ... 1 ln1 . . . lðn1Þðn1Þ 2 3 1 1 ... 1 6 1 l1 . . . lðn1Þ 7 7; we have For the inversion of Fourier matrix can be expressed as F 1 ¼ 1n  6 4... 5 ... ... ... ðn1Þ ðn1Þðn1Þ 1 l ... l 2 3 1 1 ... 1 61 7 l1 . . . lðn1Þ 7: ½a0 ; a1 ; . . . an1  ¼ ½b0 ; b1 ; . . . bn1   F 1 ¼ 1n  ½b0 ; b1 ; . . . ; bn1  6 4... ... 5 ... ... ðn1Þ ðn1Þðn1Þ 1 l ... l

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According to above expression, we have ai ¼ 1n 

Pn1 j¼0

bj  lij . h

2.2. Lagrange-power basis transformation formulae with xi = xi1  a + b P Theorem 2. Let K be a field, according to the Lagrange interpolation polynomial pðxÞ ¼ n1 i¼0 bi  li ðxÞ and the power basis Pn1 i polynomial pðxÞ ¼ i¼0 ai  x , if the pairwise distinct knots x0, x1, . . . , xn1 are generated by the recursive formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K, then the Lagrange-power basis transformation formula b = Lp(a, x) and its inverse Pn1 P i transformation formula a ¼ L1  c þ b  Si Þj and ai ¼ n1 p ðb; xÞ can be represented as bi ¼ j¼0 aj  ða Q j¼0 ðpi;j  qnj1 Þ=r, that is Q Q Q j j j iþj n1 Qn1 k t j k a = (p  0 q)/r where Si ¼ i1  k¼0 a , and xi,j given t¼0 a ; pi;j ¼ ð1Þ  bj  a  xj;ni1 = k¼1 Sk ; qj ¼ k¼0 a = k¼0 Sk ; r ¼ d below. P Pn1 j j i Proof. From Lemma 1, we have xi = ai  c + b  Si. Using bi ¼ pðxi Þ ¼ n1 j¼0 aj  xi , we get bi ¼ j¼0 aj  ða  c þ b  Si Þ , which is the expression of the Lagrange-power basis transformation formula b = Lp(a, x). Q  P P h n1 xxk n1  lj ðxÞ ¼ n1 bj  þ ð1Þ1  xj;1  xn2 þ ð1Þ2 ¼ n1 j¼0 k¼0;k–j j¼0 bj  ðx x x j k .Q i Pn1 P P n2 n1 n3 þ    þ ð1Þ  xj;n2  x þ ð1Þ  xj;n1 Þ where xj;1 ¼ k¼0;k–j xk ; xj;2 ¼ i1 >i2 i1 ;i2 –j xi1 xj;2  x k¼0;k–j ðxj  xk Þ , P P h PP ni1 xi2 ; xj;3 ¼ i1 >i2 >i3 we have ai ¼ n1  bj  j¼0 ð1Þ i1 ;i2 ;i3 –j xi1  xi2  xi3 ; . . . ; xj;n1 ¼ x0  x1  . . .  xj1  xjþ1  . . .  xn1 ; .Q i n1 xj;ni1 k¼0;k–j ðxj  xk Þ . Q Q Q Qn1 nj1 Qnj1 According to Lemma 1, we have j1 ðx  xk Þ ¼ dj  jk¼1 Sk  j1 ak and  k¼1 Sk  ðaj Þnj1 . k¼jþ1 ðxk  xj Þ ¼ d k¼0 j k¼0 On the other hand, since pðxÞ ¼

Pn1

j¼0 bj n1

 Pn1  Q Q Q iþj  xj;ni1 =dn1  jk¼1 Sk  nj1 Sk  j1 ak  ðaj Þnj1 . j¼0 bj  ð1Þ k¼1 k¼0  Qn1 k Qnj1 k . Pn1 ð1Þiþj aj bj xj;ni1 Q Q a a k j nj1 k k¼0 a  ð a Þ ¼ Q ; we have a ¼ Q  Qk¼0 dn1  n1 Since j1 nj1 i j1 nj1 j¼0 k¼0 a . k¼0 j k Thus ai ¼

a

iþj

k¼1

a

S k¼1 k

k¼0

Sk

Q Q Q Q Pn1 k  aj  bj  xj;ni1 Þ= jk¼1 Sk ; qj ¼ jk¼0 ak = jk¼0 Sk ; r ¼ dn1  n1 k¼0 a , then ai ¼ j¼0 ½pi;j  qnj1 =r,

Now we let pi;j ¼ ðð1Þ or equivalently a = (p  0 q)/r. As a note, we can calculate xj,k iteratively by the following formulae:

xj;0 ¼ 1; xj;1 ¼

n1 X

xi ¼ x1  xj  xj;0 ;

xj;2 ¼ x2  xj  xj;1 ; . . . ; xj;m ¼ xm  xj  xj;m1 ðj ¼ 0; 1; 2; . . . ; n  1Þ;

i¼0;i–j

P Pn2 Pn1 Pnm Pnmþ1 Pn1 (m = 1, 2, . . . , n  1), where x1 ¼ n1 i¼0 xi ; x2 ¼ i1 ¼0 i2 ¼i1 xi1  xi2 ; . . . ; xm ¼ i1 ¼0 i2 ¼i1 þ1 . . . im ¼im1 þ1 xi1  xi2  . . . :  xim ; . . ., and in particular, xn1 = x0  x1  . . .  xn1. Naive implementation of this would have computational complexity of order O(2n). However a more efficient method of computing xi,j is given in Section 2.3.2 where xi,j is computed via f(j, p) recursively, which has computational complexity of order O(n2). h According to the Theorem 2, we can easily get the following corollary: 2 1 6 x0 6 Corollary. Let K be a field, the n  n Vandermonde matrix be given by 4 ... xn1 0

1 x1 ... xn1 1

... ... ... ...

1

3

xn1 7 7. If the pairwise distinct knots ... 5 n1 xn1

x0, x1, . . . , xn1 are generated by the formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K, then the inversion of     Q iþj 0 0 Vandermonde matrix can be described as V 1  xi;nj1  aj = j1 S ; qj ¼ n ði; jÞ ¼ pi;j  qnj1 =r, where pi;j ¼ ð1Þ k¼0 k Qj Qj n1 Qn1 k k a = k¼0 Sk ; r ¼ d k¼0 a , with xi,j given before. k¼0 Finally, the pairwise distinct knots x0, x1, . . . , xn1 are generated either by xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K, or by xi = li (i = 0, 1, . . . , n  1) with a primitive root l from unity of degree n, for the given vectors a and x, b can be computed by b = Lp(a, x), and for the given vectors b and x, a can be computed by a ¼ L1 p ðb; xÞ. 2.3. Related algorithm 2.3.1. Algorithm 1: Lagrange-power basis transformation algorithm Let the pairwise distinct knots x0, x1, . . . , xn1 be generated by formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c).

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Input: Vector a = (a0, a1, . . . , an1) 2 Kn and three scalars a – 0, b and x0 = c in a field K. Output: b = Lp(a, x) 2 Kn. Algorithm: (1) Set S0 = 0. (2) For i = 0 to n  1 do P j i {Siþ1 ¼ Si  a þ 1; bi ¼ n1 j¼0 aj  ða  c þ b  Si Þ }. (3) Return (b).

2.3.2. Algorithm 2: The inverse Lagrange-power basis transformation algorithm Let pairwise distinct knots x0, x1, . . . , xn1 be generated by formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c). Input: A vector b = (b0, b1, . . . , bn1) 2 Kn and three scalars a – 0, b and x0 = c in a field K. n Output: a ¼ L1 p ðb; xÞ 2 K . Algorithm: (1) For i = 0 to n  1 do {setp = 0, For j = 0 to n  1 do {If j = 0 then xi,j = 1, Else xi,j = f(j, p)  xi  xi,j1}} (Note: In this step, ‘‘f(j, p)’’ will automatically call for function f(j, p) which is given below.) The computation of f(j, p) is a recursive procedure. P Qn1 Pnj If j = 1, then f ðj; pÞ ¼ n1 i¼p xi , If j = n  1 then f ðj; pÞ ¼ i¼0 xi , Else f ðj; pÞ ¼ i¼p xi  f ðj  1; i þ 1Þ. (2) Set d = (a  1)  c + b, S0 = 0. For i = 0 to n  1 do Q Q {Siþ1 ¼ Si  a þ 1; qi ¼ ik¼0 ak = ik¼0 Sk Q For j = 0 to n  1 do {pi;j ¼ ðð1Þiþj  bj  aj  xj;ni1 Þ= j1 k¼0 Sk } n1 Qn1 k }r¼d  k¼0 a (3) Compute a = (p  0 q)/r. (4) Return (a).

On the above algorithm, we can compute f(j, p) recursively in advance, then x0i;j s are computed in step (1). On step (2) the actual implementation of the computations for pi,j, qi are implemented recursively. These implementations rendered the complexity of the algorithm O(n2). This also implies the inversion of Vandermonde matrix is of order O(n2) in the corollary of Theorem 2. These algorithm and relationship can be applied to the secret sharing schemes ((k, n) threshold scheme). On such a scheme the key is generally integers where the computational accuracy can be maintained in general. 3. Lagrange–Newton transformation formulae Based on the work of Smarzewski and Kupusta [9], we have the following two results. 3.1. Lagrange–Newton transformation formula with xi = xi1  a + b P Theorem 3. Let K be a field, according to the Lagrange interpolation polynomial pðxÞ ¼ n1 i¼0 bi  li ðxÞ and the Newton Pn1 interpolation polynomial pðxÞ ¼ i¼0 ci  mi ðxÞ. If the pairwise distinct knots x0, x1, . . . , xn1 are generated by formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K, then the Lagrange–Newton transformation formulae c = LN P (b, x) is expressed as ci ¼ ij¼0 ðpj  qij Þ=r i ði ¼ 0; 1; 2; . . . ; n  1Þ, or equivalently c = (p  q)/r, where

pj ¼ bj

, j1 Y k¼0

Skþ1 ;

qij ¼ ð1Þij 

ij1 Y k¼0

,

ak

ij1 Y k¼0

Skþ1 ;

r i ¼ di 

i1 Y

ak :

k¼0

Q Qj1 k Qi j Qj1 ij Qij1 Proof. According to Lemma 1, we have j1  k¼0 Skþ1  ðaj Þij . k¼jþ1 ðxk  xj Þ ¼ d k¼0 ðxj  xk Þ ¼ d  k¼0 Skþ1  k¼0 a and .Q .Q  P  Pi P Q j1 i i Thus ci ¼ p½x0 ; x1 ; . . . ; xi  ¼ j¼0 bj ðx  xk Þ  ð1Þij  ik¼jþ1 ðxk  xj Þ ¼ ij¼0 ð1Þij k¼0;k–j ðxj  xk Þ ¼ j¼0 bj k¼0 j . Qij1 Q Q  k¼0 S Skþ1  j1 ak  ðaj Þij Þ. bj Þ di  j1 k¼0 kþ1 k¼0 . Qj1 k Q Qij1 k P Q k It is easy to check that a  ðaj Þij ¼ i1 thus ci ¼ ij¼0 ðð1Þij  bj Þ di  j1 S k¼0 a = k¼0 a., k¼0 k¼0 kþ1 .   .  i h . Pi Q Qij1 k Qj1 Qij1 ij Qij1 k Qij1 i Qi1 k k ¼  ð1Þ ðd Skþ1  i1 a a b S  a S  a Þ .  k¼0 j kþ1 k¼0 j¼0 k¼0 k¼0 k¼0 kþ1 k¼0 k¼0 Q Q Qij1 Q Pi k Now we let pj ¼ bj = j1 S ; qij ¼ ð1Þij  ij1 ak = k¼0 Skþ1 ; ri ¼ di  i1 k¼0 a , we have c i ¼ j¼0 ðpj  qij Þ=r i ði ¼ k¼0 kþ1 k¼0 0; 1; 2; . . . ; n  1Þ, or equivalently c = (p  q)/r. h

D.-j. Lu et al. / Applied Mathematics and Computation 218 (2012) 5861–5866

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3.2. The inverse Lagrange–Newton transformation formula with xi = xi1  a + b P Theorem 4. Let K be a field, according to the Lagrange interpolation polynomial pðxÞ ¼ n1 i¼0 bi  li ðxÞ and the Newton Pn1 interpolation polynomial pðxÞ ¼ i¼0 ci  mi ðxÞ. If the pairwise distinct knots x0, x1, . . . , xn1 are generated by the recursive formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K, then the inverse Lagrange–Newton transformation formula P  i b ¼ L1 is expressed as bi ¼ or equivalently b = (p  q)  r, where N ðc; xÞ j¼0 pj  qij  r i ði ¼ 0; 1; . . . ; n  1Þ, Qj1 Qi1 j Qj1 k pj ¼ cj  d  k¼0 a ; qj ¼ 1= k¼0 Skþ1 ; r i ¼ k¼0 Skþ1 . P Pi P and d = (a  1)  c + b, we have bi ¼ pðxi Þ ¼ n1 Proof. Since pðxÞ ¼ n1 i¼0 c i  mi ðxÞ; bi ¼ pðxi Þ j¼0 cj  mj ðxi Þ ¼ j¼0 c j  Pi Qj1 k Pi Qj1 k Pi  Qj1 ik k k ðx  x Þ ¼ ðc  ð a  ð a  1Þ  c þ b  a  S ÞÞ ¼ ðc  ð a  ð a  1Þ  c  S þ b  a  S ÞÞ ¼ i j j k ik ik ik j¼0 j¼0 j¼0 cj  k¼0 k¼0 Q .Q k¼0  P h   .Q i Qj1 Pi  Qj1  k i1 ij1 i ij1 j Qj1 k j Qj1 k ¼ j¼0 cj  d  k¼0 a  1 j¼0 c j  d  k¼0 Skþ1 k¼0 a ðða  1Þc þ bÞ  k¼0 Sik Þ ¼ k¼0 a  k¼0 Skþ1 k¼0 Skþ1 Q  i1 .Q P  k¼0 Skþ1 : Q Q i ak ; qj ¼ 1 j1 S ; r i ¼ i1 Let pj ¼ cj  dj  j1 k¼0 Skþ1 , we have bi ¼ j¼0 pj  qij  r i ði ¼ 0; 1; 2; . . . ; n  1Þ or equivak¼0 k¼0 kþ1 lently b ¼ ðp  qÞ  r ¼ p  1r  r. Finally, if the pairwise distinct knots x0, x1, . . . , xn1 are generated by xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K, for the given vectors b and x, the vector c can be computed by the Lagrange–Newton transformation formula c = LN(b, x). And for the given vectors c and x, the vector b can be computed by the inverse Lagrange–Newton transformation formula b ¼ L1 h N ðc; xÞ. 3.3. Related algorithm 3.3.1. Algorithm 3: Lagrange–Newton transformation algorithm Let pairwise distinct knots x0, x1, . . . , xn1 be generated by formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c) with a – 0, b and c in K. Input: A vector b = (b0, b1, . . . , bn1) 2 Kn and three scalars a – 0, b and x0 = c in a field K. Output: c = LN(b, x) 2 Kn. Algorithm: (1) Set p0 = b0, q0 = 1, S0 = 0, r0 = 1 and d = (a  . 1)  c + b. . Qi1 i Qi1 k Qi1 i Qi1 k (2) For i = 0 to n  1 do {Siþ1 ¼ Si  a þ 1; pi ¼ bi k¼0 Skþ1 ; qi ¼ ð1Þ  k¼0 a k¼0 Skþ1 ; r i ¼ d  k¼0 a }. (3) Compute c = (p  q)/r. (4) Return (c).

3.3.2. Algorithm 4: The inverse Lagrange–Newton transformation algorithm Input: A vector c = (c0, c1, . . . , cn1) 2 Kn and three scalars a – 0, b and x0 = c in a field K. n Output: b ¼ L1 N ðc; xÞ 2 K . Algorithm: (1) Set p0 = c0, r0 = 1, d = (a  1)  c + b, S0 = 0. (2) For i = 0 to n  1 do .Q Q Qi1 i1 k {Siþ1 ¼ Si  a þ 1; pi ¼ ci  di  i1 k¼0 a ; qi ¼ 1 k¼0 Skþ1 ; r i ¼ k¼0 Skþ1 }. (3) Compute b = (p  1/r)  r. (4) Return (b).

4. Example According to Theorem 2 and related algorithm, if we input a = 2, b = 3, c = 2, N = 7, we get: x[0] = 2.000000, x[1] = 7.000000, x[2] = 17.000000, x[3] = 37.000000, x[4] = 77.000000, x[5] = 157.000000, x[6] = 317.000000. If we now randomly input a = {1, 2, 3, 7, 3, 2, 1}, through our Lagrange-power basis transformation Algorithm 1 in 2.3.1, we can straightforwardly obtain the result:

b½0 ¼ 249 b½1 ¼ 161029 b½2 ¼ 27263139 b½3 ¼ 2710395559 b½4 ¼ 213944621199 b½5 ¼ 15168699699679 b½6 ¼ 1021174527437439: Conversely, given vectors b and x to find the vector a, this is equivalent the traditional inverse problem of Vandermonde matrix. Here our alternate algorithm is used to find vector a following Algorithm 2 in 2.3.2. We input values of vectors b and x,

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following the inverse Lagrange-power basis transformational Algorithm 2 given in 2.3.2, we get the value of the vectors p, q, x, and the central calculations for x0 s are displayed below as w’s. w[0][0] = 1 w[0][4] = 323061915 w[1][0] = 1 w[1][4] = 278875865 w[2][0] = 1 w[2][4] = 206501865 w[3][0] = 1 w[3][4] = 114175265 w[4][0] = 1 w[4][4] = 59239665 w[5][0] = 1 w[5][4] = 30094865 w[6][0] = 1 w[6][4] = 15158865

w[0][1] = 612 w[0][5] = 4238872092 w[1][1] = 607 w[1][5] = 2932864867 w[2][1] = 597 w[2][5] = 1374464217 w[3][1] = 577 w[3][5] = 660511117 w[4][1] = 537 w[4][5] = 323541717 w[5][1] = 457 w[5][5] = 160102117 w[6][1] = 297 w[6][5] = 79635717

w[0][2] = 120885 w[0][6] = 16873233839 w[1][2] = 117860 w[1][6] = 4820923954 w[2][2] = 111960 w[2][6] = 1985086334 w[3][2] = 100760 w[3][6] = 912066694 w[4][2] = 80760 w[4][6] = 438265814 w[5][2] = 50360 w[5][6] = 214945654 w[6][2] = 27960 w[6][6] = 106455734

w[0][3] = 9653760 w[1][3] = 9070510 w[2][3] = 7992210 w[3][3] = 6167410 w[4][3] = 3677010 w[5][3] = 1989010 w[6][3] = 1032210

From these, we get the value of vector a:

a½0 ¼ 1:000000 a½1 ¼ 2:000000 a½2 ¼ 3:000000 a½3 ¼ 7:000000 a½4 ¼ 3:000000 a½5 ¼ 2:000000 a½6 ¼ 1:000000; which is exactly what we should get, this demonstrated the correctness and accuracy of the formulae and algorithm. 5. Conclusion In this paper, we present explicit vector formulae and the following results are concluded. Firstly, the Lagrange-power basis transformation formula b = Lp(a, x) and its inverse transformation formula a ¼ L1 p ðb; xÞ are expressed as P P ij ij bi ¼ n1 and ai ¼ 1n  n1 ði ¼ 0; 1; 2; . . . ; n  1Þ, if the pairwise distinct knots x0, x1, . . . , xn1 are generated by j¼0 aj  l j¼0 bj  l xi = li (i = 0, 1, . . . , n  1) with a primitive root l from unity of degree n. Secondly, the Lagrange-power basis transformation Pn1 j i formula b = Lp(a, x) and its inverse transformation formula a ¼ L1 j¼0 aj  ða  c þ b  Si Þ and p ðb; xÞ are concluded as bi ¼ Pn1 0 ai ¼ j¼0 ðpi;j  qnj1 Þ=r, that is a = (p  q)/r, if the pairwise distinct knots x0, x1, xn1 are generated by the recursive formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c). Thirdly, the Lagrange–Newton transformation formula P  c = LN(b, x) and its inverse Pi i transformation formula b ¼ L1 N ðc; xÞ are stated as c i ¼ j¼0 ðpj  qij Þ=r i and bi ¼ j¼0 pj  qij  r i ði ¼ 0; 1; . . . ; n  1Þ, (that is c = (p  q)/r and b = (p  q)  r), if the pairwise distinct knots x0, x1, . . . , xn1 are generated by the recursive formulae xi = xi1  a + b (i = 1, 2, . . . , n  1, x0 = c). Based on these results, we also construct related algorithms. Additionally, we give an example to demonstrate the correctness of the formulae and associated algorithms. We finally note that Newton-power basis transformation and its inverse transformation can be obtained via the Lagrange–Newton, Lagrange-power basis and their inverse transformations. Acknowledgements This research is partially funded by NSFC 60863006. The first author acknowledges the support of the statistics department of the University of California, Riverside, where he is a visiting scholar during 2011–2012. References [1] A. Shamir, How to share a secret, Commun. ACM 22 (1979) 612–613. [2] M. Carpentieri, A perfect threshold secret sharing scheme to identify cheaters, Designs Codes Crypt. 5 (1995) 183–187. [3] D.J. Lu, B.R. Zhang, A efficient (t, n) threshold proxy signature scheme based on elliptic curve cryptosystems. in: 4th International Conference on Wireless Communications, Networking and Mobile Computing, 2008. [4] A. Amiraslani, D.A. Aruliah, R.M. Corless, Block LU factors of generalized companion matrix pencils, Theor. Comput. Sci. 381 (2007) 134–147. [5] G. McGrath, D. Pengelley, Lagrange and the solution of numerical equations, Historia Mathematica 28 (2001) 220–231. [6] R. Schaback, Limit problems for interpolation by analytic radial basis functions, J. Comput. Appl. Math. 212 (2008) 127–149. [7] W. Zhang, R.P. Agarwal, Construction of mappings with attracting cycles, Comput. Math. Appl. 45 (2003) 1213–1219. [8] S.K. Lodha, R. Goldman, Lattices and algorithms for bivariate Bernstein, Lagrange, Newton, and other related polynomial bases based on duality between L-Bases and B-Bases, J. Approx. Theor. 93 (1998) 59–99. [9] R. Smarzewski, J. Kapusta, Fast Lagrange–Newton transformations, J. Complex. 23 (2007) 336–345. [10] C.S. Jog, The accurate inversion of Vandermonde matrices, Comput. Math. Appl. 47 (2004) 921–929. [11] M.E.A. El-Mikkawy, Inversion of a generalized Vandermonde matrix, Int. J. Comput. Math. 80 (2003) 759–765. [12] H. Oruc, LU factorization of the Vandermonde matrix and its applications, Appl. Math. Lett. 20 (2007) 982–987. [13] A. Eisinberg, G. Fedele, On the inversion of the Vandermonde matrix, Appl. Math. Comput. 174 (2006) 1384–1397.