A new parallel polynomial division by a separable polynomial via hermite interpolation with applications, pp

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1 Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012 Article A new parallel polynomial division by a separable polynomial via hermite interpolation with applications, pp Aristides I Kechriniotis Konstantinos K Delibasis, Christos Tsonos Nicholas Petropoulos nicholas@teilamgr Follow this and additional works at: Recommended Citation Kechriniotis, Aristides I; Delibasis,, Konstantinos K; Tsonos, Christos; and Petropoulos, Nicholas (2012, "A new parallel polynomial division by a separable polynomial via hermite interpolation with applications, pp ", Electronic Journal of Linear Algebra, Volume 23 DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository For more information, please contact scholcom@uwyoedu

2 Volume 23, pp , August A NEW PARALLEL POLYNOMIAL DIVISION BY A SEPARABLE POLYNOMIAL VIA HERMITE INTERPOLATION WITH APPLICATIONS ARISTIDES I KECHRINIOTIS, KONSTANTINOS K DELIBASIS, CHRISTOS TSONOS, AND NICHOLAS PETROPOULOS Abstract A new parallel division of polynomials by a common separable divisor over a perfect field is presented and this is done by expressing the remainders as derivatives of a unique polynomial In order to get this result, a novel variant expression of the classical Lagrange Sylvester Hermite interpolating polynomial has been utilised, although any known variant may be used The above findings are utilized to obtain a number of new identities involving polynomial derivatives, including a closed formula for the semi simple part of the Jordan decomposition of a matrix Key words Euclidean polynomial division, Hermite interpolation, Semisimple part of a matrix AMS subject classifications 12Y05, 15A21, 11C08, 65F30 1 Introduction The Hermite interpolation of total degree is described in the following theorem [1]: Theorem 11 Given n distinct elements λ 0,λ 1,,λ in a perfect field K, positive integers m i, i = 0,,, and a ij K for 0 i, 0 j m i 1, then there exists one and only one polynomial r K[x] of degree less than m i, such that r (j (λ i = a ij, 0 j m i 1, 0 i (11 This polynomial r is explicitly given by, r(x = m i 1 m i j 1 k=0 Ω(x (x λ i mi j k, [ 1 1 (x λi mi a ij j! k! Ω(x Received by the editors on March 2, 2012 Accepted for publication on August 18, 2012 Handling Editor: Bryan L Shader Department of Informatics, Technological Educational Institute of Lamia, Lamia, Greece (kechrin@teilamgr Department of Computer Science and Biomedical Informatics, University of Central Greece, Lamia 35100, Greece (kdelibasis@yahoocom Department of Electronics, Technological Educational Institute of Lamia, Lamia, Greece (tsonos@teilamgr, nicholas@teilamgr 770 ] (k x=x i

3 Volume 23, pp , August A New Parallel Polynomial Division by a Separable Polynomial 771 where Ω(x = (x λ i mi, and r (k (a is the k th derivative of r at a The applications of Hermite interpolation to numerical analysis are well known A number of forms of the interpolating polynomial r(x have been reported in the literature, which require calculation of derivatives of rational polynomial functions (eg, [1], or recursive calculation of the coefficients a ij of Theorem 11 (eg, [8] We propose a closed form expression of the interpolating polynomial r of the univariate Hermite interpolation which is a variation of the classical Lagrange Sylvester formula, as presented in [11] The expression in [11] involves less computational load than the proposed Hermite interpolating polynomial except in the special case of very small values of m j We utilise the Hermite interpolating polynomial to show the main result of this work, the parallel polynomial division by a separable polynomial Let us note that the remainder r K[x] of the Euclidean division of any polynomial P K[x] of degree n by a separable polynomial Q K[x] of degree m, where n m, can be calculated in closed form using the Langrange interpolation formula as following: r(x = m P(λ i i=1 m j=1j i (x λ j (λ i λ j, where λ 1,,λ m are the roots of Q in the algebraic closure K of K In this work, we extend this simplified idea of polynomial remainder calculation by Langrange interpolation, to achieve a polynomial division by a common separable divisor, using the proposed closed form of the interpolating polynomial r of the univariate Hermite interpolation, although any expression of r may be utilized Furthermore, the above results will be used to obtain a number of new identities involving polynomial derivatives, as well as a closed form expression of the semisimple part of the Jordan decomposition of an algebraic element in an arbitrary algebra These results however are independent from the selected expression of the Hermite interpolating polynomial The restofthe paper is organizedasfollows In Section 2, we presentanew closed form for Hermite interpolation In Section 3, we present a new parallel division of polynomials by a common separable divisor over a perfect field, by expressing the remainders as derivatives of a unique polynomial In Section 4, the main result

4 Volume 23, pp , August AI Kechriniotis, KK Delibasis, C Tsonos, and N Petropoulos of Section 3 is applied to obtain a number of new identities involving polynomial derivatives, as well as a new closed form expression of the semisimple part of the Jordan decomposition of an algebraic element in an arbitrary algebra 2 A new closed form for Hermite interpolation Let λ 0,λ 1,,λ be distinct elements in a perfect field K and m 0,m 1,,m be positive integers Let us denote by L k the polynomials given by L k (x = i k (x λ i mi (λ k λ i mi K[x] (21 Furthermore, we denote by Λ k, 0 k, the m k m k lower triangular matrices [l ij ] K m k m k, 0 i,j m k 1, given by { ( i l ij := j (Lk (i j (λ k if 0 j i m k 1 0 if 0 i < j m k 1, where (L k (i j (λ k is the derivative of order (i j of the polynomial L k (x at λ k Thus, Λ k has the following representation ( 1 0 ( mk 1 0 ( 0 0 Lk (λ k 0 0 ( (Lk (1 (λ k 1 1 Lk (λ k 0 (Lk (mk 1 (λ k (Lk (mk 2 (λ k ( mk 1 1 For our purpose the following technical lemmas are required: ( mk 1 m k 1 Lk (λ k (22 Lemma 21 The matrices Λ k, 0 k n 1, are invertible with Λ 1 mk 1 (I mk Λ k i, where I mk is the m k m k unit matrix k = Proof Clearly, for 0 k holds L k (λ k = 1 Therefore, all matrices Λ k, 0 k are invertible and lower unitriangular Thus, (I mk Λ k m k = 0 and consequently Λ k mk 1 (I mk Λ k i = I mk Using Leibnitz s rule for derivatives, we easily get the following lemma: Lemma 22 For 0 i,s m k 1 and 0 j,t, the following holds: ( (x λ t s L t (x (i x=λj = s! ( i s 0 if t j 0 if t = j and i < s (Lj (i s (λ j if t = j and s i (23

5 Volume 23, pp , August A New Parallel Polynomial Division by a Separable Polynomial 773 Our proposed form of Hermite interpolation can now be presented in the following theorem Theorem 23 Given n distinct elements λ 0,λ 1,,λ in a perfect field K, positive integers m i, i = 0,,, and a ij K for 0 j, 0 i m j 1 Then there exists one and only one polynomial r K[x] of degree less than m i, such that r (i (λ j = a ij, 0 j, 0 i m j 1 (24 This polynomial r is explicitly given by, r = X j Λ 1 j A j (25 = m j 1 k=0 X j (I mj Λ j k A j, where the matrices X j and A j are given by [ (x λ X j = L j (x j 1! L j (x and (x λ j m j 1 (m j 1! L j (x ], A j = [ a 0j a ij a mj 1j ] T Proof It can be observed that (25 can be equivalently written in the following form: r(x = m j 1 (x λ j i c ij L j (x K[x], i! where c 0j c 1j c mj 1j = Λ 1 j a 0j a 1j a mj 1j, 0 j (26 Now, by calculating the derivative of order i of the polynomial r at λ j and using (23 in Lemma 22, we obtain r (i (λ j = i ( i c kj (L j (i k (λ j, (27 k k=0

6 Volume 23, pp , August AI Kechriniotis, KK Delibasis, C Tsonos, and N Petropoulos for all 0 j, 0 i m 1 Taking into consideration the definition of Λ j in (22, the system of equations (27 can be rewritten in the following matrix form r(λ j r (λ j r (mj 1 (λ j = Λ j By substituting (26 in (28, we get r(λ j r (λ j r (mj 1 (λ j c 0j c 1j c mj 1j =, 0 j (28 a 0j a 1j a mj 1j Hence, we derive that r (i (λ j = a ij, 0 j n 1,0 i m j 1 The second equality of (25 is obtained by Lemma 21 Moreover, it can be easily confirmed that any polynomial (x λ j i L j (x, 0 j n 1, 0 i m j 1 has degree less than m i, and since r is a K linear combination of these polynomials, we conclude that the degree of r is less than m i Remark 24 The inversion of matrices Λ j in Theorem 23 may be performed by any efficient numerical technique, replacing the last expression in (21 3 Division of polynomials by a separable polynomial At this point we are ready to present the following generalization of Euclidean polynomial division by a separable divisor, based on Theorem 23 Theorem 31 Let g K[x] be a separable polynomial and λ 0,,λ be the roots of g in the algebraic closure K of K Then for any polynomials f 0,f 1,,f m 1 K[x], there exists unique r K[x] of degree less than mn and unique polynomials q 0,q 1,,q m 1 K[x] such that f i = r (i +gq i, 0 i m 1 This result is optimal, in the sense that if m 1 and g is inseparable, then this result is not true The polynomial r is given by r(x = X j Λ 1 j A j K[x],

7 Volume 23, pp , August A New Parallel Polynomial Division by a Separable Polynomial 775 where [ X j = L j (x (x λ j 1! L j (x A j = [ f 0 (λ j f 1 (λ j f m 1 (λ j ] T (x λ j m 1 (m 1! L j (x ], and L j,λ j are respectively as in (21, (22 by m 0 = m 1 = = m = m Proof Let k be the dimension of the field K(λ 0,,λ as vector space over K, that is, k = [K(λ 0,,λ,K] (31 Then there exist τ 1,,τ k 1 in K(λ 0,,λ such that {1,τ 1,,τ k 1 } is a basis of K(λ 0,,λ as a vector space over K Now, using Theorem 23 by m 0 = m 1 = = m = m, we have that there is unique polynomial r K(λ 0,,λ [x] given by (25 for m 0 = m 1 = = m = m having degree less than mn such that ( r (i (λ j = f i (λ j, (32 for all 0 i m 1,0 j n 1 Therefore, there exist polynomials q i K(λ 0,,λ [x], 0 i m 1, such that r (i = f i +g q i (33 Moreover, by (31 we have that the dimension of K(λ 0,,λ [x] as free module over K[x] is k and {1,τ 1,,τ k 1 } is a basis of K(λ 0,,λ [x] over K[x] Therefore, the polynomials r, q i K(λ 0,,λ [x] can be uniquely written in the following form: and r = r + τ s r s, (34 s=1 q i = q i + τ s q si, 0 i m 1, (35 s=1 where r, r s, q i, q si K[x], with degr, degr i < mn Setting (34 and (35 in (33, we get r (i = f i +gq i (36 for all 0 i m 1

8 Volume 23, pp , August AI Kechriniotis, KK Delibasis, C Tsonos, and N Petropoulos Now from (36 we clearly get that r satisfies the identities (32, and since the polynomial of degree less than mn satisfying the identities (32 is unique, we conclude that r = r Finally, suppose that Theorem 31 is true for one polynomial g K[x] having a root λ K of multiplicity > 1 Then there exists a polynomial g 0 in K[x] such that g(x = (x λ 2 g 0 (x Applying Theorem 31 for g and f 0 = f 1 = 1, we obtain and for some r, q 0, q 1 K[x] r(x = 1+(x λ 2 g 0 (xq 0 (x (37 r (1 (x = 1+(x λ 2 g 0 (xq 1 (x (38 Differentiating (37, and setting x = λ in the resulting identity, as well as in (38, we respectively get the contradiction r (1 (λ = 0 and r (1 (λ = 1 4 Applications of Theorem 31 Firstly, we will use Theorem 31 to give a closed formula for the semi simple part of the well known Jordan decomposition, (eg, [2, 4, 7] of an algebraic element A of an algebra A over a perfect field K into a semisimple S A and a nilpotent part A proof of the existence of S A that is presented in the book of Hoffman and Kunze [4], is based on Newton s method and yields direct methods for computations An algorithm, which is essentially based on these ideas, is given by Levelt [6] The algorithm of Bourgoyne and Cushman [3] is faster, because higher derivatives are used In [9], D Schmidt has used Newton s method to construct the semi simple part of the Jordan decomposition of an algebraic element in an arbitrary algebra, showing quadratic convergence of the algorithm Another approach uses the partial fractions decomposition of the reciprocal of the minimal polynomial [5, 10, 11] An explicit construction of the spectral decomposition of a matrix using Hermite interpolation is reported in [11], which requires the use of Taylor coefficients of the reciprocal of the matrix minimal polynomial In this work, we also obtain a new closed formula for the semi simple part S A of the Jordan decomposition of an algebraic element A in an arbitrary algebra, using our proposed polynomial division by a common separable divisor The proposed closed formula requires only evaluation of the derivatives of the basic Hermite like interpolation polynomials that are associated with the eigenvalues of A, up to the

9 Volume 23, pp , August A New Parallel Polynomial Division by a Separable Polynomial 777 maximum algebraic multiplicity of the roots of the minimal polynomial of A, as well as matrix multiplication operations It has to be noted however that the semi simple part S A of A is obtained using a polynomial of higher degree than the one used in [11] In order to obtain the expression for S A we need the following lemma Lemma 41 Let g K[x] be a separable polynomial and λ 0,,λ are the roots of g in K Let f K[x,y] be a polynomial of two variables and m be a positive integer Then there exists unique r K[x] of degree less than mn such that f(x,y = r(x+y mod I, where I K[x,y] is the ideal generated from the polynomials g(x, y m Further, the polynomial r is given by r(x = X j Λ 1 j A j K[x], (41 where the matrices X j, A j are given by [ (x λ X j = L j (x j 1! L j (x [ A j = f(λ j,0 y f(λ j,0 ] (x λ j m 1 (m 1! L j (x, ] T m 1 y f(λ m 1 j,0, and L j,λ j are as in (21, (22 by m 0 = m 1 = = m = m Proof The polynomial f can be rewritten in the form for some h K[x,y] f(x,y = m 1 i 1 i! y if(x,0yi +y m h(x,y, (42 Furthermore, according to Theorem 31 there exists unique polynomial r K[x] of degree less than mn, given by (41, such that: Combining (42 with (43, we get i y if(x,0 = r(i (x mod g(x, 0 i m 1, (43 f(x,y = m 1 Furthermore, by using the Taylor formula, we have: m 1 r (i (x y i mod I (44 i! r (i (x y i = r(x+y mod y m (45 i!

10 Volume 23, pp , August AI Kechriniotis, KK Delibasis, C Tsonos, and N Petropoulos Substituting (45 in (44 completes the proof Now we will use Theorem 31 to give a closed formula for the semi simple part of the Jordan decomposition of an algebraic number of an algebra A over a perfect field K Let p K[x] be the minimal polynomialofanalgebraicelement Aofan algebraa overkwith unit 1 Let λ i, i = 0,1,,be the distinct roots ofpin K, and let k i, i = 1,2,, be their respective multiplicities We denote p(x := (x λ i and m(p := max{k 0,,k } Proposition 42 The semi simple part S A of the Jordan decomposition of A is given by S A = r(a, where r(x is the polynomial where [ X j = L j (x r(x = X j Λ 1 j A j K[x], (46 (x λ j 1! L j (x A j = [ λ j 0 0 ] T, (x λ j m 1 (m 1! L j (x ], and L j,λ j are as in (21, (22 by m 0 = m 1 = = m = m(p Proof Since S A is the semi simple part of A and N A is the nilpotent part of A, the minimal polynomials of S A and N A are respectively p(x and x m(p So we have p(s A = 0 and N m(p A = 0 Now if we apply Lemma 41 by choosing f(x,y = x, g(x = p(xandm = m(p, andtakingintoaccountthatf(x,0 = x, and i y f(x,0 = i 0 for 1 i m 1 we have that for the polynomial r given by (46 holds: x = r(x+y mod I, (47 where I is the ideal generated from the polynomials p(x and y m(p Finally setting x = S A and y = N A in (47 we get S A = r(s A +N A = r(a The next result of this section is the generalization of Theorem 31, which is expressed in the following theorem Theorem 43 Let g K[x] be separable of degree n Let Π (K[x] m m such that det(π 0 Then, for any polynomials f 0,f 1,,f m 1 K[x], there exists a unique polynomial r K[x] of degree less than mn, such that Π r r r (m 1 = E f 0 f 1 f m 1 mod g,

11 Volume 23, pp , August where E := gcd(g, det(π Proof We start from A New Parallel Polynomial Division by a Separable Polynomial 779 Π Π = ΠΠ = det(πi m, (48 where Π is the adjugate of Π and I m is the m m unit matrix Moreover, since E = gcd(g,det(π one has that there exist H,G K[x] such that Combining (48 with (49, we get H det(π+gg = E (49 HΠ Π = (E ggi m (410 Now, according to Theorem 31, there exists a unique r K[x] of degree less than mn such that r r r (m 1 = H Π f 0 f 1 f m 1 mod g (411 Multiplying (411 from the left by Π and setting (410 in the resulting identity we get the conclusion Remark 44 Choosing Π = I m in Theorem 43, we get Theorem 31 Therefore, Theorem 43 can be regarded as a generalization of Theorem 31 Now we will apply Theorem 43 to produce some formulas for polynomials involving derivatives Corollary 45 Let g, g 0, g 1,, g m 1 K[x] be polynomials Assume that g is separable and that (g i,g = 1, 0 i m 1 Then, for any f 0,f 1,,f m 1 K[x], there exists unique r K[x] of degree less than mn such that g i r (i = f i mod g, 0 i m 1

12 Volume 23, pp , August AI Kechriniotis, KK Delibasis, C Tsonos, and N Petropoulos Proof Applying Theorem 43 by g g 1 0 Π := 0 0 g m 1, and afterwards using that E = gcd(g,det(π = m 1 g i = 1, we directly get the conclusion Corollary 46 Let g, g 0,g 1,,g m 1 K[x] be as in Corollary 45 Then, for any f 0,f 1,,f m 1 K[x], there exists unique r of degree less than mn such that (g i r (i = f i mod g, 0 i m 1 Proof Let Π (K[x] m m be the matrix defined by g ( 1 (1 ( Π = 0 g g1 0 (m 1 (m 2 (0 g m 1 g m 1 g ( m 1 0 ( m 1 1 ( m 1 m 1 m 1 (412 From the assumptions (g i,g = 1, 0 i m 1, we have (det(π,g = 1 (413 Applying Theorem 43 for Π, as given in (412, using (413 and Leibnitz s rule, we get the conclusion REFERENCES [1] IS Berezin and NP Zhidkov Computing Methods (Chapter 8, Section 9, translated from Russian Pergamon, 1973 [2] N Bourbaki Elements de Mathematique; Algebre (Chapter 7, Section 5 Hermann, Paris, 1958 [3] N Burgoyne and R Cushman The decomposition of a linear mapping Linear Algebra Appl, 8: , 1974 [4] K Hoffman and R Kunze Linear Algebra, second edition (Chapter 7, Section 7 Prentice-Hall, Englewood Cliffs, NJ, 1971

13 Volume 23, pp , August A New Parallel Polynomial Division by a Separable Polynomial 781 [5] PF Hsieh, M Kohno, and Y Sibuya Construction of a fundamental matrix solution at a singular point of the first kind by means of the SN decomposition of matrices Linear Algebra Appl, 239:29 76, 1996 [6] AHM Levelt The Semi-Simple Part of a Matrix Algoritmen In De Algebra, A Seminar on Algebraic Algorithms, University of Nijmegen, Nijmegen, 1993 [7] T Mulders Computation of Normal Forms for Matrices Algoritmen In De Algebra, A Seminar on Algebraic Algorithms, University of Nijmegen, Nijmegen, 1993 [8] R Sakai and P Vertesi Hermite-Fejer interpolations of higher order III Studia Sci Math Hungar, 28:87 97, 1993 [9] D Schmidt Construction of the Jordan decomposition by means of Newton s method Linear Algebra Appl 314:75 89, 2000 [10] G Sobczyk The generalized spectral decomposition of a linear operator College Math J, 28:27 38, 1997 [11] L Verde-Star Interpolation approach to the spectral resolution of square matrices L Enseignement Mathematique, 52: , 2006