Recurrence Relations between Symmetric Polynomials of n-th Order

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Applied Mathematical Sciences, Vol. 8, 214, no. 15, 5195-522 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.214.47525 Recurrence Relations between Symmetric Polynomials of n-th Order Yuriy N. Belyayev Syktyvkar University, Syktyvkar-1671, Russia Copyright c 214 Yuriy N. Belyayev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The method of symmetric polynomials (MSP) was developed for computation analytical functions of matrices, in particular, integer powers of matrix. MSP does not require for its realization finding eigenvalues of the matrix. A new type of recurrence relations for symmetric polynomials of order n is found. Algorithm for the numerical calculation of high powers of the matrix is proposed.this computational procedure is more accurate in comparison with ordinary matrix multiplication. Mathematics Subject Classification: 11C, 34A Keywords: matrix functions, symmetric polynomials, roundoff error 1 Introduction Two groups of method play an important role in the theory of matrix functions and their practical applications. First group of approaches to the calculation of analytical function f(a) of matrix A a ij is based on the similarity transformations of matrices. For example, if a matrix A is normal, it can be represented by the formula A = UΛU 1, where U is a unitary matrix, Λ = λ i δ ij and λi are roots of the characteristic equation of matrix A: λ n p 1 λ n 1 p 2 λ n 2... p n 1 λ p n =. (1) Here p j = ( 1) j 1 σ j, j = 1,..., n, and σ j are sums of all principal minors of j-th order of det A, namely σ 1 = a 11 + a 22 +... + a nn, σ 2 = a ii a ij j>i a ji a jj,...,

5196 Yuriy N. Belyayev σ n = det A. After the matrices U, U 1, and eigenvalues λ i, i = 1,..., n, are found, matricies A k = U λ k i δ ij U 1 and hence f(a) = U f(λ i )δ ij U 1 can be calculated. Another category of methods is based on the Cayley-Hamilton theorem: each square matrix A satisfies its characteristic equation (1), in other words, A n = n i=1 p ia n i. It follows that any integer power j of matrix A can be expressed in terms of the first n powers A I, A,..., A n 1 : A j = C jl A l. (2) l= Well known formulas of Lagrange-Sylvester and Vandermonde [1, 13.4-7] use the representation (2) and express an analytic function f(a) through eigenvalues λ j of the matrix A. To resolve the main problem of the above two groups of methods, - finding the eigenvalues λ j of matrix A, algorithms of Danilevsky, Hessenberg, Krylov, Samuelson and others are applied. Features of the application of these methods are considered in the monograph [2]. In this paper, to calculate an analytic functions of matrices we develop a method of symmetric polynomials (MSP) [3]. This method is based on the representation of A j by means of the coefficients p j of characteristic equation (1), but does not require finding the eigenvalues of the matrix A. This is a major MSP advantage over the methods mentioned above. 2 Basic relations Coefficients p l of the characteristic equation (1) relate with λ j by Viète s formulas: p 1 = 6 j=1 λ j, p 2 = g<j n λ g λ j,..., p n = ( 1) n 1 λ j. In other words, the coefficients p l, l = 1,..., n, are equal, to within a sign, elementary symmetric polynomials with respect to the eigenvalues of matrix. Therefore, any function of the variables p 1,..., p n is also symmetric with respect to λ j. D e f i n i t i o n. Solutions B g (n) B g (p 1,..., p n ) of equations j=1 B g (n) = p 1 B g 1 (n) + p 2 B g 2 (n) +... + p n B g n (n), B g (n) =, g =, 1,..., n 2, B n 1 (n) = 1, are called symmetric polynomials of n-th order. (3)

Recurrence relations 5197 Obviously, that (3) allow us to calculate the values of functions B g (n) with indices g n recursively in terms of the previous n symmetric polynomials. For example, B n (n) = p 1 B n 1 (n) = p 1, B n+1 (n) = p 1 B n (n) + p 2 B n 1 (n) = p 2 1 + p 2,... If p n symmetric polynomials B g (n) with g < also exist and can be found recursively by the formula: B g (n) = 1 p n ( Bg+n (n) p 1 B g+n 1 (n)... p n 1 B g+1 (n) ), (4) namely B 1 (n) = B n 1(n) p n = 1 p n, B 2 (n) = p n 1B 1 (n)) p n = p n 1 p 2 n,... R e m a r k. Symmetric polynomials B g (n) with negative indices g for singular matrices (p n = ) are not defined. However, the following combinations of symmetric polynomials with indices g = 1, 2,..., n have meaning: p n B 1 (n) = B n 1 (n) p 1 B n 2 (n)... p n 1 B (n) = 1, 1 p n 1 B 1 (n) + p n B 2 (n) = B n 2 (n) p 1 B n 3 (n)... p n 2 B (n) =,. (5) p 2 B 1 (n) + p 2 B 2 (n) +... + p n B n+1 (n) = B 1 (n) p 1 B (n) =, p 1 B 1 (n) + p 2 B 2 (n) +... + p n B n (n) = B (n) =. These equalities follow directly from the relations (3). It was proved [3]: 1) for any integer j the coordinates C jl of the nonsingular matrix A j in the basis A, A,..., A n 1 are represented by the formulas: C jl = l p n l+g B j 1 g (n), l =,..., n 1; (6) g= 2) if det A =, then (6) determine the coefficients C jl for j. 3 Splitting of index polynomial T h e o r e m. Coordinates (6) of nonsingular matrix A j are the expansion coefficients of the polynomial B j+l (n) as a linear combination of polynomials B l (n), B l+1 (n),..., B l+n 1 (n) for any integers j and l: B j+l (n) = C jh B h+l (n). (7)

5198 Yuriy N. Belyayev If p n =, then the formula (7) holds for j + l and l n. P r o o f of this theorem we will carry out by induction. 1. Verification the relation (7), provided that l n 1. For this it is necessary to calculate the sum n 1 C jhb h+l (n). It should be noted that summand with number h = n 1 l contains B h+l (n) = B n 1 (n) = 1. Using definition (3), we find: C jh B h+l (n) = C j B l (n) +... + C j(n 2 l) B n 2 (n) +C j(n 1 l) B n 1 (n) 1 + C j(n l) B n (n) + C j(n l+1) B n+1 (n) +... + C j(n 1) B n 1+l (n). We express here the coefficients C j(n 1 l),..., C j(n 1) according to formulas (6) and recurrence relations (3): C j(n 1 l) = B j+l (n) p 1 B j+l 1 (n)... p l 1 B j+1 (n) p l B j (n), C j(n l) = B j+l 1 (n) p 1 B j+l 2 (n)... p l 2 B j+1 (n) p l 1 B j (n),. C j(n 2) = B j+1 (n) p 1 B j (n), C j(n 1) = B j (n), As a result of this C jh B h+l (n) = [ B j+l (n) p 1 B j+l 1 (n)... p l 1 B j+1 (n) p l B j (n) ] B n 1 (n) + [ B j+l 1 (n) p 1 B j+l 2 (n)... p l 2 B j+1 (n) p l 1 B j (n) ] B n (n) +... + [ B j+2 (n) p 1 B j+1 (n) p 2 B j (n) ] B n 3+l (n) + [ B j+1 (n) p 1 B j (n) ] B n 2+l (n) + [ B j (n) ] B n 1+l (n). Rearrangement of summands in the last expression gives C jh B h+l (n) = B j+l (n) B n 1 (n) +B j+l 1 (n) ( B n (n) p 1 B n 1 (n) ) 1 + B j+l 2 (n) ( B n+1 (n) p 1 B n (n) p 2 B n 1 (n) ) } {{ } p 3 B n 2 (n)+p 4 B n 3 (n)+...+p nb 1 (n)= +... + B j+1 (n) ( B n 2+l (n) p 1 B n 3+l (n)... p l 2 B n (n) p l 1 B n 1 (n) ) p l B n 2 (n)+...+p nb l 2 (n)= + B j (n) ( B n 1+l (n) p 1 B n 2+l (n)... p l B n 1 (n) ) = B j+l (n), p l+1 B n 2 (n)+...+p nb l 1 (n)=

Recurrence relations 5199 was to be shown. 2. Let (7) holds for values l = g, g + 1,..., g + n 1, where g integer. a) If g, then (7) is also true for l = g + n. Indeed, according to the definition (3) B j+g+n (n) = p 1 B j+g+n 1 (n) + p 2 B j+g+n 2 (n) +... + p n B j+g (n). Each of the symmetric polynomials of n-th order in the right-hand side of last expression we represent according to the formula (7) and transform the resulting expression using the recursive formula (3). ( B j+g+n (n) = C jh p1 B h+g+n 1 (n) +...+ p n B h+g (n) ) = C jh B h+g+n (n), was to be proved. b) If g (it is assumed that the matrix A is nonsingular, i.e. p n ), then (7) is also true for l = g 1. The proof of this is analogous to the previous one and is based on the relations of the form (4): B j+g 1 (n) = 1 p n ( Bj+g+n 1 (n) p 1 B j+g+n 2 (n)... p n 1 B j+g (n) ) = 1 ( C jh Bh+g+n 1 (n) p 1 B h+g+n 2 (n)... p n 1 B h+g (n) ) p n n 1 = C jh B h+g 1 (n). We again have obtained confirmation of the rule (7). 3. If l = 1, 2,..., n, but j+l, equality (7) holds even for symmetric polynomials B j+l (n) of singular matrices. Let us show this using relations (5). Transforming n 1 C jhb h+l (n) for the case n l 1, we find C j B l (n) + + C j( l 1) B 1 (n) + C j( l) B (n) +... + C j(n 1) B n 1+l (n) = p n B j 1 (n)b l (n) + ( p n 1 B j 1 (n) + p n B j 2 (n) ) B l+1 (n) +... + ( p n+l+2 B j 1 (n) + p n+l+3 B j 2 (n) +... + p n B j+l+1 (n) ) B 2 (n) + ( p n+l+1 B j 1 (n) + p n+l+2 B j 2 (n) +... + p n B j+l (n) ) B 1 (n) = B j 1 (n) ( p n+l+1 B 1 (n) + p n+l+2 B 2 (n) +... + p n 1 B l+1 (n) + p n B l (n) ) B n+l (n) p 1 B n+l 1 (n)... p n+l B (n)= + B j 2 (n) ( p n+l+2 B 1 (n) + p n+l+3 B 2 (n) +... + p n B l+1 (n) ) +... B n+l+1 (n) p 1 B n+l (n)... p n+l+1 B (n)= + B j+l+1 (n) ( p n 1 B 1 (n) + p n B 2 (n) ) +B j+l (n) p n B 1 (n) = B j+l (n), 1

52 Yuriy N. Belyayev which completes the proof. 4 Symmetric polynomials of doubled index Formula (7) allows us to compute the symmetric polynomial B k (n) with a large index k = j + l using two sets of polynomials B j n (n),..., B j 1 (n) and B l (n),..., B l+n 1 (n) with smaller indices. In particular, from the formulas (7) it follows that B 2j g (n) = C jh B h+j g (n), g j. (8) Consider in detail these relations by the following example. E x a m p l e. D o u b l i n g o f t h e i n d e x i n t h e s e c o n d o r d e r s y m- m e t r i c p o l y n o m i a l s. According to the formula (6) for the case n = 2 we find C j = p 2 B j 1 (2), C j1 = p 1 B j 1 (2) + p 2 B j 2 (2) = B j (2). and from (8) equations are obtained: B 2j (2) = p 2 B j 1 (2)B j (2) + B j (2)B j+1 (2) = p 2 B j 1 (2)B j (2) + B j (2) ( p 1 B j (2) + p 2 B j 1 (2) ) = p 1 B 2 j (2) + 2p 2 B j 1 (2)B j (2), (9) B 2j 1 (2) = p 2 B 2 j 1(2) + B 2 j (2) (1) = p 2 B 2 j 1(2) + ( p 1 B j 1 (2) + p 2 B j 2 (2) ) 2, (11) B 2j 2 (2) = p 2 B j 1 (2)B j 2 (2) + B j (2)B j 1 (2) = p 2 B j 1 (2)B j 2 (2) + ( p 1 B j 1 (2) + p 2 B j 2 (2) ) B j 1 (2) = p 1 B 2 j 1(2) + 2p 2 B j 1 (2)B j 2 (2). (12) In deriving these equations we used the definition of symmetric polynomials B j (2) (see formulas (3)): B j (2) = p 1 B j 1 (2) + p 2 B j 2 (2). An important feature of these formulas is that they allow to calculate a set polynomials B 2j (2), B 2j 1 (2) directly through a set B j (2), B j 1 (2) (relations (9) and (1)), or set B 2j 1 (2), B 2j 2 (2) through set B j 1 (2), B j 2 (2) (equations (11) and (12)). Recurrence relations (8) for symmetric polynomials of orders n > 2 have the same meaning. Recurrence relations (8) can significantly improve the accuracy of calculation of high powers of matrices, in particular A 2j. Consider this on the example of second-order matrices. We compare two methods of computations

Recurrence relations 521 A 2j. First approach consists in by usual j-fold squaring A 2j =(... ((A) 2 ) 2...) 2, and new one (second) is based on the formula A 2j = AB 2 j(2) + Ip 2 B 2 j 1(2), (13) which follows from (2) and (6) for the matrix A of order n = 2. Here symmetric polynomials B 2 j(2) and B 2 j 1(2) can be found by recurrence formulas (9) and (1) respectively. T a b l e. Algorithm for computing a matrix A 2j of the second order No Sequence of computing A, M 1 2 3 1. Calculation of the polynomials p 1, p 2, B 1 (2), B 2 (2) 1 p 1 = a 11 + a 22, p 2 = a 12 a 21 a 11 a 22, B 1 (2) = 1, B 2 (2) = p 1 2, 2 2. Computation of B 2 j(2), B 2 j 1(2) by formulas (9)-(1) β 1 = B2(2), 2 β 2 = p 2 B 1 (2), 2 2.1 B 3 (2) = β 1 + β 2 B 1 (2), B 4 (2) = p 1 β 1 + 2β 2 B 2 (2) 2, 4 β 1 = B4(2), 2 β 2 = p 2 B 3 (2), 2 2.2 B 7 (2) = β 1 + β 2 B 3 (2), B 8 (2) = p 1 β 1 + 2β 2 B 4 (2) 2, 4... β 1 = B 2 2 (2), β j 1 2 = p 2 B 2 j 1 1(2), 2 2.j-1 B 2 j 1(2)= β 1 + β 2 B 2 j 1 1(2), B 2 j(2)= p 1 β 1 + 2β 2 B 2 j 1(2) 2, 4 Calculation of the matrix A 2j 2j a ik by the formula (13) β 3 = p 2 B 2 j 1 1(2), 2j a 12 = a 12 B 2 j(2), 2j a 21 = a 21 B 2 j(2), 3 3 2 j a 11 = a 11 B 2 j(2) + β 3, 2j a 22 = a 22 B 2 j(2) + β 3 2, 2 Computational errors caused by rounding the results depend on the total arithmetic operations number. A time of computations also is defined by number of arithmetic operations especially multiplications. One possible sequences of calculations by formula (13) with numbers of elementary additions A and multiplications M, corresponding to each operation, is shown in the Table. This algorithm requires the fulfillment of A 2 = 2 + 2(j 1) + 2 = 2j + 2 additions and M 2 = 2 + 6(j 1) + 5 = 6j + 1 multiplications. For comparison, the number of additions and multiplications by repeated squaring of the matrix A (first method) are, respectively, A 1 = 4j and M 1 = 8j. Thus, the method of computing the second order matrix A 2j, based on the use of matrix A symmetric polynomials, is more efficient and accurate than

522 Yuriy N. Belyayev the method of repeated squaring (even for j = 1!). A similar algorithms for computing matrix A 2j of order n > 2 are based on the formulas (2), (6) and (8). 5 Conclusion The scaling and squaring method for the matrix exponential (see for example [4]) is one of the problems in which the calculations of matrices A 2j are applied. MSP solves this problem as follows. For any n-th order matrix W and scalar z: exp(w z) = [exp (W z/m)] m A m, where integer m is called the scaling parameter, ( ) [ ] l W z 1 l n+n A m l! + B j l g (n) p n l+g, (14) j! l= g= p j, j = 1,..., n, and B l (n) imply respectively the characteristic equation coefficients and n=th order symmetric polynomials for matrix W z/m. An element a jl of the matrix (14) depends from number N: a jl = a jl (N). Approximate equality in (14) is replaced by the exact if N goes to infinity. The relative truncation error ɛ A (N) max (a ik ( ) a ik (N))/a ik ( ) in computation of the matrix (14) satisfies the inequality ξ N+1 ɛ A (N) < (N + n) N i=1 (n + i), provided that ξ = (2n 1)max w ijz < 1. m Choice scaling parameter m allows to control the relative truncation error. If m > n (in particular, m = 2 j, j > n) computation A m by formulas (2), (6) and recurrence relations (3) (or (8)) minimizes roundoff error. References [1] G.A. Korn, T.M. Korn, Mathematical handbook, McGraw-Hill Company, New York, 1968. [2] D.K. Faddeev, V.N. Faddeeva, Computational methods of linear algebra, Nauka, Moscow, 1963. [3] Yu.N. Belyayev, On the calculation of functions of matrices, Mathematical Notes, 94 (213), 177-184. http://dx.doi.org/1.1134/ S14346137171 [4] N.J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM Review, 51-4 (29), 747-764. http://dx.doi.org/1.1137/9768539 Received: July 9, 214 j=n

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