Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.
On Computing the Vandermonde Matrix Inverse
Yiu-Kwong Man
n1 n2 1 Abstract — A simple and efficient approach for computing 1 1 ( ) ( ) ( ) the inverse of Vandermonde matrix is presented in this paper, 1 j 1 j 1 j which is based on synthesis division of polynomials. This j1 j1 j1 n1 n2 approach does not involve matrix multiplication, computation 2 2 1 of determinant or solving a system of linear equations for W ( ) ( ) ( ) determining the unknown elements of the inverse matrix. Some 2 j 2 j 2 j illustrative examples are provided. j2 j2 j2 n1 n2 1 Index Terms — Vandermonde matrix, matrix inverse, n n synthetic division. ( ) ( ) ( ) n j n j n j jn jn jn
I. INTRODUCTION 1 0 0 0 HE Vandermonde matrix (VDM) has important a 1 0 0 T applications in various areas such as polynomial 1 A a a 1 0 . interpolation, signal processing, curve fitting, coding theory 2 1 and control theory [2, 4, 6, 9], etc. Since the last decade, the study of more efficient approaches for computing the inverse an1 an2 an3 1 of VDM or its generalized version has been an important with a , a , a , ..., and research topic in many mathematics and sciences disciplines. 1 j 2 j m 3 j m s In this paper, we present a novel simple and efficient jm jms approach for finding the inverse of VDM, based on synthesis n an (1) j . However, the computational cost could be division of polynomials and some previous works of the high if direct matrix multiplications are applied to compute author and the others [5, 7, 8, 10]. V-1. In the next section, we will introduce a new approach to The whole paper is organized like this. The basic compute V-1 by means of synthetic division of polynomials. mathematical background is described in section 2. Then, the new computational approach is discussed in section 2, III. A NEW APPROACH followed by some numerical examples in section 3. Finally, -1 some concluding remarks are provided in section 4. Let us see how the elements of V look like, when the formula V-1 = W A is applied to the cases n = 2 and n = 3. II. MATHEMATICAL BACKGROUND (i) When n= 2, we have: Consider the polynomial: 1 f (x) (x )(x )(x ) 1 1 2 n 1 1 1 0 1 2 1 2 xn a xn1 a xn2 a V , A , W 1 2 n 1 1 2 a1 1 2 where ai, i are known constants. We are interested to find the 2 1 2 1 inverse of the following Vandermonde matrix: where a . 1 1 2 Hence, 1 1 1 1 a1 1 1 2 n 2 2 2 1 1 2 1 2 V 1 2 n V W A 2 a1 1 n1 n1 n1 2 1 2 1 1 2 n If we apply synthetic division to the [f (x) – a2](x – 1), where f (x) (x )(x ) x2 a x a , we have According to [10], the formula V-1 = W A can be applied to 1 2 1 2
compute the inverse of V, where the matrices W and A are 1 a1 defined by: 1 1 1 1+ a1 Manuscript received in Feb, 2017. This work was supported in part by the internal research grants of EdUHK. The author is an Associate Professor of We can see that the elements obtained by synthetic division is the Department of Mathematics and Information Technology, EdUHK and equal to the numerators of the elements in the first row of V-1, he is a member of IAENG. (E-mail: ykman@ eduhk.hk).
ISBN: 978-988-14047-4-9 WCE 2017 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.
2 except in the reverse order. By applying a further synthetic since 22 + a12+a2-12-13 = (2- 1)(2- 3). division to the answer obtained, we get Also, 3 1 a1 a2 2 1 1 a1 3 3 + a13 2 1 1 3+ a1 3 + a13+a2 2 1 1 + a1 3 3 -13-23 1 1 3-1-2 (3- 1)(3- 2) 1 1 2 2 since 23 + a13+a2-13-23 = (3- 1)(3- 2). Thus, the
since 21 a1 1 2. Notice that the element in bracket, synthetic division approach works equally well for -1 namely 1 2, is equal to the denominators of the elements in computing V for n = 3. the first row of V-1. Similarly, if we apply synthetic divisions UMERICAL EXAMPLES to [f(x) – a2] (x – 2), we have IV. N Example1. Find the inverse of the following Vandermonde 2 1 a1 matrix. 2 1 1 1 1 + a 2 1 V 1 2 3 . 2 1 4 9 1 2 1 Solution. Let us define the following polynomial: 3 2 since 22 a1 2 1. In other words, all the elements of f (x) (x 1)(x 2)(x 3) x 6x 11x 6. V-1 can be determined completely by using synthetic division Applying synthetic divisions, we have: of polynomials only, without having to use matrix 1 1 -6 11 multiplications or computation of determinant. 1 -5 1 -5 6 (i) When n= 3, we have: 1 -4 1 1 1 1 0 0 1 -4 2
V 1 2 3 , A a1 1 0 -1 2 2 2 a a 1 So, the first row of V is (3 -5/2 1/2). 1 2 3 2 1 Next, 2 1 1 1 2 1 -6 11 (1 2 )(1 3 ) (1 2 )(1 3 ) (1 2 )(1 3 ) 2 -8 2 1 W 2 2 1 -4 3 (2 1 )(2 3 ) (2 1 )(2 3 ) (2 1 )(2 3 ) 2 -4 2 3 3 1 1 -2 -1 (3 1 )(3 2 ) (3 1 )(3 2 ) (3 1 )(3 2 ) So, the second row of V-1 is (-3 4 -1). where a1 1 2 3 and a2 12 23 31 . Now, Also, 3 1 -6 11 2 a a a 1 1 1 1 2 1 1 3 -9 (1 2 )(1 3 ) (1 2 )(1 3 ) (1 2 )(1 3 ) 2 1 -3 2 a a a 1 V 1 2 1 2 2 2 1 3 0 (2 1 )(2 3 ) (2 1 )(2 3 ) (2 1 )(2 3 ) 2 1 0 2 a a a 1 3 1 3 2 3 1 ( )( ) ( )( ) ( )( ) 3 1 3 2 3 1 3 2 2 1 2 3 So, the third row of V-1 is (1 -3/2 1/2). Applying synthetic divisions to the [f (x) – a3] (x – 1), Hence, where f (x) (x )(x )(x ) x3 a x2 a x a , 3 5/ 2 1/ 2 1 2 3 1 2 3 we have V 1 3 4 1 . 1 1 a1 a2 1 3/ 2 1/ 2 2 1 1 + a11 2 1 1 + a1 1 + a11+a2 2 Example2. Find a quadratic interpolation polynomial which 1 1 -12-13 passes through the points (-1, 10), (1, 0) and (2, 4). 1 - - ( - )( - ) 1 2 3 1 2 1 3 since 2 2+ a +a - - = ( - )( - ). 1 1 1 2 1 2 1 3 1 2 1 3 Solution. Let the interpolation polynomial be: Next, s(x) a a x a x2. 2 1 a1 a2 1 2 3 2 2 2 + a12 Since s(-1)=10, s(1)=0, s(2)=4, so we have: 2 1 2+ a1 2 + a12+a2 2 2 2 -12-23 1 2-1-3 (2- 1)(2- 3)
ISBN: 978-988-14047-4-9 WCE 2017 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.
synthetic divisions to compute the elements of the inverse of 10 1 1 1 a1 VDM is comparatively less than that by applying direct 0 1 0 0a . 2 matrix multiplication to WA, it means this new approach 4 1 2 4 a 3 has less computational cost. More detail analysis of the Consider the following Vandermonde matrix: complexity of the synthetic division approach and its 1 1 1 applications to compute the inverse of the general VDM with V 1 0 2 . size nn will be our continued research topic and the findings will be presented or published elsewhere. 1 0 4 ACKNOWLEDGEMENT Let us define a polynomial as follows: The author would like to acknowledge that this research is 3 2 f (x) (x 1)(x 1)(x 2) x 2x x 2. supported by the Internal Research Grant awarded by Applying synthetic divisions, we have: EdUHK in 2016/17. -1 1 -2 -1 -1 3 REFERENCES 1 -3 2 [1] J. J. Rushanan, “On the Vandermonde matrix”, Amer. Math. Monthly, -1 4 vol. 96, pp, 921-924, 1989. 1 -4 6 [2] V.E. Neagoe, "Inversion of the Vandermonde matrix", IEEE Signal Processing Letters, vol. 3, pp. 119-120, 1996. -1 [3] A. Eisinberg and G. Fedele, “On the inversion of the Vandermonde So, the first row of V is (1/3 -1/2 1/6). matrix”, Appl. Math.andComput., vol. 174, pp. 1384-1397, 2006. Next, [4] M. Goldwurm and V. Lonati, “Pattern statistics and Vandermonde 1 1 -2 -1 matrices”, Theor. Comput. Sci.,vol. 356, pp. 153-169, 2006. [5] Y. K. Man, “A simple algorithm for computing partial fraction 1 -1 expansions with multiple poles,” Int. J. of Math. Educ. in Sci. and 1 -1 -2 Technol., vol. 38, pp. 247–251, 2007. 1 0 [6] H. Hakopian, K. Jetter and G. Zimmermann, “Vandermonde matrices 1 0 -2 for intersection points of curves”, J. Approx., vol. 1, 67-81, 2009. [7] Y. K. Man, “Partial fraction decomposition by synthetic division and its applications,” in Current Themes in Engineering Science 2008, A. So, the second row of V-1 is (1 1/2 -1/2). M. Korsunsky, Ed. New York: AIP, 2009, pp. 71–82. Also, [8] Y. K. Man, “A cover-up approach to partial fractions with linear or irreducible quadratic factors in the denominators,” Appl. Math. 2 1 -2 -1 andComput., vol. 219, pp. 3855–3862, 2012. 2 0 [9] A. M. Elhosary, N. Hamdy, I.A. Farag and A. E. Rohiem, “Optimum 1 0 -1 dynamic diffusion of block cipher based on maximum distance 2 4 separable matrices,” Int. J. Information & Network Security, vol. 2, pp. 327-332, 2013. 1 2 3 [10] Y. K. Man, “On the inversion of Vandermonde matrix via partial fraction decomposition,” in IAENG Transaction on Engineering So, the third row of V-1 is (-1/3 0 1/3). Sciences: Special Issue for the International Association of Engineers Conferences 2014, S. Ao, H.S. Chan, H. Katagiri and L. Xu Ed. Hence, Singapore: World Scientific, 2015, pp. 57–66. 1/ 3 1/ 2 1/ 6 V 1 1 1/ 2 1/ 2 1/ 3 0 1/ 3 Therefore, a 1/ 3 1 1/ 310 2 1 a2 1/ 2 1/ 2 0 0 5. a3 1/ 6 1/ 2 1/ 3 4 3
The required interpolation polynomial is s(x) 2 5x 3x2 .
V. CONCLUDING REMARKS In this paper, we have introduced a simple and efficient method for computing the inverse of VDM via synthetic division of polynomials, without having to involve matrix multiplication, computation of determinant or solving a system of linear equations for determining the elements of the inverse of VDM. Although we have discussed how to apply this new approach to the cases for n = 2 or 3 only, this approach can be easily adapted to deal with the inverse of VDM for n > 3 without much modifications. Also, it is not hard to see that the total number of arithmetic operations, namely multiplications and additions, involved in using
ISBN: 978-988-14047-4-9 WCE 2017 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)