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A Note on the Inversion of Sylvester Matrices in Control Systems Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2011, Article ID 609863, 10 pages doi:10.1155/2011/609863 Research Article A Note on the Inversion of Sylvester Matrices in Control Systems Hongkui Li and Ranran Li College of Science, Shandong University of Technology, Shandong 255049, China Correspondence should be addressed to Hongkui Li, [email protected] Received 21 November 2010; Revised 28 February 2011; Accepted 31 March 2011 Academic Editor: Gradimir V. Milovanovic´ Copyright q 2011 H. Li and R. Li. 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. We give a sufficient condition the solvability of two standard equations of Sylvester matrix by using the displacement structure of the Sylvester matrix, and, according to the sufficient condition, we derive a new fast algorithm for the inversion of a Sylvester matrix, which can be denoted as a sum of products of two triangular Toeplitz matrices. The stability of the inversion formula for a Sylvester matrix is also considered. The Sylvester matrix is numerically forward stable if it is nonsingular and well conditioned. 1. Introduction Let Rx be the space of polynomials over the real numbers. Given univariate polynomials ∈ f x ,g x R x , a1 / 0, where n n−1 ··· m m−1 ··· f x a1x a2x an1,a1 / 0,gx b1x b2x bm1,b1 / 0. 1.1 Let S denote the Sylvester matrix of f x and g x : ⎫ ⎛ ⎞ ⎪ ⎪ a a ··· ··· a ⎪ ⎜ 1 2 n 1 ⎟ ⎬ ⎜ ··· ··· ⎟ ⎜ a1 a2 an1 ⎟ m row ⎜ ⎟ ⎪ ⎜ .. .. .. ⎟ ⎪ ⎜ . ⎟ ⎭⎪ ⎜ ⎟ ⎜ a a ··· ··· a ⎟ ⎜ 1 2 n 1 ⎟ S ⎜ ··· ⎟ ⎫ 1.2 ⎜b1 b2 bm bm1 ⎟ ⎪ ⎜ ⎟ ⎪ ⎜ b b ··· b b ⎟ ⎪ ⎜ 1 2 m m 1 ⎟ ⎬ ⎜ ⎟ ⎝ .. .. .. .. ⎠ n row . ⎪ ⎪ b1 b2 ··· bm bm1 ⎭⎪ 2 Mathematical Problems in Engineering Sylvester matrix is applied in many science and technology fields. The solutions of Sylvester matrix equations and matrix inequations play an important role in the analysis and design of control systems. In determining the greatest common divisor of two polynomials, the Sylvester matrix plays a vital role, and the magnitude of the inverse of the Sylvester matrix is important in determining the distance to the closest polynomials which have a common root. Assuming that all principal submatrices of the matrix are nonsingular, in 1,JingYang et al. have given a fast algorithm for the inverse of Sylvester matrix by using the displacement structure of m n-order Sylvester matrix. By using the displacement structure of the Sylvester matrix, in this paper, we give asufficient condition the solvability of two standard equations of Toeplitz matrix, and, according to the sufficient condition, we derive a new fast algorithm for the inversion of a Sylvester matrix, which can be denoted as a sum of products of two triangular Toeplitz matrices. At last, the stability of the inversion formula for a Sylvester matrix is also considered. The Sylvester matrix is numerically forward stable if it is nonsingular and well conditioned. · · In this paper, 2 always denotes the Euclidean or spectral norm and F the Frobenius norm. 2. Preliminary Notes In this section, we present a lemma that is important to our main results. Lemma 2.1 ∈ n,n ∈ see 2,Section2.4.8 . Let A, B and α . Then for any floating-point arithmetic with machine precision ε, one has that √ ≤ | | ≤ | | fl αA αA E, E F ε α A F ε n α A 2, √ ≤ ≤ fl A B A B E, E F ε A B F ε n A B 2, 2.1 ≤ fl AB AB E, E F εn A F B F. As usual, one neglects the errors of Oε2,Oε2,andOεε. 3. Sylvester Inversion Formula In this section, we present our main results. Theorem 3.1. Let matrix S be a Sylvester matrix; then it satisfies the formula T T KS − SK emf − emng , 3.1 where K 0 e1 e2 ··· emn−1 is a m n × m n shift matrix, T T f b1 ··· bm bm1 − a1 ··· −an ,g 0 ··· 0 b1 ··· bm . 3.2 Mathematical Problems in Engineering 3 Proof. We have that ⎛ ⎞ ⎛ ⎞ 0 a1 ··· an1 0 ··· 0 0 a1 ··· an1 0 ··· 0 ⎜ ⎟ ⎜ ⎟ ⎜ ··· ··· ⎟ ⎜ ⎟ ⎜ a1 an1 0⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ KS − SK ⎜b ··· b 0 ··· 0⎟ − ⎜0 ··· a ··· a ⎟ ⎜ 1 m 1 ⎟ ⎜ 1 n ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ b ··· b 0 ··· 0⎟ ⎜ b ··· b ⎟ ⎜ 1 m 1 ⎟ ⎜ 1 m 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ ⎝ . ⎠ 0 ··· 0 0 ··· 0 b1 ··· bm ⎛ ⎞ 0 ··· 0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 ··· 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜b ··· b b − a −a ··· −a ⎟ ⎜ 1 m m 1 1 2 n ⎟ ⎜ ⎟ ⎜ 0 ··· 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎝ 0 .0. ⎠ 0 ··· −b1 ··· −bm ⎛ ⎞ 0 ··· 0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 ··· 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ T T ⎜ ⎟ e f − e g ⎜b ··· b b − a −a ··· −a ⎟. m m n ⎜ 1 m m 1 1 2 n ⎟ ⎜ ⎟ ⎜ 0 ··· 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎝ 0 .0. ⎠ 0 ··· −b1 ··· −bm 3.3 T T So KS − SK emf − emng . T Theorem 3.2. Let matrix S be a Sylvester matrix and x x1 x2 ··· xmn , y T T T y1 y2 ··· ymn , μ μ1 μ2 ··· μmn ,andV V1 V2 ··· Vmn the solutions of the T T systems of equations Sx em, Sy emn, S μ f,andS V g, respectively, where em and emn are both m n × 1 vectors; then −1 a S is invertible, and the column vector wj j 1, 2,...,mn of S satisfies the recurrence relation wmn y, 3.4 wi−1 Kwi μix − Viy, i m n,...,3, 2, 4 Mathematical Problems in Engineering b −1 S S1U1 S2U2 ⎛ ⎞ ··· ⎛ ⎞ ymn ymn−1 y2 y1 1 ⎜ ⎟⎜ ⎟ ⎜ . ⎟⎜− ⎟ ⎜ . ⎟⎜ Vmn 1 ⎟ ⎜ ymn . y3 y2 ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ . ⎟ ⎜ .. .. ⎟⎜ . .. ⎟ ⎜ . ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ −V −V ··· 1 ⎠ ⎝ ymn ymn−1⎠ 3 4 −V −V ··· −V 1 ymn 2 3 m n 3.5 ⎛ ⎞ ··· ⎛ ⎞ xmn xmn−1 x2 x1 0 ⎜ ⎟⎜ ⎟ ⎜ . ⎟⎜ ⎟ ⎜ . ⎟⎜μmn 0 ⎟ ⎜ xmn . x3 x2 ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ . ⎟ ⎜ .. .. ⎟⎜ . .. ⎟. ⎜ . ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ μ μ ··· 0 ⎠ ⎝ xmn xmn−1⎠ 3 4 μ μ ··· μ 0 xmn 2 3 m n Proof. By Theorem 3.1 Sx em and Sy emn,wehavethat T T T T T T KS SK emf − emng SK Sxf − Syg S K xf − yg , 3.6 so − KiS Ki 1S K xfT − ygT − Ki 2KS K xfT − ygT − 2 Ki 2S K xfT − ygT 3.7 . i S K xfT − ygT . Hence, we have that i i i T T K emn K Sy S K xf − yg y, i 0, 1,..., m n − 1. 3.8 Let i T T wmn−i K xf − yg y, i 0, 1,..., m n − 1. 3.9 Mathematical Problems in Engineering 5 i It is easy to see that K emn emn−i,andby3.8 i Swmn−i K emn emn−i,i 0, 1,..., m n − 1. 3.10 From Sy emn,wehavethatwmn y.LetX w1 w2 ··· wmn;then SX Sw1 w2 ··· wmn Sw1 Sw2 ··· Swmn Imn, 3.11 so the matrix S is invertible and the inverse of S is the matrix X. From 3.1,wehavethat −1 −1 −1 T T −1 S KS − SKS S emf − emng S , 3.12 and thus − − S 1K − KS 1 xμT − yV T . 3.13 So −1 −1 T T S K − KS ei xμ − yV ei. 3.14 Since Kei ei−1,wehavethat wi−1 Ksi μix − Viy, i m n,...,3, 2, 3.15 and hence wmn y, 3.16 wi−1 Kwi μix − Viy, i m n,...,3, 2. For b,by3.4 wmn y, wmn−1 Ky μmnx − Vmny, 2 wmn−2 K y Kμmnx − KVmny μmn−1x − Vmn−1y, 3.17 ··· mn−1 mn−2 mn−2 w1 K y K μmnx − K Vmny ··· μ2x − V2y. 6 Mathematical Problems in Engineering So −1 S w1 w2 ··· wmn ⎛ ⎞ 1 ⎜ ⎟ ⎜− ⎟ ⎜ Vmn 1 ⎟ ⎜ ⎟ ⎜ ⎟ mn−1 mn−2 ··· ⎜ . ⎟ K yK y Ky y ⎜ . .. ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ −V3 −V4 ··· 1 ⎠ −V2 −V3 ··· −Vmn 1 ⎛ ⎞ 0 ⎜ ⎟ ⎜ ⎟ ⎜μmn 0 ⎟ ⎜ ⎟ ⎜ ⎟ mn−1 mn−2 ··· ⎜ . ⎟ K xK x Kx x ⎜ . .. ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ μ3 μ4 ··· 0 ⎠ μ μ ··· μ 0 2 3 m n 3.18 ⎛ ⎞ ··· ⎛ ⎞ ymn ymn−1 y2 y1 1 ⎜ ⎟⎜ ⎟ ⎜ . ⎟⎜− ⎟ ⎜ . ⎟⎜ Vmn 1 ⎟ ⎜ ymn . y3 y2 ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ . ⎟ ⎜ .. .. ⎟⎜ . .. ⎟ ⎜ . ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ −V −V ··· 1 ⎠ ⎝ ymn ymn−1⎠ 3 4 −V −V ··· −V 1 ymn 2 3 m n ⎛ ⎞ ··· ⎛ ⎞ xmn xmn−1 x2 x1 0 ⎜ ⎟⎜ ⎟ ⎜ . ⎟⎜ ⎟ ⎜ . ⎟⎜μmn 0 ⎟ ⎜ xmn . x3 x2 ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ . ⎟ ⎜ .. .. ⎟⎜ . .. ⎟. ⎜ . ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ μ μ ··· 0 ⎠ ⎝ xmn xmn−1⎠ 3 4 μ μ ··· μ 0 xmn 2 3 m n This completes the proof. 4. Stability Analysis In this section, we will show that the Sylvester inversion formula presented in this paper is evaluation forward stable. If, for all well conditioned problems, the computed solution x is close to the true − solution x, in the sense that the relative error x x 2/ x 2 is small, then we call the algorithm forward stable the author called this weakly in 3.Round-off errors will occur in the matrix computation. Mathematical Problems in Engineering 7 Theorem 4.1. Let matrix S be a nonsingular Sylvester matrix and well conditioned; then the formula in Theorem 3.2 is forward stable. Proof. Assume that we have computed the solutions x, y, μ,andV of Sx em, Sy emn, ST μ f,andST V g in Theorem 3.2 which are perturbed by the normwise relative errors bounded by ε, ≤ ≤ ≤ ≤ x 2 x 2 1 ε , y y 1 ε , μ μ 1 ε , V V 2 1 ε . 2 2 2 2 2 4.1 Thus, we have that √ √ ≤ ≤ S1 F m n y 2, S2 F m n x 2, √ √ 4.2 ≤ 2 ≤ . U1 F m n 1 V 2, U2 F m n μ 2 The inversion formula in Theorem 3.2, using the perturbed solutions x, y, μ,andV , can be expressed as −1 S fl S1U1 S2U2 flS1 ΔS1U1 ΔU1 S2 ΔS2U2 ΔU2 4.3 −1 S ΔS1U1 S1ΔU1 ΔS2U2 S2ΔU2 E F. Here, and in the sequel, E is the matrix containing the error which results from computing the matrix products and F contains the error from subtracting the matrices.
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