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Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 782057, 11 pages doi:10.1155/2011/782057 Research Article Annular Bounds for Polynomial Zeros and Schur Stability of Difference Equations Ke Li1 and Jin Liang2 Shandong Key Laboratory of Automotive Electronic Technology, Institute of Automation, Shandong Academy of Sciences, Jinan, Shandong 250014, China Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China Correspondence should be addressed to Jin Liang, jinliang@sjtu.edu.cn Received October 2010; Accepted 30 October 2010 Academic Editor: Toka Diagana Copyright q 2011 K Li and J Liang 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 investigate the monic complex-coefficient polynomial of degree n, f z : zn an−1 zn−1 · · · a0 in the complex variable z and obtain a new annular bound for the zeros of f z, which is sharper than the previous results and has clear advantages in judging the Schur stability of difference equations In addition, examples are given to illustrate the theoretical result Introduction It is well known that many discrete-time systems in engineering are described in terms of a difference equation, and the characteristic equation for the difference equation plays a key role in the study of the behaviors of the solutions, especially the stability of the solutions, to the discrete-time systems Since the characteristic equations for difference equations are closely related to some complex polynomials, the estimates of the bound for the moduli of various complex polynomial zeros have been investigated by many researchers cf e.g., 1– and references therein In the study on this issue, one of meaningful research ideas is to indicate such a common property of a lot of polynomials by a few very special polynomials Using this idea, a good annular bound by estimating the largest nonnegative zeros of four specific polynomials is given in 8 recently As a continuation of this work and our paper 4 , in this paper we investigate further the location of the zeros of complex-coefficient polynomials on the basis of such a research idea and establish a new annular bound theorem Theorem 3.1, which improves the previous corresponding result and has clear advantages in judging the Schur stability of difference equations Examples are given to illustrate the advantages of the new result 2 Advances in Difference Equations Preliminaries Throughout this paper, we let 2.1 fz : zn an−1 zn−1 an−2 zn−2 · · · a1 z a0 with ∈ C, i ∈ {0, 1, 2, , n − 1}, and gz : −1n fzf−z z2n b2n−2 z2n−2 b2n−4 z2n−4 · · · b2 z2 b0 2.2 ≡ 0, or, equivalently, b0 / ≡ Without losing the generality, we assume that a0 / Basic notations are as follows R : {x ∈ R | x < 0}, |z|: the modulus of a complex number z, Zfz : the set of all zeros of fz, Ar, R : {z ∈ C | r ≤ |z| ≤ R} with ≤ r ≤ R, l: the smallest positive integer such that al / in fz, k: the largest positive integer such that ak / in fz, q: the smallest positive integer such that b2q / in gz, p: the largest positive integer such that b2p / in gz, m : the integer part of a real number m In order to simplify the expressions in our study, we define specially that s 2.3 yi : it for any positive integers s, t s < t, and sequence {yi ∈ C : s ≤ i ≤ t} This notation is logical and useful in the note Moreover, we write c1 x : xnl nl−1 |ai−l |xi in1 n 2l−1 |a0 | |a0 |2 i ai−l − a0 xi − x al |al | |al | i2l il1 2.4 with an and ≤ l ≤ n/2 , c2 x : x2n−k − n−1 |aik−n |xi − ik1 k in−k |aik−n − ak |xi − n−k−1 i0 |ai ak |xi 2.5 Advances in Difference Equations with n 1/2 ≤ k ≤ n − 1, d1 x : x 2n2q 2q−1 n 2i b0 b2i 2i |b0 b2i | 2i x − b2i−2q x b2i−2q − x b2q i2q in1 iq1 b2q nq−1 |b |2 b2q 2.6 with b2n and ≤ q ≤ n/2 , d2 x : x4n−2p − p n−p−1 n−1 b2ip−n x2i − b2ip−n − b2i b2p x2i − b2i b2p x2i ip1 in−p 2.7 i0 with n 1/2 ≤ p ≤ n − 1, f1 x : xn n−1 |ai |xi − |a0 |, f2 x : xn − k |ai |xi , i0 il 2.8 g1 x : x2n n−1 |b2i |x2i − |b0 |, g2 x : x2n − iq p |b2i |x2i i0 Remark 2.1 By Descartes’ rule of signs, it is easy to see that for each i ∈ {1, 2}, the polynomial ci xdi x, fi x, gi x has a unique positive zero We denote by αi , βi , γi , and δi the unique positive zero of ci x, di x, fi x, and gi x, respectively Main Result The following result is established in 8 Theorem A see 8 Zfz ⊂ Au, v , with : max{γ1 , δ1 } and v : min{γ2 , δ2 } Theorem 3.1 Let ≤ l, q ≤ n/2 , and n 1/2 ≤ k, p ≤ n − Then i Z fz ⊂ Ar, R , 3.1 where r : max{α1 , β1 } and R : min{α2 , β2 } ii Ar, R ⊆ Au, v , 3.2 where u, v are constants as in Theorem A; iii the annular bound of original polynomial fz can be further improved by iterative procedure 4 Advances in Difference Equations Proof Define a0 l cz : fz z − al znl − znl n−1 n−1 a2 a0 i a0 n z zil − z a0 zl − al al al il il nl−1 ai−l zi − i2l znl nl−1 ai−l zi in1 znl nl−1 ai−l zi in1 cz : fz zn−k − ak z2n−k k−1 n a0 i a20 z − al al il1 n n a0 a2 a0 i 2l−1 ai−l zi − z − zi − al al al i2l i2l il1 n z zin−k − k n−1 z i0 aik−n zi ik1 3.4 k aik−n z − ak zi b0 b2q dz : gz z2q − 2n2q ak zi i in−k z2n−k a0 a2 a0 i 2l−1 z − zi − , al al al il1 i0 n−1 ai−l − i2l i0 2n−k 3.3 nq−1 k aik−n − ak zi − b2i−2q z 2i n i2q in1 ak zi , i0 in−k n−k−1 b0 b2i b2i−2q − b2q z2i − 2q−1 b0 b2i 2i b02 z − , b b2q iq1 2q dz : gz z2n−2p − b2p z4n−2p n−1 ip1 b2ip−n z2i 3.5 p n−p−1 b2ip−n − b2i b2p z2i − b2i b2p z2i , in−p i0 where an b2n Then it is not difficult to see that Z fz ⊆ Z cz ∩ Zcz ∩ Z dz ∩ Z dz 3.6 This implies that for every w ∈ Zfz we have cw cw dw dw 0, 3.7 Advances in Difference Equations that is, wnl nl−1 n in1 i2l ai−l wi n−1 w2n−k aik−n wi ik1 w2n2q nq−1 w4n−2p n−1 n i2q b2ip−n w2i ip1 k 2l−1 a0 a2 a0 wi − wi − 0, al al al il1 aik−n − ak wi − n−k−1 ak wi 0, i0 in−k b2i−2q w2i in1 ai−l − b0 b2i b2i−2q − b2q 2q−1 b0 b2i 2i b02 w − 0, w2i − b b2q iq1 2q 3.8 p n−p−1 b2ip−n − b2i b2p w2i − b2i b2p w2i in−p i0 Hence, by 3.8, one has nl−1 n |a0 | i a0 i 2l−1 |a0 |2 nl i ≤ |w| |w| , |ai−l ||w| ai−l − |w| al |al | |al | in1 i2l il1 |w|2n−k ≤ n−1 |aik−n ||w|i ik1 k |aik−n − ak ||w|i n−k−1 |ai ak ||w|i , i0 in−k nq−1 2q−1 n 2i b0 b2i 2i |b0 b2i | 2i |b0 |2 2n2q ≤ |w| |w| , b2i−2q |w| b2i−2q − |w| b2q b2q in1 i2q iq1 b2q 3.9 p n−p−1 n−1 b2ip−n |w|2i b2ip−n − b2i b2p |w|2i b2i b2p |w|2i , |w|4n−2p ≤ ip1 in−p i0 which imply that Z fz ⊂ {z ∈ C : c1 |z| ≥ 0, d1 |z| ≥ 0, c2 |z| ≤ 0, d2 |z| ≤ 0} In addition, it follows from 2.4–2.7 that c1 x < 0, c1 x ≥ 0, d1 x < 0, d1 x ≥ 0, ∀x ∈ 0, α1 , ∀x ∈ α1 , ∞, ∀x ∈ 0, β1 , ∀x ∈ β1 , ∞ , 3.10 Advances in Difference Equations c2 x ≤ 0, ∀x ∈ 0, α2 , ∀x ∈ α2 , ∞, d2 x ≤ 0, ∀x ∈ 0, β2 , d2 x > 0, ∀x ∈ β2 , ∞ c2 x > 0, 3.11 Therefore, for each w ∈ Zfz we have |w| ≥ β1 , |w| ≥ α1 , |w| ≤ α2 , |w| ≤ β2 , 3.12 which imply that 3.1 is hold So i is proved Next we prove that ii holds Actually, we have nl−1 n |a0 | i |a0 |2 a0 i 2l−1 c1 γ1 γ1nl γ γ − |ai−l |γ1i ai−l − al il1 |al | |al | in1 i2l |a0 | − n−1 |ai |γ1i γ1l − nl−1 |ai−l |γ1i γ1n nl−1 |ai−l |γ1i in1 i2l |a0 | − |al | |ai−l |γ1i in1 il |a0 |γ1l nl−1 n−1 |ai |γ1i n 2l−1 |a0 | |a0 |2 i ai−l − a0 γ i − γ al il1 |al | |al | i2l n |a0 | a0 i 2l−1 ai−l − γi γ al il1 |al | i2l 3.13 il n n n |a0 | a0 i 2l−1 |a0 | i i − |ai−l |γ1i ai−l − − γ γ γ a |a | |al | l l i2l i2l il1 il1 n a0 a0 i − − ai−l − γ |ai−l | al al i2l ≤ 0, where an and ≤ l ≤ n/2 On the other hand, since the polynomial equation c1 x has a unique positive root α1 and c1 x ≤ 0, c1 x > 0, ∀x ∈ 0, α1 , ∀x ∈ α1 , ∞, we get α1 ≥ γ1 by combining3.13 and 3.14 3.14 Advances in Difference Equations In addition, we have n−1 k n−k−1 c2 γ2 γ22n−k − |aik−n |γ2i − |aik−n − ak |γ2i − |ai ak |γ2i ik1 k |ai |γ2in−k − i0 n n−1 |aik−n |γ2i − ik1 |aik−n |γ2i − in−k n−1 k k |aik−n − ak |γ2i − k |aik−n |γ2i − |aik−n |γ2i − k |aik−n − ak |γ2i − k n−k−1 |ai ak |γ2i i0 |aik−n − ak |γ2i − n−k−1 |ai ak |γ2i − |a0 ||ak | i1 in−k |aik−n | − |aik−n − |ai ak |γ2i i0 ak |γ2i − n−k−1 |ai ak |γ2i − γ2n i1 in−k n−k−1 in−k in−k |ak |γ2n k in−k ik1 |ak |γ2n i0 in−k 3.15 k i − |ai |γ2 |ak | i1 |aik−n | |ai ak | − |aik−n − ak |γ2i in−k ≥ Since c2 x < 0, ∀x ∈ 0, α2 , 3.16 ∀x ∈ α2 , ∞, c2 x ≥ 0, we have α2 ≤ γ2 In the same way, we can obtain β1 ≥ δ1 and β2 ≤ δ2 ; therefore, Ar, R ⊆ Au, v 3.17 Finally, we prove iii Set 1 1 1 c1 z : cz znl anl−1 znl−1 · · · al1 zl1 a0 , 3.18 with 1 ⎧ a , ⎪ ⎪ ⎪ i−l ⎪ ⎪ ⎪ ⎪ a0 ⎪ ⎪ , a − ⎪ ⎪ ⎨ i−l al a0 ⎪ ⎪ − , ⎪ ⎪ al ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a20 ⎪ ⎩− , al n ≤ i ≤ n l − 1; 2l ≤ i ≤ n; l ≤ i ≤ 2l − 1; i 0, 3.19 Advances in Difference Equations 1 and let l1 be the smallest positive integer such that al1 / in c1 z If l ≤ l1 ≤ n l/2 , in analogy to 3.3 and 2.4, we can define ⎛ c2 z : c1 z⎝zl1 − 1 a0 1 ⎞ ⎠ 3.20 al1 2 and c1 x, respectively It is not difficult to see that, the unique positive root of polynomial 2 2 2 2 2 c1 x, α1 ≥ α1 Similarly, we can define c2 x, d1 x, and d2 x, respectively Moreover, 2 2 2 their respective positive roots α2 , β1 , and β2 satisfy that 2 2 α2 ≤ α2 , β1 ≥ β1 , 2 β2 ≤ β2 3.21 Consequently, new annular bound of fz, namely, Ar 2 , R2 with 2 2 r 2 : max α1 , β1 , 2 2 R2 : α2 , β2 , 3.22 is better than 3.1 This procedure can be applied iteratively 2 c1 x, 2 d1 x, 3 d1 x, 3 c2 x, 2 d2 x 2 3.23 3 d2 x, 3 3.24 3 d2 x 3 3.25 4 d2 x, 4 3.26 can be further transformed into 3 c2 x, 3 c2 x, d1 x, 4 d1 x, c1 x, respectively, and c1 x, into 4 c1 x, c2 x, until the last iteration brings no practical improvement Obviously, when m increases, m m , r m : max α1 , β1 m m Rm : α2 , β2 3.27 will approach the smallest and largest modulus of polynomial zero, respectively, where m α1 m m m resp α2 , β1 , β2 3.28 Advances in Difference Equations denotes the unique positive root of m m m m c1 x resp c2 x, d1 x, d2 x 3.29 This means that iii is true Remark 3.2 a When c2 r > 0, it follows from 3.14 and 3.15 that for every w ∈ Zfz , |w| ≤ α2 < r, that is, Zfz ⊂ Br, that is, fz is r-stable c Similarly, we can draw the same conclusion when d2 r > 0, and Zfz ⊂ B r when c1 r < or d1 r < b By the similar arguments in the proof of iii of Theorem 3.1, the results in a can be improved This also provides an iterative algorithm to test the r-stability and Schur stability of polynomials c The question “What happens to Theorem 3.1 when n − ≥ l, q > n/2 , and ≤ k, p < n 1/2 ?” is worth considering further Example 3.3 Let fz z3 j z2 2jz 1, where j √ 3.30 −1 By Theorem 3.1, we obtain Z fz ⊂ A0.389, 1.647 3.31 If we start the iterative procedure given in the proof of iii of Theorem 3.1, after five iterations, we obtain Z fz ⊂ A0.390, 1.644 3.32 On the other hand, by Theorem A, one only can have Z fz ⊂ A0.387, 1.938 3.33 The following examples show the advantages of Theorems 3.1 over Theorem A in analyzing the Schur stability of difference equations discrete-time systems Example 3.4 Let the characteristic polynomial of a difference equation discrete-time system be given by √ fz z j z2 √ √ j z− − 5j , 3.34 10 Advances in Difference Equations √ where j −1 Then by Theorem 3.1, we get c2 1 7/16 > 0, which implies that all zeros of fz lie in the open unit disk, that is, this system is Schur stable However, by Theorem A, one has Z fz ⊂ A0.638, 1.175 3.35 So Theorem A cannot guarantee the stability of such a system Example 3.5 Suppose the characteristic polynomial of a difference equation discrete-time system is given by fz z3 11 1 j z2 − −j z− j , 3.36 √ where j −1 Then by Theorem 3.1, we have c1 1 −9/16 < 0, which implies that all zeros of fz are outside the open unit disk, namely, such a system is instable By Theorem A, one has Z fz ⊂ A0.824, 1.517 , 3.37 which cannot determine the instability of this system Example 3.6 Consider the following characteristic polynomial of a difference equation discrete-time system: fz z4 2z3 2z2 z √ 11 − 3.38 In this example, 1 c2 −6.316, 1 d2 0; 2 c2 −7.584, 2 d2 0.203 3.39 Consequently, such a difference equation discrete-time system is Schur stable Acknowledgments This work was supported partially by the NSF of China 10771202 and the Specialized Research Fund for the Doctoral Program of Higher Education of China 2007035805 References 1 F G Boese and W J Luther, “A note on a classical bound for the moduli of all zeros of a polynomial,” IEEE Transactions on Automatic Control, vol 34, no 9, pp 998–1001, 1989 2 G T Cargo and O Shisha, “Zeros of polynomials and fractional order differences of their coefficients,” Journal of Mathematical Analysis and Applications, vol 7, pp 176–182, 1963 3 R B Gardner and N K Govil, “On the location of the zeros of a polynomial,” Journal of Approximation Theory, vol 78, no 2, pp 286–292, 1994 Advances in Difference Equations 11 4 J Liang and K Li, “A new stability criterion for polynomials with the quasi-critical condition,” International Journal of Nonlinear Sciences and Numerical Simulation, vol 9, no 3, pp 289–292, 2008 5 M Marden, Geometry of Polynomials, Mathematical Surveys, No 3, American Mathematical Society, Providence, RI, USA, 2nd edition, 1966 6 G V Milovanovic, D S Milovanovic, and T M Rassias, Topics in Polynomials, Extremal Problems, Inequalities, Zeros, World Scientic, Singapore, 1994 7 Q I Rahman and G Schmeisser, Analytic Theory of Polynomials, Oxford University Press, Oxford, UK, 2005 8 Y.-J Sun, “New result for the annular bounds of complex-coefficient polynomial zeros,” IEEE Transactions on Automatic Control, vol 49, no 5, pp 813–814, 2004
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