To appear in International Journal of Computer Mathematics
A Note on Unimodular Eigenvalues for Palindromic Eigenvalue Problems∗ Chun-Yueh Chiang†
Eric King-wah Chu‡
Chang-Yi Weng§
Abstract We consider the occurrence of unimodular eigenvalues eigenvalue probPnfor palindromic i ∗ lems associated with the matrix polynomial Pn (λ) ≡ A λ where A i i = An−i with i=0 2 ∗ T H M ≡ M , M or P M P (P = I). From the properties of palindromic eigenvalues and their characteristic polynomials, we show that eigenvalues are not generically excluded from the unit circle, thus occurring quite often, except for the complex transpose case when Pn is complex and M ∗ ≡ M T . This behaviour is observed in numerical simulations and has important implications on several applications such as the vibration of fast trains, surface acoustic wave filters, stability of time-delay systems and crack modelling.
Keywords. palindromic eigenvalue problem, structure-preserving doubling algorithm, unimodular eigenvalue AMS subject classifications. 15A18, 15A22, 65F15
1
Introduction
We consider the palindromic eigenvalue problem (p-EVP) [3, 4] Pn (λ)x = 0 ,
x 6= 0
(1)
associated with the nth degree palindromic matrix polynomial Pn (λ) ≡
n X
A∗i = An−i ∈ Fn×n
Ai λ i ,
(2)
i=0
with the star operation ∗ defined as M ∗ ≡ M T , M H or P M P (P 2 = I) ∗ Version May 31, 2012; This research work is partially supported by a grant from the Faculty of Science, Monash University, Melboune, VIC, Australia. † Center for General Education, National Formosa University, Huwei 632, Taiwan;
[email protected] ‡ School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia; Phone: +61-399054480, FAX: +61-3-99054403;
[email protected] § School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia;
[email protected]
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2
E. K.-W. CHU & C.-Y. Chiang
Performing the star operation on Pn (λ), it is easy to deduce that the eigenvalues come in reciprocal pairs (λ, 1/λ∗ ) (with the degenerate 1 × 1 P = 1, and (0, ∞) considered to be a reciprocal pair). Palindromic eigenvalue problems occur in many important applications: (i) Vibration of fast trains [2], with ∗ = T , F = C (we refer to this type of problems as complex-T p-EVPs); (ii) Surface acoustic wave filters [3], with ∗ = T , F = C (with attenuation or damping; complexT p-EVPs); (iii) Surface acoustic wave filters, [3], with ∗ = T , F = R (without attenuation or damping; real-T p-EVPs); (iv) Time-delay systems [4], with M ∗ = P M P (P 2 = I), F = C (PCP-EVPs); (v) Crack modelling [3], with ∗ = T , F = R (real-T p-EVPs); and (vi) Computation of Crawford numbers [5], with ∗ = H, F = C (complex-H p-EVPs) These applications place different demands on the solution of the associated palindromic eigenvalue problems. For (i) and (v), eigenvalues in any part of the complex plane are of interest. For (ii), eigenvalues on or near the unit circle D are sought while (iii), the stability analysis in (iv) and (vi) require only eigenvalue on D. In addition, the structure-preserving doubling algorithm (SDA) [3, 4] has been applied, in different forms, to solve the p-EVP successfully. Theoretically, the presence of unimodular eigenvalues creates complications for the convergence of the SDA algorithm [1, 4]. Consequently, the occurrence of unimodular eigenvalues for p-EVPs has to be better understood. When creating random examples for testing numerical algorithms for p-EVPs, it has been noticed that unimodular eigenvalues occurred for real-T and complex-H p-EVPs as well as PCPEVPs quite often (see numerical simulations in Section 3), but almost never for complex-T pEVPs. (In fact it is very difficult to construct examples for complex-T p-EVPs with unimodular eigenvalues [3].) This note tries to explain this phenomenon. The result also has important implications on the applications seeking unimodular eigenvalues, which are well-posed for complex-H, real-T p-EVPs, or when M ∗ = P M P with P 2 = I. It will not be so for complex-T p-EVPs, or for PCP-EVPs with M ∗ = P M P (without the complex conjugate over M on the RHS). In other words, finding unimodular eigenvalues of p-EVPs arisen from (ii) is ill-posed and we have to look for eigenvalues nearby as well, while the similar problems from (iii), (iv) and (vi) are well-posed. It also explains, to some extent, the observation that SDAs behaves well even in the presence of (nearly) unimodular eigenvalues. Because of round-off errors, unimodular eigenvalues ˜ 1/λ ˜ ∗ ), pushing them further apart through the doubling process in a are perturbed to a pair (λ, structure-preserving manner and away from the theoretically problematic area D. In Section 2, we shall present our main results. Some numerical simulations are summarized in Section 3 and some concluding remarks are presented in Section 4.
2
Main Results
Lemma 2.1 For an arbitrary palindromic quadratic polynomial p2 (λ) = αλ2 + βλ + α with β ∈ R and α ∈ C, the roots will be unimodular if and only if the discriminant ∆ ≡ β 2 − 4|α|2 is nonpositive.
Unimodular Eigenvalues for Palindromic Eigenvalue Problems
3
Proof. From the usual formula for quadratics, we have √ −β ± ∆ λ= , ∆ ≡ β 2 − 4|α|2 2α √ When ∆ ≤ 0, |β| ≤ 2|α|, λ = (−β ± i −∆)/(2α) and we have β2 β2 |λ|2 = + 1 − = 1. (3) 4|α|2 4|α|2 √ Furthermore, when ∆ > 0, |β| > 2|α|, λ1,2 = (−β ± ∆)/(2α) satisfies |λ1 λ2 | = 1. If β < 0 then √ 4|α|2 √ β + ∆ = β− < 0 and ∆ √ √ β− ∆ β 2 − 2β ∆ + ∆ √ > 1. = |λ1 | = 4|α|2 β+ ∆ 2
Subsequently, we have |λ1 | > 1 (|λ2 | < 1) when β < 0 and |λ1 | < 1 (|λ2 | > 1) when β > 0.
Remark 2.1 For the quadratic q2 (λ) = αλ2 + βλ + α with β ∈ R and α ∈ C, the roots will be p −β ± β 2 − 4α2 λ= 2α with a complex discriminant. For such quadratics, the result in Lemma 2.1 will not hold, as the cancellation in (3) does not occur. Note that complex-T p-EVPs have characteristic polynomials with factors like q2 . Theorem 2.2 (Palindromic Factorization) An arbitrary palindromic polynomial pn (λ) of degree n n X pn (λ) = ai λi , ai = an−i (4) i=0
can be factored into the form: pn (λ)
=
m Y
(αi λ2 + βi λ + αi )
(n = 2m)
i=1
=
(α0 λ + α0 )
m Y
(αi λ2 + βi λ + αi )
(n = 2m + 1)
i=1
with βi ∈ R and αi ∈ C (i = 1, · · · , m). Proof. The result is a special case of the fundamental theorem of algebra, with the factors associated with (τ, 1/τ ) arranged into palindromic quadratic factors. For odd values of n, the palindromic nature of pn and the reciprocal nature of its roots imply at least one root on D, and the associated palindromic linear factor α0 λ + α0 = eiθ λ + e−iθ , exist. For quadratic factors, there are three possibilities. We may take out a pair of reciprocal roots not on D, or two identical or distinct roots on D:
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E. K.-W. CHU & C.-Y. Chiang (a) (λ = τ, τ −1 ; distinct, not on unit circle) (λ − τ )(τ λ − 1) = τ λ2 − (|τ |2 + 1)λ + τ with discriminant ∆ = (|τ |2 − 1)2 > 0; (b) (λ = τ = τ −1 ; double, on unit circle) (λ − τ )(τ λ − 1) = τ λ2 − 2λ + τ with discriminant ∆ = (|τ |2 − 1)2 = 0; and (c) (λ = −e−2iθ1 , −e−2iθ2 ; distinct with θ1 6= θ2 , both on unit circle) (e−iθ1 λ + eiθ1 )(e−iθ2 λ + eiθ2 ) = e−i(θ1 +θ2 ) λ2 + 2λ cos(θ1 + θ2 ) + ei(θ1 +θ2 ) with discriminant ∆ = 4[cos2 (θ1 − θ2 ) − 1] < 0.
All these possibilities give rise to palindromic quadratic factors like αλ2 + βλ + α with a real β. We next consider n n−1 X X ai λi = (eiθ λ + e−iθ ) bi λi i=0 iθ
n
iθ
= e bn−1 λ + (e bn−2 + e
−iθ
i=0
bn−1 )λ
n−1
+ · · · + (eiθ bn−i−1 + e−iθ bn−i )λn−i + · · ·
+(eiθ bi−1 + e−iθ bi )λi + · · · + (eiθ b0 + e−iθ b1 )λ + e−iθ b0 or n X i=0
ai λi = (αλ2 + βλ + α)
n−2 X
ci λi
i=0
= αcn−2 λn + (αcn−3 + βcn−2 )λn−1 + (αcn−4 + βcn−3 + αcn−2 )λn−2 + · · · +(αcn−i−2 + βcn−i−1 + αcn−i )λn−i + · · · + (αci−2 + βci−1 + αci )λi + · · · +(αc0 + βc1 + αc0 )λ2 + (βc0 + αc1 )λ + αc0 Comparing the above coefficients with ai = an−i , we can show easily that bi = bn−i , ci = cn−i and the (n − 2) degree polynomials defined by bi and ci are also palindromic. An inductive argument on the (n − 2) degree polynomials proves the palindromic factorization result.
Remark 2.2 Note that some palindromic quadratic factors in Theorem 2.2 are products of palindromic linear factors, as in Case (c) in the Proof of Theorem 2.2. Corollary 2.3 The roots of an arbitrary palindromic polynomial pn are not generically excluded from the unit circle. Furthermore, the probability of pn with n = 2m (or 2m + 1) possessing ` = 2, 4, · · · , 2m (or ` = 3, 5, · · · , 2m + 1) unimodular roots is nonzero.
Unimodular Eigenvalues for Palindromic Eigenvalue Problems
5
Proof. From the palindromic factors in Theorem 2.2, odd degree pn always have at least one eigenvalue on D. For even degree pn with n = 2m, application of Lemma 2.1 on individual palindromic quadratic factors shows that the probability of having ` = 2, 4 · · · , 2m unimodular eigenvalues is nonzero. The assertion in the Theorem follows.
Remark 2.3 For n = 2m + 1, we count from ` = 3 as pn always has at least one unimodular root. Corollary 2.4 The eigenvalues of the palindromic eigenvalue problem for the pencil Pn (λ) ≡
n X
Ai λ i
i=0
with A∗i = An−i or P Ai P = An−i , where P 2 = I and F 6= C when ∗ = T , are not generically excluded from the unit circle. Furthermore, the probability of Pn possessing (i) ` = 2, 4, · · · , n unimodular eigenvalues (with n = 2m), or (ii) ` = 3, 5, · · · , n unimodular eigenvalues (with n = 2m + 1), is nonzero. Proof. The p-EVPs all have the reciprocal pairs of eigenvalues (λ, 1/λ∗ ), thus the palindromic linear and quadratic factors in the associated characteristic polynomials pn . Theorem 2.2 and Corollary 2.3 then imply the results in the Corollary. Note that PCP-EVPs [4] with complex Q(λ) ≡ P rev(Q(λ)) P often have unimodular eigenvalues, but not those with Q(λ) ≡ P rev(Q(λ)) P .
3
Numerical Experiments
It is clear that the results in this note hold for an arbitrary Pn or its palindromic linearization λA + A∗ [5]. In this Section, we first consider the frequencies of occurrences for unimodular eigenvalues for 100 randomly generated palindromic linearizations, with A ∈ Cn×n for n = 10, 11, generated by the MATLAB [6] command: A=randn(10)+randn(10)*i; abs(eig(A,-A’)) or A=randn(11)+randn(11)*i; abs(eig(A,-A’)) The simulations are summarized in Table 1. The actual frequencies in Table 1 are not important as they only illustrate the result in Corollary 2.4 that unimodular eigenvalues occurs quite often for complex-H p-EVPs with reciprocal eigenvalue pairs (λ, 1/λ). Multiple unimodular eigenvalues have not been encountered, consistent with our finding that they occur only when the corresponding discriminants are zero and thus have zero probability of occurring in random. Similarly, the probability of ` unimodular eigenvalues occurring decreases with respect to large values of `, as the corresponding discriminants all have to be negative simultaneously. As mentioned before, these results and observations have
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E. K.-W. CHU & C.-Y. Chiang n = 10 # of Unimodular EWs 0 2 4 6 8 10 0–10
Frequency 39 28 28 4 1 0 100
n = 11 # of Unimodular EWs 1 3 5 7 9 11 1–11
Frequency 25 45 24 6 0 0 100
Table 1: Unimodular Eigenvalues from 100 Random H-Palindromic Linearizations important implications on some applications, such as the PCP-EVPs arising from time-delay systems in [4]. Changing the distribution in the random generation of A or changing the seed in randn change the actual frequencies but not the general phenomenon. Simulations for other p-EVPs with similar reciprocal eigenvalue pairs (λ, 1/λ∗ ) show similar results. In contrast, generating A by A=randn(10)+randn(10)*i; abs(eig(A,-transpose(A))) or A=randn(11)+randn(11)*i; abs(eig(A,-transpose(A))) produces random complex-T palindromic linearizations which have no (or one) unimodular eigenvalue for n = 10 and 11, respectively, throughout the 100 trials. Next, to speculate on `, the number of unimodular eigenvalues, we have repeated the simulations in Table 1 for n = 10 (five times), n = 100 (twice), and n = 40, 200, 1000 (once each). Similar to Table 1, each simulation involves 100 trials. The results are summarized in Table 2, with En (`) denoting the expected value of the number of unimodular eigenvalues for an n × n palindromic linearization. Consider the event Ei of the ith eigenpair being unimodular. If {Ei } are independent events of equal probabilities p, the probability P (k) of having k pairs of unimodular eigenvalues (k = 0, 1, · · · , m) obeys the binomial distribution B(m, p), with m P (k) = pk (1 − p)m−k k The mean of the distribution is mp. From the results for n = 10, 40, 100, 1000 in Table 2, we estimated the formulae for p and the expected number of unimodular eigenvalues En (`): r r 5 5n p= , En (`) = (5) 8n 8 √ (From the n = 10, 40, 100, 1000 results, we observed that En (l) is proportional to n and deduce the constant 5/8 from the n = 40 result.) The result has been verified by additional simulations
Unimodular Eigenvalues for Palindromic Eigenvalue Problems
` 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 En (`) Estimated p p from (5)
8 62 28 2
2.48 0.248 0.25
10 55 31 4
2.58 0.258 0.25
n = 10 15 54 28 3
2.38 0.238 0.25
7
n = 100 10 57 29 4
2.54 0.254 0.25
14 50 29 7
2.58 0.258 0.25
n = 1000
1 10 27 25 25 11
1 9 23 38 20 8
1
1
8 0.080 0.08
7.92 0.0792 0.08
n = 40 2 4 37 27 19 1
n = 200
5 0.125 0.125
11.28 0.0564 0.0559
1 4 17 29 21 14 12 2
2 11 10 14 11 14 11 10 13 3
1 25.32 0.02532 0.025
Table 2: Estimating p and En (`) with surprising accuracy (see the bottom of Table 2, in which only limited number of our simulations are displayed). We do not know how to prove the formulae in (5) but if they are valid as suggested by our simulations, En (`) will be small even for large values of n. This then implies great efficiency of the SDA in [4].
4
Concluding Remarks
In this note, we show that unimodular eigenvalues occurs frequently for complex p-EVPs with reciprocal eigenvalue pairs (λ, 1/λ) or real p-EVPs with eigenvalue pairs (λ, 1/λ). On the other hand, complex palindromic EVPs with eigenvalue pairs (λ, 1/λ) generically have no unimodular eigenvalues. The results have obvious implications in the solution of palindromic eigenvalue problems. In the analysis of train vibration [2] and the corresponding complex-T p-EVPs, it was difficult to construct examples with unimodular eigenvalues [3]. In surface acoustic wave filters with damping or attenuation [3], similar complex-T p-EVPs occur. However, we are interested in unimodular eigenvalues, making the problem ill-posed. Additional properties from the problem
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E. K.-W. CHU & C.-Y. Chiang
may make it well-posed but we may have to lower our target by considering nearly-unimodular eigenvalues. Similarly for the crack modelling problem [3], unimodular eigenvalues rarely occur and are easily perturbed away from the unit circle by computational and other errors. For the PCP-QEPs from time-delay systems [4], the search for unimodular eigenvalues is well-posed and the moderate number of unmodular eigenvalues predicted in this note implies that the SDA algorithm in [4] is efficient. For future work, unimodular eigenvalues are characterized here in terms of the palindromic factorizations of characteristic polynomials and Lemma 2.1. In most applications, this characterization will be inconvenient to check. Criteria in terms of the original matrix polynomial Pn will be more useful.
References [1] c.-y. chiang, e. k.-w. chu, c.-h. guo, t.-m. huang, w.-w. lin, and s.-f. xu, Convergence analysis of the doubling algorithm for several nonlinear matrix equations in the critical case, SIAM J. Matrix Anal. Appl., Vol. 31, pp. 227–247, 2009. [2] e.k.-w. chu, t.-m. hwang, w.-w. lin and c.-t. wu, Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms, J. Comput. Appl. Maths., Vol. 219, pp. 237–252, 2008. [3] e.k.-w. chu, t.-m. hwang, w.-w. lin and c.-t. wu, Palindromic eigenvalue problems: a brief survey, Taiwanese J. of Maths., Vol. 14, No. 3A, pp. 743-779, 2010. [4] t.-x. li, e.k.-w. chu and w.-w. lin,. A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems, J. Comput. Appl. Math., Vol. 233, pp. 1733–1745, 2010. [5] d.s. mackey, n. mackey, c. mehl and v. mehrmann, Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations , SIAM J. Matrix Anal. Appl., Vol. 28, pp. 1029–1051, 2006. [6] mathworks, MATLAB User’s Guide, 2002.
