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Directional bounds for polynomial zeros and eigenvalues
J. Math. Anal. Appl. 482 (2020) 123571
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Directional bounds for polynomial zeros and eigenvalues A. Melman Department of Applied Mathematics, School of Engineering, Santa Clara University, Santa Clara, CA 95053, United States of America
a r t i c l e
i n f o
Article history: Received 17 May 2019 Available online 3 October 2019 Submitted by V. Andrievskii Keywords: Polynomial Zero Eigenvalue Cauchy Pellet Directional bound
a b s t r a c t Directional versions of theorems by Cauchy and Pellet are obtained for both scalar and matrix polynomials, which derive bounds on the zeros of scalar polynomials and eigenvalues of matrix polynomials that depend on their arguments. Such bounds often deliver much better results than the standard theorems, which are formulated in terms of the moduli of the zeros or eigenvalues only and are therefore insensitive to their arguments. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The development of fast and reliable methods to compute the zeros of a polynomial, may, at first sight, appear to have made the many approximations and localization results for such zeros less useful, in particular the more specialized ones. However, these approximations become considerably more valuable if they can be generalized to matrix polynomials, in which case they can provide location information for polynomial eigenvalues, which are much harder to compute than polynomial zeros. Examples of results that were generalized can be found, e.g., in [3], [7], [10], and [11]. A large supply of (scalar) polynomial results can be found in the encyclopedic works [12] and [14], including several that seem to be new. One of the latter motivates the present work, namely, Theorem 8.2.4 in [14], which derives an upper bound on the modulus of a polynomial zero in terms of its argument. As such, it can be considered a directional version of a bound due to Cauchy ([5]), which is, itself, a limit case of a theorem due to Pellet ([13]). This naturally leads to the idea of deriving a directional version of the more general theorem, and we begin by doing precisely that. Pellet’s theorem can sometimes detect a gap between the moduli of groups of zeros, but often fails to do so when the moduli of just two of the zeros, one in each group, are close, so that a directional version of the E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.123571 0022-247X/© 2019 Elsevier Inc. All rights reserved.
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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theorem has a much better chance of detecting gaps. We then proceed to generalize both this theorem and an extension of Theorem 8.2.4 in [14] to matrix polynomials. Directional results of this kind for polynomial eigenvalues do not seem to be commonly encountered in the literature. The paper is organized as follows. In Section 2 we collect a few basic results that will be used throughout the paper in an effort to make it reasonably self-contained. In Section 3, the results for scalar polynomials are derived, while Section 4 is devoted to matrix polynomials. 2. Preliminaries In this section we summarize a few basic results that will be needed later on. We begin with the aforementioned bound by Cauchy. Theorem 2.1. ([5], [8, Th. (27,1), p. 122 and Exercise 1, p. 126]). All the zeros of the polynomial p(z) = n j j=0 aj z of degree n with complex coefficients lie in the disk defined by |z| ≤ ρ, where ρ is the unique n−1 positive solution of |an |z n − j=0 |aj |z j = 0. If we apply this theorem to the reverse polynomial p# (z) = z n p(1/z) (for a0 = 0), whose zeros are the reciprocals of those of p, we obtain that the polynomial p cannot have zeros with a modulus in the interval [0, σ), where σ is the unique positive solution of |an |z n + |an−1 |z n−1 + · · · + |a1 |z − |a0 | = 0 .
(1)
Theorem 2.1 can be considered as a limit case of Pellet’s theorem, which we state next. n Theorem 2.2. ([13], [8, Th. (28,1), p. 128]) Let p(z) = j=0 aj z j be a polynomial of degree n ≥ 2 with comn plex coefficients and a = 0 for some , with 1 ≤ ≤ n − 1, and let the polynomial −|a |z + j=0,j= |aj |z j have two distinct positive roots ρ1 and ρ2 with ρ1 < ρ2 . Then p has exactly zeros in or on the circle |z| = ρ1 and no zeros in the annular ring ρ1 < |z| < ρ2 . This theorem states that it is sometimes possible to detect a gap in the moduli of two groups of zeros. Both of the above theorems can be generalized to matrix polynomials, which appear in eigenvalue problems where a nonzero complex vector v and a complex number z are sought such that P (z)v = 0, with n P (z) = j=0 Aj z j and Aj ∈ C m×m . If An is singular then there are infinite eigenvalues and if A0 is singular then zero is an eigenvalue. There are nm eigenvalues, including possibly infinite ones. The finite eigenvalues are the solutions of detP (z) = 0. When An = I and n = 1, we obtain the familiar (linear) eigenvalue problem. Throughout, we assume that the matrix polynomial is regular, i.e., that det(P ) is not identically zero. For more background on matrix polynomials, we refer to [6]. We now state the matrix versions of Theorems 2.1 and 2.2. n Theorem 2.3. ([3], [7], and [9]). All the eigenvalues of the regular matrix polynomial P (z) = j=0 Aj z j of degree n with Aj ∈ C m×m and An nonsingular, lie in |z| ≤ ρ where, for any matrix norm, ρ is the unique −1 n n−1 j z − positive solution of A−1 n j=0 Aj z = 0. Analogously to (1) in the scalar case, we obtain (for A0 nonsingular) that P has no eigenvalues in the interval [0, σ), where σ is the unique positive solution of −1 =0. An z n + An−1 z n−1 + · · · + A1 z − A−1 0
(2)
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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n
Aj z j be a regular matrix polynomial of degree n ≥ 2, Aj ∈ C m×m . −1 n j z + Let A be nonsingular for some , with 1 ≤ ≤ n−1, and let the polynomial − A−1 j=0,j= Aj z have two distinct positive roots ρ1 and ρ2 with ρ1 < ρ2 , for any given matrix norm. Then P has exactly m eigenvalues in or on the disk |z| = ρ1 and no eigenvalues in the annular ring ρ1 < |z| < ρ2 . Theorem 2.4. ([3], [9]) Let P (z) =
j=0
Pellet’s theorem has been improved for both scalar and matrix polynomials ([11]), as has Cauchy’s result ([10], [14]), by using a polynomial multiplier, but these improvements still rely on moduli only and therefore also do not take the argument of the zeros into account. We note that there are many fast and efficient methods available for computing the positive solution(s) of the real scalar equations in both versions of Cauchy’s and Pellet’s theorems and the equations we will encounter below. These are not the subject of this work. We define the following quantities, that are related to the above theorems. Definition 2.1. For any given norm, the quantity ρ in the statement of both versions of Cauchy’s theorem is called the Cauchy radius of the polynomials p and P , respectively. Definition 2.2. For any given norm, the quantities ρ1 and ρ2 in the statement of both versions of Pellet’s theorem are called the Pellet -radii of the polynomials p and P , respectively. ¯ T the conjugate transpose of a complex matrix M . A matrix M is Hermitian if We denote by M ∗ = M M = M , and it is skew-Hermitian or anti-Hermitian if M ∗ = −M . If M is skew-Hermitian, then −iM is Hermitian, and if M is Hermitian, then iM is skew-Hermitian. Any square complex matrix can be written as ∗
M=
1 1 (M + M ∗ ) + (M − M ∗ ) = MH + MAH , 2 2
where MH := 12 (M + M ∗ ) and MAH := 12 (M − M ∗ ) are the Hermitian and skew-Hermitian parts of M , ∗ ∗ respectively, with MH = MH and MAH = −MAH . The eigenvalues of a Hermitian matrix M are real, and we denote their minimum and maximum values by λmin (M ) and λmax (M ), respectively. The eigenvalues of a skew-Hermitian matrix are purely imaginary. A few more standard properties are summarized in the lemma below. Lemma 2.1. Let M ∈ C m×m . (a) If M ∗ = M , then ∀u ∈ C m : u∗ M u ∈ R. (b) If M ∗ = −M , then ∀u ∈ C m : u∗ M u is purely imaginary. (c) For any u ∈ C m : u∗ M u = u∗ MH u + (−iu∗ MAH u) i, so that Re (u∗ M u) = u∗ MH u and Im (u∗ M u) = −iu∗ MAH u. (d) If M ∗ = M , then λmin (M ) =
min
{u∗ M u}
u∈C m ,u=1
and
λmax (M ) =
max
{u∗ M u}.
u∈C m ,u=1
The following notation, where x ∈ R and M is a Hermitian matrix, will be useful: [x]− = and
x if x < 0 ,
0 if x ≥ 0 ,
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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[M ]− =
M
if M is not positive semi-definite,
0 if M is positive semi-definite.
3. Scalar polynomials Pellet’s theorem (Theorem 2.2) attempts to detect a gap between the zeros that are smallest in magnitude and the remaining ones, but often fails to do so because it depends only on the moduli of the zeros. This means that, if the gap is small in one particular direction from the origin, even though it is large in another, Pellet’s theorem will either not be able to detect it at all, or it will produce only a correspondingly small gap. A result that depends on the arguments of the zeros might be able to find the larger gap, even if it does not detect the smaller one. The following theorem does precisely that and is therefore generally more successful than Pellet’s theorem. It can be considered a directional version of Pellet’s theorem. n Theorem 3.1. Let p(z) = j=1 aj z j be a polynomial of degree n ≥ 2 with complex coefficients, and let a = 0 for some , with 1 ≤ ≤ n − 1. Let ϕj = arg(aj ), and define qθ (z) := |a |z +
n
|aj | [cos (( − j)θ + ϕ − ϕj )]− z j .
j=0
j=
Then the degree of qθ is m with ≤ m ≤ n and the following holds. (a) If m > and qθ has Pellet -radii ρ1 and ρ2 , then no zeros of p with argument θ have a modulus in (ρ1 , ρ2 ), which itself contains the interval determined by the Pellet -radii of p, should the latter exist. (b) If m > , and all the coefficients of z j in qθ vanish for j < , then qθ has no Pellet -radii and p has no zeros with argument θ that have a modulus in (0, σ), where σ is the unique positive solution of −|a |z +
m
|aj | [cos (( − j)θ + ϕ − ϕj )]− z j = 0.
j=+1
(c) If m = , then qθ has no Pellet -radii and the modulus of any zero of p with argument θ is not larger than the Cauchy radius of qθ . Proof. Let ζ = reiθ , with r ≥ 0, be a zero of p. Then the real part of e−i(θ+ϕ ) p(ζ) is given by ⎞
⎛ ⎜ Re ⎝|a |r +
n
⎛
⎞
⎟ ⎜ aj e−i((−j)θ+ϕ ) rj ⎠ = Re ⎝|a |r +
n
j=0
j=0
j=
j=
= |a |r +
n
⎟ |aj |eiϕj e−i((−j)θ+ϕ ) rj ⎠
|aj | cos (( − j)θ + ϕ − ϕj )rj ,
(3)
j=0
j=
so that the real part of the equation e−i(θ+ϕ ) p(ζ) = 0 implies that |a |r = −
n
|aj | cos (( − j)θ + ϕ − ϕj )rj ≤
n
j=0
j=0
j=
j=
|aj | [cos (( − j)θ + ϕ − ϕj )]− rj .
(4)
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The degree m of qθ is less than n if one or more leading coefficients vanish. If m > and qθ has Pellet -radii ρ1 and ρ2 , then Theorem 2.2 and inequality (4) imply that r cannot lie in the interval (ρ1 , ρ2 ). This interval contains the one determined by the Pellet -radii of p, if they exist, because of the inequality −|a |r +
n
|aj |rj ≥ −|a |r +
n
j=0
j=0
j=
j=
|aj | [cos (( − j)θ + ϕ − ϕj )]− rj .
If m > , but all the coefficients of z j in qθ vanish for j < , then there cannot exist any Pellet-radii for qθ . In this case, we have from (1) that a nonzero r cannot lie in the interval (0, σ), where σ is the unique positive solution of −|a |z +
m
|aj | [cos (( − j)θ + ϕ − ϕj )]− z j = 0.
j=+1
If m = , there can be no Pellet radii for qθ , and inequality (4) then implies that r is not larger than the Cauchy radius of qθ . 2 The following theorem is a limit case of Theorem 3.1, just as Theorem 2.1 is a limit case of Theorem 2.2. It is the extension from real to complex polynomials of Theorem 8.2.4 in [14], which can be considered a directional version of Theorem 2.1. The extension amounts to a simple modification, but it is an essential one to avoid needless limitations on the matrix polynomials later on in Section 4. We mention here that the same idea as in [14] was used in [4] to derive explicit directional upper bounds on the zeros of a polynomial with complex coefficients, although they are worse than the bounds from [14]. Theorem 3.2. Let p(z) = and define
n j=1
aj z j be a polynomial of degree n with complex coefficients, let ϕj = arg(aj ),
qθ (z) := |an |z n +
n−1
|aj | [cos ((n − j)θ + ϕn − ϕj )]− z j .
j=0
Then the modulus of a zero of p with argument θ is not larger than the Cauchy radius of qθ , which is itself not larger than the Cauchy radius of p. Proof. As in the proof of Theorem 3.1, let ζ = reiθ , with r ≥ 0, be a zero of p. Then, from (3) with = n, the real part of e−i(nθ+ϕn ) p(ζ) is given by |an |rn +
n−1
|aj | cos ((n − j)θ + ϕn − ϕj )rj .
j=0
The real part of the equation e−i(nθ+ϕn ) p(ζ) = 0 therefore yields |an |r = − n
n−1 j=0
|aj | cos ((n − j)θ + ϕn − ϕj )r ≤ j
n−1
|aj | [cos ((n − j)θ + ϕn − ϕj )]− rj ,
j=0
implying that r is not larger than the Cauchy radius of qθ , which itself is not larger than the Cauchy radius of p because the magnitudes of the nonleading coefficients of qθ are not larger than those of p. 2
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Fig. 1. Zero exclusion regions based on Theorem 3.1 for p1 (left) and p2 (right).
A lower bound on the moduli of zeros with a given argument can be obtained by applying the same theorem to the reverse polynomial. We now present a few examples, whose only purpose is to illustrate the theorems of this section, and to show what kind of results one can expect from them. In practice, one would not apply the theorems for every argument as we did here, but only for those arguments one might be interested in, such as, e.g., θ = 0, π, if one were interested in real zeros. Examples. Consider the following polynomials: p1 (z) = (2 − 4i)z 8 + (3 + 4i)z 7 + 3z 6 − (1 + 3i)z 5 − (4 + i)z 4 + (10 − 20i)z 3 − (2 + 4i)z 2 − (2 + 4i)z − 4 + 5i , p2 (z) = −(4 − 5i)z 8 − (1 − 5i)z 7 − (1 − 4i)z 6 − (3 − 4i)z 5 + (1 − i)z 4 − (20 − 25i)z 3 − (3 − i)z 2 − (3 + i)z + 4 + 4i , p3 (z) = −(3 + 3i)z 8 + 3z 7 − 4z 6 + (2 − 4i)z 5 + (4 + 3i)z 4 + (3 − 2i)z 3 − (1 + 2i)z 2 + (1 + 2i)z + 2 , p4 (z) = (5 + 2i)z 8 + (5 + 4i)z 7 − 2iz 6 − (2 − 2i)z 5 − 4z 4 + (4 + 4i)z 3 − iz 2 + (1 + 3i)z + 5 − i . On the left in Fig. 1 we have plotted, for 0 ≤ θ ≤ 2π, the exclusion regions for the zeros of p1 , obtained from Theorem 3.1 with = 3. The darker shaded areas correspond to part (a), whereas the lighter shading correspond to part (b) of that theorem. There were no instances of part (c), and the polynomial p1 has no Pellet 3-radii. The large circle represents the upper bound from Theorem 2.1. The exclusion regions extending beyond this circle are not shown since they are irrelevant. The zeros of the polynomial are indicated by circled asterisks. On the right in Fig. 1, the same was done for p2 with = 3 and with the same conventions. Here, the polynomial p2 does have Pellet 3-radii, which are indicated by the two smaller concentric circles. As is clear from the figures, the directional version of Pellet’s theorem is able to detect gaps for many arguments, whereas the standard Pellet theorem either does not find a gap at all or establishes a much smaller gap. The curve inside the large circle on the left in Fig. 2 represents the upper bound on the moduli of the zeros obtained from Theorem 3.2 for p3 , whereas, on the right of that figure, the same bound is shown for p4 . The radius of the large circle is the Cauchy radius, and the circled asterisks are the zeros, as before. The directional version of Theorem 3.2 produces better bounds than the standard Theorem 2.1.
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Fig. 2. Zero inclusion regions based on Theorem 3.2 for p3 (left) and p4 (right).
4. Matrix polynomials We now generalize the theorems from the previous section to matrix polynomials. These generalizations inevitably entail a higher computational cost, and it will depend on the particular application whether this cost is justified. In many applications, the coefficient matrices are highly structured or even diagonal, which facilitates the computations, but, in any event, such considerations are beyond the scope here. n Theorem 4.1. Let P (z) = j=0 Aj z j be a regular matrix polynomial of degree n ≥ 2 with Aj ∈ C m×m . Let AHj and AAHj denote the Hermitian and skew-Hermitian parts of Aj , respectively, let AH be positive definite for some , with 1 ≤ ≤ n − 1, and define
λmin (AH ) z +
qθ (z) :=
n
λmin AHj cos ( − j)θ − iAAHj sin ( − j)θ
zj . −
j=0
j=
Then the degree of qθ is m with ≤ m ≤ n and the following holds. (a) If m > and qθ has Pellet -radii ρ1 and ρ2 , then no eigenvalue of P with argument θ can have a modulus in (ρ1 , ρ2 ). (b) If m > , and all the coefficients of z j in qθ vanish for j < , then qθ has no Pellet -radii and P has no eigenvalues with argument θ that have a modulus in (0, σ), where σ is the unique positive solution of m − λmin (AH ) z + λmin AHj cos ( − j)θ − iAAHj sin ( − j)θ
j=+1
−
j z = 0 .
(5)
(c) If m = , then qθ has no Pellet -radii and the modulus of any eigenvalue of P with argument θ is not larger than the Cauchy radius of qθ . Proof. Let ζ = reiθ , with r ≥ 0, be an eigenvalue of P with corresponding unit eigenvector u ∈ C m , so that u∗ P (ζ)u = 0. Then, with the help of Lemma 2.1, the real part of e−iθ u∗ P (ζ)u is given by ⎞
⎛ ⎜ Re ⎝(u∗ A u) r +
n j=0
j=
⎟ (u∗ Aj u) e−i(−j)θ rj ⎠
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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= (u∗ AH u) r +
n Re (u∗ Aj u) cos ( − j)θ + Im (u∗ Aj u) sin ( − j)θ rj j=0
j= ∗
= (u AH u) r +
n
∗
∗
(u AHj u) cos ( − j)θ + (−iu AAHj u) sin ( − j)θ rj
j=0
j= ∗
= (u AH u) r +
n
u
∗
AHj cos ( − j)θ − iAAHj sin ( − j)θ u rj .
(6)
j=0
j=
The matrices AHj cos ( − j)θ − iAAHj sin ( − j)θ are Hermitian since they are real linear combinations of the Hermitian matrices AHj and (−iAAHj ). The real part of the equation e−iθ u∗ P (ζ)u = 0 therefore implies that
n u∗ AHj cos ( − j)θ − iAAHj sin ( − j)θ u rj , (u∗ AH u) r = − j=0
j=
so that n ∗ λmin (AH ) r ≤ u AHj cos ( − j)θ − iAAHj sin ( − j)θ u j=0
−
j=
n ≤ λmin AHj cos ( − j)θ − iAAHj sin ( − j)θ j=0 j=
j r
−
j r .
(7)
Since qθ has Pellet -radii ρ1 and ρ2 , and λmin (AH ) > 0, the inequality in (7) implies that r cannot lie in (ρ1 , ρ2 ). Observe that λmin denotes the smallest eigenvalue, not its absolute value. Parts (b) and (c) of the theorem’s statement follow from (7) as special cases, analogously to the proof of Theorem 3.1. 2 The requirement that the real part of the coefficient of z be positive definite can always be satisfied by multiplying P by A−1 (which needs to be computed anyway to apply Pellet’s theorem), thereby making the coefficient of z the identity matrix. Computing the smallest eigenvalue for each coefficient of P , or establishing their positive definiteness may be too computationally demanding for what are, after all, mere bounds. The following corollary offers a computationally less demanding alternative, although it inevitably makes the theorem weaker. Corollary 4.1. The same conclusions as in Theorem 4.1 hold if qθ in that theorem is replaced by the matrix polynomial Qθ , defined by Qθ (z) := AH z +
n
[AHj cos ( − j)θ − iAAHj sin ( − j)θ]− z j ,
j=0
j=
the matrix version of Pellet’s theorem is used (for any matrix norm), and equation (5) is replaced (for any matrix norm), by m −1 j z + − A−1 cos ( − j)θ − iA sin ( − j)θ] [A Hj AHj H − z = 0 . j=+1
(8)
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Moreover, Theorem 4.1 can be further simplified (and thereby weakened) by replacing expressions of the form [M ]− in the above with M . Proof. The corollary is an immediate consequence of Theorem 4.1 since |λmin (M )| ≤ M for any matrix M , while for a (Hermitian) positive definite matrix M we have that ∗ −1 1 = λmax (M −1 ) = max u M u ≤ M −1 =⇒ λmin (M ) ≥ M −1 −1 , λmin (M ) u∈C m ,u=1 which, combined with (7), leads to (8). 2 We now turn to the Cauchy radius with the following theorem. n Theorem 4.2. Let P (z) = j=0 Aj z j be a regular matrix polynomial of degree n, with Aj ∈ C m×m . Let AHj and AAHj denote the Hermitian and skew-Hermitian parts of Aj , respectively, let AHn be positive definite, and define
qθ (z) :=
λmin (AHn ) z n +
n−1
λmin AHj cos (n − j)θ − iAAHj sin (n − j)θ
zj . −
j=0
Then the modulus of an eigenvalue of P with argument θ is not larger than the Cauchy radius of qθ . Proof. We proceed similarly as in the proof of Theorem 4.1. Let ζ = reiθ , with r ≥ 0, be an eigenvalue of P with corresponding unit eigenvector u ∈ C m , so that u∗ P (ζ)u = 0. Then, from (6) with = n, we obtain that the real part of (einθ )−1 u∗ P (ζ)u is given by ⎛ Re ⎝(u∗ An u) rn +
n−1
⎞ (u∗ Aj u) e−i(n−j)θ rj ⎠
j=0
∗
n
= (u AHn u) r +
n−1
u
∗
AHj cos (n − j)θ − iAAHj sin (n − j)θ u rj .
j=0
From (7) with = n, we have that the real part of the equation (einθ )−1 u∗ P (ζ)u = 0 then implies
λmin (AHn ) r ≤ n
λmin AHj cos (n − j)θ − iAAHj sin (n − j)θ
n−1 j=0
−
Therefore, r is not larger than the Cauchy radius of qθ .
j r .
2
Similarly as for Pellet’s theorem, the following corollary is a computationally less demanding alternative to Theorem 4.2. Its proof is analogous to that of Corollary 4.1 and we omit it to avoid repetition. Corollary 4.2. The same conclusions as in Theorem 4.2 hold if qθ in that theorem is replaced by the matrix polynomial Qθ , defined by Qθ (z) := AHn z n +
n−1
[AHj cos (n − j)θ − iAAHj sin (n − j)θ]− z j ,
j=0
and the matrix version of Theorem 2.1 is used (for any matrix norm). Moreover, Theorem 4.2 can be further simplified (and thereby weakened) by replacing the expression of the form [M ]− in the above with M .
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A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
Fig. 3. Zero exclusion regions based on Theorem 4.1 (left) and Corollary 4.2 (right) for the bilby problem.
Fig. 4. Zero inclusion regions based Theorem 4.2 (left) and Corollary 4.2 (right) for the butterfly problem.
We now present two examples that, as before, are meant to merely illustrate the theorems in this section. Zero exclusion and inclusion regions are shown for all arguments θ ∈ [0, 2π]. The examples are taken from the collection of nonlinear eigenvalue problems in [2]. Examples. (1) The “bilby” problem on p. 8 of [2]: a quadratic matrix polynomial A2 z 2 + A1 z + A0 with coefficients in R5×5 , where A0 and A2 are singular, so that there are both zero and infinite eigenvalues. Its parameters are described in detail in [2], except for the parameter β which here is given by β = 1.45. The problem originates in a population model for the greater bilby (Macrotis lagotis), an Australian marsupial, in [1]. The polynomial does not have Pellet radii, but both Theorem 4.1 and Corollary 4.1 establish an exclusion region for the zeros, as can be seen in Fig. 3 on the left and right, respectively, for which the polynomial was premultiplied by A−1 1 . (2) The “butterfly” problem on p. 9 of [2]: a quartic matrix polynomial A4 z 4 +A3 z 3 +A2 z 2 +A1 z+A0 with 2 2 coefficients in Rm ×m . The matrices A1 and A3 are skew-symmetric, whereas A2 and A4 are symmetric. We set m = 8, which is the default value. Fig. 4 shows bounds on the moduli of the zeros obtained from Theorem 4.2 and Corollary 4.2 on the left and right, respectively. Although, for some arguments, the result from Corollary 4.2 is worse than the Cauchy radius, represented by the circle, it is better for most. The polynomial was premultiplied by A−1 4 . As expected, the results from the corollaries are worse than those from the corresponding theorems.
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References [1] N.G. Bean, L. Bright, G. Latouche, C.E.M. Pearce, P.K. Pollett, P.G. Taylor, The quasi-stationary behavior of quasibirth-and-death processes, Ann. Appl. Probab. 7 (1997) 134–155. [2] T. Betcke, N.J. Higham, V. Mehrmann, C. Schröder, F. Tisseur, NLEVP: a collection of nonlinear eigenvalue problems, ACM Trans. Math. Software 39 (2) (2013). [3] D.A. Bini, V. Noferini, M. Sharify, Locating the eigenvalues of matrix polynomials, SIAM J. Matrix Anal. Appl. 34 (2013) 1708–1727. [4] F.G. Boese, W.J. Luther, A note on a classical bound for the moduli of all zeros of a polynomial, IEEE Trans. Automat. Control 34 (1989) 998–1001. [5] A.L. Cauchy, Sur la résolution des équations numériques et sur la théorie de l’élimination, in: Exercices de Mathématiques, Quatrième Année, de Bure frères, Paris, 1829, pp. 65–128. Also in: Oeuvres Complètes, Série 2, Tome 9, Gauthiers-Villars et fils, Paris, 1891, pp. 86–161. [6] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Computer Science and Applied Mathematics, Academic Press, Inc., New York-London, 1982. [7] N.J. Higham, F. Tisseur, Bounds for eigenvalues of matrix polynomials, Linear Algebra Appl. 358 (2003) 5–22. [8] M. Marden, Geometry of Polynomials, second edition, Mathematical Surveys, vol. 3, American Mathematical Society, Providence, R.I., 1966. [9] A. Melman, Generalization and variations of Pellet’s theorem for matrix polynomials, Linear Algebra Appl. 439 (2013) 1550–1567. [10] A. Melman, Bounds for eigenvalues of matrix polynomials with applications to scalar polynomials, Linear Algebra Appl. 504 (2016) 190–203. [11] A. Melman, Improvement of Pellet’s theorem for scalar and matrix polynomials, C. R. Math. Acad. Sci. Paris 354 (2016) 859–863. [12] G.V. Milovanović, D.S. Mitrinović, Th.M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. [13] M.A. Pellet, Sur un mode de séparation des racines des équations et la formule de Lagrange, Bull. Sci. Math. 5 (1881) 393–395. [14] Q.I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002.
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Directional bounds for polynomial zeros and eigenvalues A. Melman Department of Applied Mathematics, School of Engineering, Santa Clara University, Santa Clara, CA 95053, United States of America
a r t i c l e
i n f o
Article history: Received 17 May 2019 Available online 3 October 2019 Submitted by V. Andrievskii Keywords: Polynomial Zero Eigenvalue Cauchy Pellet Directional bound
a b s t r a c t Directional versions of theorems by Cauchy and Pellet are obtained for both scalar and matrix polynomials, which derive bounds on the zeros of scalar polynomials and eigenvalues of matrix polynomials that depend on their arguments. Such bounds often deliver much better results than the standard theorems, which are formulated in terms of the moduli of the zeros or eigenvalues only and are therefore insensitive to their arguments. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The development of fast and reliable methods to compute the zeros of a polynomial, may, at first sight, appear to have made the many approximations and localization results for such zeros less useful, in particular the more specialized ones. However, these approximations become considerably more valuable if they can be generalized to matrix polynomials, in which case they can provide location information for polynomial eigenvalues, which are much harder to compute than polynomial zeros. Examples of results that were generalized can be found, e.g., in [3], [7], [10], and [11]. A large supply of (scalar) polynomial results can be found in the encyclopedic works [12] and [14], including several that seem to be new. One of the latter motivates the present work, namely, Theorem 8.2.4 in [14], which derives an upper bound on the modulus of a polynomial zero in terms of its argument. As such, it can be considered a directional version of a bound due to Cauchy ([5]), which is, itself, a limit case of a theorem due to Pellet ([13]). This naturally leads to the idea of deriving a directional version of the more general theorem, and we begin by doing precisely that. Pellet’s theorem can sometimes detect a gap between the moduli of groups of zeros, but often fails to do so when the moduli of just two of the zeros, one in each group, are close, so that a directional version of the E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.123571 0022-247X/© 2019 Elsevier Inc. All rights reserved.
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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theorem has a much better chance of detecting gaps. We then proceed to generalize both this theorem and an extension of Theorem 8.2.4 in [14] to matrix polynomials. Directional results of this kind for polynomial eigenvalues do not seem to be commonly encountered in the literature. The paper is organized as follows. In Section 2 we collect a few basic results that will be used throughout the paper in an effort to make it reasonably self-contained. In Section 3, the results for scalar polynomials are derived, while Section 4 is devoted to matrix polynomials. 2. Preliminaries In this section we summarize a few basic results that will be needed later on. We begin with the aforementioned bound by Cauchy. Theorem 2.1. ([5], [8, Th. (27,1), p. 122 and Exercise 1, p. 126]). All the zeros of the polynomial p(z) = n j j=0 aj z of degree n with complex coefficients lie in the disk defined by |z| ≤ ρ, where ρ is the unique n−1 positive solution of |an |z n − j=0 |aj |z j = 0. If we apply this theorem to the reverse polynomial p# (z) = z n p(1/z) (for a0 = 0), whose zeros are the reciprocals of those of p, we obtain that the polynomial p cannot have zeros with a modulus in the interval [0, σ), where σ is the unique positive solution of |an |z n + |an−1 |z n−1 + · · · + |a1 |z − |a0 | = 0 .
(1)
Theorem 2.1 can be considered as a limit case of Pellet’s theorem, which we state next. n Theorem 2.2. ([13], [8, Th. (28,1), p. 128]) Let p(z) = j=0 aj z j be a polynomial of degree n ≥ 2 with comn plex coefficients and a = 0 for some , with 1 ≤ ≤ n − 1, and let the polynomial −|a |z + j=0,j= |aj |z j have two distinct positive roots ρ1 and ρ2 with ρ1 < ρ2 . Then p has exactly zeros in or on the circle |z| = ρ1 and no zeros in the annular ring ρ1 < |z| < ρ2 . This theorem states that it is sometimes possible to detect a gap in the moduli of two groups of zeros. Both of the above theorems can be generalized to matrix polynomials, which appear in eigenvalue problems where a nonzero complex vector v and a complex number z are sought such that P (z)v = 0, with n P (z) = j=0 Aj z j and Aj ∈ C m×m . If An is singular then there are infinite eigenvalues and if A0 is singular then zero is an eigenvalue. There are nm eigenvalues, including possibly infinite ones. The finite eigenvalues are the solutions of detP (z) = 0. When An = I and n = 1, we obtain the familiar (linear) eigenvalue problem. Throughout, we assume that the matrix polynomial is regular, i.e., that det(P ) is not identically zero. For more background on matrix polynomials, we refer to [6]. We now state the matrix versions of Theorems 2.1 and 2.2. n Theorem 2.3. ([3], [7], and [9]). All the eigenvalues of the regular matrix polynomial P (z) = j=0 Aj z j of degree n with Aj ∈ C m×m and An nonsingular, lie in |z| ≤ ρ where, for any matrix norm, ρ is the unique −1 n n−1 j z − positive solution of A−1 n j=0 Aj z = 0. Analogously to (1) in the scalar case, we obtain (for A0 nonsingular) that P has no eigenvalues in the interval [0, σ), where σ is the unique positive solution of −1 =0. An z n + An−1 z n−1 + · · · + A1 z − A−1 0
(2)
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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n
Aj z j be a regular matrix polynomial of degree n ≥ 2, Aj ∈ C m×m . −1 n j z + Let A be nonsingular for some , with 1 ≤ ≤ n−1, and let the polynomial − A−1 j=0,j= Aj z have two distinct positive roots ρ1 and ρ2 with ρ1 < ρ2 , for any given matrix norm. Then P has exactly m eigenvalues in or on the disk |z| = ρ1 and no eigenvalues in the annular ring ρ1 < |z| < ρ2 . Theorem 2.4. ([3], [9]) Let P (z) =
j=0
Pellet’s theorem has been improved for both scalar and matrix polynomials ([11]), as has Cauchy’s result ([10], [14]), by using a polynomial multiplier, but these improvements still rely on moduli only and therefore also do not take the argument of the zeros into account. We note that there are many fast and efficient methods available for computing the positive solution(s) of the real scalar equations in both versions of Cauchy’s and Pellet’s theorems and the equations we will encounter below. These are not the subject of this work. We define the following quantities, that are related to the above theorems. Definition 2.1. For any given norm, the quantity ρ in the statement of both versions of Cauchy’s theorem is called the Cauchy radius of the polynomials p and P , respectively. Definition 2.2. For any given norm, the quantities ρ1 and ρ2 in the statement of both versions of Pellet’s theorem are called the Pellet -radii of the polynomials p and P , respectively. ¯ T the conjugate transpose of a complex matrix M . A matrix M is Hermitian if We denote by M ∗ = M M = M , and it is skew-Hermitian or anti-Hermitian if M ∗ = −M . If M is skew-Hermitian, then −iM is Hermitian, and if M is Hermitian, then iM is skew-Hermitian. Any square complex matrix can be written as ∗
M=
1 1 (M + M ∗ ) + (M − M ∗ ) = MH + MAH , 2 2
where MH := 12 (M + M ∗ ) and MAH := 12 (M − M ∗ ) are the Hermitian and skew-Hermitian parts of M , ∗ ∗ respectively, with MH = MH and MAH = −MAH . The eigenvalues of a Hermitian matrix M are real, and we denote their minimum and maximum values by λmin (M ) and λmax (M ), respectively. The eigenvalues of a skew-Hermitian matrix are purely imaginary. A few more standard properties are summarized in the lemma below. Lemma 2.1. Let M ∈ C m×m . (a) If M ∗ = M , then ∀u ∈ C m : u∗ M u ∈ R. (b) If M ∗ = −M , then ∀u ∈ C m : u∗ M u is purely imaginary. (c) For any u ∈ C m : u∗ M u = u∗ MH u + (−iu∗ MAH u) i, so that Re (u∗ M u) = u∗ MH u and Im (u∗ M u) = −iu∗ MAH u. (d) If M ∗ = M , then λmin (M ) =
min
{u∗ M u}
u∈C m ,u=1
and
λmax (M ) =
max
{u∗ M u}.
u∈C m ,u=1
The following notation, where x ∈ R and M is a Hermitian matrix, will be useful: [x]− = and
x if x < 0 ,
0 if x ≥ 0 ,
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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[M ]− =
M
if M is not positive semi-definite,
0 if M is positive semi-definite.
3. Scalar polynomials Pellet’s theorem (Theorem 2.2) attempts to detect a gap between the zeros that are smallest in magnitude and the remaining ones, but often fails to do so because it depends only on the moduli of the zeros. This means that, if the gap is small in one particular direction from the origin, even though it is large in another, Pellet’s theorem will either not be able to detect it at all, or it will produce only a correspondingly small gap. A result that depends on the arguments of the zeros might be able to find the larger gap, even if it does not detect the smaller one. The following theorem does precisely that and is therefore generally more successful than Pellet’s theorem. It can be considered a directional version of Pellet’s theorem. n Theorem 3.1. Let p(z) = j=1 aj z j be a polynomial of degree n ≥ 2 with complex coefficients, and let a = 0 for some , with 1 ≤ ≤ n − 1. Let ϕj = arg(aj ), and define qθ (z) := |a |z +
n
|aj | [cos (( − j)θ + ϕ − ϕj )]− z j .
j=0
j=
Then the degree of qθ is m with ≤ m ≤ n and the following holds. (a) If m > and qθ has Pellet -radii ρ1 and ρ2 , then no zeros of p with argument θ have a modulus in (ρ1 , ρ2 ), which itself contains the interval determined by the Pellet -radii of p, should the latter exist. (b) If m > , and all the coefficients of z j in qθ vanish for j < , then qθ has no Pellet -radii and p has no zeros with argument θ that have a modulus in (0, σ), where σ is the unique positive solution of −|a |z +
m
|aj | [cos (( − j)θ + ϕ − ϕj )]− z j = 0.
j=+1
(c) If m = , then qθ has no Pellet -radii and the modulus of any zero of p with argument θ is not larger than the Cauchy radius of qθ . Proof. Let ζ = reiθ , with r ≥ 0, be a zero of p. Then the real part of e−i(θ+ϕ ) p(ζ) is given by ⎞
⎛ ⎜ Re ⎝|a |r +
n
⎛
⎞
⎟ ⎜ aj e−i((−j)θ+ϕ ) rj ⎠ = Re ⎝|a |r +
n
j=0
j=0
j=
j=
= |a |r +
n
⎟ |aj |eiϕj e−i((−j)θ+ϕ ) rj ⎠
|aj | cos (( − j)θ + ϕ − ϕj )rj ,
(3)
j=0
j=
so that the real part of the equation e−i(θ+ϕ ) p(ζ) = 0 implies that |a |r = −
n
|aj | cos (( − j)θ + ϕ − ϕj )rj ≤
n
j=0
j=0
j=
j=
|aj | [cos (( − j)θ + ϕ − ϕj )]− rj .
(4)
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The degree m of qθ is less than n if one or more leading coefficients vanish. If m > and qθ has Pellet -radii ρ1 and ρ2 , then Theorem 2.2 and inequality (4) imply that r cannot lie in the interval (ρ1 , ρ2 ). This interval contains the one determined by the Pellet -radii of p, if they exist, because of the inequality −|a |r +
n
|aj |rj ≥ −|a |r +
n
j=0
j=0
j=
j=
|aj | [cos (( − j)θ + ϕ − ϕj )]− rj .
If m > , but all the coefficients of z j in qθ vanish for j < , then there cannot exist any Pellet-radii for qθ . In this case, we have from (1) that a nonzero r cannot lie in the interval (0, σ), where σ is the unique positive solution of −|a |z +
m
|aj | [cos (( − j)θ + ϕ − ϕj )]− z j = 0.
j=+1
If m = , there can be no Pellet radii for qθ , and inequality (4) then implies that r is not larger than the Cauchy radius of qθ . 2 The following theorem is a limit case of Theorem 3.1, just as Theorem 2.1 is a limit case of Theorem 2.2. It is the extension from real to complex polynomials of Theorem 8.2.4 in [14], which can be considered a directional version of Theorem 2.1. The extension amounts to a simple modification, but it is an essential one to avoid needless limitations on the matrix polynomials later on in Section 4. We mention here that the same idea as in [14] was used in [4] to derive explicit directional upper bounds on the zeros of a polynomial with complex coefficients, although they are worse than the bounds from [14]. Theorem 3.2. Let p(z) = and define
n j=1
aj z j be a polynomial of degree n with complex coefficients, let ϕj = arg(aj ),
qθ (z) := |an |z n +
n−1
|aj | [cos ((n − j)θ + ϕn − ϕj )]− z j .
j=0
Then the modulus of a zero of p with argument θ is not larger than the Cauchy radius of qθ , which is itself not larger than the Cauchy radius of p. Proof. As in the proof of Theorem 3.1, let ζ = reiθ , with r ≥ 0, be a zero of p. Then, from (3) with = n, the real part of e−i(nθ+ϕn ) p(ζ) is given by |an |rn +
n−1
|aj | cos ((n − j)θ + ϕn − ϕj )rj .
j=0
The real part of the equation e−i(nθ+ϕn ) p(ζ) = 0 therefore yields |an |r = − n
n−1 j=0
|aj | cos ((n − j)θ + ϕn − ϕj )r ≤ j
n−1
|aj | [cos ((n − j)θ + ϕn − ϕj )]− rj ,
j=0
implying that r is not larger than the Cauchy radius of qθ , which itself is not larger than the Cauchy radius of p because the magnitudes of the nonleading coefficients of qθ are not larger than those of p. 2
6
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
Fig. 1. Zero exclusion regions based on Theorem 3.1 for p1 (left) and p2 (right).
A lower bound on the moduli of zeros with a given argument can be obtained by applying the same theorem to the reverse polynomial. We now present a few examples, whose only purpose is to illustrate the theorems of this section, and to show what kind of results one can expect from them. In practice, one would not apply the theorems for every argument as we did here, but only for those arguments one might be interested in, such as, e.g., θ = 0, π, if one were interested in real zeros. Examples. Consider the following polynomials: p1 (z) = (2 − 4i)z 8 + (3 + 4i)z 7 + 3z 6 − (1 + 3i)z 5 − (4 + i)z 4 + (10 − 20i)z 3 − (2 + 4i)z 2 − (2 + 4i)z − 4 + 5i , p2 (z) = −(4 − 5i)z 8 − (1 − 5i)z 7 − (1 − 4i)z 6 − (3 − 4i)z 5 + (1 − i)z 4 − (20 − 25i)z 3 − (3 − i)z 2 − (3 + i)z + 4 + 4i , p3 (z) = −(3 + 3i)z 8 + 3z 7 − 4z 6 + (2 − 4i)z 5 + (4 + 3i)z 4 + (3 − 2i)z 3 − (1 + 2i)z 2 + (1 + 2i)z + 2 , p4 (z) = (5 + 2i)z 8 + (5 + 4i)z 7 − 2iz 6 − (2 − 2i)z 5 − 4z 4 + (4 + 4i)z 3 − iz 2 + (1 + 3i)z + 5 − i . On the left in Fig. 1 we have plotted, for 0 ≤ θ ≤ 2π, the exclusion regions for the zeros of p1 , obtained from Theorem 3.1 with = 3. The darker shaded areas correspond to part (a), whereas the lighter shading correspond to part (b) of that theorem. There were no instances of part (c), and the polynomial p1 has no Pellet 3-radii. The large circle represents the upper bound from Theorem 2.1. The exclusion regions extending beyond this circle are not shown since they are irrelevant. The zeros of the polynomial are indicated by circled asterisks. On the right in Fig. 1, the same was done for p2 with = 3 and with the same conventions. Here, the polynomial p2 does have Pellet 3-radii, which are indicated by the two smaller concentric circles. As is clear from the figures, the directional version of Pellet’s theorem is able to detect gaps for many arguments, whereas the standard Pellet theorem either does not find a gap at all or establishes a much smaller gap. The curve inside the large circle on the left in Fig. 2 represents the upper bound on the moduli of the zeros obtained from Theorem 3.2 for p3 , whereas, on the right of that figure, the same bound is shown for p4 . The radius of the large circle is the Cauchy radius, and the circled asterisks are the zeros, as before. The directional version of Theorem 3.2 produces better bounds than the standard Theorem 2.1.
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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Fig. 2. Zero inclusion regions based on Theorem 3.2 for p3 (left) and p4 (right).
4. Matrix polynomials We now generalize the theorems from the previous section to matrix polynomials. These generalizations inevitably entail a higher computational cost, and it will depend on the particular application whether this cost is justified. In many applications, the coefficient matrices are highly structured or even diagonal, which facilitates the computations, but, in any event, such considerations are beyond the scope here. n Theorem 4.1. Let P (z) = j=0 Aj z j be a regular matrix polynomial of degree n ≥ 2 with Aj ∈ C m×m . Let AHj and AAHj denote the Hermitian and skew-Hermitian parts of Aj , respectively, let AH be positive definite for some , with 1 ≤ ≤ n − 1, and define
λmin (AH ) z +
qθ (z) :=
n
λmin AHj cos ( − j)θ − iAAHj sin ( − j)θ
zj . −
j=0
j=
Then the degree of qθ is m with ≤ m ≤ n and the following holds. (a) If m > and qθ has Pellet -radii ρ1 and ρ2 , then no eigenvalue of P with argument θ can have a modulus in (ρ1 , ρ2 ). (b) If m > , and all the coefficients of z j in qθ vanish for j < , then qθ has no Pellet -radii and P has no eigenvalues with argument θ that have a modulus in (0, σ), where σ is the unique positive solution of m − λmin (AH ) z + λmin AHj cos ( − j)θ − iAAHj sin ( − j)θ
j=+1
−
j z = 0 .
(5)
(c) If m = , then qθ has no Pellet -radii and the modulus of any eigenvalue of P with argument θ is not larger than the Cauchy radius of qθ . Proof. Let ζ = reiθ , with r ≥ 0, be an eigenvalue of P with corresponding unit eigenvector u ∈ C m , so that u∗ P (ζ)u = 0. Then, with the help of Lemma 2.1, the real part of e−iθ u∗ P (ζ)u is given by ⎞
⎛ ⎜ Re ⎝(u∗ A u) r +
n j=0
j=
⎟ (u∗ Aj u) e−i(−j)θ rj ⎠
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
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= (u∗ AH u) r +
n Re (u∗ Aj u) cos ( − j)θ + Im (u∗ Aj u) sin ( − j)θ rj j=0
j= ∗
= (u AH u) r +
n
∗
∗
(u AHj u) cos ( − j)θ + (−iu AAHj u) sin ( − j)θ rj
j=0
j= ∗
= (u AH u) r +
n
u
∗
AHj cos ( − j)θ − iAAHj sin ( − j)θ u rj .
(6)
j=0
j=
The matrices AHj cos ( − j)θ − iAAHj sin ( − j)θ are Hermitian since they are real linear combinations of the Hermitian matrices AHj and (−iAAHj ). The real part of the equation e−iθ u∗ P (ζ)u = 0 therefore implies that
n u∗ AHj cos ( − j)θ − iAAHj sin ( − j)θ u rj , (u∗ AH u) r = − j=0
j=
so that n ∗ λmin (AH ) r ≤ u AHj cos ( − j)θ − iAAHj sin ( − j)θ u j=0
−
j=
n ≤ λmin AHj cos ( − j)θ − iAAHj sin ( − j)θ j=0 j=
j r
−
j r .
(7)
Since qθ has Pellet -radii ρ1 and ρ2 , and λmin (AH ) > 0, the inequality in (7) implies that r cannot lie in (ρ1 , ρ2 ). Observe that λmin denotes the smallest eigenvalue, not its absolute value. Parts (b) and (c) of the theorem’s statement follow from (7) as special cases, analogously to the proof of Theorem 3.1. 2 The requirement that the real part of the coefficient of z be positive definite can always be satisfied by multiplying P by A−1 (which needs to be computed anyway to apply Pellet’s theorem), thereby making the coefficient of z the identity matrix. Computing the smallest eigenvalue for each coefficient of P , or establishing their positive definiteness may be too computationally demanding for what are, after all, mere bounds. The following corollary offers a computationally less demanding alternative, although it inevitably makes the theorem weaker. Corollary 4.1. The same conclusions as in Theorem 4.1 hold if qθ in that theorem is replaced by the matrix polynomial Qθ , defined by Qθ (z) := AH z +
n
[AHj cos ( − j)θ − iAAHj sin ( − j)θ]− z j ,
j=0
j=
the matrix version of Pellet’s theorem is used (for any matrix norm), and equation (5) is replaced (for any matrix norm), by m −1 j z + − A−1 cos ( − j)θ − iA sin ( − j)θ] [A Hj AHj H − z = 0 . j=+1
(8)
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Moreover, Theorem 4.1 can be further simplified (and thereby weakened) by replacing expressions of the form [M ]− in the above with M . Proof. The corollary is an immediate consequence of Theorem 4.1 since |λmin (M )| ≤ M for any matrix M , while for a (Hermitian) positive definite matrix M we have that ∗ −1 1 = λmax (M −1 ) = max u M u ≤ M −1 =⇒ λmin (M ) ≥ M −1 −1 , λmin (M ) u∈C m ,u=1 which, combined with (7), leads to (8). 2 We now turn to the Cauchy radius with the following theorem. n Theorem 4.2. Let P (z) = j=0 Aj z j be a regular matrix polynomial of degree n, with Aj ∈ C m×m . Let AHj and AAHj denote the Hermitian and skew-Hermitian parts of Aj , respectively, let AHn be positive definite, and define
qθ (z) :=
λmin (AHn ) z n +
n−1
λmin AHj cos (n − j)θ − iAAHj sin (n − j)θ
zj . −
j=0
Then the modulus of an eigenvalue of P with argument θ is not larger than the Cauchy radius of qθ . Proof. We proceed similarly as in the proof of Theorem 4.1. Let ζ = reiθ , with r ≥ 0, be an eigenvalue of P with corresponding unit eigenvector u ∈ C m , so that u∗ P (ζ)u = 0. Then, from (6) with = n, we obtain that the real part of (einθ )−1 u∗ P (ζ)u is given by ⎛ Re ⎝(u∗ An u) rn +
n−1
⎞ (u∗ Aj u) e−i(n−j)θ rj ⎠
j=0
∗
n
= (u AHn u) r +
n−1
u
∗
AHj cos (n − j)θ − iAAHj sin (n − j)θ u rj .
j=0
From (7) with = n, we have that the real part of the equation (einθ )−1 u∗ P (ζ)u = 0 then implies
λmin (AHn ) r ≤ n
λmin AHj cos (n − j)θ − iAAHj sin (n − j)θ
n−1 j=0
−
Therefore, r is not larger than the Cauchy radius of qθ .
j r .
2
Similarly as for Pellet’s theorem, the following corollary is a computationally less demanding alternative to Theorem 4.2. Its proof is analogous to that of Corollary 4.1 and we omit it to avoid repetition. Corollary 4.2. The same conclusions as in Theorem 4.2 hold if qθ in that theorem is replaced by the matrix polynomial Qθ , defined by Qθ (z) := AHn z n +
n−1
[AHj cos (n − j)θ − iAAHj sin (n − j)θ]− z j ,
j=0
and the matrix version of Theorem 2.1 is used (for any matrix norm). Moreover, Theorem 4.2 can be further simplified (and thereby weakened) by replacing the expression of the form [M ]− in the above with M .
10
A. Melman / J. Math. Anal. Appl. 482 (2020) 123571
Fig. 3. Zero exclusion regions based on Theorem 4.1 (left) and Corollary 4.2 (right) for the bilby problem.
Fig. 4. Zero inclusion regions based Theorem 4.2 (left) and Corollary 4.2 (right) for the butterfly problem.
We now present two examples that, as before, are meant to merely illustrate the theorems in this section. Zero exclusion and inclusion regions are shown for all arguments θ ∈ [0, 2π]. The examples are taken from the collection of nonlinear eigenvalue problems in [2]. Examples. (1) The “bilby” problem on p. 8 of [2]: a quadratic matrix polynomial A2 z 2 + A1 z + A0 with coefficients in R5×5 , where A0 and A2 are singular, so that there are both zero and infinite eigenvalues. Its parameters are described in detail in [2], except for the parameter β which here is given by β = 1.45. The problem originates in a population model for the greater bilby (Macrotis lagotis), an Australian marsupial, in [1]. The polynomial does not have Pellet radii, but both Theorem 4.1 and Corollary 4.1 establish an exclusion region for the zeros, as can be seen in Fig. 3 on the left and right, respectively, for which the polynomial was premultiplied by A−1 1 . (2) The “butterfly” problem on p. 9 of [2]: a quartic matrix polynomial A4 z 4 +A3 z 3 +A2 z 2 +A1 z+A0 with 2 2 coefficients in Rm ×m . The matrices A1 and A3 are skew-symmetric, whereas A2 and A4 are symmetric. We set m = 8, which is the default value. Fig. 4 shows bounds on the moduli of the zeros obtained from Theorem 4.2 and Corollary 4.2 on the left and right, respectively. Although, for some arguments, the result from Corollary 4.2 is worse than the Cauchy radius, represented by the circle, it is better for most. The polynomial was premultiplied by A−1 4 . As expected, the results from the corollaries are worse than those from the corresponding theorems.
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