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Extrait d'une lettre de Mr. Ch. Hermite de Paris à Mr. Borchardt de Berlin sur le nombre des racines d'une équation algébrique comprises entre des limites données
Journal für die reine und angewandte Mathematik
... Interlacing patterns of the zeros of two polynomials have occurred already in the middle of the 19th century. The work of Sturm, Cauchy, and Hermite [9] established a connection between the stability 1 of a polynomial p in terms of interlacing of the real zeros of two associated polynomials, now known as the Hermite-Biehler Theorem. This criterion can be found in different versions in the work of Biehler [3] and Hurwitz [12]. ...
... In addition to that, it is bounded on the imaginary axis, which means thatg kl is constant by the Phragmén-Lindelöf principle ( [15]). This implies g kl (z) = c kl + d kl z with some constants c kl , d kl , which proves representation (9). ...
... Representation (9) can now be obtained in the same way as for the implication (iii) ⇒ (i). Moreover, the matrices A j are the residues of −Q. ...
Scalar-valued meromorphic Herglotz-Nevanlinna functions are characterized by the interlacing property of their poles and zeros together with some growth properties. We give a characterization of matrix-valued Herglotz-Nevanlinna functions by means of a higher-order interlacing property. As an application we deduce a matrix version of the classical Hermite-Biehler Theorem for entire functions.
... , g m ) are defined above. To extract the formula Φ characterizing this condition, the algorithm applies to each relation f i = 0 ∧ g ≥ 0 the real root counting method described in [13], based on Hermite's quadratic form [18]. ...
... Then, there exists a non-empty Zariski open set M ⊂ GL n (C) such that for all M ∈ M ∩ GL n (Q), there exists a non-empty Zariski open set Y M ⊂ C t such that for all y ∈ Y M ∩ Q t , for any connected component C ⊂ D r (y, ·) ∩ R n , the results of Proposition 3.2 hold.Proof. The proof of Proposition 3.2 follows from[17, Proposition 17,18]. As the proof of[17, Proposition 17] is purely algebraic and thus valid over the base field C(y), we can then invert the order of choosing y and M as in[20, Proposition 6]. ...
We consider linear matrix inequalities (LMIs) with the 's being symmetric matrices, with entries in a ring . When , the feasibility problem consists in deciding whether the 's can be instantiated to obtain a positive semidefinite matrix. When , the problem asks for a formula on the parameters , which describes the values of the parameters for which the specialized LMI is feasible. This problem can be solved using general quantifier elimination algorithms, with a complexity that is exponential in n. In this work, we leverage the LMI structure of the problem to design an algorithm that computes a formula describing a dense subset of the feasible region of parameters, under genericity assumptions. The complexity of this algorithm is exponential in and t but becomes polynomial in n when m is fixed. We apply the algorithm to a parametric sum-of-squares problem and to the convergence analyses of certain first-order optimization methods, which are both known to be equivalent to the feasibility of certain parametric LMIs, hence demonstrating its practical interest.
... To do that, we slightly generalize the notion of Hermite quadratic forms, a classical tool for counting solutions of zerodimensional systems, to parametric systems. Originally introduced by Hermite [106] for counting real or complex solutions of univariate polynomials, Hermite quadratic forms were then extended in [160] to multivariate systems; their definition is as follows. ...
... In [106], Hermite introduced a method for counting the solutions of a given univariate polynomial by associating to it a quadratic form. Later on, in [160], Hermite's quadratic forms were generalized to multivariate zero-dimensional systems. ...
Solving polynomial systems is an active research area located betweencomputer sciences and mathematics. It finds many applications invarious fields of engineering and sciences (robotics, biology,cryptography, imaging, optimal control). In symbolic computation, onestudies and designs efficient algorithms that compute exact solutionsto those applications, which could be very delicate for numericalmethods because of the non-linearity of the given systems.Most applications in engineering are interested in the real solutionsto the system. The development of algorithms to deal with polynomialsystems over the reals is based on the concepts of effective realalgebraic geometry in which the class of semi-algebraic setsconstitute the main objects.This thesis focuses on three problems below, which appear in manyapplications and are widely studied in computer algebra and effectivereal algebraic geometry:- Classify the real solutions of a parametric polynomial system according to the values of the parameters;- One-block quantifier elimination, which is also the computation of the projection of a semi-algebraic set- Computation of the isolated points of a semi-algebraic set.We designed new symbolic algorithms with better complexity than thestate-of-the-art. In practice, our efficient implementations of thesealgorithms are capable of solving applications beyond the reach of thestate-of-the-art software.
... Hermite matrices and Hermite bilinear forms were introduced by Hermite in 1850 [14] for univariate polynomials and were extended to the multivariate zero-dimensional setting in [7,23]. Hermite matrices have many applications, including counting real roots [7,15,16] and locating them [4]. Assume that we are given the ideal I := f 1 , . . . ...
... First we define the signature of a matrix. The classical univariate Hermite Theorem [16] was generalized to the multivariate case by Pedersen, Roy and Szpirglas [20], and was also proved in [7] and [11]: ...
Let I=<f_1, ..., f_m> be a zero dimensional radical ideal Q[x_1,...,x_n]. Assume that we are given approximations {z_1,...,z_k} in C^n for the common roots V(I)={xi_1,...,xi_k}. In this paper we show how to construct and certify the rational entries of Hermite matrices for I from the approximate roots {z_1, ...,z_k}. When I is non-radical, we give methods to construct and certify Hermite matrices for the radical of I from approximate roots. Furthermore, we use signatures of these Hermite matrices to give rational certificates of non-negativity of a given polynomial over a (possibly positive dimensional) real variety, as well as certificates that there is a real root within an epsilon distance from a given point z in Q^n.
... As a consequence, Hurwitz polynomials are widely studied in the literature (see, for instance, [20,21]), and there are many criteria to determine if a given polynomial with real coefficients is Hurwitz without explicitly computing its zeros. Among many others, we have the Routh-Hurwitz criterion [20], the stability test [22], the continued fraction method [20], and the Hermite-Biehler theorem [23]. The latter determines the Hurwitz character of a polynomial f (x) by verifying some properties of two polynomials associated with f . ...
... Proof. If P n (x) and Q n−1 (x) have simple and interlaced roots in (−2, 2), then the polynomial φ 2n−1 (z) obtained from Equation (20) is a Schur (and thus orthogonal) monic polynomial with real coefficients, from Theorem 3. By using Equations (23) and (24) for φ 2n−1 (z) we obtain again P n (x) and Q n−1 (x), and therefore P n (x) is orthogonal with respect to some µ supported on [−2, 2] and Q n−1 (x) is orthogonal with respect to (x 2 − 4)dµ. The polynomials P n−1 , . . . ...
A new criterion for Schur stability is derived by using basic results of the theory of orthogonal polynomials. In particular, we use the relation between orthogonal polynomials on the real line and on the unit circle known as the Szego transformation. Some examples are presented.
... The problem of determining the number of zeros of a polynomial in a given region of the complex plane is very classical and goes back to Descartes, Gauss, Cauchy [4], Routh [22,23], Hermite [9], Hurwitz [14], and many others. The entire second volume of the delightful Problems and Theorems in Analysis by Pólya and Szegő [21] is devoted to this and related problems. ...
... We shall discuss obstacles to the converse statement for M > 2 in Sect. 9. Section 2 is devoted to a generalized Euclidean algorithm, which is crucial to our proof. ...
Given a polynomial with positive
coefficients , and a positive integer , we define a(n infinite)
generalized Hurwitz matrix . We prove that the
polynomial f(z) does not vanish in the sector whenever the matrix is totally
nonnegative. This result generalizes the classical Hurwitz' Theorem on stable
polynomials (M=2), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials
with negative real roots (M=1), and the Cowling-Thron theorem (M=n). In
this connection, we also develop a generalization of the classical Euclidean
algorithm, of independent interest per se.
... a 0 a г a 2 a 3 a 4 aк (a t a 2a 0 a 3 )ja 1 (a x a A -a 0 a 5 )ja 1 (a x a 6a 0 a 1 )ja 1 ... In this part we consider polynomials with real coefficients. The coefficients of the related (even primary) polynomials calculated by (5.4) will be real as well. ...
... a 0 a г a 2 a 3 a 4 aк (a t a 2a 0 a 3 )ja 1 (a x a A -a 0 a 5 )ja 1 (a x a 6a 0 a 1 )ja 1 ... In this part we consider polynomials with real coefficients. The coefficients of the related (even primary) polynomials calculated by (5.4) will be real as well. ...
... Since e −i θ 2 M 2 (z)p(z − h) has all zeros in the upper half-plane (Im h > 0) or in the lower half-plane (Im h < 0), the polynomial ∞ k=0 c k z k has only real zeros by Hermite-Biehler theorem [9,2] (see also [13] or [14, Chapter VII]). ...
We completely describe all finite difference operators of the form preserving the Laguerre-P\'olya class of entire functions. Here and are some complex functions and h is a nonzero complex number.
... A polynomial f is stable if the condition f (z) = 0 implies Re z < 0. The following is a version of the Hermite-Biehler theorem ( [9], [3]). ...
Simple proofs of the Hermite-Biehler and Routh-Hurwitz theorems are presented. The total nonnegativity of the Hurwitz matrix of a stable real polynomial follows as an immediate corollary.
... Furthermore, much information has been generated because of the importance of these polynomials in the study of systems' stability: if the characteristic polynomial of a linear system has all its roots with negative real part, that is if the polynomial is Hurwitz, then the origin is an asymptotically stable equilibrium point of the corresponding linear system. The Routh-Hurwitz criterion (see Hurwitz [1]) is undeniably the most popular criterion, but it is also worth mentioning the Hermite-Biehler theorem (see Hermite [2]), the Lienard-Chipart conditions (see Gantmacher [3]) or the stability test (see Bhattacharyya et al. [4]). ...
In this paper, we study the Hurwitz stability of polynomials. By considering that a degree Hurwitz polynomial has its corresponding Markov parameters, we define the set in Section 3, we also define . Based on properties of the Hankel matrices and the stability test, as well as by using ideas of differential topology, we show that is a fiber bundle with a base. This result allows us to obtain an interesting application: Given a Hurwitz polynomial, we can generate two families of positive definite Hankel matrices.
... Then, in [98], Sylvester generalized Sturm's method to problem (2.2). Finally, in [58], C. Hermite developed a general theory giving the number of real roots as the signature of a Hankel matrix. However, many drawbacks are present in using Sturm sequence for computing (2.2). ...
In this dissertation, we study the computation of the L∞-norm of finite-dimensional linear time-invariant systems. This problem is first reduced to the computation of the maximal y-projection of the real solutions (x, y) of a bivariate polynomial equations system Σ = {P = 0, ∂P/∂x = 0}, where P ∈ Z[x, y]. Then, we use standard computer algebra methods to solve this problem. In particular, we alternatively study a method based on rational univariate representations, a method based on root separation, and finally a method first based on the sign variation of the leading coefficients of a signed subresultant sequence (Sturm-Habicht) and on the identification of an isolating interval for the maximal y-projection of the real solutions of Σ. We then compute the worst-case bit complexity of each method and compare their theoretical behavior. We also implement each method in Maple and compare their practical behavior (average complexity). A generalization of the above algorithms is finally proposed to the case where the polynomial P also depends on a set of parameters α = [α1, . . . , αd] ∈ Rd. To do that, we solve the problem using the notion of the Cylindrical Algebraic Decomposition, well-known in algebraic geometry.
... The idea is to write the PGF of 2X/3 as g 0 (x 2 ) + xg 1 (x 2 ), where g 0 and g 1 have interlacing roots. By the Hermite-Biehler theorem [12,53], such polynomials are Hurwitz stable. This means that the PGF of 2X/3 can be factorized into polynomials with positive coefficients of degrees no greater than 2. Thus, 2X/3 is a Poisson multinomial variable, that is the sum of independent random variables with values in {0, 1, 2}. ...
This is an expository article on the Poisson binomial distribution.We review lesser known results and recent progress on this topic, including geometry of polynomials and distribution learning. We also provide examples to illustrate the use of the Poisson binomial machinery. Some open questions of approximating rational fractions of the Poisson binomial are presented.
... Among the most widely known criteria for determining the stability of a given polynomial, we can mention the Hurwitz-Routh test [4] and the stability test [5]. Other important characterizations are the Hermite-Biehler theorem [6] and Markov's parameters theorem [2]. e latter two characterizations play a central role in the relation between Hurwitz polynomials and orthogonal polynomials, as it will be discussed in the sequel. ...
In this contribution, we use the connection between stable polynomials and orthogonal polynomials on the real line to construct sequences of Hurwitz polynomials that are robustly stable in terms of several uncertain parameters. These sequences are constructed by using properties of orthogonal polynomials, such as the well-known three-term recurrence relation, as well as by considering linear combinations of two orthogonal polynomials with consecutive degree. Some examples are presented.
... To do that, we rely on well-known properties of Hermite's quadratic form to count the real roots of zero-dimensional ideals; see [7]. Basically, given a zero-dimensional ...
Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may ask whether the corresponding linear series contains an effective divisor with totally real support. This translates into a particular type of parametrized real root counting problem that we wish to solve exactly. On the other hand, it is known that for a given genus and number of real connected components, any linear series of sufficiently large degree contains a totally real effective divisor. Using the algorithms described in this paper, we solve a number of examples, which we can compare to the best known bounds for the required degree.
... To do that, we rely on well-known properties of Hermite quadratic forms to count the real roots of zero-dimensional ideals ; see (Hermite, 1856). Basically, given a zero-dimensional ideal I ⊂ Q[x], Hermite's quadratic form operates on the finite dimensional Q-vector space A := Q[x]/I as follows ...
We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let with and , be the algebraic set defined by f and be the projection . Under the assumptions that f admits finitely many complex roots for generic values of y and that the ideal generated by f is radical, we solve the following problem. On input f, we compute semi-algebraic formulas defining semi-algebraic subsets of the y-space such that is dense in and the number of real points in is invariant when varies over each . This algorithm exploits properties of some well chosen monomial bases in the algebra where I is the ideal generated by f in and the specialization property of the so-called Hermite matrices. This allows us to obtain compact representations of the sets by means of semi-algebraic formulas encoding the signature of a symmetric matrix. When f satisfies extra genericity assumptions, we derive complexity bounds on the number of arithmetic operations in and the degree of the output polynomials. Let d be the maximal degree of the 's and , we prove that, on a generic , one can compute those semi-algebraic formulas with operations in and that the polynomials involved have degree bounded by D. We report on practical experiments which illustrate the efficiency of our algorithm on generic systems and systems from applications. It allows us to solve problems which are out of reach of the state-of-the-art.
... The idea is to write the PGF of 2X/3 as g 0 (x 2 )+xg 1 (x 2 ), where g 0 and g 1 have interlacing roots. By the Hermite-Biehler theorem [10,51], such polynomials are Hurwitz stable. This means that the PGF of 2X/3 can be factorized into polynomials with positive coefficients of degrees no greater than 2. Thus, 2X/3 is a Poisson multinomial variable, that is the sum of independent random variables with values in {0, 1, 2}. ...
This is an expository article on the Poisson binomial distribution. We review lesser known results and recent progress on this topic, including geometry of polynomials and distribution learning. We also provide examples to illustrate the use of the Poisson binomial machinery. Some open questions of approximating rational fractions of the Poisson binomial are presented.
... Information about Hurwitz polynomials can be found in [2,3]. The Routh-Hurwitz cri- terion is probably the most famous method for checking if a polynomial is a Hurwitz polynomial, but it is worth mentioning the Hermite-Biehler theorem [4,5], the Stabil- ity Test [6] and the method of Continued Fractions (Stieltjes's Theorem, see [2]). Some generalizations of Hurwitz polynomials have been discussed in [7]. ...
In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.
... For this purpose, we recall the results of Grommer and Chebotarev in a special form adjusted to the situation concerning the entire functions of order 1, which is the case of interest here. Chebotarev generalized the theorem of Grommer which, in its turn, is a generalization of an analogous statement known for polynomials and attributed to Hermite [9]. For further research on this account, the reader may also consult [4,20,21]. ...
We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated.
... Determining the stability of real polynomials is of fundamental importance in the study of dynamical systems and as such, several equivalent characterisations have been given. One such characterisation is the Hermite-Biehler Theorem [6,3], a proof of which can also be found in [7]. The Hermite-Biehler Theorem has been instrumental in the study of the "robust parametric stability problem", that is, the problem of guaranteeing that stability is preserved by real coefficient perturbations (see [8,2]). ...
If a real polynomial is Hurwitz stable (every root if f lies in the open left half-plane), then the Hermite-Biehler Theorem says that the polynomials and have interlacing real roots. We extend this result to general polynomials by giving a lower bound on the number of real roots of and and showing that these real roots interlace. This bound depends on the number of roots of f which lie in the left half plane. Another classical result in the theory of polynomials is Descartes' Rule of Signs, which bounds the number of positive roots of a polynomial in terms of the number of sign changes in its coefficients. We use our extension of the Hermite-Biehler Theorem to give an inverse rule of signs for polynomials with one positive root.
... Since e −i θ 2 M 2 (z)p(z − h) has all zeros in the upper half-plane (Im h > 0) or in the lower half-plane (Im h < 0), the polynomial ∞ k=0 c k z k has only real zeros by Hermite-Biehler theorem [9,2] (see also [13] or [14, Chapter VII]). ...
We prove that the difference operator of the following form:
where and are some complex functions and h is a non-zero complex number, preserves the Laguerre–Pólya class of entire functions if and only if, the step h is pure imaginary, , and the function is entire of order at most 2 and belongs to a subclass of the Hermite–Biehler class of entire functions.
... The approach to problems of root location via the use of quadratic forms goes back to Jacobi. The greatest impetus to this line of research was given by Hermite (1856). A very thorough exposition of the method of quadratic forms is given in the classic paper Krein and Naimark (1936). ...
In this chapter we discuss both the general theory of quadratic forms, the notion of congruence and its invariants, and the applications of the theory to the analysis of special forms. We focus on quadratic forms induced by rational functions, most notably the Hankel and Bezout forms, because of their connection to system-theoretic problems like stability and signature-symmetric realizations. These forms use as their data different representations of rational functions, power series, and coprime factorizations, respectively. But we also will discuss the partial fraction representation in relation to the computation of the Cauchy index of a rational function, the proof of the Hermite-Hurwitz theorem, and the continued fraction representation as a tool in the computation of signatures of Hankel matrices as well as in the problem of Hankel matrix inversion. Thus, different representations of rational functions, that is, different encodings of the information carried by a rational function, provide efficient starting points for different methods. The results obtained for rational functions are applied to root location problems for polynomials in the next chapter.
... In this paper we will be concerned with the following general question: Question 1.1. Given a finite group G, which quadratic forms over K are trace forms of G-Galois extensions L/K? Question 1.1 was studied in the mid-19th century; in particular, Sylvester [14], Jacobi [7], and Hermite [5], [6] independently proved that the number of real roots of a polynomial p(x) ∈ R[x] equals the signature of the trace form of the Galois algebra R[x]/(p(x)); see [1,Section 1]. There has been a resurgence of interest in this topic at the end of the twentieth century, due in part, to an influential paper of Serre [13], relating the trace form to the extension problem in inverse Galois theory. ...
Let G be a finite group containing a non-abelian Sylow 2-subgroup. We elementarily show that every G-Galois field extension L/K has a hyperbolic trace form in the presence of root of unity.
... Theorem 2.5 (Hermite-Biehler theorem, see [11,19]). In order for the polynomial ...
Inverse problem of recovering masses, coefficients of damping and lengths of the intervals between the masses using two spectra of boundary value problems and the total length of the Stieltjes string (an elastic thread bearing point masses) is considered. For the case of point-wise damping at the first counting from the right end mass the problem of recovering the masses, the damping coefficient and the lengths of the subintervals by one spectrum and the total length of the string is solved.
... While a q > 0 for all q is necessary for stability, it is not sufficient [5]. The Routh-Hurwitz conditions [3][4][5], which are mathematically equivalent to the original criteria formulated by Hermite and the related criteria embodied in Lyapunov's second method [9,[15][16][17][18][19], provide both necessary and sufficient conditions for steady-state stability. While these conditions have traditionally been obtained through the use of arcane mathematics, a remarkably simple proof has been found more recently requiring only basic algebra and continuity arguments [8]. ...
In analogy to chemical reaction networks, I demonstrate the utility of expressing the governing equations of an arbitrary dynamical system (interaction network) as sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries) for analysis of its stability. The reaction stoichiometries and first derivatives define the network's "influence topology", a signed directed bipartite graph. Parameter reduction of the influence topology permits simplified expression of the principal minors (sums of products of non-overlapping bipartite cycles) and Hurwitz determinants (sums of products of the principal minors or the bipartite cycles directly) for assessing the network's steady state stability. Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution). Any further explicit algebraic specification of the network will project onto this stability phase space. Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.
... It was introduced in the middle of the 19th century and appears in the context of the theory of equations as a basic tool in the study of location of zeros for real and complex polynomials, in stability LINEAR ALGEBRA AND ITS APPLICATIONS 122/123,'124:1039-1097 (1989) @ Elsevier Science Publishing Co., Inc., 1989 theory as well as in classical elimination theory. The naming of Bezoutians goes back to Sylvester (1853) and it has been used to great advantage among others by Sylvester, by Cayley, and in a very powerful way by Hermite (1856). In spite of the fact that it is an extremely useful tool, the Bezoutian is hardly mentioned any more in general algebra texts, due to the trend in mathematics towards more abstraction. ...
We survey the theory of Béoutians with a special emphasis on its relation to system theoretic problems. Some instances are the connections with realization theory in particular signature symmetric realizations, the Cauchy index, stability, and the characterization of output feedback invariants. We describe canonical forms and invariants for the action of static output feedback on scalar linear systems of McMillan degree n. Previous results on this subject are obtained in a new and unified way, by making use of only a few elementary properties of Bézout matrices. As new results we obtain a minimal complete set of 2n-2 independent invariants, an explicit example of a continuous canonical form for the case of odd McMillan degree, and finally a canonical form which induces a cell decomposition of the quotient space for output feedback
... The most known criteria for Hurwitz and Schur polynomials are the Routh-Hurwitz criterion (Hurwitz (1985)), and the Jury's Test (Jury (1977)), respectively. In both cases there exist the corresponding Hermite-Biehler Theorems (see Hermite (1856), and Bose (1993)) and the Stability Tests (Bhattacharyya et al. (1995)). Some good references about Hurwitz and Schur polynomials are the books Gantmacher (1959), Bhattacharyya et al. (1995) and Jury (1982). ...
In this paper we study a kind of even degree polynomials of a special form. Necessary and sufficient conditions are given in order to decide if such polynomials have all of their roots on the unit circle. Next, we apply this results to obtain sufficient conditions to have the Schur stability of a segment of polynomials.
... Theorem 2.1. (Hermite-Biehler theorem, see [20], [21]). In order that the polynomial ...
... This is a problem in algebra. The classical result is a theorem of Fujiwara [2], proved in 1926 by a method which goes back to Hermite [3] (the celebrated "Hermitian forms"): ...
... The approach to problems of root location via the use of quadratic forms goes back to Jacobi. The greatest impetus to this line of research was given by Hermite (1856). A very thorough exposition of the method of quadratic forms is given in the classic paper Krein and Naimark (1936). ...
Scalar-valued meromorphic Herglotz-Nevanlinna functions are characterized by the interlacing property of their poles and zeros together with some growth properties. We give a characterization of matrix-valued Herglotz-Nevanlinna functions by means of a higher-order interlacing property. As an application we deduce a matrix version of the classical Hermite-Biehler Theorem for entire functions.
We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of the parameters. More precisely, let f=(f1,…,fm)⊂Q[y][x] with y=(y1,…,yt) and x=(x1,…,xn), V⊂Ct×Cn be the algebraic set defined by the simultaneous vanishing of the fi's and π be the projection (y,x)→y. Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the ideal generated by f is radical, we solve the following algorithmic problem. On input f, we compute semi-algebraic formulas defining open semi-algebraic sets S1,…,Sℓ in the parameters' space Rt such that ∪i=1ℓSi is dense in Rt and, for 1≤i≤ℓ, the number of real points in V∩π−1(η) is invariant when η ranges over Si.
This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra Q(y)[x]/I where I⊂Q(y)[x] is the ideal generated by f in Q(y)[x] as well as the specialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain “compact” representations of the semi-algebraic sets Si by means of semi-algebraic formulas encoding the signature of a given symmetric matrix.
When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gröbner bases to derive complexity bounds both on the number of arithmetic operations in Q and the degree of the output polynomials. More precisely, letting d be the maximal degrees of the fi's and D=n(d−1)dn, we prove that, on a generic input f=(f1,…,fn), one can compute those semi-algebraic formulas using O˜((t+Dt)23tn2t+1d3nt+2(n+t)+1) arithmetic operations in Q and that the polynomials involved in these formulas have degree bounded by D.
We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametric systems coming from applications since it allows us to solve systems which were out of reach on the current state-of-the-art.
Let be univariate polynomials with rational coefficients and be the ideal they generate. Assume that we are given approximations for the common roots . In this study, we describe a symbolic-numeric algorithm to construct a rational matrix, called Hermite matrix, from the approximate roots and certify that this matrix is the true Hermite matrix corresponding to the roots . Applications of Hermite matrices include counting and locating real roots of the polynomials and certifying their existence.
Simple conditions based on generalisations of the Routh-Hurwitz and Mikhailov criteria that ensure the absence of polynomial roots in an RHP sector straddling the positive real semi--axis (S-stability) are presented. In particular, it is shown that S-stability is ensured if the phase variation of a suitable power of the original n-th degree characteristic polynomial is equal to n right angles, which implies that the zeros of the real and imaginary parts of this power must satisfy an interlacing property similar to the interlacing property satisfied by Hurwitz polynomials according to the classic Hermite-Biehler theorem. The condition can be checked by means of Sturm sequences. Examples show how the proposed methods operate.
Mathematical structures often start from simple ones and are extended by various constructions to structures of increasing complexity. This process is to be controlled, and the guiding lines should include, among other things, applicability to problems of interest. The present chapter is devoted to a circle of ideas from abstract linear algebra that covers several topics of interest to us because of their applicability to the study of linear systems. These topics include bilinear forms defined on vector spaces, module homomorphisms over various rings, and the analysis of classes of special structured matrices such as Bezoutian, Hankel, and Toeplitz matrices.
The method of stability analysis presented in this study allows a definition to be made of a reduced order model of a high-order, Lurie-Post-nikov system. This reduced order system has two main properties for our purpose: linear conjecture can be applied to it, and the stability of its equilibrium involves the same property for the equilibrium of the original system. Moreover these stability conditions may be easily checked and they directly determine an admissible sector for variations of the non-linear static gain. A common algebraic condition appears in the different stated theorems. A section deals with solving this condition, and a method based on the use of a particular algebraic array is proposed. In order to sum up and apply these results, a methodology for systems analysis is implemented, which both computes the expression of the reduced order system and the associated stability conditions. An example of a fourth-order system is then presented to illustrate the different steps of the study.
In considering the stability of mechanical systems we are led to the characteristic equation p(λ)=0. Continuous-time systems are stable if all the roots of this equation are in the left half-plane (Hurwitz stability), while discrete-time systems require all |λ|<1 (Schur stability). Hurwitz stability has been treated by the Cauchy index and Sturm sequences, leading to various determinantal criteria and Routh's array, and several other methods. We also have to consider the question of robust stability, i.e. whethera system remains stable when its coefficients vary. In the Hurwitz case Kharitonov's theorem reduces the answer to the consideration of 4 extreme polynomials, and other authors consider cases where the coefficients depend on parameters in various ways. Schur stability is notably dealt with by the Schur-Cohn algorithm, which constructs a sequence of polynomials and tests whether all their constant terms are negative. Methods are described which reduce overflow in this process. Robust Schur stability is harder to deal with than Hurwitz, but several partial solutions are described.
Recursive filters Stability criteria Algorithms used in stability tests Linear predictive coding Appendix A: demonstration of the Schur-Cohn criterion Appendix B: optimum 2-D stability criteria Bibliography
Let f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m=3 we can approximate within ε all the roots of f in the interval [0,R] using just arithmetic operations. In particular, we can count the number of roots in any bounded interval using just arithmetic operations. Our speed-ups are significant and near-optimal: The asymptotically sharpest previous complexity upper bounds for both problems were super-linear in D, while our algorithm has complexity close to the respective complexity lower bounds. We also discuss conditions under which our algorithms can be extended to general m, and a connection to a real analogue of Smale's 17th Problem.
The relationship between the topology of a parameter set of candidate systems and the convergence analysis of recursive identification algorithms on this parameter set is studied using, for example, the “ODE approach” of Ljung together with classical differential-topological methods for the qualitative analysis of differential equations. Combined with recent studies ([4], [8], [10], [12], [19] and [23]) on the topology of certain spaces of systems this analysis suggests that, topologically, the defining conditions for the candidate systems in a globally convergent identification scheme have the effect of forcing the parameter space X of candidates to be Euclidean. This interpretation of such defining properties as stable, miniphase, constant controllability indices, etc., is supported by examples from the literature (esp. [1], [3], [16]).
The problem of solid body dynamics in the central Newton field of forces is considered. This motion is described by the well-known system of Euler-Poisson equations. By original change of variables the system is reduced to the normal form with the first integral of norm type. The solution of this system is considered as the non-perturbed motion and its stability is investigated. The procedure is outlined for obtaining the steady motion in general case. The controlling force nature is defined. The methodology is applied to three cases with special restrictions on the body’s inertia moments, so-called generalized classical cases of Euler, Lagrange and Kovalevskaya.
Let x 1 ,...,x n be real numbers, P(x)=p n (x-x 1 )⋯(x-x n ), and Q(x) be a polynomial of order less than or equal to n. Denote by Δ(Q) the matrix of generalized divided differences of Q(x) with nodes x 1 ,...,x n and by B(P,Q) the Bézoutian matrix of P and Q. A relationship between the corresponding principal minors of the matrices B(P,Q) and Δ(Q) counted from the right lower corner is established. It implies that if the principal minors of the matrix of divided differences of a function g(x) are positive or have alternating signs then the roots of the Newton’s interpolation polynomial of g are real and separated by the nodes of interpolation.
A deep and detailed historiographical analysis of a particular case study
concerning the so-called Lee-Yang theorem of theoretical statistical mechanics
of phase transitions, has emphasized what real historical roots underlie such a
case study. To be precise, it turned out that some well-determined aspects of
entire function theory have been at the primeval origins of this important
formal result of statistical physics.
The Cauchy index of a real rational function can be computed by evaluating the signature of a certain Hankel matrix. Alternative procedures for its computation are presented here, one of which offers greater computational simplicity.
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