On a Class of Matrix Pencils and ℓ -ifications Equivalent to a Given Matrix Polynomial
OON A CLASS OF MATRIX PENCILS AND (cid:96) -IFICATIONSEQUIVALENT TO A GIVEN MATRIX POLYNOMIAL ˚ DARIO A. BINI : AND
LEONARDO ROBOL ; Abstract.
A new class of linearizations and (cid:96) -ifications for m ˆ m matrix polynomials P p x q ofdegree n is proposed. The (cid:96) -ifications in this class have the form A p x q “ D p x q`p e b I m q W p x q where D is a block diagonal matrix polynomial with blocks B i p x q of size m , W is an m ˆ qm matrix polynomialand e “ p , . . . , q t P C q , for a suitable integer q . The blocks B i p x q can be chosen a priori, subjectedto some restrictions. Under additional assumptions on the blocks B i p x q the matrix polynomial A p x q is a strong (cid:96) -ification, i.e., the reversed polynomial of A p x q defined by A p x q : “ x deg A p x q A p x ´ q is an (cid:96) -ification of P p x q . The eigenvectors of the matrix polynomials P p x q and A p x q are relatedby means of explicit formulas. Some practical examples of (cid:96) -ifications are provided. A strategy forchoosing B i p x q in such a way that A p x q is a well conditioned linearization of P p x q is proposed. Somenumerical experiments that validate the theoretical results are reported. AMS classification:
Keywords:
Matrix polynomials, matrix pencils, linearizations, companion matrix,tropical roots.
1. Introduction.
A standard way to deal with an m ˆ m matrix polynomial P p x q “ ř ni “ P i x i is to convert it to a linear pencil, that is to a linear matrix poly-nomial of the form L p x q “ Ax ´ B where A and B are mn ˆ mn matrices such thatdet P p x q “ det L p x q . This process, known as linearization , has been considered in[17].In certain cases, like for matrix polynomials modeling Non-Skip-Free stochasticprocesses [4], it is more convenient to reduce the matrix polynomial to a quadraticpolynomial of the form Ax ` Bx ` C , where A, B, C are matrices of suitable size[4]. The process that we obtain this way is referred to as quadratization . If P p x q isa matrix power series, like in M/G/1 Markov chains [25, 26], the quadratization of P p x q can be obtained with block coefficients of infinite size [28]. In this framework,the quadratic form is desirable since it is better suited for an effective solution ofthe stochastic model; in fact it corresponds to a QBD process for which there existefficient solution algorithms [4], [22]. In other situations it is preferable to reduce thematrix polynomial P p x q of degree n to a matrix polynomial of lower degree (cid:96) . Thisprocess is called (cid:96) -ification in [15].Techniques for linearizing a matrix polynomial have been widely investigated.Different companion forms of a matrix polynomial have been introduced and analyzed,see for instance [2, 14, 24] and the literature cited therein. A wide literature existson matrix polynomials with contribution of many authors [1, 10, 12, 13, 14, 17, 19,20, 21, 30, 33], motivated both by the theoretical interest of this subject and by themany applications that matrix polynomials have [4, 22, 23, 25, 25, 32]. Techniques forreducing a matrix polynomial, or a matrix power series into quadratic form, possiblywith coefficients of infinite size, have been investigated in [4, 28]. Reducing a matrixpolynomial to a polynomial of degree (cid:96) is analyzed in [15].Denote by C r x s m ˆ m the set of m ˆ m matrix polynomials over the complex field C . If P p x q “ ř ni “ P i x i P C r x s m ˆ m and P n ‰ P p x q has degree n . Ifdet P p x q is not identically zero we say that P p x q is regular . Throughout the paper ˚ Work supported by Gruppo Nazionale di Calcolo Scientifico (GNCS) of INdAM : Dipartimento di Matematica, Universit`a di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa( [email protected] ) ; Scuola Normale Superiore, Piazza dei Cavalieri, Pisa ( [email protected] )1 a r X i v : . [ m a t h . NA ] J un e assume that P p x q is a regular polynomial of degree n . The following definition isuseful in our framework. Definition 1.1.
Let P p x q P C r x s m ˆ m be a matrix polynomial of degree n . Let q be an integer such that ă q ď n . We say that a matrix polynomial Q p x q P C r x s qm ˆ qm is equivalent to P p x q , and we write P p x q « Q p x q if there exist two matrix polynomials E p x q , F p x q P C r x s qm ˆ qm such that det E p x q and det F p x q are nonzero constants, thatis E p x q and F p x q are unimodular , and E p x q Q p x q F p x q “ „ I m p q ´ q P p x q “ : I m p q ´ q ‘ P p x q . Denote P p x q “ x n P p x ´ q the reversed polynomial obtained by reverting the order ofthe coefficients. We say that the polynomials P p x q and Q p x q are strongly equivalent if P p x q « Q p x q and P p x q « Q p x q . If the degree of Q p x q is 1 and P p x q « Q p x q we say that Q p x q is a linearization of P p x q . Similarly, we say that Q p x q is a stronglinearization if Q p x q is strongly equivalent to P p x q and deg Q p x q “ . If Q p x q hasdegree (cid:96) we use the terms (cid:96) -ification and strong (cid:96) -ification . It is clear from the definition that P p x q « Q p x q implies det P p x q “ κ det Q p x q where κ is some nonzero constant, but the converse is not generally true. The equiva-lence property is actually stronger because it preserves also the eigenstructure of thematrix polynomial, and not only the eigenvalues. For a more in-depth view of thissubject see [15].In the literature, a number of different linearizations have been proposed. Themost known are probably the Frobenius and the Fiedler linearizations [13]. One ofthem is, for example, xA ´ B “ x »———– I m . . . I m P n fiffiffiffifl ´ »———– ´ P I m ´ P . . . ... I m ´ P n ´ fiffiffiffifl , (1.1)where I m denotes the identity matrix of size m . In this paper we provide a general way to transforma given m ˆ m matrix polynomial P p x q “ ř ni “ P i x i of degree n into a stronglyequivalent matrix polynomial A p x q of lower degree (cid:96) and larger size endowed with astrong structure. The technique relies on representing P p x q with respect to a basisof matrix polynomials of the form C i p x q “ ś qj “ , j ‰ i B j p x q , i “ , . . . , q , where B i p x q such that deg B i p x q “ d i satisfy the following requirements:1. For every i, j , B i p x q B j p x q ´ B j p x q B i p x q “
0, i.e., the B i p x q commute;2. B i p x q and B j p x q are right coprime for every i ‰ j . This implies that thereexist α i,j p x q , β i,j p x q appropriate matrix polynomials such that B i p x q α i,j p x q` B j p x q β i,j p x q “ I .The above conditions are sufficient to obtain an (cid:96) -ification. In order to provide astrong (cid:96) -ification, we need the following additional assumptions1. deg B i p x q “ d for i “ , . . . , q ;2. B i p x q and B j p x q are right coprime for every i ‰ j .According to the choice of the basis we arrive at different (cid:96) -ifications A p x q , where (cid:96) ě r n { q s is determined by the degree of the B i p x q , represented as a q ˆ q blockdiagonal matrix with m ˆ m blocks plus a matrix of rank at most m . oreover, we provide an explicit version of right and left eigenvectors of A p x q inthe general case.An example of (cid:96) -ification A p x q is given by A p x q “ D p x q ` p e b I m qr W , . . . , W q s ,D p x q “ diag p B p x q , . . . , B q p x qq , e “ p , . . . , q t P C q ,B i p x q “ b i p x q I m , for i “ , . . . , q ´ ,B q p x q “ b q p x q P n ` sI m , d q “ deg b q p x q ,W i p x q P C r x s m ˆ m , deg W i p x q ă deg B i p x q , where b p x q , . . . , b q p x q are pairwise co-prime monic polynomials of degree d , . . . , d q ,respectively, such that n “ d ` ¨ ¨ ¨ ` d q , and s is such that λb q p ξ q ` s ‰ λ of P n and for any root ξ of b i p x q for i “ , . . . , q ´
1. The matrixpolynomial A p x q has degree (cid:96) “ max t d , . . . , d q u ě r nq s and size mq ˆ mq .If b i p x q “ x ´ β i are linear polynomials then deg A p x q “ P p x q can be viewedas the generalized eigenvalues of the matrix pencil A p x q A p x q “ x »———– I . . . I P n fiffiffiffifl ´ »———– β I . . . β n ´ I β n P n ´ sI fiffiffiffifl ` »————– I ...... I fiffiffiffiffifl r W . . . W n s where W i “ $’&’% P p β i q ś n ´ j “ , j ‰ i p β i ´ β j q pp β i ´ β n q P n ` sI m q ´ for i ă n, P p β n q ś n ´ j “ p β n ´ β j q ´ sI m ´ s ř n ´ j “ W i β n ´ β j otherwise . (1.2)If P p x q is a scalar polynomial then det A p x q “ ś ni “ p x ´ β i qp ř nj “ W j x ´ β j ` q so that the eigenvalue problem can be rephrased in terms of the secular equation ř nj “ W j x ´ β j ` “
0. Motivated by this fact, we will refer to this linearization as secular linearization .Observe that this kind of linearization relies on the representation of P p x q ´ ś ni “ B i p x q in the Lagrange basis formed by C i p x q “ ś nj “ , j ‰ i B j p x q , i “ , . . . , n which is different from the linearization given in [2] where the pencil A p x q has an ar-rowhead structure. Unlike the linearization of [2], our linearization does not introduceeigenvalues at infinity.This secular linearization has some advantages with respect to the Frobeniuslinearization (1.1). For a monic matrix polynomial we show that with the linearizationobtained by choosing β i “ ω in , where ω n is a principal n th root of 1, our linearizationis unitarily similar to the block Frobenius pencil associated with P p x q . By choosing β i “ αω in , we obtain a pencil unitarily similar to the scaled Frobenius one. With thesechoices, the eigenvalues of the secular linearization have the same condition numberas the eigenvalues of the (scaled) Frobenius matrix.This observation leads to better choices of the nodes β i performed according tothe magnitude of the eigenvalues of P p x q . In fact, by using the information providedby the tropical roots in the sense of [6], we may compute at a low cost particular valuesof the nodes β i which greatly improve the condition number of the eigenvalues. From n experimental analysis we find that in most cases the conditioning of the eigenvaluesof the linearization obtained this way is lower by several orders of magnitude withrespect to the conditioning of the eigenvalues of the Frobenius matrix even if it isscaled with the optimal parameter.Our experiments, reported in Section 6 are based on some randomly generatedpolynomials and on some problems taken from the repository NLEVP [3].We believe that the information about the tropical roots, used in [16] for providingbetter numerically conditioned problems, can be more effectively used with our (cid:96) -ification. This analysis is part of our future work.Another advantage of this representation is that any matrix in the form “diagonalplus low-rank” can be reduced to Hessenberg form H by means of Givens rotation witha low number of arithmetic operations provided that the diagonal is real. Moreover,the function p p x q “ det p xI ´ H q as well as the Newton correction p p x q{ p p x q can becomputed in O p nm q operations [8]. This fact can be used to implement the Aberthiteration in O p n m q ops instead of O p nm ` n m q of [5]. This complexity boundseems optimal in the sense that for each one of the mn eigenvalues all the m p n ` q data are used at least once.The paper is organized as follows. In Section 2 we provide the reduction of anymatrix polynomial P p x q to the equivalent form A p x q “ D p x q ` p e b I m qr W p x q , . . . , W q p x qs , that is, the (cid:96) -ification of P p x q . In Section 2.1 we show that P p x q is strongly equivalentto A p x q in the sense of Definition 1.1. In Section 3 we provide the explicit form ofleft and right eigenvectors of A p x q relating them to the corresponding eigenvectors of P p x q . In Section 4 we analyze the case where B i p x q “ b i p x q I for i “ , . . . , q ´ B q p x q “ b q p x q P n ` sI for scalar polynomials b i p x q . Section 5 outlines an algorithmfor computing the (cid:96) -ifications. Finally, in Section 6 we present the results of somenumerical experiments.
2. A diagonal plus low rank (cid:96) -ification.
Here we recall a known companion-like matrix for scalar polynomials represented as a diagonal plus a rank-one matrix,provide a more general formulation and then extend it to the case of matrix polynomi-als. This form was introduced by B.T. Smith in [29], as a tool for providing inclusionregions for the zeros of a polynomial, and used by G.H. Golub in [18] in the analysisof modified eigenvalue problems.Let p p x q “ ř ni “ p i x i be a polynomial of degree n with complex coefficients,assume p p x q monic, i.e., p n “
1, consider a set of pairwise different complex numbers β , . . . , β n and set e “ p , . . . , q t . Then it holds that [18], [29] p p x q “ det p xI ´ D ` ew t q ,D “ diag p β , . . . , β n q , w “ p w i q , w i “ p p β i q ś j ‰ i p β i ´ β j q . (2.1)Now consider a monic polynomial b p x q of degree n factored as b p x q “ ś qi “ b i p x q ,where b i p x q , i “ , . . . , q are monic polynomials of degree d i which are co-prime, thatis, gcd p b i p x q , b j p x qq “ i ‰ j , where gcd denotes the monic greatest commondivisor. Recall that given a pair u p x q , v p x q of polynomials there exist unique polyno-mials s p x q , r p x q such that deg s p x q ă deg v p x q , deg r p x q ă deg u p x q , and u p x q s p x q ` v p x q r p x q “ gcd p u p x q , v p x qq . From this property it follows that if u p x q and v p x q are o-prime, there exists s p x q such that s p x q u p x q ” v p x q . This polynomial canbe viewed as the reciprocal of u p x q modulo v p x q . Here and hereafter we denote u p x q mod v p x q the remainder of the division of u p x q by v p x q .This way, we may uniquely represent any polynomial of degree n in terms of thegeneralized Lagrange polynomials c i p x q “ b p x q{ b i p x q , i “ , . . . , q as follows. Lemma 2.1.
Let b i p x q , i “ , . . . , q be co-prime monic polynomials such that deg b i p x q “ d i and b p x q “ ś qi “ b i p x q has degree n . Define c i p x q “ b p x q{ b i p x q .Then there exist polynomials s i p x q such that s i p x q c i p x q ” b i p x q , moreover,any monic polynomial p p x q of degree n can be uniquely written as p p x q “ b p x q ` q ÿ i “ w i p x q c i p x q ,w i p x q ” p p x q s i p x q mod b i p x q , i “ , . . . , q, (2.2) where deg w i p x q ă d i .Proof . Since gcd p b i p x q , b j p x qq “ i ‰ j then b i p x q and c i p x q “ ś j ‰ i b j p x q are co-prime. Therefore there exists s i p x q ” { c i p x q mod b i p x q . Moreover, setting w i p x q ” p p x q s i p x q mod b i p x q for i “ , . . . , q , it turns out that the equation p p x q “ b p x q ` ř qi “ w i p x q c i p x q is satisfied modulo b i p x q for i “ , . . . , q . For the primality of b p x q , . . . , b q p x q , this means that the polynomial ψ p x q : “ p p x q´ b p x q´ ř qi “ w i p x q c i p x q is a multiple of ś qi “ b i p x q which has degree n . Since ψ p x q has degree at most n ´ ψ p x q “
0. That is (2.2) provides a representation of p p x q . Thisrepresentation is unique since another representation, say, given by ˜ w i p x q , i “ , . . . , q ,would be such that ř qi “ p ˜ w i p x q ´ w i p x qq c i p x q “
0, whence p ˜ w i p x q ´ w i p x qq c i p x q ” b i p x q . That is, for the co-primality of b i p x q and c i p x q , the polynomial ˜ w i p x q ´ w i p x q would be multiple of b i p x q . The property deg p b i p x qq ă deg p ˜ w i p x q ´ w i p x qq implies that ˜ w i p x q ´ w i p x q “ p p x q in Lemma 2.1 can be represented by means of the deter-minant of a (not necessarily linear) matrix polynomial as expressed by the followingresult which provides a generalization of (2.1) Theorem 2.2.
Under the assumptions of Lemma 2.1 we have p p x q “ det A p x q , A p x q “ D p x q ` e r w p x q , . . . , w q p x qs for D “ diag p b p x q , . . . , b q p x qq and e “ r , . . . , s t .Proof . Formally, one has A p x q “ D p x qp I ` D p x q ´ e r w p x q , . . . , w q p x qsq so thatdet A p x q “ det D p x q det p I q ` D p x q ´ e r w p x q , . . . , w q p x qsq“ b p x qp ` r w p x q , . . . , w q p x qs D p x q ´ e q , where b p x q “ ś qi “ b i p x q . Whence, we find that det A p x q “ b p x q ` ř qi “ w i p x q c i p x q “ p p x q , where the latter equality holds in view of Lemma 2.1.Observe that for d i “ w i are constantpolynomials. From the computational point of view, the polynomials w i p x q are ob-tained by performing a polynomial division since w i p x q is the remainder of the divisionof p p x q s i p x q by b i p x q . (cid:96) -ifications for matrix polynomials. The following technicalLemma is needed to prove the next Theorem 2.5.
Lemma 2.3.
Let B p x q , B p x q P C r x s m ˆ m be regular and such that B p x q B p x q “ B p x q B p x q . Assume that B p x q and B p x q are right co-prime, that is, there exist p x q , β p x q P C r x s m ˆ m such that B p x q α p x q ` B p x q β p x q “ I m . Then the ˆ block-matrix polynomial F p x q “ ” α p x q B p x q´ β p x q B p x q ı is unimodular.Proof . From the decomposition „ I m B p x q ´ B p x q „ α p x q B p x q´ β p x q B p x q “ „ α p x q B p x q I m we have ´ det B p x q det F p x q “ ´ det B p x q . Since B p x q is regular then det F p x q “ Lemma 2.4.
Let P p x q , Q p x q and T p x q be pairwise commuting right co-primematrix polynomials. Then P p x q Q p x q and T p x q are also right co-prime.Proof . We know that there exists α P p x q , β P p x q , α Q p x q , β Q p x q , matrix polynomialssuch that P p x q α P p x q ` T p x q β P p x q “ I, Q p x q α Q p x q ` T p x q β Q p x q “ I. We shall prove that there exist appropriate α p x q , β p x q matrix polynomials such that P p x q Q p x q α p x q ` T p x q β p x q “ I . We have P p x q Q p x qp α Q p x q α P p x qq ` T p x qp P p x q β Q p x q α P p x q ` β P p x qq “ P p x qp Q p x q α Q p x q ` T p x q β Q p x qq α P p x q ` T p x q β P p x q “ P p x q α P p x q ` T p x q β P p x q “ I, where the first equality holds since T p x q P p x q “ P p x q T p x q . So we can conclude thatalso P p x q Q p x q and T p x q are right coprime, and a possible choice for α p x q and β p x q is: α p x q “ α Q p x q α P p x q , β p x q “ P p x q β Q p x q α P p x q ` β P p x q . Now we are ready to prove the main result of this section, which provides an (cid:96) -ification of a matrix polynomial P p x q which is not generally strong. Conditions underwhich this (cid:96) -ification is strong are given in Theorem 2.7. Theorem 2.5.
Let P p x q “ ř ni “ P i x i , B p x q , . . . , B q p x q , and W p x q , . . . , W q p x q be polynomials in C r x s m ˆ m . Let C i p x q “ ś qj “ , j ‰ i B j p x q and suppose that the fol-lowing conditions hold:1. P p x q “ ś qi “ B i p x q ` ř qi “ W i p x q C i p x q ;2. the polynomials B i p x q are regular, commute, i.e., B i p x q B j p x q´ B j p x q B i p x q “ for any i, j , and are pairwise right co-prime.Then the matrix polynomial A p x q defined as A p x q “ D p x q ` p e b I m qr W p x q , . . . , W q p x qs , D p x q “ diag p B p x q , . . . , B q p x qq (2.3) is equivalent to P p x q , i.e., there exist unimodular q ˆ q matrix polynomials E p x q , F p x q such that E p x q A p x q F p x q “ I m p q ´ q ‘ P p x q .Proof . Define E p x q the following (constant) matrix: E p x q “ »—————– I m ´ I m I m ´ I m . . . . . . I m ´ I m I m fiffiffiffiffiffifl . direct inspection shows that E p x q A p x q “ »—————– B p x q ´ B p x q B p x q ´ B p x q . . . . . . B q ´ p x q ´ B q p x q W p x q W p x q ¨ ¨ ¨ W q ´ p x q B q p x q ` W q p x q fiffiffiffiffiffifl . Using the fact that the polynomials B i p x q are right co-prime, we transform the lattermatrix into block diagonal form. We start by cleaning B p x q . Since B p x q , B p x q areright co-prime, there exist polynomials α p x q , β p x q such that B p x q α p x q` B p x q β p x q “ I m . For the sake of brevity, from now on we write α, β , W i and B i in place of α p x q , β p x q , W i p x q and B i p x q , respectively. Observe that the matrix F p x q “ „ α B ´ β B ‘ I m p q ´ q . is unimodular in view of Lemma 2.3, moreover E p x q A p x q F p x q “ »—————– I m ´ B β B B ´ B . . . . . . B q ´ ´ B q W α ´ W β W B ` W B ¨ ¨ ¨ W m ´ B q ` W q fiffiffiffiffiffifl . Using row operations we transform to zero all the elements in the first column of thismatrix (by just adding multiples of the first row to the others). That is, there existsa suitable unimodular matrix E p x q such that E p x q E p x q A p x q F p x q “ »—————– I m B B ´ B . . . . . . B q ´ ´ B q W B ` W B ¨ ¨ ¨ W q ´ B q ` W q fiffiffiffiffiffifl . In view of Lemma 2.4, B B is right coprime with B . Thus, we can recursively applythe same process until we arrive at the final reduction step: E q ´ p x q . . . E p x q A p x q F p x q . . . F q ´ p x q “ I m p q ´ q ‘ ˜ q ź i “ B i p x q ` q ÿ i “ W i p x q C i p x q ¸ where the last diagonal block is exactly P p x q in view of assumption 1.Observe that if m “ B i p x q “ x ´ β i then we find that (2.1) provides thesecular linearization for P p x q .Now, we can prove that the polynomial equivalence that we have just presentedis actually a strong equivalence, under the following additional assumptionsdeg B i p x q “ d, i “ , . . . , q,n “ dq, deg W i p x q ă deg B i p x q B i p x q , are pairwise right co-prime . (2.4) emark 2.6. Recall that, according to Theorem 7.5 of [15] in an algebraicallyclosed field there exists a strong (cid:96) -ification for a degree n regular matrix polynomial P p x q P C m ˆ m r x s if and only if (cid:96) | nm . Conditions (2.4) satisfy this requirement, since (cid:96) “ d and d | n . To accomplish this task we will show that the reversed matrix polynomial of A p x q “ D p x q ` p e b I m q W , W “ r W , . . . , W q s , has the same structure as A p x q itself. Theorem 2.7.
Under the assumptions of Theorem 2.5 and of (2.4) the secular (cid:96) -ification given in Theorem 2.5 is strong.Proof . Consider A p x q “ x d A p x ´ q . We have A p x q “ diag p x d B p x ´ q , . . . , x d B q p x ´ qq ` p e b I m qr x d W p x ´ q , ¨ ¨ ¨ , x d W q p x ´ qs . This matrix polynomial is already in the same form as A p x q of (2.3) and verifies theassumptions of Theorem 2.5 since the polynomials x d B p x ´ q are pairwise right co-prime and commute with each other in view of equation (2.4). Thus, Theorem 2.5implies that A p x q is an (cid:96) -ification for q ź i “ x d B i p x ´ q ` q ÿ i “ x d W i p x ´ q ź j ‰ i x d B j p x ´ q“ x dq ˜ q ź i “ B i p x ´ q ` q ÿ i “ W i p x ´ q ź j ‰ i B j p x ´ q ¸ “ x n P p x ´ q “ P p x q , where the first equality follows from the fact that n “ dq in view of equation (2.4).This concludes the proof. Remark 2.8.
Note that in the case where the B i p x q do not have the same degree,the secular (cid:96) -ification might not be strong: the finite eigenstructure is preserved butsome infinite eigenvalues not present in P p x q will be artificially introduced.
3. Eigenvectors.
In this section we provide an explicit expression of right andleft eigenvectors of the matrix polynomial A p x q . Theorem 3.1.
Let P p x q be a matrix polynomial, A p x q its secular (cid:96) -ificationdefined in Theorem 2.5, λ P C such that det P p λ q “ , and assume that det B i p λ q ‰ for all i “ , . . . , q . If v A “ p v t , . . . , v tq q t P C mq is such that A p λ q v A “ , v A ‰ then P p λ q v “ where v “ ´ ś qi “ B i p λ q ´ ř qj “ W j v j ‰ . Conversely, if v P C m is anonzero vector such that P p λ q v “ , then the vector v A defined by v i “ ś j ‰ i B j p λ q v , i “ , . . . , q is nonzero and such that A p λ q v A “ .Proof . Let v A ‰ A p λ q v A “
0, so that B i p λ q v i ` q ÿ j “ W j p λ q v j “ , i “ , . . . , q. (3.1)Let v “ ´p ś qi “ B i p λ q ´ q ř qj “ W j p λ q v j . Combining the latter equation and (3.1)yields v i “ ´ B i p λ q ´ ˜ q ÿ j “ W j p λ q v j ¸ “ q ź j “ , j ‰ i B j p λ q v. (3.2)Observe that if v “ v , one has ř qj “ W j p λ q v j “ B i p λ q v i “
0. Since det B i p λ q ‰ i “ i so that v A “ P p λ q v “
0. In view of (3.2) we have P p λ q v “ q ź j “ B j p λ q v ` q ÿ i “ W i p λ q q ź j “ , j ‰ i B j p λ q v “ q ź j “ B j p λ q v ` q ÿ i “ W i p λ q v i . Moreover, by definition of v we get P p λ q v “ ´ q ź j “ B j p λ qp q ź i “ B i p λ q ´ q q ÿ i “ W i p λ q v i ` q ÿ i “ W i p λ q v i “ . Similarly, we can prove the opposite implication.A similar result can be proven for left eigenvectors. The following theorem relatesleft eigenvectors of A p x q and left eigenvectors of P p x q . Theorem 3.2.
Let P p x q be a matrix polynomial, A p x q its secular (cid:96) -ificationdefined in Theorem 2.5, λ P C such that det P p λ q “ , and assume that det B i p λ q ‰ .If u tA “ p u t , . . . , u tq q P C mq is such that u tA A p λ q “ , u A ‰ , then u t P p λ q “ where u “ ř qi “ u i ‰ . Conversely, if u t P p λ q “ for a nonzero vector u P C m then u tA A p λ q “ , where u A is a nonzero vector defined by u ti “ ´ u t W i p λ q B i p λ q ´ for i “ , . . . , q .Proof . If u tA A p λ q “ A p x q given in Theorem 2.5 wehave u ti B i p λ q ` ˜ q ÿ j “ u tj ¸ W i p λ q “ , i “ , . . . , q. (3.3)Assume that u “ ř qj “ u j “
0. Then from the above expression we obtain, for any i , u ti B i p λ q “ u i “ i since det B i p λ q ‰
0. This is in contradiction with u A ‰
0. From (3.3) we obtain u ti “ ´ u t W i p λ q B i p λ q ´ . Moreover, multiplying (3.3)to the right by ś qj “ , j ‰ i B j yields0 “ u ti q ź j “ B j p λ q ` u t W i p λ q q ź j “ , j ‰ i B j p λ q . Taking the sum of the above expression for i “ , . . . , q yields0 “ ˜ q ÿ i “ u ti ¸ q ź j “ B j p λ q ` u t q ÿ i “ W i p λ q q ź j “ , j ‰ i B j p λ q “ u t P p λ q . Conversely, assuming that u t P p λ q “
0, from the representation P p x q “ n ź j “ B j p x q ` q ÿ i “ W i p x q q ź j “ , j ‰ i B i p x q , defining u ti “ ´ u t W i p λ q B i p λ q ´ we obtain q ÿ i “ u ti “ ´ u t q ÿ i “ W i p λ q B i p λ q ´ “ ´ u t p P p λ q q ź j “ B j p λ q ´ ´ I q “ u t and therefore from (3.3) we deduce that u tA A p λ q “ B i p λ q “ i . .1. A sparse (cid:96) -ification. Consider the block bidiagonal matrix L having I m on the block diagonal and ´ I m on the block subdiagonal. It is immediate to verifythat L p e b I m q “ e b I m , where e “ p , , . . . , q t . This way, the matrix polynomial H p x q “ LA p x q is a sparse (cid:96) -ification of the form H p x q “ »——————– B p x q ` W p x q W p x q . . . W q ´ p x q W q p x q´ B p x q B p x q´ B p x q . . .. . . B q ´ p x q´ B q ´ p x q B q p x q fiffiffiffiffiffiffifl
4. A particular case.
In the previous section we have provided (strong) (cid:96) -ifications of a matrix polynomial P p x q under the assumption of the existence of therepresentation P p x q “ q ź i “ B i p x q ` q ÿ i “ W i p x q ź j ‰ i B j p x q (4.1)and under suitable conditions on B i p x q . In this section we show that a specific choiceof the blocks B i p x q satisfies the above assumptions and implies the existence of therepresentation (4.1). Moreover, we provide explicit formulas for the computation of W i p x q given P p x q and B i p x q .We provide also some additional conditions in order to make the resulting (cid:96) -ification strong. Assumption 1.
The matrix polynomials B i p x q are defined as follows B i p x q “ b i p x q I i “ , . . . , q ´ B q p x q “ b q p x q P n ` sI otherwisewhere b i p x q are scalar polynomials such that deg b i p x q “ d i and ř qi “ d i “ n ; thepolynomials b i p x q , i “ , . . . , q , are pairwise co-prime; s is a constant such that λb q p ξ q ` s ‰ for any eigenvalue λ of P n and for any root ξ of b i p x q , for i “ , . . . , q ´ . In this case it is possible to prove the existence of the representation (4.1). Werely on the Chinese remainder theorem that here we rephrase in terms of matrixpolynomials.
Lemma 4.1.
Let b i p x q , i “ , . . . , q be co-prime polynomials of degree d , . . . , d q ,respectively, such that ř qi “ d i “ n . If P p x q , P p x q are matrix polynomials of degreeat most n ´ then P p x q “ P p x q if and only if P p x q ´ P p x q ” b i p x q , for i “ , . . . , q .Proof . The implication P p x q ´ P p x q “ ñ P p x q ´ P p x q ” b i p x q istrivial. Conversely, if P p x q ´ P p x q ” b i p x q for every b i then the entries of P p x q´ P p x q are multiples of ś qi “ b i p x q for the co-primality of the polynomials b i p x q .But this implies that P p x q ´ P p x q “ P p x q ´ P p x q is at most n ´ ś qi “ b i p x q has degree n .We have the following Theorem 4.2.
Let P p x q “ ř ni “ x i P i be an m ˆ m matrix polynomial over analgebraically closed field. Under Assumption 1, set C i p x q “ ś j ‰ i B j p x q . Then there xists a unique decomposition P p x q “ B p x q ` q ÿ i “ W i p x q C i p x q , B p x q “ q ź i “ B i p x q , (4.2) where W i p x q are matrix polynomials of degree less than d i for i “ , . . . , q defined by W i p x q “ P p x q ś q ´ j “ , j ‰ i b j p x q p b q p x q P n ` sI m q ´ mod b i p x q , i “ , . . . , q ´ W q p x q “ ś q ´ j “ b j p x q P p x q ´ sI m ´ s q ´ ÿ j “ W j p x q b j p x q mod b q p x q . (4.3) Proof . We show that there exist matrix polynomials W i p x q of degree less than d i such that P p x q ´ B p x q ” ř qi “ W i p x q C i p x q mod b i p x q for i “ , . . . , q . Then weapply Lemma 4.1 with P p x q “ P p x q ´ B p x q that by construction has degree atmost n ´
1, and with P p x q “ ř qi “ W i p x q C i p x q , and conclude that P p x q “ B p x q ` ř qi “ W i p x q C i p x q . Since for i “ , . . . , q ´ b i p x q divides every entryof B p x q and of C j p x q for j ‰ i , we find that P p x q ” W i p x q C i p x q mod b i p x q , i “ , . . . , q. Moreover, for i ă q we have C i p x q “ ´ś j ‰ i,j ă q b j p x q I m ¯ p b q p x q P n ` sI m q .The first term is invertible modulo b i p x q since by assumption b i p x q is co-prime with b j for every j ‰ i . We need to prove that the matrix on the right is invertible modulo b i p x q , that is, its eigenvalues µ are such that b i p µ q ‰
0. Now, since the eigenvaluesof b q p x q P n ` sI m have the form µ “ b q p x q λ ` s , where λ is an eigenvalue of P n , it isenough to ensure that for every ξ which is a root of b i p x q the value λb q p ξ q` s is differentfrom 0 for i “ , . . . , q ´
1. This is guaranteed by hypothesis, and so we obtain theexplicit formula for W i p x q , i “ , . . . , q ´ W q p x q . We have W q p x q C q p x q “ P p x q ´ ř q ´ j “ W j p x q C j p x q , where theright-hand side is made by known polynomials. This way, taking the latter expressionmodulo b q p x q we can compute W q p x q since C q p x q “ ś q ´ j “ b j p x q I m is invertible modulo b q p x q in view of the co-primality of the polynomials b p x q , . . . , b q p x q . This way we getthe expression of W q in (4.3). Remark 4.3.
Note that in the case where P n “ I it is possible to choose s “ so that Equations (4.3) take a simpler form. We observe that the matrix polynomials B i p x q which satisfy Assumption 1 verifythe hypotheses of Theorem 2.5. Therefore there exists an (cid:96) -ification of P p x q whichcan be computed. In view of Theorem 2.7 we have that this (cid:96) -ification is also strongif the following conditions are satisfied:1. The B i p x q have the same degree d . In our case this implies that deg b i p x q “ d for every i “ , . . . , q .2. The matrix polynomials x d B i p x ´ q are right coprime. It can be seen thatunder Assumption 1 this is equivalent to asking that b i p q ‰ i “ , . . . , q and that either P n ‰ I or ξ d b q p ξ q ` s ‰ ξ root of b i p x q , for i ă q .Here we provide an example of an (cid:96) -ification of degree 2 for a 2 ˆ P p x q of degree 4. Example 4.4.
Let P p x q “ „ x ` ´ x x ´ , b p x q “ x ´ , b p x q “ x ` , s “ . pplying the above formulas we obtain W p x q “ „ ´ x ´ ` x , W p x q “ „ ´ ´ x ´ ` x . Then we have that A p x q is a degree 2 (cid:96) -ification for P p x q , that is, a quadratization,by setting A p x q “ »——– x ´ x ´ x ` fiffiffifl ` »——– ´ ´ x ´ ` x ´ x ´ ` x ´ ´ x ´ ` x ´ x ´ ` x fiffiffifl . In the case where b i p x q , i “ , . . . , q are linear polynomials we have the following: Corollary 4.5. If b i p x q “ x ´ β i , i “ , . . . , q , then q “ n and W i “ P p β i q ś n ´ j “ , j ‰ i p β i ´ β j q pp β i ´ β n q P n ` sI m q ´ , i “ , . . . , n ´ ,W n “ P p β n q ś n ´ j “ p β n ´ β j q ´ sI m ´ s n ´ ÿ j “ W j β n ´ β j . Moreover, if P p x q is monic then with s “ the expression for W i turns simply into W i “ P p β i q{ ś nj “ , j ‰ i p β i ´ β j q , for i “ , . . . , n .Proof . It follows from Theorem 4.2 and from the property v p x q mod x ´ β “ v p β q valid for any polynomial v p x q .Given n and q ď n , let (cid:96) “ r nq s . We may choose polynomials b i p x q of degree d i in between (cid:96) ´ (cid:96) such that ř qi “ d i “ n . This way A p x q is an mq ˆ mq matrixpolynomial of degree (cid:96) . For instance, if (cid:96) “ P p x q .If P p x q is monic, that is P n “ I , we can handle another particular case of (cid:96) -ification by choosing diagonal matrix polynomials B i p x q .Let B i p x q “ diag p d p i q p x q , . . . , d p i q m p x qq “ : D i p x q be monic matrix polynomials suchthat the corresponding diagonal entries of D i p x q and D j p x q are pairwise co-prime forany i ‰ j so that the second assumption of Theorem 2.5 is satisfied. Let us provethat there exist matrix polynomials W i p x q such that deg W i p x q ă deg D i p x q and P p x q “ q ź i “ D i p x q ` q ÿ i “ W i p x q C i p x q , C i p x q “ q ź j “ , j ‰ i D i p x q , (4.4)so that Theorem 2.5 can be applied. Observe that equating the coefficients of x i in(4.4) for i “ , . . . , n ´ m n equations in m n unknowns.Equating the j th columns of both sides of (4.4) modulo d p i q j p x q yields P p x q e j mod d p i q j p x q “ n ź s “ , s ‰ i d p s q j W i p x q e j mod d p i q j p x q , i “ , . . . , m. The above equation allows one to compute the coefficients of the polynomials of degreeat most deg D j p x q ´ j th column of W j p x q by means of the Chinese remaindertheorem. . Computational issues. In the previous section we have given explicit formu-las for the (strong) (cid:96) -ification of a matrix polynomial satisfying Assumption 1. Herewe describe some algorithms for the computation of the matrix coefficients W i p x q .In the case where d “
1, the equations given in Corollary 4.5 provide a straight-forward algorithm for the computation of the matrices W i for i “ , . . . , n . In the casewhere d i “ deg b i p x q ą i we have to apply (4.3) which involveoperations modulo scalar polynomials b i p x q for i “ , . . . , q .The main computational issues in this case are the evaluation of a scalar polyno-mial modulo a given b i p x q , the evaluation of the inverse of a scalar polynomial modulo b i p x q and the more complicated task of evaluating the inverse of a matrix polynomialmodulo b i p x q .In general we recall that, given polynomials v p x q and b p x q such that v p x q is co-prime with b p x q , there exist polynomials α p x q and β p x q such that α p x q v p x q ` β p x q b p x q “ . (5.1)This way, we have α p x q “ { v p x q mod b p x q .There are algorithms for computing the coefficients of α p x q given the coefficientsof v p x q and b p x q . We refer the reader to the book [7] and to any textbook in computeralgebra for the design and analysis of algorithms for this computation. Here, we recalla simple numerical technique, based on the evaluation-interpolation strategy, whichcan be also directly applied to the matrix case.Observe that from (5.1) it turns out that α p ξ j q “ { v p ξ j q for j “ , . . . , d , where ξ j are the zeros of the polynomial b p x q of degree d . Since α p x q has degree at most d ´
1, it is enough to compute the values 1 { v p ξ j q in order to recover the coefficientsof α p x q through an interpolation process. This procedure amounts to d evaluationsof a polynomial at a point and to solving an interpolation problem for the overallcost of O p d q arithmetic operations. If the polynomial b p x q is chosen in such a waythat its roots are multiples of the d th roots of unity then the evaluation/interpolationproblem is well conditioned, and it can be performed by means of FFT which is a fastand numerically stable procedure.The evaluation/interpolation technique can be extended to the case of matrixpolynomials. For instance, the computation of the coefficients of the matrix polyno-mial F p x q “ V p x q ´ mod b p x q , where V p x q is a given matrix polynomial co-primewith b p x q I , can be performed in the following way:1. compute Y k “ V p ξ k q ´ , for k “ , . . . , d ;2. for any pair p i, j q , interpolate the entries y p k q i,j , k “ , . . . , d of the matrix Y k and find the coefficients of the polynomial f i,j p x q , where F p x q “ p f i,j p x qq .This procedure requires the evaluation of m polynomials at d points, followed bythe inversion of d matrices of order m and the solution of m interpolation problems.The cost turns to O p m d q ops for the evaluation stage, O p m d q ops for the inversionstage, and O p m d q for the interpolation stage. In the case where the polynomial b p x q is such that its roots are multiple of the d roots of the unity, the evaluation and theinterpolation stage have cost O p m d log d q if performed by means of FFT.Observe that, in the case of polynomials b i p x q of degree one, the above procedurecoincides with the one provided directly by equations in Corollary (4.5).
6. Numerical issues.
Let ω n be a principal n th root of the unity, define Ω n “ ? n p ω ijn q i,j “ ,n the Fourier matrix such that Ω ˚ n Ω n “ I n and observe that Ω n e “ e n where e “ p , . . . , q t , e n “ p , . . . , , q t . Assume for simplicity P n “ I m . For the inearization obtained with β i “ ω in , i “ , . . . , n , we have, following Corollary 4.5, A p x q “ xI mn ´ diag p ω n I m , ω n I m , . . . , ω nn I m q ` p e b I m qr W , . . . , W n s with W i “ n ω in P p ω in q . The latter equation follows from W i “ P p ω in q{p ś j ‰ i p ω in ´ ω jn qq since ś j ‰ i p ω in ´ ω jn q coincides with the first derivative of x n ´ “ ś nj “ p x ´ ω jn q evaluated at x “ ω in , that is ś j ‰ i p ω in ´ ω jn q “ nω ´ in . It is easy to verify that thepencil p Ω ˚ n b I m q A p x qp Ω n b I m q has the form xI mn ´ F, F “ p C b I m q ´ »———– P ` I m P ... P n ´ fiffiffiffifl p e tn b I m q where C “ p c i,j q is the unit circulant matrix defined by c i,j “ p δ i,j ` n q . Thatis, F is the block Frobenius matrix associated with the matrix polynomial P p x q .This shows that our linearization includes the companion Frobenius matrix witha specific choice of the nodes. In particular, since Ω n is unitary, the condition numberof the eigenvalues of A p x q coincides with the condition number of the eigenvalues of F .Observe also that if we choose β i “ αω in with α ‰
0, then p Ω ˚ n b I m q A p x qp Ω n b I m q “ xI ´ D ´ α F D α for D α “ diag p , α, . . . , α n ´ q . That is, we obtain a scaled Frobeniuspencil.Here, we present some numerical experiments to show that in many interestingcases a careful choice of the B i p x q can lead to linearizations (or (cid:96) -ifications) wherethe eigenvalues are much better conditioned than in the original problem. Here weare interested in measuring the conditioning of the eigenvalues of a pencil built usingthese different strategies. Recall that the conditioning of an eigenvalue λ of a matrixpencil xA ´ B can be bounded by κ λ ď (cid:107) v (cid:107)(cid:107) w (cid:107) | w ˚ Av | where v and w are the right andleft eigenvectors relative to λ , respectively [31]. This is the quantity measured bythe condeig function in MATLAB that we have used in the experiments. The abovebound can be extended to a matrix polynomial P p x q “ ř ni “ P i x i . In particular,the conditioning number of an eigenvalue λ of P can be bounded by κ λ ď (cid:107) v (cid:107)(cid:107) w (cid:107) | w ˚ P p λ q v | where v and w are the right and left eigenvectors relative to λ , respectively, [31].Observe that, if xA ´ B is the Frobenius linearization of a matrix polynomial P p x q ,then the condition number of the eigenvalues of the linearization is larger than theone concerning P p x q , since the perturbation to the input data on xA ´ B can be seenas a larger set with respect to the perturbation to the coefficients of P p x q . This isnot true, in general, for a linearization in a different basis, as in our case, since thereis not a direct correspondence between the perturbations on the original coefficientsand the perturbations on the linearization. An analysis of the condition number foreigenvalue problems of matrix polynomials represented in different basis is given in[11].The code used to generate these examples can be downloaded from http://numpi.dm.unipi.it/software/secular-linearization/ . As a first example, consider a monic scalar poly-nomial p p x q “ ř ni “ p i x i where the coefficients p i have unbalanced moduli. In thiscase, we generate p i using the MATLAB command p = exp(12 * randn(1,n+1));p(n+1)=1; igure 6.1 . Conditioning of the eigenvalues of different linearizations of a degree scalarpolynomial with random unbalanced coefficients. Then we build our linearization by means of the function seccomp(b,p) thattakes a vector b together with the coefficients of the polynomial and generates thelinearization A p x q where B i p x q “ x ´ β i for β i “ b p i q . Finally, we measure theconditioning of the eigenvalues of A p x q by means of the Matlab function condeig .We have considered three different linearizations: ‚ The Frobenius linearization obtained by compan(p) ; ‚ the secular linearization obtained by taking as β i some perturbed values ofthe roots; these values have been obtained by multiplying the roots by p ` (cid:15) q with (cid:15) chosen randomly with Gaussian distribution (cid:15) „ ´ ¨ N p , q . ‚ the secular linearization with nodes given by the tropical roots of the poly-nomial multiplied by unit complex numbers.The results are displayed in Figure 6.1. One can see that in the first case thecondition numbers of the eigenvalues are much different from each other and can beas large as 10 for the worst conditioned eigenvalue. In the second case the conditionnumber of all the eigenvalues is close to 1, while in the third linearization the conditionnumbers are much smaller than those of the Frobenius linearization and have analmost uniform distribution.These experimental results are a direct verification of a conditioning result of [9,Sect. 5.2] that is at the basis of the secsolve algorithm presented in that paper.These tests are implemented in the function files Example1.m and
Experiment1.m included in the MATLAB source code for the experiments. A similar behavior of theconditioning for the eigenvalue problem holds in the matrix case.
Consider now a matrix polynomial P p x q “ ř ni “ P i x i .As in the previous case, we start by considering monic matrix polynomials. As a firstexample, consider the case where the coefficients P i have unbalanced norms. Here isthe Matlab code that we have used to generate this test: n = 5; m = 64;P = {};for i = 1 : nP { i } = exp (12 * randn ) * randn ( m );endP { n +1} = eye ( m ); e can give reasonable estimates to the modulus of the eigenvalues using thePellet theorem or the tropical roots. See [16, 27], for some insight on these tools.The same examples given in the scalar case have been replicated for matrix poly-nomials relying on the Matlab script published on the website reported above byissuing the following commands: >> P = Example2 ();>> Experiment2 ( P ); −5 Moduli of the eigenvalues C ond i t i on i ng Random unbalanced matrix polynomial Frobenius linearizationSecular linearization − eivalsSecular linearization − tropical −4 −3 −2 −1 Moduli of the eigenvalues C ond i t i on i ng Random unbalanced matrix polynomial Frobenius linearizationSecular linearization − eivalsSecular linearization − tropical −4 −2 Moduli of the eigenvalues C ond i t i on i ng Random unbalanced matrix polynomial Frobenius linearizationSecular linearization − eivalsSecular linearization − tropical Moduli of the eigenvalues C ond i t i on i ng Random unbalanced matrix polynomial Frobenius linearizationSecular linearization − eivalsSecular linearization − tropical
Figure 6.2 . Conditioning of the eigenvalues of different linearizations for some matrix poly-nomials with random coefficients having unbalanced norms.
We have considered three linearizations: the standard Frobenius companion lin-earization, and two versions of our secular linearizations. In the first version the nodes β i are the mean of the moduli of set of eigenvalues with close moduli multiplied byunitary complex numbers. In the second, the values of β i are obtained by the Pelletestimates delivered by the tropical roots.In Figure 6.2 we report the conditioning of the eigenvalues, measured with Mat-lab’s condeig .It is interesting to note that the conditioning of the secular linearization is, inevery case, not exceeding 10 . Moreover it can be observed that no improvement is btained on the conditioning of the eigenvalues that are already well-conditioned. Incontrast, there is a clear improvement on the ill-conditioned ones. In this particularcase, this class of linearizations seems to give an almost uniform bound to the conditionnumber of all the eigenvalues.Further examples come from the NLEVP collection of [3]. We have selected someproblems that exhibit bad conditioning.As a first example we consider the problem orr sommerfeld . Using the tropicalroots we can find some values inside the unique annulus that is identified by thePellet theorem. In this example the values obtained only give a partial picture of theeigenvalues distribution. The Pellet theorem gives about and as lowerand upper bound to the moduli of the eigenvalues, but the tropical roots are rathersmall and near to the lower bound. More precisely, the tropical roots are and with multiplicities 3 and 1, respectively. −4 −3 −2 −1 Moduli of the eigenvalues C ond i t i on i ng orr_sommerfeld with tropical roots as b i Frobenius linearizationSecular linearization − eivalsSecular linearization − tropical −4 −3 −2 −1 Moduli of the eigenvalues C ond i t i on i ng orr_sommerfeld with b i given by Pellet theorem Frobenius linearizationSecular linearization − eivalsSecular linearization − Pellet Figure 6.3 . On the left we report the conditioning of the Frobenius and of the secular lineariza-tion with the choices of b i as mean of subsets of eigenvalues with close moduli and as the estimatesgiven by the tropical roots. On the right the tropical roots are coupled with estimates given by thePellet theorem. This leads to a linearization A p x q that is well-conditioned for the smaller eigen-values but with a higher conditioning on the eigenvalues of bigger modulus as can beseen in Figure 6.3 on the left (the eigenvalues are ordered in nonincreasing order withrespect to their modulus). It can be seen, though, that coupling the tropical rootswith the standard Pellet theorem and altering the b i by adding a value slightly smallerthan the upper bound (in this example we have chosen 5 but the result is not verysensitive to this choice) leads to a much better result that is reported in Figure 6.3on the right. In the right figure we have used b = [ 1.7e-4, 1.4e-3, -1.4e-3, 5] . This seems to justify that there exists a link between the quality of the approx-imations obtained through the tropical roots and the conditioning properties of thesecular linearization.We analyzed another example problem from the NLEVP collection that is called planar waveguide . The results are shown in Figure 6.4. This problem is a PEP ofdegree 4 with two tropical roots approximately equal to 127 . .
24. Again, it canbe seen that for the eigenvalues of smaller modulus (that will be near the tropical root1 .
24) the Frobenius linearization and the secular one behave in the same way, whilstfor the bigger ones the secular linearization has some advantage in the conditioning.This may be justified by the fact that the Frobenius linearization is similar to a secular −1 Moduli of the eigenvalues C ond i t i on i ng Planar waveguide problem Frobenius linearizationSecular linearization − eivalsSecular linearization − tropical
Figure 6.4 . Conditioning of the eigenvalues for three different linearizations on the planar waveguide problem. linearization on the roots of the unity.Note that in this case the information obtained by the tropical roots seems moreaccurate than in the orr sommerfeld case, so the secular linearization built usingthe tropical roots and the one built using the block-mean of the eigenvalues behaveapproximately in the same way. −4 −2 −16 −14 −12 −10 −8 −6 Moduli of the eigenvalues R e l a t i v e e rr o r Relative error of eigenvalues computed with different linearizations PolyeigSecularFrobenius
Figure 6.5 . The accuracy of the computed eigenvalues using polyeig, the Frobenius linearizationand the secular linearization with the b i obtained through the computation of the tropical roots. As a last example, we have tried to find the eigenvalues of a matrix polynomialdefined by integer coefficients. We have used polyeig and our secular linearization(using the tropical roots as b i ) and the QZ method. We have chosen the polynomial p x q “ P x ` P x ` P x ` P where P “ »——– fiffiffifl , P “ »——– fiffiffifl , P “ P t , P “ »——– fiffiffifl . In this case the tropical roots are good estimates of the blocks of eigenvalues of thematrix polynomial. We obtain the tropical roots 1 . ¨ , 0 . . ¨ ´ with multiplicities 2, 7 and 2, respectively. We have computed the eigenvalues witha higher precision and we have compared them with the results of polyeig and of eig applied to the secular linearization and to the standard Frobenius linearization.Here, the secular linearization has been computed with the standard floating pointarithmetic. As shown in Figure 6.5 we have achieved much better accuracy with thelatter choice. The secular linearization has achieved a relative error of the order ofthe machine precision on all the eigenvalues except the smaller block (with modulusabout 10 ´ ). In that case the relative error is about 10 ´ but the absolute error is,again, of the order of the machine precision. Moreover, polyeig fails to detect theeigenvalues with bigger modules, and marks them as eigenvalues at infinity. This canbe noted by the fact that the circles relative to the bigger eigenvalues are missing in polyeig plot of Figure 6.5. Acknowledgments.
We wish to thank the referees for their careful commentsand useful remarks that helped us to improve the presentation.
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