Minimization of norms and logarithmic norms by diagonal similarities

Ström, Torsten

doi:10.1007/bf02242378

Cited by 15 publications

(6 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , 1 + |an−2| + |an−1|}, if n is odd, which again coincides with the upper bound in (18). Although, Mσ 1 (p) and Mσ 2 (p) look almost the same and, as a consequence, Mσ 1 (p) ∞ and Mσ 2 (p) ∞ have also the same flavor, there are relevant differences.…”

Section: Lower and Upper Bounds From ∞-Norms Of Fiedler Matricessupporting

confidence: 48%

“…Observe that in the statement of Theorem 4.1 we have not imposed a0 = 0, which, strictly speaking, is necessary for obtaining the lower bound in (18). However, if a0 = 0, then the lower bound can be taken to be zero and this is consistent with the fact that p(z) has at least one root equal to zero.…”

Section: Lower and Upper Bounds From ∞-Norms Of Fiedler Matricesmentioning

confidence: 62%

“…For instance, if a0 = 10 −16 , a2 = 1, and the rest of the ai are all equal to zero, then C2(p) C2(p ) ∞ −1 ) and Cauchy's upper (i.e., C2(p) ∞) bounds by a factor larger than 2. In this sense, the classical Cauchy's bounds in Theorem 1.1 are optimal, up to a factor 2, among those obtained from (18) and, in fact, we will see that they are strictly optimal for a large subclass of Fiedler matrices.…”

Section: Lower and Upper Bounds From ∞-Norms Of Fiedler Matricesmentioning

confidence: 93%

“…The first one is Lemma 7.2, which merges Theorem 1, Corollary 1, and Corollary 2 in [18]. The concepts mentioned in the statement of Lemma 7.2 are contained in [13].…”

Section: Lower and Upper Bounds From Frobenius Norms Of Inverses Of Fmentioning

confidence: 99%

See 3 more Smart Citations

New bounds for roots of polynomials based on Fiedler companion matrices

Terán

Dopico

Pérez

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

Several matrix norms of the classical Frobenius companion matrices of a monic polynomial p(z) have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots λ of p(z). Recently, M. Fiedler has introduced a new family of companion matrices of p(z) (Lin. Alg. Appl., 372 (2003) 325-331) that has received considerable attention and it is natural to investigate if matrix norms of Fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on |λ|. The development of such bounds requires first to know simple expressions for some relevant matrix norms of Fiedler matrices and we obtain them in the case of the 1-and ∞-matrix norms. With these expressions at hand, we will show that norms of Fiedler matrices produce many new bounds, but that none of them improves significatively the classical bounds obtained from the Frobenius companion matrices. However, we will prove that if the norms of the inverses of Fiedler matrices are used, then another family of new bounds on |λ| is obtained and some of the bounds in this family improve significatively the bounds coming from the Frobenius companion matrices for certain polynomials.

show abstract

Section: Lower and Upper Bounds From ∞-Norms Of Fiedler Matricessupporting

confidence: 48%

Section: Lower and Upper Bounds From ∞-Norms Of Fiedler Matricesmentioning

confidence: 62%

Section: Lower and Upper Bounds From ∞-Norms Of Fiedler Matricesmentioning

confidence: 93%

“…The first one is Lemma 7.2, which merges Theorem 1, Corollary 1, and Corollary 2 in [18]. The concepts mentioned in the statement of Lemma 7.2 are contained in [13].…”

Section: Lower and Upper Bounds From Frobenius Norms Of Inverses Of Fmentioning

confidence: 99%

See 2 more Smart Citations

New bounds for roots of polynomials based on Fiedler companion matrices

Terán

Dopico

Pérez

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…Refs. [1][2][3][4]6,7,[9][10][11][12]24,[27][28][29][30][31][32][34][35][36] are given even though they are not directly used in this paper in order to provide the reader with some additional material useful in the discussed subject.…”

Section: Introductionmentioning

confidence: 99%

Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

Kohaupt

2008

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

The new idea is to study the stability behavior of the solution x = x(t) of the initial value problemẋ = Ax, t t 0 , x(t 0 ) = x 0 , with A ∈ C n×n , in a weighted (semi-) norm · R where R is taken as an appropriate solution of the matrix eigenvalue problem RA + A * R = R, rather than as the solution of the algebraic Lyapunov matrix equation RA + A * R = −S with given positive (semi-) definite matrix S. Substantially better results are obtained by the new method. For example, if A is diagonalizable and all eigenvalues, which is much more than the old result, which only states that lim t→∞ x(t) = 0. Especially, the semi-norms · R i have a decoupling and filter effect on x(t). Further, new two-sided bounds (depending on x 0 ) for the asymptotic behavior can be derived. The best constants in the bounds are obtained by the differential calculus of norms. Applications are made to free linear dynamical systems, and computations underpin the theoretical findings. : Matrix eigenvalue problem VA + A * V = V ; Algebraic Lyapunov equation RA + A * R = −S; Linear dynamical system; Decoupling and filter effect of weighted semi-norms; Stability; Two-sided bounds depending on the initial conditions; Differential calculus of norms

show abstract

Computational methods of linear algebra

Faddeev¹,

Faddeeva²

1981

J Math Sci

View full text Add to dashboard Cite

Minimization of norms and logarithmic norms by diagonal similarities

Cited by 15 publications

References 5 publications

New bounds for roots of polynomials based on Fiedler companion matrices

New bounds for roots of polynomials based on Fiedler companion matrices

Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

Computational methods of linear algebra

Contact Info

Product

Resources

About