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Volker Mehrmann - One of the best experts on this subject based on the ideXlab platform.
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Low-Rank Perturbation of Regular Matrix Pencils with Symmetry Structures
Foundations of Computational Mathematics, 2021Co-Authors: Fernando De Terán, Christian Mehl, Volker MehrmannAbstract:The generic change of the Weierstraß canonical form of Regular complex structured Matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered, and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a different behavior for the eigenvalues 0 and $$\infty $$ ∞ , respectively, $$+1$$ + 1 and $$-1$$ - 1 . The differences arise in those cases where the parity of the partial multiplicities in the perturbed Matrix pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-1 and rank-2 cases that were obtained in Batzke (Linear Algebra Appl 458:638–670, 2014, Oper Matrices 10:83–112, 2016) for the case of special structure-preserving perturbations. As the main tool, we use decompositions of Matrix pencils with symmetry structure into sums of rank-1 Matrix pencils, as those allow a parametrization of the set of Matrix pencils with a given symmetry structure and a given rank.
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Low rank perturbation of Regular Matrix pencils with symmetry structures
arXiv: Spectral Theory, 2019Co-Authors: Fernando De Terán, Christian Mehl, Volker MehrmannAbstract:The generic change of the Weierstrass Canonical Form of Regular complex structured Matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a different behavior for the eigenvalues $0$ and $\infty$, respectively $+1$ and $-1$. The differences arise in those cases where the parity of the partial multiplicities in the perturbed pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-$1$ and rank-$2$ cases that were obtained in [L. Batzke, Generic Low-Rank Perturbations of Structured Regular Matrix Pencils and Structured Matrices, PhD Thesis, TU Berlin, Berlin, Germany, 2015] and [L. Batzke, Generic rank-two perturbations of structured Regular Matrix pencils, Oper. Matrices,10:83-112, 2016] for the case of special structure-preserving perturbations. As the main tool, we use decompositions of Matrix pencils with symmetry structure into sums of rank-one pencils, as those allow a parametrization of the set of Matrix pencils with a given symmetry structure and a given rank.
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On the Nearest Singular Matrix Pencil
SIAM Journal on Matrix Analysis and Applications, 2017Co-Authors: Nicola Guglielmi, Christian Lubich, Volker MehrmannAbstract:Given a Regular Matrix pencil $A + \mu E$, we consider the problem of determining the nearest singular Matrix pencil with respect to the Frobenius norm. We present new approaches based on the solut...
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jordan structures of alternating Matrix polynomials
Linear Algebra and its Applications, 2010Co-Authors: Steven D. Mackey, Christian Mehl, Niloufer Mackey, Volker MehrmannAbstract:Alternating Matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating Matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating Matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating Matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as Regular Matrix polynomials.
Fernando De Terán - One of the best experts on this subject based on the ideXlab platform.
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Low-Rank Perturbation of Regular Matrix Pencils with Symmetry Structures
Foundations of Computational Mathematics, 2021Co-Authors: Fernando De Terán, Christian Mehl, Volker MehrmannAbstract:The generic change of the Weierstraß canonical form of Regular complex structured Matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered, and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a different behavior for the eigenvalues 0 and $$\infty $$ ∞ , respectively, $$+1$$ + 1 and $$-1$$ - 1 . The differences arise in those cases where the parity of the partial multiplicities in the perturbed Matrix pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-1 and rank-2 cases that were obtained in Batzke (Linear Algebra Appl 458:638–670, 2014, Oper Matrices 10:83–112, 2016) for the case of special structure-preserving perturbations. As the main tool, we use decompositions of Matrix pencils with symmetry structure into sums of rank-1 Matrix pencils, as those allow a parametrization of the set of Matrix pencils with a given symmetry structure and a given rank.
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Low rank perturbation of Regular Matrix pencils with symmetry structures
arXiv: Spectral Theory, 2019Co-Authors: Fernando De Terán, Christian Mehl, Volker MehrmannAbstract:The generic change of the Weierstrass Canonical Form of Regular complex structured Matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a different behavior for the eigenvalues $0$ and $\infty$, respectively $+1$ and $-1$. The differences arise in those cases where the parity of the partial multiplicities in the perturbed pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-$1$ and rank-$2$ cases that were obtained in [L. Batzke, Generic Low-Rank Perturbations of Structured Regular Matrix Pencils and Structured Matrices, PhD Thesis, TU Berlin, Berlin, Germany, 2015] and [L. Batzke, Generic rank-two perturbations of structured Regular Matrix pencils, Oper. Matrices,10:83-112, 2016] for the case of special structure-preserving perturbations. As the main tool, we use decompositions of Matrix pencils with symmetry structure into sums of rank-one pencils, as those allow a parametrization of the set of Matrix pencils with a given symmetry structure and a given rank.
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generic change of the partial multiplicities of Regular Matrix pencils under low rank perturbations
SIAM Journal on Matrix Analysis and Applications, 2016Co-Authors: Fernando De Terán, Froilan M DopicoAbstract:We describe the generic change of the partial multiplicities at a given eigenvalue $\lambda_0$ of a Regular Matrix pencil $A_0+\lambda A_1$ under perturbations with low normal rank. More precisely, if the pencil $A_0+\lambda A_1$ has exactly $g$ nonzero partial multiplicities at $\lambda_0$, then for most perturbations $B_0+\lambda B_1$ with normal rank $r
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fiedler companion linearizations and the recovery of minimal indices
SIAM Journal on Matrix Analysis and Applications, 2010Co-Authors: Fernando De Terán, Froilan M Dopico, Steven D. MackeyAbstract:A standard way of dealing with a Matrix polynomial $P(\lambda)$ is to convert it into an equivalent Matrix pencil—a process known as linearization. For any Regular Matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when $P(\lambda)$ is a singular square Matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial $P(\lambda)$ from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a Regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications.
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linearizations of singular Matrix polynomials and the recovery of minimal indices
Electronic Journal of Linear Algebra, 2009Co-Authors: Fernando De Terán, Froilan M Dopico, Steven D. MackeyAbstract:A standard way of dealing with a Regular Matrix polynomial P (λ) is to convert it into an equivalent Matrix pencil - a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P ) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ )w hen P is Regular. The goal of this work is to show that most of the pencils in L1(P )a ndL2(P )a re stil l linearizations when P (λ) is a singular square Matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ )f rom the minimalindices and bases of l in L1(P )a ndL2(P ). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P )= L1(P ) ∩ L2(P )w il l never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases.
Bo Kagstrom - One of the best experts on this subject based on the ideXlab platform.
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parallel and blocked algorithms for reduction of a Regular Matrix pair to hessenberg triangular and generalized schur forms
Parallel Computing, 2002Co-Authors: Bjorn Adlerborn, Krister Dackland, Bo KagstromAbstract:A parallel three-stage algorithm for reduction of a Regular Matrix pair (A, B) to generalized Schur from (S, T) is presented. The first two stages transform (A, B) to upper Hessenberg-triangular form (H, T) using orthogonal equivalence transformations. The third stage iteratively reduces the Matrix in (H, T) form to generalized Schur form. Algorithm and implementation issues regarding the single-/double-shift QZ algorithm are discussed. We also describe multishift strategies to enhance the performance in blocked as well as in parallell variants of the QZ method.
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parallel two stage reduction of a Regular Matrix pair to hessenberg triangular form
Parallel Computing, 2000Co-Authors: Bjorn Adlerborn, Krister Dackland, Bo KagstromAbstract:A parallel two-stage algorithm for reduction of a Regular Matrix pair (A,B) to Hessenberg-triangular form (H, T) is presented. Stage one reduces the Matrix pair to a block upper Hessenberg-triangular form (Hr, T), where Hr is upper r-Hessenberg with r > 1 subdiagonals and T is upper triangular. In stage two, the desired upper Hessenberg-triangular form is computed using two-sided Givens rotations. Performance results for the ScaLAPACK-style implementations show that the parallel algorithms can be used to solve large scale problems effectively.
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blocked algorithms and software for reduction of a Regular Matrix pair to generalized schur form
ACM Transactions on Mathematical Software, 1999Co-Authors: Krister Dackland, Bo KagstromAbstract:A two-stage blocked algorithm for reduction of a Regular Matrix pair (A , B ) to upper Hessenberg-triangular form is presented. In stage 1 (A, B is reduced to block upper Hessenberg-triangular form using mainly level 3 (Matrix-Matrix) operations that permit data reuse in the higher levels of a memory hierarchy. In the second stage all but one of the r subdiagonals of the block Hessenberg A-part are set to zero using Givens rotations. The algorithm proceeds in a sequence of supersweeps, each reducing m columns. The updates with respect to row and column rotations are organized to reference consecutive columns of A and B. To further improve the data locality, all rotations produced in a supersweep are stored to enable a left-looking reference pattern, i.e., all updates are delayed until they are required for the continuation of the supersweep. Moreover, we present a blocked variant of the single-diagonal double-shift QZ method for computing the generalized Schur form of (A, B in upper Hessenberg-triangular form. The blocking for improved data locality is done similarly, now by restructuring the reference pattern of the updates associated with the bulge chasing in the QZ iteration. Timing results show that our new blocked variants outperform the current LAPACK routines, including drivers for the generalized eigenvalue problem, by a factor 2-5 for sufficiently large problems.
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a scalapack style algorithm for reducing a Regular Matrix pair to block hessenberg triangular form
Parallel Computing, 1998Co-Authors: Krister Dackland, Bo KagstromAbstract:A parallel algorithm for reduction of a Regular Matrix pair (A, B) to block Hessenberg-triangular form is presented. It is shown how a sequential elementwise algorithm can be reorganized in terms of blocked factorizations and Matrix-Matrix operations. Moreover, this LAPACK-style algorithm is straightforwardly extended to a parallel algorithm for a rectangular 2D processor grid using parallel kernels from ScaLAPACK. A hierarchical performance model is derived and used for algorithm analysis and selection of optimal blocking parameters and grid sizes.
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computing eigenspaces with specified eigenvalues of a Regular Matrix pair a b and condition estimation theory algorithms and software
Numerical Algorithms, 1996Co-Authors: Bo Kagstrom, Peter PoromaaAbstract:Theory, algorithms and LAPACK-style software for computing a pair of deflating subspaces with specified eigenvalues of a Regular Matrix pair (A, B) and error bounds for computed quantities (eigenva ...
L Lerer - One of the best experts on this subject based on the ideXlab platform.
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quasi commutativity of Regular Matrix polynomials resultant and bezoutian
Operator Theory: Advances and Applications, 2010Co-Authors: M A Kaashoek, L LererAbstract:To Israel Gohberg, an outstanding mathematician, an inspiring teacher and a wonderful friend, on the occasion of his 80th birthday. Abstract. In a recent paper of I. Gohberg and the authors necessary and sufficient conditions are obtained in order that for two Regular Matrix polynomials L and M the dimension of the null space of the associate square resultant Matrix is equal to the sum of the multiplicities of the common zeros of L and M, infinity included. The conditions are stated in terms of quasi commutativity. In the case of commuting Matrix polynomials, in particular, in the scalar case, these conditions are automatically fulfilled. The proofs in the above paper are heavily based on the spectral theory of Matrix polynomials. In the present paper a new proof is given of the sufficiency part of the result mentioned above. Here we use the connections between the Bezout and resultant matrices and a general abstract scheme for determining the null space of the Bezoutian of Matrix polynomials which is based on a state space analysis of Bezoutians.
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the resultant for Regular Matrix polynomials and quasi commutativity
Indiana University Mathematics Journal, 2008Co-Authors: Israel Gohberg, M A Kaashoek, L LererAbstract:Necessary and sufficient conditions are presented in order that for two Regular Matrix polynomials the null space of the straightforward analogue of the classical Sylvester resultant Matrix is completely determined by the common spectral data of the polynomials involved. The conditions are stated in terms of a quasi commutativity property. In the scalar case, and more generally in the case of commuting Matrix polynomials, the conditions are automatically satisfied.
Steven D. Mackey - One of the best experts on this subject based on the ideXlab platform.
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Linearizations of singular Matrix polynomials and the recovery of minimal indices
2015Co-Authors: O De Teran, Frolian Dopico, Steven D. Mackey, Mims Eprint, Fernando De TeránAbstract:Abstract. A standard way of dealing with a Regular Matrix polynomial P (λ) is to convert it into an equivalent Matrix pencil – a process known as linearization. Two vector spaces of pencils L1(P) and L2(P) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ) when P is Regular. The goal of this work is to show that most of the pencils in L1(P) and L2(P) are still linearizations when P (λ) is a singular square Matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ) from the minimal indices and bases of linearizations in L1(P) and L2(P). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P) = L1(P) ∩ L2(P) will never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases
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LINEARIZATIONS OF SINGULAR Matrix POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES ∗
2013Co-Authors: M. Dopico, Steven D. MackeyAbstract:Abstract. A standard way of dealing with a Regular Matrix polynomial P (λ) is to convert it into an equivalent Matrix pencil – a process known as linearization. Two vector spaces of pencils L1(P) and L2(P) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ) when P is Regular. The goal of this work is to show that most of the pencils in L1(P)andL2(P)arestill linearizations when P (λ) is a singular square Matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ) fromthe minimalindices and bases of linearizations in L1(P)andL2(P). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P)=L1(P) ∩ L2(P)will never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases. Key words
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jordan structures of alternating Matrix polynomials
Linear Algebra and its Applications, 2010Co-Authors: Steven D. Mackey, Christian Mehl, Niloufer Mackey, Volker MehrmannAbstract:Alternating Matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating Matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating Matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating Matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as Regular Matrix polynomials.
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fiedler companion linearizations and the recovery of minimal indices
SIAM Journal on Matrix Analysis and Applications, 2010Co-Authors: Fernando De Terán, Froilan M Dopico, Steven D. MackeyAbstract:A standard way of dealing with a Matrix polynomial $P(\lambda)$ is to convert it into an equivalent Matrix pencil—a process known as linearization. For any Regular Matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when $P(\lambda)$ is a singular square Matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial $P(\lambda)$ from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a Regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications.
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linearizations of singular Matrix polynomials and the recovery of minimal indices
Electronic Journal of Linear Algebra, 2009Co-Authors: Fernando De Terán, Froilan M Dopico, Steven D. MackeyAbstract:A standard way of dealing with a Regular Matrix polynomial P (λ) is to convert it into an equivalent Matrix pencil - a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P ) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ )w hen P is Regular. The goal of this work is to show that most of the pencils in L1(P )a ndL2(P )a re stil l linearizations when P (λ) is a singular square Matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ )f rom the minimalindices and bases of l in L1(P )a ndL2(P ). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P )= L1(P ) ∩ L2(P )w il l never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases.
