The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part I

Demmel, James; Kågström, Bo

doi:10.1145/152613.152615

Cited by 162 publications

(103 citation statements)

References 19 publications

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“…For the general case, one can proceed by upper triangularizing the matrix pencil using the QZ algorithm (or the GUPTRI algorithm [7,20] to deal with singular pencils), and then solving generalized Sylvester equations [12,Sec. 7.7] to block diagonalize the matrices, detect the Jordan block sizes and find the corresponding transformations for each block.…”

Section: Characterizing Bounded Qcqp Via the Canonical Formmentioning

confidence: 99%

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Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Adachi

Nakatsukasa

2017

Math. Program.

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A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or Moré's algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the (usually black-box) iterations for computing the eigenpair. Numerical experiments illustrate the efficiency and accuracy of our algorithm. We also analyze the QCQP solution extensively, including difficult cases, and show that the canonical form of a matrix pair gives a complete classification of the QCQP in terms of boundedness and attainability, and explain how to obtain a global solution whenever it exists.Keywords QCQP · Generalized eigenvalue problem · Canonical form for symmetric matrix pair Mathematics Subject Classification 49M37 · 65K05 · 90C20 · 90C30

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Section: Characterizing Bounded Qcqp Via the Canonical Formmentioning

confidence: 99%

“…Using the GUPTRI algorithm [7,20] for the canonical form, the worstcase complexity is O(n 4 ). We repeat that most QCQP that are solvable in practice are solved by Algorithm 3.2, which is O(n 3 ) or faster.…”

Section: Solution Process For Nongeneric Qcqpmentioning

confidence: 99%

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Adachi

Nakatsukasa

2017

Math. Program.

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“…XII]. This canonical form can be stably computed through unitary transformations that lead to the GUPTRI form [8,9,13,28]. Therefore it is natural to look for relationships (if any) between the minimal indices of a singular matrix polynomial P and the minimal indices of a given linearization, since this would provide a numerical method for computing the minimal indices of P .…”

Section: Minimal Indices and Minimal Bases Eigenstructure Of A Singumentioning

confidence: 99%

Linearizations of singular matrix polynomials and the recovery of minimal indices

Terán¹,

Dopico²,

Mackey³

2009

ELA

147

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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 L 1 (P ) and L 2 (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 L 1 (P ) and L 2 (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 L 1 (P ) and L 2 (P ). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P ) = L 1 (P ) ∩ L 2 (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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“…The minimal reducing subspace is unique and can be numerically computed in a stable way from the generalized upper triangular form (GUPTRI), see, e.g., [6,7]. It is trivial to construct a basis for the kernel of the tensor product A ⊗ D from the kernels of A and D. The task is much harder if we take a difference of two tensor products, which is the form of the operator determinants (1.3).…”

Section: Elamentioning

confidence: 99%

On the singular two-parameter eigenvalue problem

Muhič¹,

Plestenjak²

2009

ELA

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Abstract. In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above relation to singular two-parameter eigenvalue problems is extended, and it is shown that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the associated system of generalized eigenvalue problems agree. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems.

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The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part I

Cited by 162 publications

References 19 publications

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Linearizations of singular matrix polynomials and the recovery of minimal indices

On the singular two-parameter eigenvalue problem

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