Finding a positive definite linear combination of two Hermitian matrices

Crawford, Charles R.; Moon, Yiu Sang

doi:10.1016/0024-3795(83)90148-9

Cited by 27 publications

(30 citation statements)

References 5 publications

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“…Similarly, SeDuMi and SDPT3 are not shown here, as their speed remained about the same as in the dense case for n ≤ 700, hence impractical for n ≥ 10 3 . Here we examine the runtime and accuracy of our algorithm Eig for varying matrix size n from 10 3 to as large as 10 6 . We test with two types of sparse matrices: tridiagonal and random sparse (generated using MATLAB's sprandsym).…”

Section: Resultsmentioning

confidence: 99%

“…Such algorithms include [6,15], which also provide a value t 0 for which A sin t 0 + B cos t 0 0 if (A, B) is definite. If the pair (A, B) is determined not to be definite then QCQP is not definite feasible.…”

Section: Checking Definite Feasibilitymentioning

confidence: 99%

“…To this end we used the algorithm in [6,15] to findλ, as discussed in Sect. 2.4.1, to obtain the interval D. Although any value in D is allowed to beλ, we choseλ as the middle point of D to avoid ill-conditioning of A +λB.…”

Section: Computingλmentioning

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: Resultsmentioning

confidence: 99%

Section: Checking Definite Feasibilitymentioning

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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“…4) and then computing an extremal eigenvalue of each solvent. A definiteness test on Q evaluated at the average of the two extremal eigenvalues finally determines whether Q is overdamped.…”

Section: Q(x) = Axmentioning

confidence: 99%

“…The second approach is to apply to (A, B) an algorithm of Crawford and Moon [4] for detecting definiteness of Hermitian matrix pairs. Although only linearly convergent, this algorithm usually terminates within 30 iterations with a message of "definite," "indefinite," or "fail" (denoting failure of the algorithm to make a determination).…”

Section: Operationsmentioning

confidence: 99%

Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems

Guo¹,

Higham²,

Tisseur³

2009

SIAM J. Matrix Anal. & Appl.

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Abstract. Hyperbolic quadratic matrix polynomials Q(λ) = λ 2 A + λB + C are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with nonpositive eigenvalues. Neither the definition of overdamped nor any of the standard characterizations provides an efficient way to test if a given Q has this property. We show that a quadratically convergent matrix iteration based on cyclic reduction, previously studied by Guo and Lancaster, provides necessary and sufficient conditions for Q to be overdamped. For weakly overdamped Q the iteration is shown to be generically linearly convergent with constant at worst 1/2, which implies that the convergence of the iteration is reasonably fast in almost all cases of practical interest. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar μ < 0 is provided that lies in the gap between the n largest and n smallest eigenvalues of the n × n quadratic eigenvalue problem (QEP) Q(λ)x = 0. Once such a μ is known, the QEP can be solved by linearizing to a definite pencil that can be reduced, using already available Cholesky factorizations, to a standard Hermitian eigenproblem. By incorporating an initial preprocessing stage that shifts a hyperbolic Q so that it is overdamped, we obtain an efficient algorithm that identifies and solves a hyperbolic or overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues.

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Matrix Eigenvalue Problems

Björck

2014

Texts in Applied Mathematics

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Finding a positive definite linear combination of two Hermitian matrices

Cited by 27 publications

References 5 publications

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

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

Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems

Matrix Eigenvalue Problems

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