Algorithms for Solving a Unilateral Quadratic Matrix Equation and the Model Updating Problem

Larin, V. B.

doi:10.1007/s10778-014-0635-9

Cited by 6 publications

(2 citation statements)

References 18 publications

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“…There is no unambiguous method to solve this quadratic matrix equation (see e.g., Highman andKim, 2000 andLarin, 2014) and the solution might not be symmetric. However, we will use the known properties of the Riccati equations (see e.g., Dym, 2007 pp.…”

Section: The Proposed Methodsmentioning

confidence: 99%

Precision Matrix Estimation With ROPE

Kuismin

Kemppainen

Sillanpää

2017

Journal of Computational and Graphical Statistics

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It is known that the accuracy of the maximum likelihood based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article we propose a Ridge-type Operator for the Precision matrix Estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context.

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Section: The Proposed Methodsmentioning

confidence: 99%

Precision Matrix Estimation With ROPE

Kuismin

Kemppainen

Sillanpää

2017

Journal of Computational and Graphical Statistics

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“…Bini et al [11] proposed an algorithm by complementing the transformation with the shrink-and-shift technique of Ramaswami for finding the solution of the UQME. Larin [29] generalized the Schur and doubling methods to the UQME. For the unstructured QME, which has a wide application in the quasi-birth-death process [6,30], the minimal nonnegative solution is of importance.…”

Section: Introductionmentioning

confidence: 99%

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

2018

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We consider a matrix polynomial equation (MPE) A n X n + A n−1 X n−1 + · · · + A 0 = 0, where A n , A n−1 , . . . , A 0 ∈ R m×m are the coefficient matrices, and X ∈ R m×m is the unknown matrix. A sufficient condition for the existence of the minimal nonnegative solution is derived, where minimal means that any other solution is componentwise no less than the minimal one. The explicit expressions of normwise, mixed and componentwise condition numbers of the matrix polynomial equation are obtained. A backward error of the approximated minimal nonnegative solution is defined and evaluated. Some numerical examples are given to show the sharpness of the three kinds of condition numbers.

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Stabilization of the Motion of a Nonlinear System with Interval Initial Conditions

Babenko

Martynuyk

2016

Int Appl Mech

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Algorithms for Solving a Unilateral Quadratic Matrix Equation and the Model Updating Problem

Cited by 6 publications

References 18 publications

Precision Matrix Estimation With ROPE

Precision Matrix Estimation With ROPE

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

Stabilization of the Motion of a Nonlinear System with Interval Initial Conditions

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