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Optimal Parameters for Doubling Algorithms

Optimal Parameters for Doubling Algorithms
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Year:    2017

Author:    Tsung-Ming Huang, Ren-Cang Li, Wen-Wei Lin, Linzhang Lu

Journal of Mathematical Study, Vol. 50 (2017), Iss. 4 : pp. 339–357

Abstract

In using the structure-preserving algorithm (SDA) [Linear Algebra Appl., 2005, vol. 396, pp. 55–80] to solve a continuous-time algebraic Riccati equation, a parameter-dependent linear fractional transformation $z→(z−\gamma)/(z+\gamma)$ is first performed in order to bring all the eigenvalues of the associated Hamiltonian matrix in the open left half-plane into the open unit disk. The closer the eigenvalues are brought to the origin by the transformation via judiciously selected parameter $\gamma,$ the faster the convergence of the doubling iteration will be later on. As the first goal of this paper, we consider several common regions that contain the eigenvalues of interest and derive the best $\gamma$ so that the images of the regions under the transform are closest to the origin. For our second goal, we investigate the same problem arising in solving an $M$-matrix algebraic Riccati equation by the alternating-directional doubling algorithm (ADDA) [SIAM J. Matrix Anal. Appl., 2012, vol. 33, pp. 170–194] which uses the product of two linear fractional transformations $(z_1,z_2)→[(z_2−\gamma_2)/(z_2+\gamma_1)][(z_1−\gamma_1)/$$(z_1+\gamma_2)]$ that involves two parameters. Illustrative examples are presented to demonstrate the efficiency of our parameter selection strategies.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v50n4.17.04

Journal of Mathematical Study, Vol. 50 (2017), Iss. 4 : pp. 339–357

Published online:    2017-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Doubling algorithm SDA ADDA optimal parameter algebraic Riccati equation.

Author Details

Tsung-Ming Huang

Ren-Cang Li

Wen-Wei Lin

Linzhang Lu

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