TY - JOUR

T1 - A note on golden means, nonlinear matrix equations and structured doubling algorithms

AU - Chiang, Chun Yueh

AU - Chu, Eric King Wah

AU - Lin, Wen-Wei

PY - 2010/1/1

Y1 - 2010/1/1

N2 - Several beautiful formulae for the solutions of some nonlinear matrix equations were proposed by Yongdo Lim in 2007, in terms of the matrix golden means. Numerically, these formulae will not be applicable when some matrices involved are ill-conditioned. In this note, we propose to partially fill in this gap of applicability with the structured doubling algorithm, under some favourable conditions. We also discuss how some pre-processing or scaling procedures can be applied to the matrix equations, to improve their condition. More generally, we also explore the possibility of computing the matrix golden mean using structured doubling algorithms. Some numerical examples will be presented for illustrative purposes.

AB - Several beautiful formulae for the solutions of some nonlinear matrix equations were proposed by Yongdo Lim in 2007, in terms of the matrix golden means. Numerically, these formulae will not be applicable when some matrices involved are ill-conditioned. In this note, we propose to partially fill in this gap of applicability with the structured doubling algorithm, under some favourable conditions. We also discuss how some pre-processing or scaling procedures can be applied to the matrix equations, to improve their condition. More generally, we also explore the possibility of computing the matrix golden mean using structured doubling algorithms. Some numerical examples will be presented for illustrative purposes.

KW - Algebraic riccati equation

KW - Condition

KW - Matrix golden mean

KW - Nonlinear matrix equation

KW - Structured doubling algorithm

UR - http://www.scopus.com/inward/record.url?scp=77954167897&partnerID=8YFLogxK

U2 - 10.11650/twjm/1500405876

DO - 10.11650/twjm/1500405876

M3 - Article

AN - SCOPUS:77954167897

VL - 14

SP - 955

EP - 972

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

SN - 1027-5487

IS - 3 A

ER -