Ergodicity Coefficients Defined by Vector Norms

2011 ◽  
Vol 32 (1) ◽  
pp. 153-200 ◽  
Author(s):  
Ilse C. F. Ipsen ◽  
Teresa M. Selee
Keyword(s):  
Ergodicity Coefficients ◽  
Vector Norms

Coefficients of ergodicity with respect to vector norms

1983 ◽  
Vol 20 (2) ◽  
pp. 277-287 ◽  
Author(s):  
Choon-Peng Tan
Keyword(s):  
Transition Matrix ◽  
Ergodic Markov Chain ◽  
Numerical Examples ◽  
Localization Property ◽  
New Class ◽  
Geometric Convergence ◽  
Functional Forms ◽  
Ergodicity Coefficients ◽  
Spectrum Localization ◽  
Vector Norms

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.

Coefficients of ergodicity with respect to vector norms

1983 ◽  
Vol 20 (02) ◽  
pp. 277-287 ◽  
Author(s):  
Choon-Peng Tan
Keyword(s):  
Transition Matrix ◽  
Ergodic Markov Chain ◽  
Numerical Examples ◽  
Localization Property ◽  
New Class ◽  
Geometric Convergence ◽  
Functional Forms ◽  
Ergodicity Coefficients ◽  
Spectrum Localization ◽  
Vector Norms

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l 1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.

scholarly journals The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

2021 ◽  
Vol 88 (1) ◽  
Author(s):  
Antoine Gautier ◽  
Matthias Hein ◽  
Francesco Tudisco
Keyword(s):  
Global Convergence ◽  
Convergence Theorem ◽  
Matrix Norm ◽  
Nonnegative Matrices ◽  
Np Hard ◽  
Power Method ◽  
Contraction Ratio ◽  
Matrix Norms ◽  
Vector Norms

AbstractWe analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

Stability, stabilization, regulation using vector norms

1996 ◽  
pp. 45-90 ◽  
Author(s):  
P. Borne ◽  
J. P. Richard ◽  
N. E. Radhy
Keyword(s):  
Vector Norms

Matrix and Vector Norms

1975 ◽  
pp. 41-53
Author(s):  
C. G. Broyden
Keyword(s):  
Vector Norms
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