Using the QR Factorization and Group Inversion to Compute, Differentiate, and Estimate the Sensitivity of Stationary Probabilities for Markov Chains

DOI10.1137/0607031zbMath0594.60072OpenAlexW1972521646MaRDI QIDQ3725280

Gene H. Golub, Carl D. jun. Meyer

Publication date: 1986

Published in: SIAM Journal on Algebraic Discrete Methods (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/0607031

zbMATH Keywords

perturbationsstationary distributionQR factorizationhomogeneous ergodic Markov chainestimates for the sensitivity

Mathematics Subject Classification ID

Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Theory of matrix inversion and generalized inverses (15A09) Markov chains (discrete-time Markov processes on discrete state spaces) (60J10) Numerical computation of matrix norms, conditioning, scaling (65F35) Stochastic matrices (15B51) Conditioning of matrices (15A12) Orthogonalization in numerical linear algebra (65F25)

Related Items

The group inverse of finite homogeneous QBD processes, Group inverses of \(M\)-matrices associated with nonnegative matrices having few eigenvalues, An Arnoldi-type algorithm for computing Page Rank, On the Smallest Positive Singular Value of a Singular M-Matrix with Applications to Ergodic Markov Chains, Nonnegative alternating circulants leading to \(M\)-matrix group inverses, Acute perturbation of the group inverse, Aggregation Methods for Computing Steady States in Statistical Physics, Sensitivity analysis of discrete Markov chains via matrix calculus, Optimizing Weighted Ensemble Sampling of Steady States, Stratification as a General Variance Reduction Method for Markov Chain Monte Carlo, Perturbation bounds for the group inverse and its oblique projection, Unnamed Item, A direct projection method for Markov chains, Improved bounds for a condition number for Markov chains, Perturbation analysis of continuous‐time absorbing Markov chains, A note on the perturbation of positive matrices by normal and unitary matrices, Hopf bifurcation in a mean-field model of spiking neurons, On optimal condition numbers for Markov chains, Blockwise perturbation theory for nearly uncoupled Markov chains and its application, New perturbation bounds for denumerable Markov chains, Expressions for the drazin inverse of a 2×2 Block Matrix, An improvement on the perturbation of the group inverse and oblique projection, An oblique projection iterative method to compute Drazin inverse and group inverse, Influence of Preconditioning and Blocking on Accuracy in Solving Markovian Models, A new decision making model based on rank centrality for GDM with fuzzy preference relations, The Drazin inverse of updating of a square matrix with application to perturbation formula, Single sample path-based optimization of Markov chains, On the perturbation of the group inverse and oblique projection, Markov chain sensitivity measured by mean first passage times, Sharp Entrywise Perturbation Bounds for Markov Chains, Sensitivity of the stationary distribution vector for an ergodic Markov chain

Cites Work

• Unnamed Item
• Unnamed Item
• Matrix Methods for Queuing Problems
• The Use of Positive Matrices for the Analysis of the Large Time Behavior of the Numerical Solution of Reaction-Diffusion Systems
• Solution of Homogeneous Systems of Linear Equations Arising from Compartmental Models
• The Condition of a Finite Markov Chain and Perturbation Bounds for the Limiting Probabilities
• The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains
• Perturbation theory and finite Markov chains