### Abstract

The short term behavior of a Markov chain can be inferred from its fundamental matrix F. One method of computing the parts of F that are needed is to compute Fy for a given vector y. It is shown that all forward stable algorithms that solve a particular least squares problem lead to forward stable algorithms for computing Fy. This in turn leads to a class of algorithms that compute Fy accurately whenever the underlying problem is well-conditioned. One algorithm from this class is based upon the Grassman-Taksar-Heyman variant of Gaussian elimination. Other such algorithms include one based upon orthogonal factorization and one based upon the conjugate gradient least squares algorithm.

Original language | English (US) |
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Pages (from-to) | 230-241 |

Number of pages | 12 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

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*SIAM Journal on Matrix Analysis and Applications*, vol. 22, no. 1, pp. 230-241. https://doi.org/10.1137/S0895479898334538

**Stable computation with the fundamental matrix of a Markov chain.** / Barlow, Jesse L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Stable computation with the fundamental matrix of a Markov chain

AU - Barlow, Jesse L.

PY - 2000/1/1

Y1 - 2000/1/1

N2 - The short term behavior of a Markov chain can be inferred from its fundamental matrix F. One method of computing the parts of F that are needed is to compute Fy for a given vector y. It is shown that all forward stable algorithms that solve a particular least squares problem lead to forward stable algorithms for computing Fy. This in turn leads to a class of algorithms that compute Fy accurately whenever the underlying problem is well-conditioned. One algorithm from this class is based upon the Grassman-Taksar-Heyman variant of Gaussian elimination. Other such algorithms include one based upon orthogonal factorization and one based upon the conjugate gradient least squares algorithm.

AB - The short term behavior of a Markov chain can be inferred from its fundamental matrix F. One method of computing the parts of F that are needed is to compute Fy for a given vector y. It is shown that all forward stable algorithms that solve a particular least squares problem lead to forward stable algorithms for computing Fy. This in turn leads to a class of algorithms that compute Fy accurately whenever the underlying problem is well-conditioned. One algorithm from this class is based upon the Grassman-Taksar-Heyman variant of Gaussian elimination. Other such algorithms include one based upon orthogonal factorization and one based upon the conjugate gradient least squares algorithm.

UR - http://www.scopus.com/inward/record.url?scp=0034353660&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034353660&partnerID=8YFLogxK

U2 - 10.1137/S0895479898334538

DO - 10.1137/S0895479898334538

M3 - Article

AN - SCOPUS:0034353660

VL - 22

SP - 230

EP - 241

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 1

ER -