### Abstract

The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e. when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. The results of numerical experiments are provided to demonstrate the effectiveness of the probing method.

Original language | English (US) |
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Pages (from-to) | 485-501 |

Number of pages | 17 |

Journal | Numerical Linear Algebra with Applications |

Volume | 19 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2012 |

### Keywords

- Covariance matrices
- Graph theory
- Green's functions
- Krylov-subspace methods
- Matrix diagonal extraction
- Probing
- Sparse approximate inverses

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## Cite this

*Numerical Linear Algebra with Applications*,

*19*(3), 485-501. https://doi.org/10.1002/nla.779