## 2005.9: A Schur-Newton Method for the Matrix p'th Root and its Inverse

2005.9:
Chun-Hua Guo and Nicholas J. Higham
(2006)
*A Schur-Newton Method for the Matrix p'th Root and its Inverse.*

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## Abstract

Newton's method for the inverse matrix $p$th root, $A^{-1/p}$, has the attraction that it involves only matrix multiplication. We show that if the starting matrix is $c^{-1}I$ for $c\in\R^+$ then the iteration converges quadratically to $A^{-1/p}$ if the eigenvalues of $A$ lie in a wedge-shaped convex set containing the disc $\{\, z: |z-c^p| < c^p\,\}$. We derive an optimal choice of $c$ for the case where $A$ has real, positive eigenvalues. An application is described to roots of transition matrices from Markov models, in which for certain problems the convergence condition is satisfied with $c=1$. Although the basic Newton iteration is numerically unstable, a coupled version is stable and a simple modification of it provides a new coupled iteration for the matrix $p$th root. For general matrices we develop a hybrid algorithm that computes a Schur decomposition, takes square roots of the upper (quasi)triangular factor, and applies the coupled Newton iteration to a matrix for which fast convergence is guaranteed. The new algorithm can be used to compute either $A^{1/p}$ or $A^{-1/p}$, and for large $p$ that are not highly composite it is more efficient than the method of Smith based entirely on the Schur decomposition.

Item Type: | MIMS Preprint |
---|---|

Uncontrolled Keywords: | Matrix $p$th root, principal $p$th root, matrix logarithm, inverse, Newton's method, preprocessing, Schur decomposition, numerical stability, convergence, Markov model, transition matrix |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2005.9 |

Deposited By: | Nick Higham |

Deposited On: | 16 February 2006 |

### Available Versions of this Item

- A Schur-Newton Method for the Matrix p'th Root and its Inverse (deposited 27 October 2006)
- A Schur-Newton Method for the Matrix p'th Root and its Inverse (deposited 09 October 2006)
- A Schur-Newton Method for the Matrix p'th Root and its Inverse (deposited 16 February 2006)
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