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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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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

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