Residual methods for the large-scale matrix \(p\)th root and some related problems
The authors present and analyze a low-cost residual algorithm for computing the matrix \(p\)th root, \(X\), of a nonlinear matrix equation \(X^p-A=0\), where \(A\in{\mathbb R} ^{n\times n}\) is symmetric and positive definite and \(p\geq 2\). This is the best possible scenario for Smith's method [\textit{M. I. Smith}, SIAM J. Matrix Anal. Appl. 24, No.~4, 971--989 (2003; Zbl 1040.65038)]. This algorithm has local convergence properties in its pure form and global convergence in its globalized form. It is more efficient and requires fewer iterations for convergence when \(p\) is small since the condition number of the derivative of the residual map, \(X^p-A\), increases with \(p\). It is also robust in general and suitable for large-scale problems for which high accuracy is not required.
scientific article; zbMATH DE number 5804921
scientific article