A one parameter method for the matrix inverse square root
An inverse square root of a matrix \(A\in L(\mathbb C^m) \) is a solution \(X\in L(\mathbb C^m) \) of the matrix equation \(AX^2=I\) and it is denoted by \(A^{-\frac {1}{2}}\). The inverse square root of \(A\) always exists for a nonsingular matrix \(A\). The author proposes two one-parameter iterative methods and proves that the sequence of iterations constructed by one of this methods converges to the principal \(A^{-\frac {1}{2}}\) if \(A\) has non-negative eigenvalues. Choice of a parameter and numerical stability is discussed. Presented and earlier algorithms published by \textit{N. Sherif} [Computing 46, No. 4, 295-305 (1991; Zbl 0741.65039)] are tested. The four numerical examples show the advantages of parametrized methods.
scientific article; zbMATH DE number 1450133