On the computation of a matrix inverse square root (Q1179549)

The problem of finding an inverse square root \(X\) of a non-singular \(C\in \mathbb{C}^{n\times n}\) (i.e. such that \(X^ 2C=I\)) is considered. Two iterative methods are proposed to compute the matrix \(X\). One of them provides a sequence \((X_ r)\) defined by \(X_ 0=I\), \(X_{r+1}=2X_ r(I+CX^ 2_ r)^{-1}\) which converges to the principal square root of \(C^{-1}\). A sufficient condition which ensures the local stability of this scheme is emphasized. An alternative locally stable scheme for diagonalizable matrices is given. Other possible methods are enumerated, as the matrix sign function method, the matrix continued-fraction method, the Schur method and the Newton-Raphson method. The performances of the suggested techniques are illustrated by four numerical examples and the application of the inverse square root to computing the optimal symmetric orthogonalization of a set of vectors is emphasized.