The matrix unwinding function, with an application to computing the matrix exponential (Q2877080)

Let \(\mathcal{U}(A)=(A-\log e^A)/2\pi i \) be a unwinding number of \(A \in \mathbb{C}^{n\times n}\). Basic properties of \(\mathcal{U}(A)\) are derived in the paper. Bounds for the norm and the condition number of \(\mathcal{U}(A)\) are given and also discussed. A number of matrix identities involving the functions \(\log z\) and \(z^\alpha\) are derived. Connections with the matrix sign function are explored. A Schur-Parlett algorithm with a special reordering for computing \(\mathcal{U}(A)\) is given. Also given is some analysis connecting the conditioning of the Sylvester equations to the conditioning of \(\mathcal{U}\). Numerical experiments show that the algorithm performs well in practice. The use of the unwinding function for argument reduction with the matrix exponential is investigated.