2014/01/01 by Mary Aprahamian, Nicholas J. Higham · 1 citation
Computer Science · Mathematics · #Matrix Theory and Algorithms #Numerical methods for differential equations #Mathematical functions and polynomials #Mathematics #Matrix function #Matrix (chemical analysis) #Function (biology) #Eigenvalues and eigenvectors #Logarithm #Square root of a 2 by 2 matrix #Exponential function #Combinatorics #Scaling #Matrix exponential #Sign function #Symmetric matrix #Mathematical analysis #Physics #Geometry #Quantum mechanics
paper · doi:10.1137/130920137
openalex publication_date 2014/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
A new matrix function corresponding to the scalar unwinding number of Corless, Hare, and Jeffrey is introduced. This matrix unwinding function, U, is shown to be a valuable tool for deriving identities involving the matrix logarithm and fractional matrix powers, revealing, for example, the precise relation between log Aα and α log A. The unwinding function is also shown to be closely connected with the matrix sign function. An algorithm for computing the unwinding function based on the Schur--Parlett method with a special reordering is proposed. It is shown that matrix argument reduction using the function mod(A) = A-2π i U(A), which has eigenvalues with imaginary parts in the interval (-π,π] and for which eA = emod(A), can give significant computational savings in the evaluation of the exponential by scaling and squaring algorithms.