THE MATRIX UNWINDING FUNCTION, WITH AN APPLICATION TO COMPUTING THE MATRIX EXPONENTIAL

摘要

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 A~α and α logA. 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-2πi U(A), which has eigenvalues with imaginary parts in the interval (-π, π] and for which e~A = emod(A), can give significant computational savings in the evaluation of the exponential by scaling and squaring algorithms.