On real logarithms of nearby matrices and structured matrix interpolation
Two different, but related problems are considered: The computation of the real logarithm of a matrix, given the logarithm of a nearby matrix; the second problem is to interpolate given structured matrix data by a smooth matrix function, within a class of matrices having the same structure (structured here means symplectic, positive definite, or orthogonal). The logarithm of a nearby matrix is obtained by updating the known logarithm with a Taylor series expansion or a Newton iteration method. The Fréchet derivative can be given an integral representation which could be evaluated with a quadrature formula. However, it might be advantageous to not compute the principal logarithm because this could give a discontinuity if we want to keep the imaginary parts of the eigenvalues of the logarithm in \((-\pi,\pi)\). For the interpolation problem, the structure of the interpolant can be guaranteed by transforming the data, interpolating the transformed data by a structured polynomial matrix, and then transforming back. This method guarantees the smoothness of the interpolant. The transform could be a Cayley transform, but also a log transform with inverse transform exp. This relates the two problems. Theoretical analysis and error estimates are given, but also implementation details and several numerical examples are included.
scientific article; zbMATH DE number 1311283