Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

Existing algorithms for computing the matrix cosine are tightly coupled to a specific precision of floating-point arithmetic optimal efficiency so they do not conveniently extend an arbitrary environment. We develop algorithm that takes unit roundoff working as input, and works in precision. The employs Taylor approximation with scaling recovering it can be used Schur decomposition or decomposition-free manner. also derive framework Fréchet derivative, construct efficient evaluation scheme its derivative simultaneously precision, show how this extended compute sine, cosine, their derivatives all together. Numerical experiments new behave forward stable way over wide range precisions. transformation-free version is competitive accuracy state-of-the-art double surpasses existing alternatives both speed precisions higher than double.