Computing matrix-valued Nevanlinna-Pick interpolation

We describe a computational method, known as the Nevanlinna algorithm, for the matrix-valued Nevanlinna-Pick interpolation. The original interpolation problem formulated using the Carathéodory class of matrix-valued rational functions is first converted to an equivalent setting using the Schur class of rational functions. As a result, the necessary and sufficient Pick's condition for the interpolation becomes consistent with the scalar-valued formulation, so that some efficient techniques developed for the scalar-valued interpolation can be employed or modified for the matrix-valued case. We give a brief, yet sufficiently clear, derivation and a detailed arithmetic complexity analysis for the algorithm. We show that an n-point matrix-valued Nevanlinna-Pick interpolation using the new algorithm requires approximately 95nm 3 complex arithmetic operations, where m is the matrix dimension.