Universal approximations for flavor models

We develop a systematic analytical approximation scheme for the singular value decompositions of arbitrary complex three dimensional matrices Y with non-degenerate singular values. We derive exact expressions for the errors of this approximation and show that they are bounded from above by very simple ratios of the form (yi/yj )2n where yi< yj are singular values of Y and n is the order of the approximation. The applications we have in mind are the analytical and numerical treatments of arbitrary theories of flavor. We also compute upper bounds for the errors of the Cabbibo Kobayashi Maskawa (CKM) matrix that only depend on the ratios of the masses and the physical CKM angles.