  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.
