When phase-function and differential-sweep methods (in the real and complex case, respectively) are applied to spectral problems for linear ordinary differential equations or to reduce the dimensions for non-linear autonomous differential systems, the need arises for a numerical solution of an auxiliary problem of finding the discrete eigenvalue spectrum. The use of an effective modification of shooting algorithms, boundary-condition parametrization (BCP), where the required eigenvalues are used as the ''missing initial parameters'', is proposed for this purpose. For an arbitrarily chosen bounded vector function of parametrization of the boundary conditions, a theorem stating that the iterative process of moving along the length of the integration interval is finite and the sequence of approximate solutions to the exact value is uniformly convergent is proved.
