We consider the problems of multiple interpolation of analytic functions $f(z)=f_0+f_1z+\dots$ in the unit disk with node $z=0$ by means of simple partial fractions (logarithmic derivatives of algebraic polynomials) with free poles and with all poles on the circle $|z|=1$. We obtain estimates of the interpolation errors under a condition of the form $|f_{m-1}|<C/\sqrt{m}$, $m=1,2,\dots$.