An algorithm for the calculation of the net induction energy and induced multipole tensors in a set of charge distributions with non-linear contributions of all potentials of external fields and both permanent and induced multipoles Abstract: This algorithm for the calculation of the induced multipole tensors in a set of charge distributions includes the contributions of partial derivatives of arbitrary order of the potentials defined by both induced and permanent multipoles as well as non-uniform external fields of arbitrary strength. Specific equations are given for both systems with and without translational order. The algorithm, which is based on the Maxwell invariant form, uses direct extensions of algorithms previously developed and tested for the calculation of permanent multipole energies and induced dipole vectors (when the non-uniformity of the field is neglected). The induced tensor components are calculated iteratively. The first approximation, which gives the components as the solution of a set of simultaneous linear equations, includes all non-linear, non-uniform contributions of permanent multipoles and external fields as well as contributions linear in derivatives of the induced potentials. The induced tensor components are then used to calculate the net induction energy. The general relation between the polarizability tensors with respect to a centre and the moments of polarizability densities about the centre are derived.