APE--100 parallel supercomputers which are organized in the single instruction multiple data (SIMD) regime offer fast data exchange between neighbouring nodes without latency time to initiate communication. The nodes are topologically ordered to form a 3--dimensional cube with periodic boundary conditions thereby allowing physical real space problems to be modelled conveniently.   The setup of the system matrix A is performed by the commercial software MAFIA on a workstation whereas the APE--100 computes the eigenpairs of A using aLanczos algorithm without reorthogonalization. The implementation of the matrix vector multiplication and the bisection algorithm for the diagonalization of the tridiagonal Lanczos matrix on APE--100 are described in detail. The influence of the burst length for memory to register transfers on the total computing performance is discussed.   The method has been tested on several examples, for instance up to 30 eigenmodes have been calculated for a 3--dimensional model of a 9--cell TESLA cavity and a TESLA bunch compressor. The results are briefly reported.