  We show that the time evolution operator of kicked quantum systems, although
a full matrix of size NxN, can be diagonalized with the help of a new method
based on a suitable combination of fast Fourier transform and Lanczos algorithm
in just N^2 ln(N) operations. It allows the diagonalization of matrizes of
sizes up to N\approx 10^6 going far beyond the possibilities of standard
diagonalization techniques which need O(N^3) operations. We have applied this
method to the kicked Harper model revealing its intricate spectral properties.
