  In this article we introduce algorithms which compute iterations of
Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed
Hodge structure of affine varieties of dimension $n$ in terms of differential
forms. In the case $n=1$ such computations have many applications in
differential equations and counting their limit cycles. For $n>3$, these
computations give us an explicit definition of Hodge cycles.
