Abstract: A fast method is presented to compute high order approximate inverses for multidiagonal banded matrices of strong pivot. It is based on several concepts such as the order of an element in a strong pivot matrix, the order matrix and the influence areas of elimination steps.Under the condition of strong pivot, this method can be applied to unsymmetric and indefinite matrices. Coupling with the block preconditioned conjugate gradient, it is used for the numerical solution of elliptic partial differential equations and similar problems. It is demonstrated by the computational experiments that the method reported here has a higher efficiency in computations, comparing with similar methods.