A perturbation-expansion-at-zero-interval-limit based numeric al algorithm to calculate the eigenpairs of Hilbert-Schmidt integral operators having symmetric kernels is developed in the present work. We have developed the perturbation expansion only for the most dominant eigenvalue and relevant eigenfunction. The less important eigenpairs have been determined by using the most dominant spectral component extraction recursively over the kernel restrictions. The main lines of the formulation and certain related technicalities are presented here. The confirmation of the presented theory via certain illustrative implementations and the convergence discussion for the obtained perturbation series as well as the numerical comparison with some mostly considered methods are given in the next companion of this paper.