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A fast algorithm is used to study the transient behavior due to a step-like pulse applied to a nano-chip. This algorithm is carried out on a computer and consists of two parts: The algorithm I reduces the computational complexity to T0N3 for large systems as long as T< N; The algorithm II employs the fast multipole technique and achieves scaling T0N3 whenever T< N2 beyond which it becomes T log2 N for even longer time. Hence it is of order O (1) if T< N2. Benchmark calculation has been done on graphene nanoribbons with N= 104 and T= 108. This new algorithm allows many large scale transient problems to be solved, including magnetic tunneling junctions and ferroelectric tunneling junctions that could not be achieved before, and using less computing capacity.