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Every student knows Newton’s iteration method from a textbook, which is widely used in numerical simulation, what few may know is that its ancient Chinese partner, Ying Buzu Shu, in about second century BC has much advantages over Newton’s method. The purpose of this paper is to introduce the ancient Chinese algorithm and its modifications for numerical simulation.

An example is given to show that the ancient Chinese algorithm is insensitive to initial guess, while a fast convergence rate is predicted.

Two new algorithms, which are suitable for numerical simulation, are introduced by absorbing the advantages of the Newton iteration method and the ancient Chinese algorithm.

This paper focuses on a single algebraic equation; however, it is easy to extend the theory to algebraic systems.

The Newton iteration method can be updated in numerical simulation.

The ancient Chinese algorithm is elucidated to have modern applications in various numerical methods.

Analytical methods are widely used in heat and fluid flow; the purpose of this paper is to couple Taylor series method and Bubbfil algorithm to solve nonlinear differential equations.

A series solution is obtained with some unknown constants, which can be determined by incorporating boundary conditions, and the constants are calculated by the Bubbfil algorithm.

This paper gives an analytical approach to a nonlinear equation arising in porous catalyst using Taylor series and Bubbfil algorithm, and a high accuracy can be achieved.

The coupled method of Taylor series and Bubbfil algorithm is a powerful method for nonlinear differential equations.

The proposed technology can be used in various numerical methods.

A new analytical method is proposed based on Taylor series and Bubbfil algorithm, which is a development of Newton’s iteration method and an ancient Chinese algorithm. The solution process is simple and easy to follow.