Newton's Iteration Method

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Newton's iteration method (also known as the Newton-Raphson method) is a classical numerical method for finding the roots of a nonlinear equation $f(x) = 0$ . Its fundamental idea is to expand the nonlinear equation in a Taylor series around the current iterate, retain the linear part as an approximation, and successively approach the true solution.

Core Elements

Basic Formula

Assuming $f(x)$ is continuously differentiable on an interval containing $x_k$ , the linear approximation of the Taylor expansion yields the iteration formula:

x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}, \quad k = 0, 1, 2, \ldots

Geometric Interpretation

The geometric meaning of Newton's iteration method is: draw the tangent to $f(x)$ at the point $(x_k, f(x_k))$ , and the intersection of this tangent with the $x$ -axis gives the next iterate $x_{k+1}$ . For this reason, it is also called the tangent method.

Application in Trajectory Design

Trajectory design problems can be reduced to finding the roots of a nonlinear system $F(x) = 0$ . For a ballistic missile, the design variables are the launch azimuth $A_0$ and the pitch program angle rate $\dot{\varphi}_{pr}$ , and the terminal constraint is zero impact point deviation:

\begin{bmatrix} \dot{\varphi}_{pr}^{(k+1)} \\ A_0^{(k+1)} \end{bmatrix} = \begin{bmatrix} \dot{\varphi}_{pr}^{(k)} \\ A_0^{(k)} \end{bmatrix} + \begin{bmatrix} \frac{\partial \Delta L}{\partial \dot{\varphi}_{pr}} & \frac{\partial \Delta L}{\partial A_0} \\ \frac{\partial \Delta H}{\partial \dot{\varphi}_{pr}} & \frac{\partial \Delta H}{\partial A_0} \end{bmatrix}^{-1} \begin{bmatrix} -\Delta L^{(k)} \\ -\Delta H^{(k)} \end{bmatrix}

Partial Derivative Computation

The accuracy of partial derivatives significantly affects convergence. Common methods:

Method	Characteristics	Computational Cost
Finite difference	Simple to implement, sensitive to step size selection	Low (1 extra trajectory per parameter)
Richardson extrapolation	High accuracy, insensitive to step size	High (4 extra trajectories per parameter)

The finite-difference partial derivative approximation:

\frac{\partial f}{\partial x_i} \approx \frac{f(x_i + \delta x_i) - f(x_i)}{\delta x_i}

Convergence Properties

Property	Description
Fast convergence	Local superlinear convergence
Sensitive to initial guess	Poor initial guess may cause divergence
Finds a feasible solution	Only locates a solution satisfying the constraints, not necessarily the optimal one

Application Value

Newton's iteration method is the core numerical method for powered-phase trajectory design of ballistic missiles and launch vehicles. By reformulating the trajectory design problem as a nonlinear equation system, it can rapidly determine the flight program angle and launch azimuth that satisfy terminal constraints. The method is simple to implement and converges quickly, making it widely used in engineering.

Sequential Quadratic Programming

References

Zheng Wei, An Xueying, Zhou Xiang, He Ruizhi. Aerospace Flight Mechanics (空天飞行力学)[M]. National University of Defense Technology, 2026.
Jia Peiran, Chen Kejun, et al. Long-Range Rocket Ballistics (远程火箭弹道学)[M]. National University of Defense Technology Press.