Non-Linear Least Squares

Levernberg-Marquatdt Method

This is a second-order method.

We define an indicator $\rho$ to describe the degree of approximation: $\rho =\frac{f(x+\Delta x)-f(x)}{J(x{)}^T\Delta x}$

Abstract

1. Give an initial value $x_0$.
2. For $k$-th iteration, we find an incremental value of$\Delta x_k$, such that the object function $\Vert f(x_k+\Delta x_k){\Vert }_2^2$ reaches a smaller value.
3. Compute $\rho$ using the above equation.
4. If $\rho >\frac{3}{4}$ , set $\mu =2\mu$ .
5. Otherwise, if $\rho <\frac{1}{4}$, set $\mu =0.5\mu$
6. If $\rho$ is larger than a given threshold, set $x_{k+1}=x_k+\Delta x_k$.
7. Go back to step 2 if not converged, otherwise return the result.

We use

Related

• Gauss-Newton Method