NonlinearSolver::newton_raphson Namespace Reference

Items related to the NewtonRaphson nonlinear solver. More...

Classes
struct	NewtonRaphson
	A Newton-Raphson correction scheme for nonlinear systems of equations $A_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}(x)=b$. More...

Functions
double	next_step_length (size_t globalization_iteration_id, double step_length, double prev_step_length, double residual, double residual_slope, double next_residual, double prev_residual)
	Find the next step length for the line-search globalization. More...

Detailed Description

Items related to the NewtonRaphson nonlinear solver.

Function Documentation

next_step_length()

double NonlinearSolver::newton_raphson::next_step_length	(	size_t	globalization_iteration_id,
		double	step_length,
		double	prev_step_length,
		double	residual,
		double	residual_slope,
		double	next_residual,
		double	prev_residual
	)

Find the next step length for the line-search globalization.

The next step length is chosen such that it minimizes the quadratic (first globalization step, i.e., when globalization_iteration_id is 0) or cubic (subsequent globalization steps) polynomial interpolation. This function implements Algorithm A6.3.1 in [50] (p. 325). This is how argument names map to symbols in that algorithm:

• step_length: $\lambda$
• prev_step_length: ${\lambda }_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{v}}$
• residual: $f_c$
• residual_slope: $g^{T}p<0$
• next_residual: $f_{+}$
• prev_residual: $f_{+,\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{v}}$

Note that the argument residual_slope is the derivative of the residual function $f$ w.r.t. the step length, i.e. $\frac{\mathrm{d}f}{\mathrm{d}\lambda }$, which must be negative. For the common scenario where $f(x)=|\mathit{r}(x){|}^2$, i.e. the residual function is the L2 norm of a residual vector $\mathit{r}(x)$, and where that in turn is the residual of a nonlinear equation $\mathit{r}(x)=b-A_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}(x)$ in a Newton-Raphson step as described in NonlinearSolver::newton_raphson::NewtonRaphson, then the residual_slope reduces to

$$\begin{array}{}\text{(1)}& \frac{\mathrm{d}f}{\mathrm{d}\lambda }=\frac{\mathrm{d}f}{\mathrm{d}{x}^i}\frac{\mathrm{d}{x}^i}{\mathrm{d}\lambda }=2\mathit{r}(x)\cdot \frac{\mathrm{d}\mathit{r}}{\mathrm{d}{x}^i}\frac{\mathrm{d}{x}^i}{\mathrm{d}\lambda }=-2|\mathit{r}(x){|}^2=-2f(x)\equiv -2f_c\text{.}\end{array}$$

Here we have used the relation

$$\begin{array}{}\text{(2)}& \frac{\mathrm{d}\mathit{r}}{\mathrm{d}{x}^i}\frac{\mathrm{d}{x}^i}{\mathrm{d}\lambda }=-\frac{\delta A_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}}{\delta x}\cdot \delta x=-r\end{array}$$

of a Newton-Raphson step of full length $\delta x$.