Solve App and Piecewise function

[Han]

(10-22-2015 01:46 PM)cclinus Wrote:  Ok, but, why the second example work well?

In the second example, your piecewise function consists of two functions that are each continuous on either half-open or half-closed intervals. In the first example, your piecewise function has a piece that is nowhere continuous.

A lot of numerical solving algorithms tend to use some sort of Newton process, which requires that the auxiliary function for the system to be continuous over some interval. As an example,

2x+3y = 5
5x-4y = 1

This system is equivalent to solving \( \vec{F}(x,y) = 0 \) where
\[ \vec{F}(x,y) = \begin{bmatrix} 2x+3y-5 \\ 5x-4y-1 \end{bmatrix} \]

Generally speaking, the zeroes of \( \vec{F}(x,y) \) can be estimated by Newton's method; however \( \vec{F} \) must be differentiable (and hence continuous) near the zero. This is why systems with piecewise functions where the solution itself occurs at a point of discontinuity will likely never be found by numerical methods (that use Newton-like processes).