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Absolute Value Functions

Syntax
Purpose
x
y
Derivative
Directional Derivative
Example

Syntax
y = abs(x)

Purpose
Sets y equal to the absolute value of $x$.

x
The argument x can be an instance of float, an a_float, an a2float, or an numpy.array of such objects.

y
If x is an instance of float, y will also be an instance of float . Otherwise y will have the same type as x .  In the case where x is an array, y will the same shape as x and the elements of y will have the same type as the elements of x .

Derivative
pycppad defines the derivative of the absolute value function by $$\R{abs}^{(1)} (x) = \R{sign} (x) = \left\{ \begin{array}{ll} 1 & \R{if} \; x > 0 \\ 0 & \R{if} \; x = 0 \\ -1 & \R{if} \; x < 0 \end{array} \right.$$

Directional Derivative
Prior to 2011-12-30 , forward mode computed the directional derivative of the absolute value function which is defined by $$\R{abs}^\circ ( x , d ) = \lim_{\lambda \downarrow 0 } \frac{\R{abs}(x + \lambda d) - \R{abs}(x) }{ \lambda }$$ For $x \neq 0$, $$\R{abs}^\circ ( x , d ) = \R{abs}^{(1)} ( x ) * d$$ and $\R{abs}^\circ (0 , d) = |d|$.

Example
The file abs.py contains an example and test of this function.