\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

abs

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AD Absolute Value Functions: abs, fabs

Syntax

y = abs ( x )

y = fabs ( x )

x, y

See the Possible Types for a unary standard math function.

Atomic

In the case where x is an AD type, this is an atomic operation .

Complex Types

The functions abs and fabs are not defined for the base types std::complex<float> or std::complex<double> because the complex abs function is not complex differentiable (see complex types faq ).

Derivative

CppAD defines the derivative of the abs function is the sign function; i.e.,

\[\begin{split}{\rm abs}^{(1)} ( x ) = {\rm sign} (x ) = \left\{ \begin{array}{rl} +1 & {\rm if} \; x > 0 \\ 0 & {\rm if} \; x = 0 \\ -1 & {\rm if} \; x < 0 \end{array} \right.\end{split}\]

The result for x == 0 used to be a directional derivative.

Example

The file fabs.cpp contains an example and test of this function.