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

Jacobian

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Jacobian: Driver Routine

Syntax

jac = f . Jacobian ( x )

Purpose

We use \(F : \B{R}^n \rightarrow \B{R}^m\) to denote the AD Function corresponding to f . The syntax above sets jac to the Jacobian of F evaluated at x ; i.e.,

\[jac = F^{(1)} (x)\]

f

The object f has prototype

ADFun < Base > f

Note that the ADFun object f is not const (see Forward or Reverse below).

x

The argument x has prototype

const Vector & x

(see Vector below) and its size must be equal to n , the dimension of the Domain space for f . It specifies that point at which to evaluate the Jacobian.

jac

The result jac has prototype

Vector jac

(see Vector below) and its size is \(m * n\); i.e., the product of the Domain and Range dimensions for f . For \(i = 0 , \ldots , m - 1\) and \(j = 0 , \ldots , n - 1\)

\[jac[ i * n + j ] = \D{ F_i }{ x_j } ( x )\]

Vector

The type Vector must be a SimpleVector class with elements of type Base . The routine CheckSimpleVector will generate an error message if this is not the case.

Forward or Reverse

This will use order zero Forward mode and either order one Forward or order one Reverse to compute the Jacobian (depending on which it estimates will require less work). After each call to Forward , the object f contains the corresponding Taylor coefficients . After a call to Jacobian , the zero order Taylor coefficients correspond to f . Forward (0, x ) and the other coefficients are unspecified.

Example

The routine Jacobian is both an example and test. It returns true , if it succeeds and false otherwise.