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

forward_order

View page source

Multiple Order Forward Mode

Syntax

yq = f . Forward ( q , xq )

yq = f . Forward ( q , xq , s )

Purpose

We use \(F : \B{R}^n \rightarrow \B{R}^m\) to denote the AD Function corresponding to f . Given a function \(X : \B{R} \rightarrow \B{R}^n\), defined by its Taylor coefficients , forward mode computes the Taylor coefficients for the function

\[Y (t) = F [ X(t) ]\]

Function Values

If you are using forward mode to compute values for \(F(x)\), forward_zero is simpler to understand than this explanation of the general case.

Derivative Values

If you are using forward mode to compute values for \(F^{(1)} (x) * dx\), forward_one is simpler to understand than this explanation of the general case.

Notation

n

We use n to denote the dimension of the Domain space for f .

m

We use m to denote the dimension of the Range space for f .

f

The ADFun object f has prototype

ADFun < Base > f

Note that the ADFun object f is not const . After this call we will have

      f . size_order () == q + 1

      f . size_direction () == 1

One Order

If xq . size () == n , then we are only computing one order. In this case, before this call we must have

      f . size_order () >= q

      f . size_direction () == 1

q

The argument q has prototype

size_t q

and specifies the highest order of the Taylor coefficients to be calculated.

xq

The argument xq has prototype

const BaseVector & xq

(see BaseVector below). As above, we use n to denote the dimension of the Domain space for f . The size of xq must be either n or n * ( q +1) . After this call we will have

f . size_order () == q + 1

One Order

If xq . size () == n , the q-th order Taylor coefficient for \(X(t)\) is defined by

      \(x^{(q)} =\) xq . For \(k = 0 , \ldots , q-1\), the Taylor coefficient \(x^{(k)}\) is defined by xk in the previous call to

f . Forward ( k , xk )

Multiple Orders

If xq . size () == n * ( q +1) , For \(k = 0 , \ldots , q\), \(j = 0 , \ldots , n-1\), the j-th component of the k-th order Taylor coefficient for \(X(t)\) is defined by

      \(x_j^{(k)} =\) xq [ ( q +1) * j + k ]

Restrictions

Note if f uses atomic_one functions, the size of xq must be n .

s

If the argument s is not present, std::cout is used in its place. Otherwise, this argument has prototype

std::ostream& s

If order zero is begin calculated, s specifies where the output corresponding to PrintFor will be written. If order zero is not being calculated, s is not used

X(t)

The function \(X : \B{R} \rightarrow \B{R}^n\) is defined using the Taylor coefficients \(x^{(k)} \in \B{R}^n\):

\[X(t) = x^{(0)} * t^0 + x^{(1)} * t^1 + \cdots + x^{(q)} * t^q\]

Note that for \(k = 0 , \ldots , q\), the k-th derivative of \(X(t)\) is related to the Taylor coefficients by the equation

\[x^{(k)} = \frac{1}{k !} X^{(k)} (0)\]

Y(t)

The function \(Y : \B{R} \rightarrow \B{R}^m\) is defined by \(Y(t) = F[ X(t) ]\). We use \(y^{(k)} \in \B{R}^m\) to denote the k-th order Taylor coefficient of \(Y(t)\); i.e.,

\[Y(t) = y^{(0)} * t^0 + y^{(1)} * t^1 + \cdots + y^{(q)} * t^q + o( t^q )\]

where \(o( t^q ) * t^{-q} \rightarrow 0\) as \(t \rightarrow 0\). Note that \(y^{(k)}\) is related to the k-th derivative of \(Y(t)\) by the equation

\[y^{(k)} = \frac{1}{k !} Y^{(k)} (0)\]

yq

The return value yq has prototype

BaseVector yq

(see BaseVector below).

One Order

If xq . size () == n , the vector yq has size m . The q-th order Taylor coefficient for \(Y(t)\) is returned as

yq

\(= y^{(q)}\).

Multiple Orders

If xq . size () == n * ( q +1) , the vector yq has size m * ( q +1) . For \(k = 0 , \ldots , q\), for \(i = 0 , \ldots , m-1\), the i-th component of the k-th order Taylor coefficient for \(Y(t)\) is returned as

yq [ ( q +1) * i + k ]

\(= y_i^{(k)}\)

BaseVector

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

Zero Order

The case where \(q = 0\) and xq . size () == n , corresponds to the zero order Special Case .

First Order

The case where \(q = 1\) and xq . size () == n , corresponds to the first order Special Case .

Second Order

The case where \(q = 2\) and xq . size () == n , corresponds to the second order Special Case .

Example

The files forward.cpp and forward_order.cpp contain examples and tests of using forward mode with one order and multiple orders respectively. They return true if they succeed and false otherwise.