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Up-> CppAD ADFun FunEval Reverse reverse_two

ADFun-> Independent FunConstruct Dependent abort_recording seq_property FunEval Drivers FunCheck optimize

FunEval-> Forward Reverse Sparse

Reverse-> reverse_one reverse_two reverse_any

reverse_two-> reverse_two.cpp HesTimesDir.cpp

Headings-> Syntax Purpose x^(k) W f w dw ---..First Order Partials ---..Second Order Partials Vector Hessian Times Direction Example

Second Order Reverse Mode
Syntax
dw = f.Reverse(2, w)

Purpose
We use $\mathit{F}:{\mathit{B}}^{\mathit{n}}\to {\mathit{B}}^{\mathit{m}}$ to denote the AD function corresponding to f . Reverse mode computes the derivative of the Forward mode Taylor coefficients with respect to the domain variable $\mathit{x}$ .

x^(k)
For $\mathit{k}=0,1$ , the vector ${\mathit{x}}^{(\mathit{k})}\in {\mathit{B}}^{\mathit{n}}$ is defined as the value of x_k in the previous call (counting this call) of the form
     f.Forward(k, x_k)
If there is no previous call with $\mathit{k}=0$ , ${\mathit{x}}^{(0)}$ is the value of the independent variables when the corresponding AD of Base operation sequence was recorded.

W
The functions ${\mathit{W}}_0:{\mathit{B}}^{\mathit{n}}\to \mathit{B}$ and ${\mathit{W}}_1:{\mathit{B}}^{\mathit{n}}\to \mathit{B}$ are defined by $$\begin{array}{ccc}\hfill {\mathit{W}}_0(\mathit{u})& \hfill =\hfill & {\mathit{w}}_0*{\mathit{F}}_0(\mathit{u})+\cdots +{\mathit{w}}_{\mathit{m}-1}*{\mathit{F}}_{\mathit{m}-1}(\mathit{u})\hfill \\ \hfill {\mathit{W}}_1(\mathit{u})& \hfill =\hfill & {\mathit{w}}_0*{\mathit{F}}_0^{(1)}(\mathit{u})*{\mathit{x}}^{(1)}+\cdots +{\mathit{w}}_{\mathit{m}-1}*{\mathit{F}}_{\mathit{m}-1}^{(1)}(\mathit{u})*{\mathit{x}}^{(1)}\hfill \end{array}$$ This operation computes the derivatives $$\begin{array}{ccc}\hfill {\mathit{W}}_0^{(1)}(\mathit{u})& \hfill =\hfill & {\mathit{w}}_0*{\mathit{F}}_0^{(1)}(\mathit{u})+\cdots +{\mathit{w}}_{\mathit{m}-1}*{\mathit{F}}_{\mathit{m}-1}^{(1)}(\mathit{u})\hfill \\ \hfill {\mathit{W}}_1^{(1)}(\mathit{u})& \hfill =\hfill & {\mathit{w}}_0*{\left({\mathit{x}}^{(1)}\right)}^{\mathrm{T}}*{\mathit{F}}_0^{(2)}(\mathit{u})+\cdots +{\mathit{w}}_{\mathit{m}-1}*{\left({\mathit{x}}^{(1)}\right)}^{\mathrm{T}}{\mathit{F}}_{\mathit{m}-1}^{(2)}(\mathit{u})\hfill \end{array}$$ at $\mathit{u}={\mathit{x}}^{(0)}$ .

f
The object f has prototype
     const ADFun<Base> f
Before this call to Reverse, the value returned by
     f.size_taylor()
must be greater than or equal two (see size_taylor ).

w
The argument w has prototype
     const Vector &w
(see Vector below) and its size must be equal to m , the dimension of the range space for f .

dw
The result dw has prototype
     Vector dw
(see Vector below). It contains both the derivative ${\mathit{W}}^{(1)}(\mathit{x})$ and the derivative ${\mathit{U}}^{(1)}(\mathit{x})$ . The size of dw is equal to $\mathit{n}\times 2$ , where $\mathit{n}$ is the dimension of the domain space for f .

First Order Partials
For $\mathit{j}=0,\dots ,\mathit{n}-1$ , $$\mathit{dw}[\mathit{j}*2+0]=\frac{\partial {\mathit{W}}_0}{\partial {\mathit{u}}_{\mathit{j}}}\left({\mathit{x}}^{(0)}\right)={\mathit{w}}_0*\frac{\partial {\mathit{F}}_0}{\partial {\mathit{u}}_{\mathit{j}}}\left({\mathit{x}}^{(0)}\right)+\cdots +{\mathit{w}}_{\mathit{m}-1}*\frac{\partial {\mathit{F}}_{\mathit{m}-1}}{\partial {\mathit{u}}_{\mathit{j}}}\left({\mathit{x}}^{(0)}\right)$$ This part of dw contains the same values as are returned by reverse_one .

Second Order Partials
For $\mathit{j}=0,\dots ,\mathit{n}-1$ , $$\mathit{dw}[\mathit{j}*2+1]=\frac{\partial {\mathit{W}}_1}{\partial {\mathit{u}}_{\mathit{j}}}\left({\mathit{x}}^{(0)}\right)=\sum _{\ell =0}^{\mathit{n}-1}{\mathit{x}}_{\ell }^{(1)}\left[{\mathit{w}}_0*\frac{{\partial }^2{\mathit{F}}_0}{\partial {\mathit{u}}_{\ell }\partial {\mathit{u}}_{\mathit{j}}}\left({\mathit{x}}^{(0)}\right)+\cdots +{\mathit{w}}_{\mathit{m}-1}*\frac{{\partial }^2{\mathit{F}}_{\mathit{m}-1}}{\partial {\mathit{u}}_{\ell }\partial {\mathit{u}}_{\mathit{j}}}\left({\mathit{x}}^{(0)}\right)\right]$$
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.

Hessian Times Direction
Suppose that $\mathit{w}$ is the i-th elementary vector. It follows that for $\mathit{j}=0,\dots ,\mathit{n}-1$ $$\begin{array}{ccc}\hfill \mathit{dw}[\mathit{j}*2+1]& \hfill =\hfill & {\mathit{w}}_{\mathit{i}}\sum _{\ell =0}^{\mathit{n}-1}\frac{{\partial }^2{\mathit{F}}_{\mathit{i}}}{\partial {\mathit{u}}_{\mathit{j}}\partial {\mathit{u}}_{\ell }}\left({\mathit{x}}^{(0)}\right){\mathit{x}}_{\ell }^{(1)}\hfill \\ \hfill & \hfill =\hfill & {\left[{\mathit{F}}_{\mathit{i}}^{(2)}\left({\mathit{x}}^{(0)}\right)*{\mathit{x}}^{(1)}\right]}_{\mathit{j}}\hfill \end{array}$$ Thus the vector $(\mathit{dw}[1],\mathit{dw}[3],\dots ,\mathit{dw}[\mathit{n}*\mathit{p}-1])$ is equal to the Hessian of ${\mathit{F}}_{\mathit{i}}(\mathit{x})$ times the direction ${\mathit{x}}^{(1)}$ . In the special case where ${\mathit{x}}^{(1)}$ is the l-th elementary vector , $$\mathit{dw}[\mathit{j}*2+1]=\frac{{\partial }^2{\mathit{F}}_{\mathit{i}}}{\partial {\mathit{x}}_{\mathit{j}}\partial {\mathit{x}}_{\ell }}\left({\mathit{x}}^{(0)}\right)$$
Example
The files reverse_two.cpp and HesTimesDir.cpp contain a examples and tests of reverse mode calculations. They return true if they succeed and false otherwise.

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Input File: omh/reverse.omh