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Up-> CppAD AD ADValued atomic atomic_base atomic_reverse

ADValued-> Arithmetic unary_standard_math binary_math CondExp Discrete numeric_limits atomic

atomic-> checkpoint atomic_base

atomic_base-> atomic_ctor atomic_option atomic_afun atomic_forward atomic_reverse atomic_for_sparse_jac atomic_rev_sparse_jac atomic_for_sparse_hes atomic_rev_sparse_hes atomic_base_clear atomic_get_started.cpp atomic_norm_sq.cpp atomic_reciprocal.cpp atomic_set_sparsity.cpp atomic_tangent.cpp atomic_eigen_mat_mul.cpp atomic_eigen_mat_inv.cpp atomic_eigen_cholesky.cpp atomic_mat_mul.cpp

atomic_reverse-> atomic_reverse.cpp

Headings-> Syntax Purpose Implementation q tx ty F G, H py ---..px ok Examples

@(@\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}} }@)@ Atomic Reverse Mode
Syntax
ok = afun.reverse(q, tx, ty, px, py)

Purpose
This function is used by reverse to compute derivatives.

Implementation
If you are using reverse mode, this virtual function must be defined by the atomic_user class. It can just return ok == false (and not compute anything) for values of q that are greater than those used by your reverse mode calculations.

q
The argument q has prototype
     size_t q
It specifies the highest order Taylor coefficient that computing the derivative of.

tx
The argument tx has prototype
     const CppAD::vector<Base>& tx
and tx.size() == (q+1)*n . For @(@ j = 0 , \ldots , n-1 @)@ and @(@ k = 0 , \ldots , q @)@, we use the Taylor coefficient notation @[@ \begin{array}{rcl} x_j^k & = & tx [ j * ( q + 1 ) + k ] \\ X_j (t) & = & x_j^0 + x_j^1 t^1 + \cdots + x_j^q t^q \end{array} @]@ Note that superscripts represent an index for @(@ x_j^k @)@ and an exponent for @(@ t^k @)@. Also note that the Taylor coefficients for @(@ X(t) @)@ correspond to the derivatives of @(@ X(t) @)@ at @(@ t = 0 @)@ in the following way: @[@ x_j^k = \frac{1}{ k ! } X_j^{(k)} (0) @]@

ty
The argument ty has prototype
     const CppAD::vector<Base>& ty
and tx.size() == (q+1)*m . For @(@ i = 0 , \ldots , m-1 @)@ and @(@ k = 0 , \ldots , q @)@, we use the Taylor coefficient notation @[@ \begin{array}{rcl} Y_i (t) & = & f_i [ X(t) ] \\ Y_i (t) & = & y_i^0 + y_i^1 t^1 + \cdots + y_i^q t^q + o ( t^q ) \\ y_i^k & = & ty [ i * ( q + 1 ) + k ] \end{array} @]@ where @(@ o( t^q ) / t^q \rightarrow 0 @)@ as @(@ t \rightarrow 0 @)@. Note that superscripts represent an index for @(@ y_j^k @)@ and an exponent for @(@ t^k @)@. Also note that the Taylor coefficients for @(@ Y(t) @)@ correspond to the derivatives of @(@ Y(t) @)@ at @(@ t = 0 @)@ in the following way: @[@ y_j^k = \frac{1}{ k ! } Y_j^{(k)} (0) @]@

F
We use the notation @(@ \{ x_j^k \} \in B^{n \times (q+1)} @)@ for @[@ \{ x_j^k \W{:} j = 0 , \ldots , n-1, k = 0 , \ldots , q \} @]@ We use the notation @(@ \{ y_i^k \} \in B^{m \times (q+1)} @)@ for @[@ \{ y_i^k \W{:} i = 0 , \ldots , m-1, k = 0 , \ldots , q \} @]@ We define the function @(@ F : B^{n \times (q+1)} \rightarrow B^{m \times (q+1)} @)@ by @[@ y_i^k = F_i^k [ \{ x_j^k \} ] @]@ Note that @[@ F_i^0 ( \{ x_j^k \} ) = f_i ( X(0) ) = f_i ( x^0 ) @]@ We also note that @(@ F_i^\ell ( \{ x_j^k \} ) @)@ is a function of @(@ x^0 , \ldots , x^\ell @)@ and is determined by the derivatives of @(@ f_i (x) @)@ up to order @(@ \ell @)@.

G, H
We use @(@ G : B^{m \times (q+1)} \rightarrow B @)@ to denote an arbitrary scalar valued function of @(@ \{ y_i^k \} @)@. We use @(@ H : B^{n \times (q+1)} \rightarrow B @)@ defined by @[@ H ( \{ x_j^k \} ) = G[ F( \{ x_j^k \} ) ] @]@

py
The argument py has prototype
     const CppAD::vector<Base>& py
and py.size() == m * (q+1) . For @(@ i = 0 , \ldots , m-1 @)@, @(@ k = 0 , \ldots , q @)@, @[@ py[ i * (q + 1 ) + k ] = \partial G / \partial y_i^k @]@

px
The px has prototype
     CppAD::vector<Base>& px
and px.size() == n * (q+1) . The input values of the elements of px are not specified (must not matter). Upon return, for @(@ j = 0 , \ldots , n-1 @)@ and @(@ \ell = 0 , \ldots , q @)@, @[@ \begin{array}{rcl} px [ j * (q + 1) + \ell ] & = & \partial H / \partial x_j^\ell \\ & = & ( \partial G / \partial \{ y_i^k \} ) \cdot ( \partial \{ y_i^k \} / \partial x_j^\ell ) \\ & = & \sum_{k=0}^q \sum_{i=0}^{m-1} ( \partial G / \partial y_i^k ) ( \partial y_i^k / \partial x_j^\ell ) \\ & = & \sum_{k=\ell}^q \sum_{i=0}^{m-1} py[ i * (q + 1 ) + k ] ( \partial F_i^k / \partial x_j^\ell ) \end{array} @]@ Note that we have used the fact that for @(@ k < \ell @)@, @(@ \partial F_i^k / \partial x_j^\ell = 0 @)@.

ok
The return value ok has prototype
     bool ok
If it is true, the corresponding evaluation succeeded, otherwise it failed.

Examples
The file atomic_reverse.cpp contains an example and test that uses this routine. It returns true if the test passes and false if it fails.

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Input File: cppad/core/atomic_base.hpp