Changeset 3348

Ignore:
Timestamp:
Sep 21, 2014 5:24:45 AM (6 years ago)
Message:

Fix some errors in the documentation for the general reverse mode sweep.

Location:
trunk
Files:
3 edited

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Unmodified
 r3301 where \f$n \f$ is the number of independent variables and \f$m \f$ is the number of dependent variables. We define \f$u^{(k)} \f$ as the value of x_k in the previous call of the form f.Forward(k, x_k) We define \f$X : {\bf R}^{n \times d} \rightarrow {\bf R}^n \f$ by \f[ X(t, u) =  u^{(0)} + u^{(1)} t + \cdots + u^{(d)} t^d \f] We define \f$Y : {\bf R}^{n \times d} \rightarrow {\bf R}^m \f$ by \f[ Y(t, u) =  F[ X(t, u) ] \f] We define the function \f$G : {\bf R}^{n \times d} \rightarrow {\bf R} \f$ by \f$W : {\bf R}^{n \times d} \rightarrow {\bf R} \f$ by \f[ G( u ) = \frac{1}{d !} \frac{ \partial^d }{ \partial t^d } \left[ \sum_{i=1}^m w_i  F_i ( u^{(0)} + u^{(1)} t + \cdots + u^{(d)} t^d ) \right]_{t=0} W(u) = \sum_{k=0}^{d} ( w^{(k)} )^{\rm T} \frac{1}{k !} \frac{\partial^k}{\partial t^k} Y(0, u) \f] Note that the scale factor  1 / a d  converts the \a d-th partial derivative to the \a d-th order Taylor coefficient. This routine computes the derivative of \f$G(u) \f$ (The matrix \f$w \in {\bf R}^m \f$, is defined below under the heading Partial.) Note that the scale factor  1 / k  converts the k-th partial derivative to the k-th order Taylor coefficient. This routine computes the derivative of \f$W(u) \f$ with respect to all the Taylor coefficients \f$u^{(k)} \f$ for \f$k = 0 , ... , d \f$. The vector \f$w \in {\bf R}^m \f$, and value of \f$u \in {\bf R}^{n \times d} \f$ at which the derivative is computed, are defined below. \n \n \b Input: The last \f$m \f$ rows of \a Partial are inputs. The vector \f$v \f$, used to define \f$G(u) \f$, The matrix \f$w \f$, used to define \f$W(u) \f$, is specified by these rows. For i = 0 , ... , m - 1, \a Partial [ ( \a numvar - m + i ) * K + d ] = v_i. For i = 0 , ... , m - 1 and for k = 0 , ... , d - 1, \a Partial [ ( \a numvar - m + i ) * K + k ] = 0. For i = 0 , ... , m - 1, for k = 0 , ... , d, Partial [ (numvar - m + i ) * K + k ] = w[i,k]. \n \n For j = 1 , ... , n and for k = 0 , ... , d, \a Partial [ j * K + k ] is the partial derivative of \f$G( u ) \f$ with is the partial derivative of \f$W( u ) \f$ with respect to \f$u_j^{(k)} \f$.
 r3214 $head Notation$$subhead x^(k)$$$subhead u^(k)$$For latex k = 0, \ldots , q-1$$, the vector $latex x^{(k)} \in B^n$$is defined as the value of the vector latex u^{(k)} \in B^n$$ is defined as the value of$icode x_k$$in the previous calls of the form codei% %$$ If there is no previous call with $latex k = 0$$, latex x^{(0)}$$ is the value of the independent variables when the$latex u^{(0)}$$is the value of the independent variables when the corresponding AD of icode Base$$
 r3311 assist you in learning about changes between various versions of CppAD. $head 10-21$$Fix a typo in documentation for cref/any order reverse/reverse_any/$$. To be specific,$latex x^{(k)}$$was changed to be latex u^{(k)}$$. \$head 05-28$$list number$$