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Up-> CppAD ADFun sparsity_pattern ForSparseHes

CppAD-> Install Introduction AD ADFun preprocessor multi_thread utility ipopt_solve Example speed Appendix

ADFun-> record_adfun drivers Forward Reverse sparsity_pattern sparse_derivative optimize abs_normal FunCheck check_for_nan

sparsity_pattern-> for_jac_sparsity ForSparseJac rev_jac_sparsity RevSparseJac rev_hes_sparsity RevSparseHes for_hes_sparsity ForSparseHes dependency.cpp rc_sparsity.cpp subgraph_sparsity

ForSparseHes-> for_sparse_hes.cpp

Headings-> Syntax Purpose f x r s h VectorSet Algorithm Example

@(@\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}} }@)@Hessian Sparsity Pattern: Forward Mode
Syntax
h = f.ForSparseHes(r, s)

Purpose
We use @(@ F : \B{R}^n \rightarrow \B{R}^m @)@ to denote the AD function corresponding to f . we define @[@ \begin{array}{rcl} H(x) & = & \partial_x \left[ \partial_u S \cdot F[ x + R \cdot u ] \right]_{u=0} \\ & = & R^\R{T} \cdot (S \cdot F)^{(2)} ( x ) \cdot R \end{array} @]@ Where @(@ R \in \B{R}^{n \times n} @)@ is a diagonal matrix and @(@ S \in \B{R}^{1 \times m} @)@ is a row vector. Given a sparsity pattern for the diagonal of @(@ R @)@ and the vector @(@ S @)@, ForSparseHes returns a sparsity pattern for the @(@ H(x) @)@.

f
The object f has prototype
     const ADFun<Base> f

x
If the operation sequence in f is independent of the independent variables in @(@ x \in B^n @)@, the sparsity pattern is valid for all values of (even if it has CondExp or VecAD operations).

r
The argument r has prototype
     const VectorSet& r
(see VectorSet below) If it has elements of type bool, its size is @(@ n @)@. If it has elements of type std::set<size_t>, its size is one and all the elements of s[0] are between zero and @(@ n - 1 @)@. It specifies a sparsity pattern for the diagonal of @(@ R @)@. The fewer non-zero elements in this sparsity pattern, the faster the calculation should be and the more sparse @(@ H(x) @)@ should be.

s
The argument s has prototype
     const VectorSet& s
(see VectorSet below) If it has elements of type bool, its size is @(@ m @)@. If it has elements of type std::set<size_t>, its size is one and all the elements of s[0] are between zero and @(@ m - 1 @)@. It specifies a sparsity pattern for the vector S . The fewer non-zero elements in this sparsity pattern, the faster the calculation should be and the more sparse @(@ H(x) @)@ should be.

h
The result h has prototype
     VectorSet& h
(see VectorSet below). If h has elements of type bool, its size is @(@ n * n @)@. If it has elements of type std::set<size_t>, its size is @(@ n @)@ and all the set elements are between zero and n-1 inclusive. It specifies a sparsity pattern for the matrix @(@ H(x) @)@.

VectorSet
The type VectorSet must be a SimpleVector class with elements of type bool or std::set<size_t>; see sparsity pattern for a discussion of the difference. The type of the elements of VectorSet must be the same as the type of the elements of r .

Algorithm
See Algorithm II in Computing sparse Hessians with automatic differentiation by Andrea Walther. Note that s provides the information so that 'dead ends' are not included in the sparsity pattern.

Example
The file for_sparse_hes.cpp contains an example and test of this operation. It returns true if it succeeds and false otherwise.

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Input File: cppad/core/for_sparse_hes.hpp