Prev

Next

Index-> contents reference index search external

Up-> CppAD Appendix speed speed_utility sparse_evaluate

Appendix-> Faq speed Theory glossary Bib Bugs WishList whats_new deprecated License

speed-> speed_main speed_utility speed_double speed_adolc speed_cppad speed_fadbad speed_sacado

speed_utility-> uniform_01 det_of_minor det_by_minor det_by_lu det_33 det_grad_33 mat_sum_sq ode_evaluate sparse_evaluate

sparse_evaluate-> sparse_evaluate.cpp sparse_evaluate.hpp

Headings-> Syntax Purpose Inclusion Float x i j m fm ---..Function ---..Gradient ---..Hessian Example Source Code

Evaluate a Function That Has a Sparse Hessian
Syntax
# include <cppad/speed/sparse_evaluate.hpp>
sparse_evaluate(x, i, j, m, fm)

Purpose
This routine evaluates $\mathit{f}(\mathit{x})$ , ${\mathit{f}}^{(1)}(\mathit{x})$ , or ${\mathit{f}}^{(2)}(\mathit{x})$ where the Hessian ${\mathit{f}}^{(2)}(\mathit{x})$ is sparse. The function $\mathit{f}:{\mathbf{R}}^{\mathit{n}}\to \mathbf{R}$ depends on the index vectors i and j . The non-zero entries in the Hessian of this function have one of the following forms: $$\frac{{\partial }^2\mathit{f}}{\partial \mathit{x}[\mathit{i}[\mathit{k}]]\partial \mathit{x}[\mathit{i}[\mathit{k}]]},\frac{{\partial }^2\mathit{f}}{\partial \mathit{x}[\mathit{i}[\mathit{k}]]\partial \mathit{x}[\mathit{j}[\mathit{k}]]},\frac{{\partial }^2\mathit{f}}{\partial \mathit{x}[\mathit{j}[\mathit{k}]]\partial \mathit{x}[\mathit{i}[\mathit{k}]]},\frac{{\partial }^2\mathit{f}}{\partial \mathit{x}[\mathit{j}[\mathit{k}]]\partial \mathit{x}[\mathit{j}[\mathit{k}]]}$$ for some $\mathit{k}$ between zero and $\ell -1$ . All the other terms of the Hessian are zero.

Inclusion
The template function sparse_evaluate is defined in the CppAD namespace by including the file cppad/speed/sparse_evaluate.hpp (relative to the CppAD distribution directory). It is only intended for example and testing purposes, so it is not automatically included by cppad.hpp .

Float
The type Float must be a NumericType . In addition, if y and z are Float objects,
     y = exp(z)
must set the y equal the exponential of z , i.e., the derivative of y with respect to z is equal to y .

x
The argument x has prototype
     const CppAD::vector<Float> &x
It contains he argument value for which the function, or its derivative, is being evaluated. We use $\mathit{n}$ to denote the size of the vector x .

i
The argument i has prototype
      const CppAD::vector<size_t> &i
It specifies one of the first index of $\mathit{x}$ for each non-zero Hessian term (see purpose above). All the elements of i must be between zero and n-1 . We use $\ell$ to denote the size of the vector i .

j
The argument j has prototype
      const CppAD::vector<size_t> &j
and is a vector with size $\ell$ . It specifies one of the second index of $\mathit{x}$ for each non-zero Hessian term. All the elements of j must be between zero and n-1 .

m
The argument m has prototype
     size_t m
It is between zero and two and specifies the order of the derivative of $\mathit{f}$ that is being evaluated, i.e., ${\mathit{f}}^{(\mathit{m})}(\mathit{x})$ is evaluated.

fm
The argument fm has prototype
     CppAD::vector<Float> &fm
The input value of the elements of fm does not matter.

Function
If m is zero, fm has size one and fm[0] is the value of $\mathit{f}(\mathit{x})$ .

Gradient
If m is one, fm has size n and for $\mathit{j}=0,\dots ,\mathit{n}-1$ $$\frac{\partial \mathit{f}}{\partial \mathit{x}[\mathit{j}]}=\mathit{fm}[\mathit{j}]$$
Hessian
If m is two, fm has size n * n and for $\mathit{k}=0,\dots ,\mathit{n}-1$ , $\mathit{m}=0,\dots ,\mathit{n}-1$ $$\frac{{\partial }^2\mathit{f}}{\partial \mathit{x}[\mathit{k}]\partial \mathit{x}[\mathit{m}]}=\mathit{fm}[\mathit{k}*\mathit{n}+\mathit{m}]$$
Example
The file sparse_evaluate.cpp contains an example and test of sparse_evaluate.hpp. It returns true if it succeeds and false otherwise.

Source Code
The file sparse_evaluate.hpp contains the source code for this template function.

---

Input File: cppad/speed/sparse_evaluate.hpp