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cppad_py-> setup.py library whats_new_2018

library-> py_lib cpp_lib

py_lib-> py_fun py_sparse py_utility more_py

py_sparse-> py_sparse_rc py_sparse_rcv py_jac_sparsity py_hes_sparsity py_sparse_jac py_sparse_hes

py_sparse_hes-> sparse_hes_xam.py

Headings-> Syntax Purpose f subset x r pattern work n_sweep Uses Forward Example

@(@\newcommand{\B}[1]{ {\bf #1} } \newcommand{\R}[1]{ {\rm #1} }@)@Computing Sparse Hessians
Syntax
work = cppad_py.sparse_hes_work()
n_sweep = f.sparse_hes(subset, x, r, pattern, work)

Purpose
We use @(@ F : \B{R}^n \rightarrow \B{R}^m @)@ to denote the function corresponding to f . Given a vector @(@ r \in \B{R}^m @)@, define @[@ H(x) = (r^\R{T} F)^{(2)} ( x ) @]@ This routine takes advantage of sparsity when computing elements of the Hessian @(@ H(x) @)@.

f
This object must have been returned by a previous call to the python d_fun constructor. Note that the Taylor coefficients stored in f are affected by this operation; see uses forward below.

subset
This argument must have be a matrix returned by the sparse_rcv constructor. Its row size and column size is n ; i.e., subset.nr() == n and subset.nc() == n . It specifies which elements of the Hessian are computed. The input value of its value vector subset.val() does not matter. Upon return it contains the value of the corresponding elements of the Jacobian. All of the row, column pairs in subset must also appear in pattern ; i.e., they must be possibly non-zero.

x
This argument is a numpy vector with float elements and size n . It specifies the point at which to evaluate the Hessian @(@ H(x) @)@.

r
This argument is a numpy vector with float elements and size m . It specifies the multiplier for each component of @(@ F(x) @)@; i.e., @(@ r_i @)@ is the multiplier for @(@ F_i (x) @)@.

pattern
This argument must have be a pattern returned by the sparse_rc constructor. Its row size and column sizes are n ; i.e., pattern.nr() == n and pattern.nc() == n . It is a sparsity pattern for the Hessian @(@ H(x) @)@. This argument is not used (and need not satisfy any conditions), when work is non-empty.

work
This argument must have been constructed by the call
  work = cppad_py.sparse_hes_work()
We refer to its initial value, and its value after work.clear() , as empty. If it is empty, information is stored in work . This can be used to reduce computation when a future call is for the same object f , and the same subset of the Hessian. If either of these values change, use work.clear() to empty this structure.

n_sweep
The return value n_sweep has prototype
     int n_sweep
It is the number of first order forward sweeps used to compute the requested Hessian values. Each first forward sweep is followed by a second order reverse sweep so it is also the number of reverse sweeps. This is proportional to the total computational work, not counting the zero order forward sweep, or combining multiple columns and rows into a single sweep.

Uses Forward
After each call to py_fun_forward , the object f contains the corresponding Taylor coefficients for all the variables in the operation sequence.. After a call to sparse_hes the zero order coefficients correspond to
     f.forward(0, x)
All the other forward mode coefficients are unspecified.

Example
sparse_hes_xam.py

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Input File: lib/python/sparse_hes.py