1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
|
// SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
// SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
// SPDX-FileContributor: 2003-24 Bradley M. Bell
// ----------------------------------------------------------------------------
/*
{xrst_begin dependency.cpp}
{xrst_spell
rl
}
Computing Dependency: Example and Test
######################################
Discussion
**********
The partial of an dependent variable with respect to an independent variable
might always be zero even though the dependent variable depends on the
value of the dependent variable. Consider the following case
.. math::
f(x) = {\rm sign} (x) =
\left\{ \begin{array}{rl}
+1 & {\rm if} \; x > 0 \\
0 & {\rm if} \; x = 0 \\
-1 & {\rm if} \; x < 0
\end{array} \right.
In this case the value of :math:`f(x)` depends on the value of :math:`x`
but CppAD always returns zero for the derivative of the :ref:`sign-name` function.
Dependency Pattern
******************
If the *i*-th dependent variables depends on the
value of the *j*-th independent variable,
the corresponding entry in the dependency pattern is non-zero (true).
Otherwise it is zero (false).
CppAD uses :ref:`sparsity patterns<glossary@Sparsity Pattern>`
to represent dependency patterns.
Computation
***********
The *dependency* argument to
:ref:`for_jac_sparsity<for_jac_sparsity@dependency>` and
:ref:`RevSparseJac<RevSparseJac@dependency>` is a flag that signals
that the dependency pattern (instead of the sparsity pattern) is computed.
{xrst_literal
// BEGIN C++
// END C++
}
{xrst_end dependency.cpp}
*/
// BEGIN C++
# include <cppad/cppad.hpp>
namespace {
double heavyside(const double& x)
{ if( x <= 0.0 )
return 0.0;
return 1.0;
}
CPPAD_DISCRETE_FUNCTION(double, heavyside)
}
bool dependency(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
typedef CPPAD_TESTVECTOR(size_t) SizeVector;
typedef CppAD::sparse_rc<SizeVector> sparsity;
// VecAD object for use later
CppAD::VecAD<double> vec_ad(2);
vec_ad[0] = 0.0;
vec_ad[1] = 1.0;
// domain space vector
size_t n = 5;
CPPAD_TESTVECTOR(AD<double>) ax(n);
for(size_t j = 0; j < n; j++)
ax[j] = AD<double>(j + 1);
// declare independent variables and start tape recording
CppAD::Independent(ax);
// some AD constants
AD<double> azero(0.0), aone(1.0);
// range space vector
size_t m = n;
size_t m1 = n - 1;
CPPAD_TESTVECTOR(AD<double>) ay(m);
// Note that ay[m1 - j] depends on ax[j]
ay[m1 - 0] = sign( ax[0] );
ay[m1 - 1] = CondExpLe( ax[1], azero, azero, aone);
ay[m1 - 2] = CondExpLe( azero, ax[2], azero, aone);
ay[m1 - 3] = heavyside( ax[3] );
ay[m1 - 4] = vec_ad[ ax[4] - AD<double>(4.0) ];
// create f: x -> y and stop tape recording
CppAD::ADFun<double> f(ax, ay);
// sparsity pattern for n by n identity matrix
size_t nr = n;
size_t nc = n;
size_t nnz = n;
sparsity pattern_in(nr, nc, nnz);
for(size_t k = 0; k < nnz; k++)
{ size_t r = k;
size_t c = k;
pattern_in.set(k, r, c);
}
// compute dependency pattern
bool transpose = false;
bool dependency = true; // would transpose dependency pattern
bool internal_bool = true; // does not affect result
sparsity pattern_out;
f.for_jac_sparsity(
pattern_in, transpose, dependency, internal_bool, pattern_out
);
const SizeVector& row( pattern_out.row() );
const SizeVector& col( pattern_out.col() );
SizeVector col_major = pattern_out.col_major();
// check result
ok &= pattern_out.nr() == n;
ok &= pattern_out.nc() == n;
ok &= pattern_out.nnz() == n;
for(size_t k = 0; k < n; k++)
{ ok &= row[ col_major[k] ] == m1 - k;
ok &= col[ col_major[k] ] == k;
}
// -----------------------------------------------------------
// RevSparseJac and set dependency
CppAD::vector< std::set<size_t> > eye_set(m), depend_set(m);
for(size_t i = 0; i < m; i++)
{ ok &= eye_set[i].empty();
eye_set[i].insert(i);
}
depend_set = f.RevSparseJac(n, eye_set, transpose, dependency);
for(size_t i = 0; i < m; i++)
{ std::set<size_t> check;
check.insert(m1 - i);
ok &= depend_set[i] == check;
}
dependency = false;
depend_set = f.RevSparseJac(n, eye_set, transpose, dependency);
for(size_t i = 0; i < m; i++)
ok &= depend_set[i].empty();
return ok;
}
// END C++
|
