@(@\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}} }@)@This is cppad-20221105 documentation. Here is a link to its
current documentation
.
Preferred Sparsity Patterns: Row and Column Indices: Example and Test
Purpose
This example show how to use row and column index sparsity patterns
sparse_rc
to compute sparse Jacobians and Hessians.
This became the preferred way to represent sparsity on
2017-02-09
.
# include <cppad/cppad.hpp>
namespace {
using CppAD::sparse_rc;
using CppAD::sparse_rcv;
using CppAD::NearEqual;
//typedefCPPAD_TESTVECTOR(bool) b_vector;
typedefCPPAD_TESTVECTOR(size_t) s_vector;
typedefCPPAD_TESTVECTOR(double) d_vector;
typedefCPPAD_TESTVECTOR( CppAD::AD<double> ) a_vector;
//
double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
// -----------------------------------------------------------------------// function f(x) that we are computing sparse results for// -----------------------------------------------------------------------
a_vector fun(const a_vector& x)
{ size_t n = x.size();
a_vector ret(n + 1);
for(size_t i = 0; i < n; i++)
{ size_t j = (i + 1) % n;
ret[i] = x[i] * x[i] * x[j];
}
ret[n] = 0.0;
return ret;
}
// -----------------------------------------------------------------------// Jacobian// -----------------------------------------------------------------------
bool check_jac(
const d_vector& x ,
const sparse_rcv<s_vector, d_vector>& subset )
{ bool ok = true;
size_t n = x.size();
//
ok &= subset.nnz() == 2 * n;
const s_vector& row( subset.row() );
const s_vector& col( subset.col() );
const d_vector& val( subset.val() );
s_vector row_major = subset.row_major();
for(size_t i = 0; i < n; i++)
{ size_t j = (i + 1) % n;
size_t k = 2 * i;
//
ok &= row[ row_major[k] ] == i;
ok &= row[ row_major[k+1] ] == i;
//
size_t ck = col[ row_major[k] ];
size_t ckp = col[ row_major[k+1] ];
double vk = val[ row_major[k] ];
double vkp = val[ row_major[k+1] ];
//// put diagonal element firstif( j < i )
{ std::swap(ck, ckp);
std::swap(vk, vkp);
}
// diagonal element
ok &= ck == i;
ok &= NearEqual( vk, 2.0 * x[i] * x[j], eps99, eps99 );
// off diagonal element
ok &= ckp == j;
ok &= NearEqual( vkp, x[i] * x[i], eps99, eps99 );
}
return ok;
}
// Use forward mode for Jacobian and sparsity pattern
bool forward_jac(CppAD::ADFun<double>& f)
{ bool ok = true;
size_t n = f.Domain();
//// sparsity pattern for identity matrix
sparse_rc<s_vector> pattern_in(n, n, n);
for(size_t k = 0; k < n; k++)
pattern_in.set(k, k, k);
//// sparsity pattern for Jacobian
bool transpose = false;
bool dependency = false;
bool internal_bool = false;
sparse_rc<s_vector> pattern_out;
f.for_jac_sparsity(
pattern_in, transpose, dependency, internal_bool, pattern_out
);
//// compute entire Jacobian
size_t group_max = 1;
std::string coloring = "cppad";
sparse_rcv<s_vector, d_vector> subset( pattern_out );
CppAD::sparse_jac_work work;
d_vector x(n);
for(size_t j = 0; j < n; j++)
x[j] = double(j + 2);
size_t n_sweep = f.sparse_jac_for(
group_max, x, subset, pattern_out, coloring, work
);
//// check Jacobian
ok &= check_jac(x, subset);
ok &= n_sweep == 2;
//return ok;
}
// Use reverse mode for Jacobian and sparsity pattern
bool reverse_jac(CppAD::ADFun<double>& f)
{ bool ok = true;
size_t n = f.Domain();
size_t m = f.Range();
//// sparsity pattern for identity matrix
sparse_rc<s_vector> pattern_in(m, m, m);
for(size_t k = 0; k < m; k++)
pattern_in.set(k, k, k);
//// sparsity pattern for Jacobian
bool transpose = false;
bool dependency = false;
bool internal_bool = false;
sparse_rc<s_vector> pattern_out;
f.rev_jac_sparsity(
pattern_in, transpose, dependency, internal_bool, pattern_out
);
//// compute entire Jacobian
std::string coloring = "cppad";
sparse_rcv<s_vector, d_vector> subset( pattern_out );
CppAD::sparse_jac_work work;
d_vector x(n);
for(size_t j = 0; j < n; j++)
x[j] = double(j + 2);
size_t n_sweep = f.sparse_jac_rev(
x, subset, pattern_out, coloring, work
);
//// check Jacobian
ok &= check_jac(x, subset);
ok &= n_sweep == 2;
//return ok;
}
// ------------------------------------------------------------------------// Hessian// ------------------------------------------------------------------------
bool check_hes(
size_t i ,
const d_vector& x ,
const sparse_rcv<s_vector, d_vector>& subset )
{ bool ok = true;
size_t n = x.size();
size_t j = (i + 1) % n;
//
ok &= subset.nnz() == 3;
const s_vector& row( subset.row() );
const s_vector& col( subset.col() );
const d_vector& val( subset.val() );
s_vector row_major = subset.row_major();
//
double v0 = val[ row_major[0] ];
double v1 = val[ row_major[1] ];
double v2 = val[ row_major[2] ];
if( j < i )
{ ok &= row[ row_major[0] ] == j;
ok &= col[ row_major[0] ] == i;
ok &= NearEqual( v0, 2.0 * x[i], eps99, eps99 );
//
ok &= row[ row_major[1] ] == i;
ok &= col[ row_major[1] ] == j;
ok &= NearEqual( v1, 2.0 * x[i], eps99, eps99 );
//
ok &= row[ row_major[2] ] == i;
ok &= col[ row_major[2] ] == i;
ok &= NearEqual( v2, 2.0 * x[j], eps99, eps99 );
}
else
{ ok &= row[ row_major[0] ] == i;
ok &= col[ row_major[0] ] == i;
ok &= NearEqual( v0, 2.0 * x[j], eps99, eps99 );
//
ok &= row[ row_major[1] ] == i;
ok &= col[ row_major[1] ] == j;
ok &= NearEqual( v1, 2.0 * x[i], eps99, eps99 );
//
ok &= row[ row_major[2] ] == j;
ok &= col[ row_major[2] ] == i;
ok &= NearEqual( v2, 2.0 * x[i], eps99, eps99 );
}
return ok;
}
// Use forward mode for Hessian and sparsity pattern
bool forward_hes(CppAD::ADFun<double>& f)
{ bool ok = true;
size_t n = f.Domain();
size_t m = f.Range();
//
b_vector select_domain(n);
for(size_t j = 0; j < n; j++)
select_domain[j] = true;
sparse_rc<s_vector> pattern_out;
//for(size_t i = 0; i < m; i++)
{ // select i-th component of range
b_vector select_range(m);
d_vector w(m);
for(size_t k = 0; k < m; k++)
{ select_range[k] = k == i;
w[k] = 0.0;
if( k == i )
w[k] = 1.0;
}
//
bool internal_bool = false;
f.for_hes_sparsity(
select_domain, select_range, internal_bool, pattern_out
);
//// compute Hessian for i-th component function
std::string coloring = "cppad.symmetric";
sparse_rcv<s_vector, d_vector> subset( pattern_out );
CppAD::sparse_hes_work work;
d_vector x(n);
for(size_t j = 0; j < n; j++)
x[j] = double(j + 2);
size_t n_sweep = f.sparse_hes(
x, w, subset, pattern_out, coloring, work
);
//// check Hessianif( i == n )
ok &= subset.nnz() == 0;
else
{ ok &= check_hes(i, x, subset);
ok &= n_sweep == 1;
}
}
return ok;
}
// Use reverse mode for Hessian and sparsity pattern
bool reverse_hes(CppAD::ADFun<double>& f)
{ bool ok = true;
size_t n = f.Domain();
size_t m = f.Range();
//// n by n identity matrix
sparse_rc<s_vector> pattern_in(n, n, n);
for(size_t j = 0; j < n; j++)
pattern_in.set(j, j, j);
//
bool transpose = false;
bool dependency = false;
bool internal_bool = true;
sparse_rc<s_vector> pattern_out;
//
f.for_jac_sparsity(
pattern_in, transpose, dependency, internal_bool, pattern_out
);
//for(size_t i = 0; i < m; i++)
{ // select i-th component of range
b_vector select_range(m);
d_vector w(m);
for(size_t k = 0; k < m; k++)
{ select_range[k] = k == i;
w[k] = 0.0;
if( k == i )
w[k] = 1.0;
}
//
f.rev_hes_sparsity(
select_range, transpose, internal_bool, pattern_out
);
//// compute Hessian for i-th component function
std::string coloring = "cppad.symmetric";
sparse_rcv<s_vector, d_vector> subset( pattern_out );
CppAD::sparse_hes_work work;
d_vector x(n);
for(size_t j = 0; j < n; j++)
x[j] = double(j + 2);
size_t n_sweep = f.sparse_hes(
x, w, subset, pattern_out, coloring, work
);
//// check Hessianif( i == n )
ok &= subset.nnz() == 0;
else
{ ok &= check_hes(i, x, subset);
ok &= n_sweep == 1;
}
}
return ok;
}
}
// driver for all of the cases above
bool rc_sparsity(void)
{ bool ok = true;
//// record the funcion
size_t n = 20;
size_t m = n + 1;
a_vector x(n), y(m);
for(size_t j = 0; j < n; j++)
x[j] = CppAD::AD<double>(j+1);
CppAD::Independent(x);
y = fun(x);
CppAD::ADFun<double> f(x, y);
//// run the example / tests
ok &= forward_jac(f);
ok &= reverse_jac(f);
ok &= forward_hes(f);
ok &= reverse_hes(f);
//return ok;
}
