Prev Next Index-> contents reference index search external Up-> CppAD AD ADValued atomic atomic_base atomic_norm_sq.cpp ADValued-> Arithmetic unary_standard_math binary_math CondExp Discrete numeric_limits atomic atomic-> checkpoint atomic_base atomic_base-> atomic_ctor atomic_option atomic_afun atomic_forward atomic_reverse atomic_for_sparse_jac atomic_rev_sparse_jac atomic_for_sparse_hes atomic_rev_sparse_hes atomic_base_clear atomic_get_started.cpp atomic_norm_sq.cpp atomic_reciprocal.cpp atomic_set_sparsity.cpp atomic_tangent.cpp atomic_eigen_mat_mul.cpp atomic_eigen_mat_inv.cpp atomic_eigen_cholesky.cpp atomic_mat_mul.cpp atomic_norm_sq.cpp Headings-> Theory sparsity Start Class Definition Constructor forward reverse for_sparse_jac rev_sparse_jac rev_sparse_hes End Class Definition Use Atomic Function ---..Constructor ---..Recording ---..forward ---..reverse ---..for_sparse_jac ---..rev_sparse_jac ---..rev_sparse_hes

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
Atomic Euclidean Norm Squared: Example and Test

Theory
This example demonstrates using atomic_base to define the operation $f : \B{R}^n \rightarrow \B{R}^m$ where $n = 2$, $m = 1$, where $$f(x) = x_0^2 + x_1^2$$

sparsity
This example only uses bool sparsity patterns.

Start Class Definition
# include <cppad/cppad.hpp>
namespace {           // isolate items below to this file
using CppAD::vector;  // abbreviate as vector
//
class atomic_norm_sq : public CppAD::atomic_base<double> {

Constructor
public:
// constructor (could use const char* for name)
atomic_norm_sq(const std::string& name) :
// this example only uses boolean sparsity patterns
{ }
private:

forward
     // forward mode routine called by CppAD
virtual bool forward(
size_t                    p ,
size_t                    q ,
const vector<bool>&      vx ,
vector<bool>&      vy ,
const vector<double>&    tx ,
vector<double>&    ty
)
{
# ifndef NDEBUG
size_t n = tx.size() / (q+1);
size_t m = ty.size() / (q+1);
# endif
assert( n == 2 );
assert( m == 1 );
assert( p <= q );

// return flag
bool ok = q <= 1;

// Variable information must always be implemented.
// y_0 is a variable if and only if x_0 or x_1 is a variable.
if( vx.size() > 0 )
vy[0] = vx[0] | vx[1];

// Order zero forward mode must always be implemented.
// y^0 = f( x^0 )
double x_00 = tx[ 0*(q+1) + 0];        // x_0^0
double x_10 = tx[ 1*(q+1) + 0];        // x_10
double f = x_00 * x_00 + x_10 * x_10;  // f( x^0 )
if( p <= 0 )
ty[0] = f;   // y_0^0
if( q <= 0 )
return ok;
assert( vx.size() == 0 );

// Order one forward mode.
// This case needed if first order forward mode is used.
// y^1 = f'( x^0 ) x^1
double x_01 = tx[ 0*(q+1) + 1];   // x_0^1
double x_11 = tx[ 1*(q+1) + 1];   // x_1^1
double fp_0 = 2.0 * x_00;         // partial f w.r.t x_0^0
double fp_1 = 2.0 * x_10;         // partial f w.r.t x_1^0
if( p <= 1 )
ty[1] = fp_0 * x_01 + fp_1 * x_11; // f'( x^0 ) * x^1
if( q <= 1 )
return ok;

// Assume we are not using forward mode with order > 1
assert( ! ok );
return ok;
}

reverse
     // reverse mode routine called by CppAD
virtual bool reverse(
size_t                    q ,
const vector<double>&    tx ,
const vector<double>&    ty ,
vector<double>&    px ,
const vector<double>&    py
)
{
# ifndef NDEBUG
size_t n = tx.size() / (q+1);
size_t m = ty.size() / (q+1);
# endif
assert( px.size() == n * (q+1) );
assert( py.size() == m * (q+1) );
assert( n == 2 );
assert( m == 1 );
bool ok = q <= 1;

double fp_0, fp_1;
switch(q)
{     case 0:
// This case needed if first order reverse mode is used
// F ( {x} ) = f( x^0 ) = y^0
fp_0  =  2.0 * tx[0];  // partial F w.r.t. x_0^0
fp_1  =  2.0 * tx[1];  // partial F w.r.t. x_0^1
px[0] = py[0] * fp_0;; // partial G w.r.t. x_0^0
px[1] = py[0] * fp_1;; // partial G w.r.t. x_0^1
assert(ok);
break;

default:
// Assume we are not using reverse with order > 1 (q > 0)
assert(!ok);
}
return ok;
}

for_sparse_jac
     // forward Jacobian bool sparsity routine called by CppAD
virtual bool for_sparse_jac(
size_t                                p ,
const vector<bool>&                   r ,
vector<bool>&                   s ,
const vector<double>&                 x )
{     // This function needed if using f.ForSparseJac
size_t n = r.size() / p;
# ifndef NDEBUG
size_t m = s.size() / p;
# endif
assert( n == x.size() );
assert( n == 2 );
assert( m == 1 );

// sparsity for S(x) = f'(x) * R
// where f'(x) = 2 * [ x_0, x_1 ]
for(size_t j = 0; j < p; j++)
{     s[j] = false;
for(size_t i = 0; i < n; i++)
{     // Visual Studio 2013 generates warning without bool below
s[j] |= bool( r[i * p + j] );
}
}
return true;
}

rev_sparse_jac
     // reverse Jacobian bool sparsity routine called by CppAD
virtual bool rev_sparse_jac(
size_t                                p  ,
const vector<bool>&                   rt ,
vector<bool>&                   st ,
const vector<double>&                 x  )
{     // This function needed if using RevSparseJac or optimize
size_t n = st.size() / p;
# ifndef NDEBUG
size_t m = rt.size() / p;
# endif
assert( n == x.size() );
assert( n == 2 );
assert( m == 1 );

// sparsity for S(x)^T = f'(x)^T * R^T
// where f'(x)^T = 2 * [ x_0, x_1]^T
for(size_t j = 0; j < p; j++)
for(size_t i = 0; i < n; i++)
st[i * p + j] = rt[j];

return true;
}

rev_sparse_hes
     // reverse Hessian bool sparsity routine called by CppAD
virtual bool rev_sparse_hes(
const vector<bool>&                   vx,
const vector<bool>&                   s ,
vector<bool>&                   t ,
size_t                                p ,
const vector<bool>&                   r ,
const vector<bool>&                   u ,
vector<bool>&                   v ,
const vector<double>&                 x )
{     // This function needed if using RevSparseHes
# ifndef NDEBUG
size_t m = s.size();
# endif
size_t n = t.size();
assert( x.size() == n );
assert( r.size() == n * p );
assert( u.size() == m * p );
assert( v.size() == n * p );
assert( n == 2 );
assert( m == 1 );

// There are no cross term second derivatives for this case,
// so it is not necessary to use vx.

// sparsity for T(x) = S(x) * f'(x)
t[0] = s[0];
t[1] = s[0];

// V(x) = f'(x)^T * g''(y) * f'(x) * R  +  g'(y) * f''(x) * R
// U(x) = g''(y) * f'(x) * R
// S(x) = g'(y)

// back propagate the sparsity for U
size_t j;
for(j = 0; j < p; j++)
for(size_t i = 0; i < n; i++)
v[ i * p + j] = u[j];

// include forward Jacobian sparsity in Hessian sparsity
// sparsity for g'(y) * f''(x) * R  (Note f''(x) has same sparsity
// as the identity matrix)
if( s[0] )
{     for(j = 0; j < p; j++)
for(size_t i = 0; i < n; i++)
{     // Visual Studio 2013 generates warning without bool below
v[ i * p + j] |= bool( r[ i * p + j] );
}
}

return true;
}

End Class Definition

}; // End of atomic_norm_sq class
}  // End empty namespace


Use Atomic Function
bool norm_sq(void)
{     bool ok = true;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();

Constructor

// --------------------------------------------------------------------
// Create the atomic reciprocal object
atomic_norm_sq afun("atomic_norm_sq");


Recording
     // Create the function f(x)
//
// domain space vector
size_t  n  = 2;
double  x0 = 0.25;
double  x1 = 0.75;
ax[0]      = x0;
ax[1]      = x1;

// declare independent variables and start tape recording

// range space vector
size_t m = 1;

// call user function and store norm_sq(x) in au[0]
afun(ax, ay);        // y_0 = x_0 * x_0 + x_1 * x_1

// create f: x -> y and stop tape recording
f.Dependent (ax, ay);

forward
     // check function value
double check = x0 * x0 + x1 * x1;
ok &= NearEqual( Value(ay[0]) , check,  eps, eps);

// check zero order forward mode
size_t q;
vector<double> x_q(n), y_q(m);
q      = 0;
x_q[0] = x0;
x_q[1] = x1;
y_q    = f.Forward(q, x_q);
ok &= NearEqual(y_q[0] , check,  eps, eps);

// check first order forward mode
q      = 1;
x_q[0] = 0.3;
x_q[1] = 0.7;
y_q    = f.Forward(q, x_q);
check  = 2.0 * x0 * x_q[0] + 2.0 * x1 * x_q[1];
ok &= NearEqual(y_q[0] , check,  eps, eps);

reverse
     // first order reverse mode
q     = 1;
vector<double> w(m), dw(n * q);
w[0]  = 1.;
dw    = f.Reverse(q, w);
check = 2.0 * x0;
ok &= NearEqual(dw[0] , check,  eps, eps);
check = 2.0 * x1;
ok &= NearEqual(dw[1] , check,  eps, eps);

for_sparse_jac
     // forward mode sparstiy pattern
size_t p = n;
CppAD::vectorBool r1(n * p), s1(m * p);
r1[0] = true;  r1[1] = false; // sparsity pattern identity
r1[2] = false; r1[3] = true;
//
s1    = f.ForSparseJac(p, r1);
ok  &= s1[0] == true;  // f[0] depends on x[0]
ok  &= s1[1] == true;  // f[0] depends on x[1]

rev_sparse_jac
     // reverse mode sparstiy pattern
q = m;
CppAD::vectorBool s2(q * m), r2(q * n);
s2[0] = true;          // compute sparsity pattern for f[0]
//
r2    = f.RevSparseJac(q, s2);
ok  &= r2[0] == true;  // f[0] depends on x[0]
ok  &= r2[1] == true;  // f[0] depends on x[1]

rev_sparse_hes
     // Hessian sparsity (using previous ForSparseJac call)
}