bool PowInt(void)
{ bool ok = true;
using CppAD::pow;
using CppAD::exp;
using CppAD::log;
using namespace CppAD;
// independent variable vector, indices, values, and declaration
CPPAD_TESTVECTOR(AD<double>) U(1);
U[0] = 2.;
Independent(U);
// dependent variable vector and indices
CPPAD_TESTVECTOR(AD<double>) Z(2);
// dependent variable values
Z[0] = pow(U[0], 5); // x = u^5
Z[1] = pow(U[0], -5); // y = u^{-5}
// create f: U -> Z and vectors used for derivative calculations
ADFun<double> f(U, Z);
CPPAD_TESTVECTOR(double) v( f.Domain() );
CPPAD_TESTVECTOR(double) w( f.Range() );
/*
x_u = 5 * u^4
y_u = - 5 * u^{-6}
*/
// check function values values
double u = Value(U[0]);
ok &= NearEqual(Z[0] , exp( log(u) * 5.), 1e-10 , 1e-10);
ok &= NearEqual(Z[1] , exp( - log(u) * 5.), 1e-10 , 1e-10);
// forward computation of partials
v[0] = 1.;
w = f.Forward(1, v);
ok &= NearEqual(w[0] , 5. * exp( log(u) * 4.), 1e-10 , 1e-10);
ok &= NearEqual(w[1] , - 5. * exp( - log(u) * 6.), 1e-10 , 1e-10);
return ok;
}
bool Erf(void)
{ bool ok = true;
using namespace CppAD;
using CppAD::atan;
using CppAD::exp;
using CppAD::sqrt;
// Construct function object corresponding to erf
CPPAD_TEST_VECTOR< AD<double> > X(1);
CPPAD_TEST_VECTOR< AD<double> > Y(1);
X[0] = 0.;
Independent(X);
Y[0] = erf(X[0]);
ADFun<double> Erf(X, Y);
// vectors to use with function object
CPPAD_TEST_VECTOR<double> x(1);
CPPAD_TEST_VECTOR<double> y(1);
CPPAD_TEST_VECTOR<double> dx(1);
CPPAD_TEST_VECTOR<double> dy(1);
// check value at zero
x[0] = 0.;
y = Erf.Forward(0, x);
ok &= NearEqual(0., y[0], 4e-4, 0.);
// check the derivative of error function
dx[0] = 1.;
double pi = 4. * atan(1.);
double factor = 2. / sqrt( pi );
int i;
for(i = -10; i <= 10; i++)
{ x[0] = i / 4.;
y = Erf.Forward(0, x);
// check derivative
double derf = factor * exp( - x[0] * x[0] );
dy = Erf.Forward(1, dx);
ok &= NearEqual(derf, dy[0], 0., 2e-3);
// test using erf with AD< AD<double> >
AD< AD<double> > X0 = x[0];
AD< AD<double> > Y0 = erf(X0);
ok &= ( y[0] == Value( Value(Y0) ) );
}
return ok;
}
bool VecUnary(void)
{
using namespace CppAD;
using CppAD::abs;
using CppAD::sin;
using CppAD::atan;
using CppAD::cos;
using CppAD::exp;
using CppAD::log;
using CppAD::sqrt;
bool ok = true;
size_t n = 8;
size_t i;
CPPAD_TEST_VECTOR< AD<double> > X(n);
VecAD<double> Y(n);
CPPAD_TEST_VECTOR< AD<double> > Z(n);
for(i = 0; i < n; i++)
X[i] = int(i); // some compilers require the int here
Independent(X);
AD<double> j;
j = 0.;
Y[j] = X[0];
Z[0] = -Y[j];
j = 1.;
Y[j] = X[1];
Z[1] = sin( Y[j] );
j = 2.;
Y[j] = X[2];
Z[2] = abs( Y[j] );
j = 3.;
Y[j] = X[3];
Z[3] = atan( Y[j] );
j = 4.;
Y[j] = X[4];
Z[4] = cos( Y[j] );
j = 5.;
Y[j] = X[5];
Z[5] = exp( Y[j] );
j = 6.;
Y[j] = X[6];
Z[6] = log( Y[j] );
j = 7.;
Y[j] = X[7];
Z[7] = sqrt( Y[j] );
ADFun<double> f(X, Z);
CPPAD_TEST_VECTOR<double> x(n);
CPPAD_TEST_VECTOR<double> z(n);
for(i = 0; i < n; i++)
x[i] = 2. / double(i + 1);
x[7] = abs( x[7] );
z = f.Forward(0, x);
ok &= NearEqual(z[0], - x[0], 1e-10, 1e-10);
ok &= NearEqual(z[1], sin( x[1] ), 1e-10, 1e-10);
ok &= NearEqual(z[2], abs( x[2] ), 1e-10, 1e-10);
ok &= NearEqual(z[3], atan(x[3] ), 1e-10, 1e-10);
ok &= NearEqual(z[4], cos( x[4] ), 1e-10, 1e-10);
ok &= NearEqual(z[5], exp( x[5] ), 1e-10, 1e-10);
ok &= NearEqual(z[6], log( x[6] ), 1e-10, 1e-10);
ok &= NearEqual(z[7], sqrt(x[7] ), 1e-10, 1e-10);
return ok;
}
bool ForHess(void)
{ bool ok = true;
using namespace CppAD;
using CppAD::exp;
using CppAD::sin;
using CppAD::cos;
using CppAD::NearEqual;
double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
size_t i;
// create independent variable vector with assigned values
CPPAD_TESTVECTOR(double) u0(3);
CPPAD_TESTVECTOR(AD<double>) U(3);
for(i = 0; i < 3; i++)
U[i] = u0[i] = double(i+1);
Independent( U );
// define the function
CPPAD_TESTVECTOR(AD<double>) Y(2);
Y[0] = U[0] * exp( U[1] );
Y[1] = U[1] * sin( U[2] );
// create the function y = F(u)
ADFun<double> F(U, Y);
// formulas for the upper triangle of Hessian of F_0
CPPAD_TESTVECTOR(double) H0(9);
H0[0] = 0.; // d^2 y[0] / d_u[0] d_u[0]
H0[1] = exp( u0[1] ); // d^2 y[0] / d_u[0] d_u[1]
H0[2] = 0.; // d^2 y[0] / d_u[0] d_u[2]
H0[4] = u0[0] * exp( u0[1] ); // d^2 y[0] / d_u[1] d_u[1]
H0[5] = 0.; // d^2 y[0] / d_u[1] d_u[2]
H0[8] = 0.; // d^2 y[0] / d_u[2] d_u[2]
// formulas for the upper triangle of Hessian of F_1
CPPAD_TESTVECTOR(double) H1(9);
H1[0] = 0.; // d^2 Y[1] / d_U[0] d_U[0]
H1[1] = 0.; // d^2 Y[1] / d_U[0] d_U[1]
H1[2] = 0.; // d^2 Y[1] / d_U[0] d_U[2]
H1[4] = 0.; // d^2 Y[1] / d_U[1] d_U[1]
H1[5] = cos( u0[2] ); // d^2 Y[1] / d_U[1] d_U[2]
H1[8] = - u0[1] * sin( u0[2] );// d^2 Y[1] / d_U[2] d_U[2]
// Define U(t) = u0 + u1 t + u2 t^2 / 2
CPPAD_TESTVECTOR(double) u1(3);
CPPAD_TESTVECTOR(double) u2(3);
for(i = 0; i < 3; i++)
u1[i] = u2[i] = 0.;
size_t j;
for(i = 0; i < 3; i++)
{ // diagonal of Hessians in i-th coordiante direction
u1[i] = 1.;
F.Forward(1, u1);
CPPAD_TESTVECTOR(double) Di = F.Forward(2, u2);
ok &= NearEqual( 2. * Di[0] , H0[ i + 3 * i], eps99, eps99);
ok &= NearEqual( 2. * Di[1] , H1[ i + 3 * i], eps99, eps99);
//
for(j = i+1; j < 3; j++)
{ // cross term in i and j direction
u1[j] = 1.;
F.Forward(1, u1);
CPPAD_TESTVECTOR(double) Cij = F.Forward(2, u2);
// diagonal of Hessian in j-th coordinate direction
u1[i] = 0.;
F.Forward(1, u1);
CPPAD_TESTVECTOR(double) Dj = F.Forward(2, u2);
// (i, j) elements of the Hessians
double H0ij = Cij[0] - Di[0] - Dj[0];
ok &= NearEqual( H0ij, H0[j + 3 * i], eps99, eps99);
double H1ij = Cij[1] - Di[1] - Dj[1];
ok &= NearEqual( H1ij, H1[j + 3 * i], eps99, eps99);
// reset all components of u1 to zero
u1[j] = 0.;
}
}
return ok;
}