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Not Complex Differentiable: Example and Test

Not Complex Differentiable
If x is complex, the functions real(x), imag(x), conj(x), and abs(x) are examples of functions that are not complex differentiable.

See Also
ComplexPoly.cpp

Poly
Select this link to view specifications for Poly :

# include <cppad/cppad.hpp>
# include <complex>

bool not_complex_ad(void)
{    bool ok    = true;
     size_t deg = 4;

     using CppAD::AD;
     using CppAD::Poly;
     typedef std::complex<double>              Complex; 
     typedef std::complex< CppAD::AD<double> > ComplexAD; 

     // polynomial coefficients
     CPPAD_TEST_VECTOR< Complex >   a   (deg + 1); // coefficients for p(z)
     CPPAD_TEST_VECTOR< ComplexAD > A   (deg + 1); 
     size_t i;
     for(i = 0; i <= deg; i++)
     {    a[i] = Complex(i, i);
          A[i] = ComplexAD( AD<double>(i) , AD<double>(i) );
     }

     // declare independent variables and start taping
     CPPAD_TEST_VECTOR< AD<double> > Z_real(1);
     double z_real = 1.;
     Z_real[0]     = z_real;
     Independent(Z_real);

     // complex calculations
     double z_imag = 2.;
     ComplexAD Z = ComplexAD( Z_real[0], AD<double>(z_imag) );
     ComplexAD P = Poly(0, A, Z);

     // range space vector
     CPPAD_TEST_VECTOR< AD<double> > P_real(1);
     P_real[0] = P.real();   // real() is not complex differentiable

     // create f: Z_real -> P_real  and stop taping
     CppAD::ADFun<double> f(Z_real, P_real);

     // check first derivative w.r.t z
     CPPAD_TEST_VECTOR<double> v( f.Domain() );
     CPPAD_TEST_VECTOR<double> w( f.Range() );
     v[0]      = 1.;
     w         = f.Forward(1, v);
     Complex z = Complex(z_real, z_imag);
     Complex p = Poly(1, a, z);
     ok &= ( w[0]  == p.real() );

     // second derivative w.r.t z is 2 times its second order Taylor coeff
     v[0] = 0.;
     w    = f.Forward(2, v);
     p    = Poly(2, a, z);
     ok &= ( 2. * w[0]  == p.real() );

     return ok;
}


Input File: example/not_complex_ad.cpp