bool TShape3DB2::evaluate_with_hess( const MsqMatrix<3,3>& T,
double& result,
MsqMatrix<3,3>& wrt_T,
MsqMatrix<3,3> second[6],
MsqError& err )
{
double f = sqr_Frobenius(T);
double g = sqr_Frobenius(adj(T));
double d = det(T);
if (invalid_determinant(d)) {
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
const double den = 1.0/(9*d*d);
result = f*g*den- 1;
MsqMatrix<3,3> dg = 2 * (f * T - T * transpose(T) * T);
MsqMatrix<3,3> df = 2 * T;
MsqMatrix<3,3> dtau = transpose_adj(T);
wrt_T = g*df + f*dg - 2*f*g/d * transpose_adj(T);
wrt_T *= den;
set_scaled_2nd_deriv_norm_sqr_adj( second, den*f, T );
pluseq_scaled_I( second, 2*den*g );
pluseq_scaled_sum_outer_product( second, den, dg, df );
pluseq_scaled_sum_outer_product( second, -2*den*g/d, df, dtau );
pluseq_scaled_sum_outer_product( second, -2*den*f/d, dg, dtau );
pluseq_scaled_outer_product( second, 6*den*f*g/(d*d), dtau );
pluseq_scaled_2nd_deriv_of_det( second, -2*den*f*g/d, T );
return true;
}
bool TShapeSizeB1::evaluate_with_hess( const MsqMatrix<3,3>& T,
double& result,
MsqMatrix<3,3>& deriv_wrt_T,
MsqMatrix<3,3> second_wrt_T[6],
MsqError& err )
{
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
result = 0.0;
return false;
}
const MsqMatrix<3,3> adjt = transpose_adj(T);
const double nT = sqr_Frobenius(T);
const double nadj = sqr_Frobenius(adjt);
const double f = 1/(tau*tau);
result = nT + f*nadj - 6;
//! \f$ \frac{\partial}{\partial T} |adj T|^2 \f$
const MsqMatrix<3,3> dNadj_dT = 2 * (nT * T - T * transpose(T) * T);
deriv_wrt_T = T;
deriv_wrt_T -= f/tau * nadj * adjt;
deriv_wrt_T *= 2;
deriv_wrt_T += f * dNadj_dT;
// calculate negative of 2nd wrt T of (|adj T|^2 / tau^2) (sec 3.2.2)
set_scaled_2nd_deriv_norm_sqr_adj( second_wrt_T, f, T );
pluseq_scaled_2nd_deriv_of_det( second_wrt_T, -2*f*f*nadj*tau, T );
pluseq_scaled_outer_product( second_wrt_T, 6*f*f*nadj, adjt );
pluseq_scaled_sum_outer_product( second_wrt_T, -2*f*f *tau, adjt, dNadj_dT );
// calculate 2nd wrt T of this metric
pluseq_scaled_I( second_wrt_T, 2.0 );
return true;
}
bool Target3DShapeSizeBarrierAlt1::evaluate_with_hess( const MsqMatrix<3,3>& A,
const MsqMatrix<3,3>& W,
double& result,
MsqMatrix<3,3>& wrt_A,
MsqMatrix<3,3> second[6],
MsqError& err )
{
const MsqMatrix<3,3> Winv = inverse(W);
const MsqMatrix<3,3> T = A * Winv;
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
result = 0.0;
return false;
}
const double f = sqr_Frobenius(T);
const double g = sqr_Frobenius(adj(T));
result = (f + g)/(6 * tau);
MsqMatrix<3,3> dtau = transpose_adj(T);
MsqMatrix<3,3> dg = -transpose(T) * T;
dg(0,0) += f;
dg(1,1) += f;
dg(2,2) += f;
dg = T * dg;
dg *= 2;
wrt_A = T;
wrt_A += 0.5*dg;
wrt_A *= 1.0/3.0;
wrt_A -= result * dtau;
wrt_A *= 1.0/tau;
wrt_A = wrt_A * transpose(Winv);
set_scaled_2nd_deriv_norm_sqr_adj( second, 1.0/6.0, T );
pluseq_scaled_I( second, 1.0/3.0 );
pluseq_scaled_sum_outer_product( second, -1./3./tau, T, dtau );
pluseq_scaled_sum_outer_product( second, -1./6./tau, dg, dtau );
pluseq_scaled_outer_product( second, 2*result/tau, dtau );
pluseq_scaled_2nd_deriv_of_det( second, -result, T );
hess_scale( second, 1.0/tau );
second_deriv_wrt_product_factor( second, Winv );
result -= 1.0;
return true;
}
bool TShapeSize3DB2::evaluate_with_hess( const MsqMatrix<3,3>& T,
double& result,
MsqMatrix<3,3>& wrt_T,
MsqMatrix<3,3> second[6],
MsqError& err )
{
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
result = 0.0;
return false;
}
const double f = sqr_Frobenius(T);
const double g = sqr_Frobenius(adj(T));
result = (f + g)/(6 * tau);
MsqMatrix<3,3> dtau = transpose_adj(T);
MsqMatrix<3,3> dg = -transpose(T) * T;
dg(0,0) += f;
dg(1,1) += f;
dg(2,2) += f;
dg = T * dg;
dg *= 2;
wrt_T = T;
wrt_T += 0.5*dg;
wrt_T *= 1.0/3.0;
wrt_T -= result * dtau;
wrt_T *= 1.0/tau;
set_scaled_2nd_deriv_norm_sqr_adj( second, 1.0/6.0, T );
pluseq_scaled_I( second, 1.0/3.0 );
pluseq_scaled_sum_outer_product( second, -1./3./tau, T, dtau );
pluseq_scaled_sum_outer_product( second, -1./6./tau, dg, dtau );
pluseq_scaled_outer_product( second, 2*result/tau, dtau );
pluseq_scaled_2nd_deriv_of_det( second, -result, T );
hess_scale( second, 1.0/tau );
result -= 1.0;
return true;
}
