template <unsigned DIM> static inline
bool grad( const MsqMatrix<DIM,DIM>& T,
double& result,
MsqMatrix<DIM,DIM>& deriv_wrt_T )
{
const double tr = trace(T);
const double f = DimConst<DIM>::inv() * fabs(tr);
result = sqr_Frobenius( T ) - f * tr;
deriv_wrt_T = T;
pluseq_scaled_I( deriv_wrt_T, -f );
deriv_wrt_T *= 2;
return true;
}
bool TShape3DB2::evaluate_with_grad( const MsqMatrix<3,3>& T,
double& result,
MsqMatrix<3,3>& wrt_T,
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;
}
result = (f*g) / (9*d*d) - 1;
wrt_T = T;
wrt_T *= (g + f*f);
wrt_T -= f * (T * transpose(T) * T);
wrt_T -= f * g / d * transpose_adj(T);
wrt_T *= 2 / (9*d*d);
return true;
}
bool TShapeOrientB2::evaluate( const MsqMatrix<2,2>& T,
double& result,
MsqError& err )
{
double tau = det(T);
if (TMetric::invalid_determinant(tau)) {
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
const double tr = trace(T);
result = (0.5/tau) * (sqr_Frobenius( T ) - 0.5 * tr * fabs(tr));
return true;
}
bool TShapeB1::evaluate( const MsqMatrix<2,2>& T,
double& result,
MsqError& err)
{
const double d = det(T);
if (invalid_determinant(d)) { // barrier
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
result = 0.5 * sqr_Frobenius(T) / d - 1;
return true;
}
bool TShapeSize2DNB2::evaluate_with_hess( const MsqMatrix<2,2>& T,
double& result,
MsqMatrix<2,2>& deriv_wrt_T,
MsqMatrix<2,2> second[3],
MsqError& err )
{
double frob_sqr = sqr_Frobenius(T);
double psi = sqrt( frob_sqr + 2.0*det(T) );
double a = 1e-12;
while (!Mesquite::divide(frob_sqr+2,2*psi,result)) {
MsqMatrix<2,2> Tdelta(T);
Tdelta(0,0) += a;
Tdelta(1,1) += a;
a *= 2.0;
frob_sqr = sqr_Frobenius(Tdelta);
psi = sqrt( frob_sqr + 2.0*det(Tdelta) );
if (psi > 1e-50)
result = (frob_sqr+2) / 2*psi;
}
const double inv_psi = 1.0/psi;
MsqMatrix<2,2> d_psi = T + transpose_adj(T);
d_psi *= inv_psi;
deriv_wrt_T = d_psi;
deriv_wrt_T *= -result;
deriv_wrt_T += T;
deriv_wrt_T *= inv_psi;
set_scaled_2nd_deriv_wrt_psi( second, -result*inv_psi, psi, T );
pluseq_scaled_outer_product( second, 2*result*inv_psi*inv_psi, d_psi );
pluseq_scaled_sum_outer_product( second, -inv_psi*inv_psi, d_psi, T );
pluseq_scaled_I( second, inv_psi );
result -= 1.0;
return true;
}
/** \f$ \mu(T) = \frac{|T|^2+2}{2\psi(T)} - 1 \f$
* \f$ \psi(T) = \sqrt{|T|^2 + 2 \tau}\f$
* \f$ \tau = det(T) \f$
*/
bool TShapeSize2DNB2::evaluate( const MsqMatrix<2,2>& T,
double& result,
MsqError& )
{
double frob_sqr = sqr_Frobenius(T);
double psi = sqrt( frob_sqr + 2.0*det(T) );
double a = 1e-12;
while (!Mesquite::divide(frob_sqr+2,2*psi,result)) {
MsqMatrix<2,2> Tdelta(T);
Tdelta(0,0) += a;
Tdelta(1,1) += a;
a *= 2.0;
frob_sqr = sqr_Frobenius(Tdelta);
psi = sqrt( frob_sqr + 2.0*det(Tdelta) );
if (psi > 1e-50)
result = (frob_sqr+2) / 2*psi;
}
result -= 1.0;
return true;
}
bool TShapeSizeB3::evaluate( const MsqMatrix<2,2>& T,
double& result,
MsqError& err )
{
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
result = sqr_Frobenius(T) - 2.0 * std::log(tau) - 2;
return true;
}
template <unsigned DIM> static inline
bool hess( const MsqMatrix<DIM,DIM>& A,
const MsqMatrix<DIM,DIM>& W,
double& result,
MsqMatrix<DIM,DIM>& deriv,
MsqMatrix<DIM,DIM>* second )
{
deriv = A - W;
result = sqr_Frobenius( deriv );
deriv *= 2.0;
set_scaled_I( second, 2.0 );
return true;
}
bool TShapeSize2DB2::evaluate( const MsqMatrix<2,2>& T,
double& result,
MsqError& err )
{
const double two_det = 2.0 * det(T);
if (invalid_determinant(two_det)) { // barrier
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
const double frob_sqr = sqr_Frobenius(T);
result = (frob_sqr - 2.0 * sqrt( frob_sqr + two_det ) + 2.0)/two_det;
return true;
}
bool AWShape2DNB1::evaluate_with_grad( const MsqMatrix<2,2>& A,
const MsqMatrix<2,2>& W,
double& result,
MsqMatrix<2,2>& deriv_wrt_A,
MsqError& )
{
const double alpha = det(A);
const double omega = det(W);
const MsqMatrix<2,2> AtA = transpose(A) * A;
const MsqMatrix<2,2> WtW = transpose(W) * W;
const MsqMatrix<2,2> AWt = A * transpose(W);
const double alphaWtW = alpha * sqr_Frobenius(WtW);
const double omegaAtA = omega * sqr_Frobenius(AtA);
const double omegaAWt = omega * sqr_Frobenius(AWt);
result = omega*omegaAtA;
result += alpha*alphaWtW;
result -= 2*alpha*omegaAWt;
deriv_wrt_A = 4*omega*omega * A * transpose(A) * A;
deriv_wrt_A -= 4*alpha*omega * AWt * W;
deriv_wrt_A += 2*(alphaWtW - omegaAWt) * transpose_adj(A);
return true;
}
bool Target3DShapeOrientBarrierAlt1::evaluate_with_Hess( const MsqMatrix<3,3>& A,
const MsqMatrix<3,3>& W,
double& result,
MsqMatrix<3,3>& deriv_wrt_A,
MsqMatrix<3,3> second_wrt_A[6],
MsqError& err )
{
MsqMatrix<3,3> Winv = inverse(W);
MsqMatrix<3,3> T = A * Winv;
double tau = det(T);
if (invalid_determinant(tau)) {
result = 0.0;
return false;
}
const double b = 0.5/tau;
// calculate non-barrier value (ShapeOrientAlt1)
const double tr = trace(T);
const double f = MSQ_ONE_THIRD * fabs(tr);
result = sqr_Frobenius( T ) - f * tr;
// calculate non-barrier first derivatives
deriv_wrt_A = T;
deriv_wrt_A(0,0) -= f;
deriv_wrt_A(1,1) -= f;
deriv_wrt_A(2,2) -= f;
deriv_wrt_A *= 2;
// calculate barrier second derivs
const MsqMatrix<3,3> adjt = transpose_adj(T);
set_scaled_sum_outer_product( second_wrt_A, -b/tau, deriv_wrt_A, adjt );
pluseq_scaled_outer_product( second_wrt_A, result/(tau*tau*tau), adjt );
pluseq_scaled_2nd_deriv_of_det( second_wrt_A, -result * b / tau, T );
// calculate non-barrier barrier portion of second derivs
pluseq_scaled_I( second_wrt_A, 1/tau );
pluseq_scaled_outer_product_I_I( second_wrt_A, MSQ_ONE_THIRD/tau * (tr < 0 ? 1 : -1) );
// calculate barrier derivs from non-barrier
deriv_wrt_A *= tau;
deriv_wrt_A -= result * adjt;
deriv_wrt_A *= b/tau;
// barrier value from non-barrier
result *= b;
// convert from derivs wrt T to derivs wrt A
deriv_wrt_A = deriv_wrt_A * transpose(Winv);
second_deriv_wrt_product_factor( second_wrt_A, Winv );
return true;
}
bool Target2DShapeBarrier::evaluate( const MsqMatrix<2,2>& A,
const MsqMatrix<2,2>& W,
double& result,
MsqError& )
{
const MsqMatrix<2,2> T = A * inverse(W);
const double d = det(T);
if (invalid_determinant(d)) { // barrier
result = 0.0;
return false;
}
result = 0.5 * sqr_Frobenius(T) / d - 1;
return true;
}
bool Target3DShapeOrientBarrierAlt1::evaluate( const MsqMatrix<3,3>& A,
const MsqMatrix<3,3>& W,
double& result,
MsqError& )
{
const MsqMatrix<3,3> T = A * inverse(W);
double tau = det(T);
if (invalid_determinant(tau)) {
result = 0.0;
return false;
}
const double tr = trace(T);
result = (0.5/tau) * (sqr_Frobenius( T ) - MSQ_ONE_THIRD * tr * fabs(tr));
return true;
}
bool TShapeSizeB1::evaluate( const MsqMatrix<2,2>& T,
double& result,
MsqError& )
{
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
result = 0.0;
return false;
}
const double nT = sqr_Frobenius(T);
const double f = 1/(tau*tau);
result = (1 + f) * nT - 4;
return true;
}
bool TShapeSizeOrientB1::evaluate( const MsqMatrix<2,2>& T,
double& result,
MsqError& err )
{
double tau = det(T);
if (TMetric::invalid_determinant(tau)) {
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
MsqMatrix<2,2> T_I(T);
pluseq_scaled_I( T_I, -1 );
result = sqr_Frobenius( T_I ) / (2*tau);
return true;
}
template <unsigned DIM> static inline
bool hess( const MsqMatrix<DIM,DIM>& T,
double& result,
MsqMatrix<DIM,DIM>& deriv_wrt_T,
MsqMatrix<DIM,DIM>* second_wrt_T )
{
const double tr = trace(T);
const double f = DimConst<DIM>::inv() * fabs(tr);
result = sqr_Frobenius( T ) - f * tr;
deriv_wrt_T = T;
pluseq_scaled_I( deriv_wrt_T, -f );
deriv_wrt_T *= 2;
set_scaled_I( second_wrt_T, 2.0 );
pluseq_scaled_outer_product_I_I( second_wrt_T, DimConst<DIM>::inv() * (tr < 0 ? 2 : -2) );
return true;
}
bool InverseMeanRatio2D::evaluate( const MsqMatrix<2,2>& A,
const MsqMatrix<2,2>& W,
double& result,
MsqError& err )
{
const MsqMatrix<2,2> T = A * inverse(W);
const double d = det( T );
if (invalid_determinant(d)) {
result = 0.0;
return false;
}
else {
result = sqr_Frobenius(T) / (2 * d) - 1;
return true;
}
}
bool TShapeSizeB3::evaluate_with_grad( const MsqMatrix<2,2>& T,
double& result,
MsqMatrix<2,2>& deriv_wrt_T,
MsqError& err )
{
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
result = sqr_Frobenius(T) - 2.0 * std::log(tau) - 2;
deriv_wrt_T = T;
deriv_wrt_T -= 1/tau * transpose_adj(T);
deriv_wrt_T *= 2;
return true;
}
bool TShapeOrientB2::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 )
{
double tau = det(T);
if (TMetric::invalid_determinant(tau)) {
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
result = 0.0;
return false;
}
const double b = 0.5/tau;
// calculate non-barrier value (ShapeOrientAlt1)
const double tr = trace(T);
const double f = MSQ_ONE_THIRD * fabs(tr);
result = sqr_Frobenius( T ) - f * tr;
// calculate non-barrier first derivatives
deriv_wrt_T = T;
pluseq_scaled_I( deriv_wrt_T, -f );
deriv_wrt_T *= 2;
// calculate barrier second derivs
const MsqMatrix<3,3> adjt = transpose_adj(T);
set_scaled_sum_outer_product( second_wrt_T, -b/tau, deriv_wrt_T, adjt );
pluseq_scaled_outer_product( second_wrt_T, result/(tau*tau*tau), adjt );
pluseq_scaled_2nd_deriv_of_det( second_wrt_T, -result * b / tau, T );
// calculate non-barrier barrier portion of second derivs
pluseq_scaled_I( second_wrt_T, 1/tau );
pluseq_scaled_outer_product_I_I( second_wrt_T, MSQ_ONE_THIRD/tau * (tr < 0 ? 1 : -1) );
// calculate barrier derivs from non-barrier
deriv_wrt_T *= tau;
deriv_wrt_T -= result * adjt;
deriv_wrt_T *= b/tau;
// barrier value from non-barrier
result *= b;
return true;
}
bool TShapeB1::evaluate_with_grad( const MsqMatrix<2,2>& T,
double& result,
MsqMatrix<2,2>& deriv_wrt_T,
MsqError& err )
{
const double d = det(T);
if (invalid_determinant(d)) { // barrier
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
double inv_d = 1.0/d;
result = 0.5 * sqr_Frobenius(T) * inv_d;
deriv_wrt_T = T;
deriv_wrt_T -= result * transpose_adj(T);
deriv_wrt_T *= inv_d;
result -= 1.0;
return true;
}
bool TShapeSizeOrientB1::evaluate_with_grad( const MsqMatrix<3,3>& T,
double& result,
MsqMatrix<3,3>& deriv_wrt_T,
MsqError& err )
{
const double d = det(T);
if (TMetric::invalid_determinant(d)) { // barrier
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
deriv_wrt_T = T;
pluseq_scaled_I( deriv_wrt_T, -1 );
double inv_d = 1.0/d;
result = 0.5 * sqr_Frobenius(deriv_wrt_T) * inv_d;
deriv_wrt_T -= result * transpose_adj(T);
deriv_wrt_T *= inv_d;
return true;
}
bool TShapeSizeB1::evaluate_with_grad( const MsqMatrix<2,2>& T,
double& result,
MsqMatrix<2,2>& deriv_wrt_T,
MsqError& )
{
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
result = 0.0;
return false;
}
const MsqMatrix<2,2> adjt = transpose_adj(T);
const double nT = sqr_Frobenius(T);
const double f = 1/(tau*tau);
result = (1 + f) * nT - 4;
deriv_wrt_T = T;
deriv_wrt_T *= 2 + 2*f;
deriv_wrt_T -= 2 * f/tau * nT * adjt;
return true;
}
bool Target2DShapeBarrier::evaluate_with_grad( const MsqMatrix<2,2>& A,
const MsqMatrix<2,2>& W,
double& result,
MsqMatrix<2,2>& deriv_wrt_A,
MsqError& err )
{
const MsqMatrix<2,2> Winv = inverse(W);
const MsqMatrix<2,2> T = A * Winv;
const double d = det(T);
if (invalid_determinant(d)) { // barrier
result = 0.0;
return false;
}
double inv_d = 1.0/d;
result = 0.5 * sqr_Frobenius(T) * inv_d;
deriv_wrt_A = T;
deriv_wrt_A -= result * transpose_adj(T);
deriv_wrt_A *= inv_d;
deriv_wrt_A = deriv_wrt_A * transpose(Winv);
result -= 1.0;
return true;
}
bool TShapeSize2DB2::evaluate_with_grad( const MsqMatrix<2,2>& T,
double& result,
MsqMatrix<2,2>& deriv_wrt_T,
MsqError& err )
{
const double d = det(T);
if (invalid_determinant(d)) { // barrier
MSQ_SETERR(err)( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
const double frob_sqr = sqr_Frobenius(T);
const double psi = sqrt( frob_sqr + 2.0*det(T) );
const double v = frob_sqr - 2.0 * psi + 2.0;
result = v / (2*d);
// deriv of V wrt T
MsqMatrix<2,2> adjt = transpose_adj(T);
MsqMatrix<2,2> v_wrt_T(T);
if (psi > 1e-50)
{
v_wrt_T *= (1.0 - 1.0/psi);
v_wrt_T -= 1.0/psi * adjt;
v_wrt_T *= 2;
}
else
{
std::cout << "Warning: Division by zero avoided in TShapeSize2DB2::evaluate_with_grad()" << std::endl;
}
// deriv of mu wrt T
deriv_wrt_T = v_wrt_T;
deriv_wrt_T *= 0.5/d;
deriv_wrt_T -= v / (2*d*d) * adjt;
return true;
}
template <unsigned DIM> static inline
bool eval( const MsqMatrix<DIM,DIM>& T, double& result)
{
result = sqr_Frobenius( T );
return true;
}
