Set<MultipleDeriv> applyZx(const Set<MultipleDeriv>& W,
const MultiIndex& x)
{
Set<MultipleDeriv> rtn;
TEUCHOS_TEST_FOR_EXCEPTION(x.order() < 0 || x.order() > 1, std::logic_error,
"invalid multiindex " << x << " in this context");
for (Set<MultipleDeriv>::const_iterator i=W.begin(); i!=W.end(); i++)
{
const MultipleDeriv& md = *i;
TEUCHOS_TEST_FOR_EXCEPTION(md.order() != 1, std::logic_error,
"Only first-order multiple functional derivatives "
"should appear in this function. The derivative "
<< md << " is not first-order.");
const Deriv& d = *(md.begin());
if (d.isFunctionalDeriv())
{
/* */
TEUCHOS_TEST_FOR_EXCEPTION(!d.canBeDifferentiated(),
std::logic_error, "function signature " << d << " cannot be "
"differentiated further spatially");
/* accept a functional derivative if the associated function
* is not identically zero */
const SymbolicFuncElement* sfe = d.symbFuncElem();
TEUCHOS_TEST_FOR_EXCEPTION(sfe==0, std::logic_error,
"can't cast function in "
<< d << " to a SymbolicFuncElement");
if (sfe && !sfe->evalPtIsZero()) rtn.put(md);
}
}
return rtn;
}
Set<MultipleDeriv> Xx(const MultiIndex& x)
{
Set<MultipleDeriv> rtn;
TEUCHOS_TEST_FOR_EXCEPTION(x.order() < 0 || x.order() > 1, std::logic_error,
"invalid multiindex " << x << " in this context");
MultipleDeriv xmd = makeMultiDeriv(coordDeriv(x.firstOrderDirection()));
rtn.put(xmd);
return rtn;
}
Expr SpectralPreprocessor::takeDeriv(const Expr& f, const MultiIndex& mi)
{
TEUCHOS_TEST_FOR_EXCEPT(mi.order() > 1);
const SpectralExpr* se
= dynamic_cast<const SpectralExpr*>(f.ptr().get());
Expr d = new Derivative(mi.firstOrderDirection());
int n = se->getSpectralBasis().nterms();
if (se)
{
Array<Expr> c(n);
for (int i=0; i<n; i++)
{
c[i] = d*se->getCoeff(i);
}
return new SpectralExpr(se->getSpectralBasis(), c);
}
else
{
return d*f;
}
}
void CubicHermite::evalOnTriangle(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
result.resize(10);
ADReal x = ADReal(pt[0], 0, 2);
ADReal y = ADReal(pt[1], 1, 2);
ADReal one(1.0, 2);
Array<ADReal> tmp(10);
SUNDANCE_OUT(this->verb() > 3, "x=" << x.value() << " y="
<< y.value());
tmp[0] = 1 - 3*x*x + 2 * x*x*x - 13*x*y + 13*x*x*y - 3*y*y + 13 *x*y*y + 2 *y*y*y;
tmp[1] = x - 2 *x*x + x*x*x - 3*x*y + 3*x*x*y + 2*x*y*y;
tmp[2] = y - 3 *x *y + 2* x*x* y - 2* y*y + 3* x*y*y + y*y*y;
tmp[3] = 3 * x*x - 2* x*x*x - 7* x* y + 7* x*x *y + 7* x*y*y;
tmp[4] = -x*x + x*x*x + 2*x *y - 2* x*x* y - 2* x* y*y;
tmp[5] = -x* y + 2* x*x* y + x* y*y;
tmp[6] = -7* x* y + 7* x*x*y + 3* y*y + 7* x* y*y - 2* y*y*y;
tmp[7] = -x *y + x*x* y + 2* x* y*y;
tmp[8] = 2 *x *y - 2* x*x* y - y*y - 2* x* y*y + y*y*y;
tmp[9] = 27* x *y - 27* x*x* y - 27* x* y*y;
for (int i=0; i<tmp.length(); i++)
{
if (deriv.order()==0) result[i] = tmp[i].value();
else
result[i] = tmp[i].gradient()[deriv.firstOrderDirection()];
}
}
void CubicHermite::evalOnLine(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
result.resize(4);
ADReal x = ADReal(pt[0],0,1);
Array<ADReal> tmp(4);
tmp[0] = 1 + x * x * ( -3 + 2 * x );
tmp[1] = x * ( 1 + (x - 2 ) * x );
tmp[2] = ( 3 - 2*x ) * x * x;
tmp[3] = (-1+x)*x*x;
for (int i=0; i<tmp.length(); i++)
{
if (deriv.order()==0)
{
result[i] = tmp[i].value();
}
else
{
result[i] = tmp[i].gradient()[0];
}
}
return;
}
void EdgeLocalizedBasis::evalOnTriangle(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
ADReal x = ADReal(pt[0], 0, 2);
ADReal y = ADReal(pt[1], 1, 2);
ADReal one(1.0, 2);
ADReal zero(0.0, 2);
Array<ADReal> tmp;
SUNDANCE_OUT(this->verb() > 3, "x=" << x.value() << " y="
<< y.value());
result.resize(3);
tmp.resize(3);
bool onEdge2 = std::fabs(pt[1]) < 1.0e-14;
bool onEdge0 = std::fabs(1.0-pt[0]-pt[1]) < 1.0e-14;
bool onEdge1 = std::fabs(pt[0]) < 1.0e-14;
TEUCHOS_TEST_FOR_EXCEPTION(!(onEdge0 || onEdge1 || onEdge2),
std::runtime_error,
"EdgeLocalizedBasis should not be evaluated at points not on edges");
TEUCHOS_TEST_FOR_EXCEPTION((onEdge0 && onEdge1) || (onEdge1 && onEdge2)
|| (onEdge2 && onEdge0), std::runtime_error,
"Ambiguous edge in EdgeLocalizedBasis::evalOnTriangle()");
if (onEdge0)
{
tmp[0] = one;
tmp[1] = zero;
tmp[2] = zero;
}
if (onEdge1)
{
tmp[0] = zero;
tmp[1] = one;
tmp[2] = zero;
}
if (onEdge2)
{
tmp[0] = zero;
tmp[1] = zero;
tmp[2] = one;
}
for (int i=0; i<tmp.length(); i++)
{
SUNDANCE_OUT(this->verb() > 3,
"tmp[" << i << "]=" << tmp[i].value()
<< " grad=" << tmp[i].gradient());
if (deriv.order()==0) result[i] = tmp[i].value();
else
result[i] = tmp[i].gradient()[deriv.firstOrderDirection()];
}
}
void Legendre::evalOnLine(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
result.resize(2+nrDOF_edge_);
ADReal x = ADReal(pt[0],0,1);
Array<ADReal> refAll(7);
refAll[0] = 1-x;
refAll[1] = x;
refAll[2] = 2.44948974278318 * ( (2*x-1)*(2*x-1) - 1 ) / 4;
refAll[3] = 3.16227766016838 * ( (2*x-1)*(2*x-1) - 1 ) * (2*x-1) / 4;
refAll[4] = 3.74165738677394 * ( 5*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) - 6*(2*x-1)*(2*x-1) + 1) / 16;
refAll[5] = 4.24264068711929 * (2*x-1) * (7*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) - 10*(2*x-1)*(2*x-1) + 3) / 16;
refAll[6] = 4.69041575982343 * (21*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) -
35*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) + 15*(2*x-1)*(2*x-1) - 1) / 32;
for (int i=0; i<result.length(); i++)
{
if (deriv.order()==0)
{
result[i] = refAll[i].value();
}
else
{
result[i] = refAll[i].gradient()[0];
}
}
//SUNDANCE_OUT( true , "Legendre::evalOnLine result.length():" << result.length() );
return;
}
void Bernstein::evalOnTet(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
ADReal x = ADReal(pt[0], 0, 3);
ADReal y = ADReal(pt[1], 1, 3);
ADReal z = ADReal(pt[2], 2, 3);
ADReal one(1.0, 3);
Array<ADReal> tmp(result.length());
if(order_==0)
{
tmp.resize(1);
result.resize(1);
tmp[0] = one;
}
else
{
}
for (int i=0; i<tmp.length(); i++)
{
if (deriv.order()==0) result[i] = tmp[i].value();
else
result[i] = tmp[i].gradient()[deriv.firstOrderDirection()];
}
}
double TestEvalMediator::evalDummyBasis(int m, const MultiIndex& mi) const
{
TEUCHOS_TEST_FOR_EXCEPTION(mi.order() > 1, std::runtime_error,
"TestEvalMediator::evalDummyBasis found multiindex "
"order > 1. The bad multiindex was " << mi.toString());
ADReal result = fields_[m].basis().evaluate(ADField::evalPoint());
SUNDANCE_MSG3(verb(), "basis.value() " << result.value());
SUNDANCE_MSG3(verb(), "basis.gradient() " << result.gradient());
if (mi.order()==0)
{
return result.value();
}
else
{
return result.gradient()[mi.firstOrderDirection()];
}
}
void EdgeLocalizedBasis::evalOnLine(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
ADReal one(1.0, 1);
result.resize(1);
Array<ADReal> tmp(result.length());
tmp[0] = one;
for (int i=0; i<tmp.length(); i++)
{
if (deriv.order()==0) result[i] = tmp[i].value();
else result[i] = tmp[i].gradient()[deriv.firstOrderDirection()];
}
}
SpatialDerivSpecifier SpatialDerivSpecifier::derivWrtMultiIndex(const MultiIndex& mi) const
{
if (isPartial() || isIdentity())
{
return SpatialDerivSpecifier(mi_+mi);
}
else if (mi.order()==0)
{
return *this;
}
else
{
TEST_FOR_EXCEPTION(true, InternalError, "cannot take an arbitrary "
"spatial derivative of SDS=" << *this);
return *this; // -Wall
}
}
void Bernstein::evalOnLine(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
ADReal x = ADReal(pt[0], 0, 1);
ADReal one(1.0, 1);
result.resize(order()+1);
Array<ADReal> tmp(result.length());
Array<double> x0(order()+1);
if (order_ == 0)
{
tmp[0] = one;
}
else
{
double binom_cur = 1.0;
for (int i=0; i<=order_; i++)
{
tmp[i] = one;
for (int j=0;j<order_-i;j++)
{
tmp[i] *= (1-x);
}
for (int j=0;j<i;j++)
{
tmp[i] *= x;
}
tmp[i] *= binom_cur;
binom_cur *= double(order()-i) / double(i+1);
}
}
for (int i=0; i<tmp.length(); i++)
{
if (deriv.order()==0) result[i] = tmp[i].value();
else result[i] = tmp[i].gradient()[0];
}
}
void Legendre::evalOnQuad(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
result.resize( 4 + 4*nrDOF_edge_ + nrDOF_face_);
ADReal x = ADReal(pt[0], 0, 2);
ADReal y = ADReal(pt[1], 1, 2);
ADReal one(1.0, 2);
Array<ADReal> refAllx(7);
Array<ADReal> refAlly(7);
refAllx[0] = 1-x;
refAllx[1] = x;
refAllx[2] = 2.44948974278318 * ( (2*x-1)*(2*x-1) - 1 ) / 4;
refAllx[3] = 3.16227766016838 * ( (2*x-1)*(2*x-1) - 1 ) * (2*x-1) / 4;
refAllx[4] = 3.74165738677394 * ( 5*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) - 6*(2*x-1)*(2*x-1) + 1) / 16;
refAllx[5] = 4.24264068711929 * (2*x-1) * (7*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) - 10*(2*x-1)*(2*x-1) + 3) / 16;
refAllx[6] = 4.69041575982343 * (21*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) -
35*(2*x-1)*(2*x-1)*(2*x-1)*(2*x-1) + 15*(2*x-1)*(2*x-1) - 1) / 32;
refAlly[0] = 1-y;
refAlly[1] = y;
refAlly[2] = 2.44948974278318 * ( (2*y-1)*(2*y-1) - 1 ) / 4;
refAlly[3] = 3.16227766016838 * ( (2*y-1)*(2*y-1) - 1 ) * (2*y-1) / 4;
refAlly[4] = 3.74165738677394 * ( 5*(2*y-1)*(2*y-1)*(2*y-1)*(2*y-1) - 6*(2*y-1)*(2*y-1) + 1) / 16;
refAlly[5] = 4.24264068711929 * (2*y-1) * (7*(2*y-1)*(2*y-1)*(2*y-1)*(2*y-1) - 10*(2*y-1)*(2*y-1) + 3) / 16;
refAlly[6] = 4.69041575982343 * (21*(2*y-1)*(2*y-1)*(2*y-1)*(2*y-1)*(2*y-1) -
35*(2*y-1)*(2*y-1)*(2*y-1)*(2*y-1) + 15*(2*y-1)*(2*y-1) - 1) / 32;
SUNDANCE_OUT(this->verb() > 3, "x=" << x.value() << " y="
<< y.value());
int sideIndex[4][2] = { {0,0} , {1,0} , {0,1} , {1,1}};
int edgeIndex[4] = { 0 , 1 , 1 , 0};
int edgeMultI[4] = { 0 , 0 , 1 , 1};
int offs = 0;
Array<ADReal> tmp(4 + 4*nrDOF_edge_ + nrDOF_face_);
// loop over vertexes
for (int i=0; i < 4 ; i++, offs++){
tmp[offs] = refAllx[sideIndex[i][0]] * refAlly[sideIndex[i][1]];
}
// loop over edges
for (int i=0; i < 4 ; i++){
// loop over each DOF on the edge
if (edgeIndex[i] == 0){
for (int j = 0 ; j < nrDOF_edge_ ; j++ , offs++){
tmp[offs] = refAllx[2+j] * refAlly[edgeMultI[i]];
}
}
else
{
for (int j = 0 ; j < nrDOF_edge_ ; j++ , offs++){
tmp[offs] = refAllx[edgeMultI[i]] * refAlly[2+j];
}
}
}
// loop over all internal DOFs
if ( nrDOF_face_ > 0 ){
// loop for each hierarchical layer
for (int hierarch = 0 ; hierarch < order_ - 3 ; hierarch++)
{
//SUNDANCE_OUT( true , "Legendre::evalOnQuad hierarch:" << hierarch );
// for each layer add the basis function
for (int i=0 ; i < hierarch+1 ; i++ , offs++)
{
//SUNDANCE_OUT( true , "Legendre::evalOnQuad offs:" << offs << " 2+i:" << 2+i << " , 2+(hierarch-1-i):" << 2+(hierarch-i));
tmp[offs] = refAllx[2+i] * refAlly[2+(hierarch-i)];
}
}
}
// compute the results
for (int i=0; i<result.length(); i++)
{
if (deriv.order()==0) result[i] = tmp[i].value();
else
result[i] = tmp[i].gradient()[deriv.firstOrderDirection()];
}
//SUNDANCE_OUT( true , "Legendre::evalOnQuad result.length():" << result.length() );
}
void CurveEvalMediator
::evalDiscreteFuncElement(const DiscreteFuncElement* expr,
const Array<MultiIndex>& multiIndices,
Array<RCP<EvalVector> >& vec) const
{
int verbo = dfVerb();
Tabs tab1;
SUNDANCE_MSG2(verbo , tab1
<< "CurveEvalMediator evaluating Discrete Function expr " << expr->toString());
const DiscreteFunctionData* f = DiscreteFunctionData::getData(expr);
TEUCHOS_TEST_FOR_EXCEPTION(f==0, std::logic_error,
"QuadratureEvalMediator::evalDiscreteFuncElement() called "
"with expr that is not a discrete function");
SUNDANCE_MSG2(verbo , tab1 << "After casting DiscreteFunctionData" << expr->toString());
RCP<Array<Array<double> > > localValues;
Array<int> cellLIDs_tmp(1);
Array<Point> phyPoints;
Array<Point> refPoints;
Array<Point> refDevs;
Array<Point> refNormal;
int nCells = cellLID()->size();
RCP<const MapStructure> mapStruct;
int myIndex = expr->myIndex();
int nQuad = numQuadPtsForMaxCell_;
Array<int> k(multiIndices.size(),0);
Teuchos::BLAS<int,double> blas;
SUNDANCE_MSG2(verbo , tab1 << "After declaring BLAS: " << expr->toString());
// resize correctly the result vector
for (int i=0; i<multiIndices.size(); i++)
{
vec[i]->resize(nCells*nQuad);
}
// loop over each cell
for (int c=0; c<nCells; c++)
{
int maxCellLID = (*cellLID())[c];
localValues = rcp(new Array<Array<double> >());
SUNDANCE_MSG2(verbo , tab1 << "Cell:" << c << " of " << nCells << " , maxCellLID:" << maxCellLID );
cellLIDs_tmp[0] = (*cellLID())[c];
SUNDANCE_MSG2(verbo , tab1 << " Before calling f->getLocalValues:" << cellLIDs_tmp.size() <<
" f==0 : " << (f==0) << " tmp:" << f->mesh().spatialDim());
// - get local values from the DiscreteFunctionElementData
mapStruct = f->getLocalValues(maxCellDim(), cellLIDs_tmp , *localValues);
SUNDANCE_MSG2(verbo , tab1 << " After getting mapStruct:" << maxCellLID );
SUNDANCE_MSG2(verbo , tab1 << " mapStruct->numBasisChunks():" << mapStruct->numBasisChunks() );
int chunk = mapStruct->chunkForFuncID(myIndex);
int funcIndex = mapStruct->indexForFuncID(myIndex);
int nFuncs = mapStruct->numFuncs(chunk);
SUNDANCE_MSG2(verbo , tab1 << " chunk:" << chunk );
SUNDANCE_MSG2(verbo , tab1 << " funcIndex:" << funcIndex );
SUNDANCE_MSG2(verbo , tab1 << " nFuncs:" << nFuncs );
// the chunk of the function
BasisFamily basis = rcp_dynamic_cast<BasisFamilyBase>(mapStruct->basis(chunk));
int nNodesTotal = basis.nReferenceDOFsWithFacets(maxCellType(), maxCellType());
// - get intersection (reference)points from the mesh (if not existent than compute them)
if ( mesh().hasCurvePoints( maxCellLID , paramcurve_.myID() ))
{
mesh().getCurvePoints( maxCellLID , paramcurve_.myID() , refPoints , refDevs , refNormal );
}
else // we have to calculate now the points
{
// calculate the intersection points
CurveIntegralCalc::getCurveQuadPoints( maxCellType_ , maxCellLID , mesh() , paramcurve_ , quad_ ,
refPoints, refDevs , refNormal);
// store the intersection point in the mesh
mesh().setCurvePoints( maxCellLID , paramcurve_.myID() , refPoints, refDevs , refNormal );
}
// loop over each multi-index
SUNDANCE_MSG2(verbo , tab1 << " multiIndices.size()" << multiIndices.size() );
for (int i=0; i<multiIndices.size(); i++)
{
int nDerivResults = 1;
if ( multiIndices[i].order() == 1 ) nDerivResults = maxCellDim();
int pDir = 0;
int derivNum = 1;
MultiIndex mi;
SUNDANCE_MSG2(verbo , tab1 << " before asking anything i = " << i);
SUNDANCE_MSG2(verbo , tab1 << " multiindex order : " << multiIndices[i].order());
SUNDANCE_MSG2(verbo , tab1 << " multiindex : " << multiIndices[i] );
if (multiIndices[i].order() > 0){
pDir = multiIndices[i].firstOrderDirection();
mi[pDir] = 1;
derivNum = mesh().spatialDim();
}
Array<Array<double> > result(nQuad*derivNum);
Array<Array<Array<double> > > tmp;
int offs = nNodesTotal;
// resize the result vector
for (int deriv = 0 ; deriv < derivNum ; deriv++)
{
// test weather we have to compute derivative
if (multiIndices[i].order() > 0){
// in case of derivatives we set one dimension
MultiIndex mi_tmp;
mi_tmp[deriv] = 1;
SpatialDerivSpecifier deriv(mi_tmp);
SUNDANCE_MSG2(verbo , tab1 << "computing derivatives : " << deriv << " on reference cell ");
basis.refEval( maxCellType_ , refPoints , deriv, tmp , verbo );
} else {
SpatialDerivSpecifier deriv(mi);
// --- end eval basis functions
SUNDANCE_MSG2(verbo , tab1 << "computing values reference cell ");
basis.refEval( maxCellType_ , refPoints , deriv, tmp , verbo );
}
SUNDANCE_MSG2(verbo , tab1 << "resize result vector , offs:" << offs);
for (int q=0; q<nQuad; q++){
result[nQuad*deriv + q].resize(offs);
}
// copy the result in an other format
SUNDANCE_MSG2(verbo , tab1 << "copy results ");
int offs1 = 0;
for (int q=0; q<nQuad; q++)
{
offs1 = 0;
for (int d=0; d<basis.dim(); d++)
{
int nNodes = tmp[d][q].size();
for (int n=0; n<nNodes; n++ , offs1++ )
{
result[nQuad*deriv + q][offs1] = tmp[d][q][n];
}
}
}
}// loop over all dimensional derivative
// multiply the local results with the coefficients, (matrix vector OP)
SUNDANCE_MSG2(verbo , tab1 << "summing up values , funcIndex:" << funcIndex << " offs:" << offs);
for (int deriv = 0 ; deriv < derivNum ; deriv++)
{
for (int q=0; q<nQuad; q++)
{
double sum = 0.0;
// sum over nodes
for (int n = 0 ; n < offs ; n++){
sum = sum + result[nQuad*deriv + q][n] * (*localValues)[chunk][funcIndex*offs + n];
}
// sum up the result in the 0th element
result[nQuad*deriv + q][0] = sum;
}
}
// multiply the result if necesary with the inverse of the Jacobian
const CellJacobianBatch& J = JTrans();
if (mi.order()==1)
{
Tabs tab1;
Tabs tab2;
SUNDANCE_MSG2(verbo, tab2 << "Jacobian batch nCells=" << J.numCells());
SUNDANCE_MSG2(verbo, tab2 << "Jacobian batch cell dim=" << J.cellDim());
SUNDANCE_MSG2(verbo, tab2 << "Jacobian batch spatial dim=" << J.spatialDim());
// we just multiply the derivative direction component
Array<double> invJ;
J.getInvJ(c, invJ);
for (int q=0; q<nQuad; q++)
{
double sum = 0.0;
for (int deriv = 0 ; deriv < derivNum ; deriv++)
{
// multiply one row from the J^{-T} matrix with the gradient vector
sum = sum + result[nQuad*deriv + q][0] * invJ[derivNum*pDir + deriv];
}
// the resulting derivative on the physical cell in the "pDir" direction
result[q][0] = sum;
}
}
// --- just copy the result to the "vec" back, the result should be in the "result[q][0]" place----
//SUNDANCE_MSG2(verbo , tab1 << "copy results back ");
double* vecPtr = vec[i]->start();
for (int q=0; q<nQuad; q++, k[i]++)
{
vecPtr[k[i]] = result[q][0];
}
SUNDANCE_MSG2(verbo , tab1 << " END copy results back ");
} // --- end loop multiindex
SUNDANCE_MSG2(verbo , tab1 << " END loop over multiindex ");
}// --- end loop over cells
SUNDANCE_MSG2(verbo , tab1 << " END loop over cells ");
}
void Bernstein::evalOnTriangle(const Point& pt,
const MultiIndex& deriv,
Array<double>& result) const
{
ADReal x = ADReal(pt[0], 0, 2);
ADReal y = ADReal(pt[1], 1, 2);
ADReal one(1.0, 2);
Array<ADReal> tmp;
SUNDANCE_OUT(this->verb() > 3, "x=" << x.value() << " y="
<< y.value());
if(order_==0) {
result.resize(1);
tmp.resize(1);
tmp[0] = one;
}
else {
int N = (order()+1)*(order()+2)/2;
result.resize(N);
tmp.resize(N);
// these are the barycentric coordinates
ADReal b1 = 1.0 - x - y;
ADReal b2 = x;
ADReal b3 = y;
// will hold \binom{n}{\alpha_1}
int bfcur = 0;
for (int alpha1=order();alpha1>=0;alpha1--)
{
for (int alpha2 = order()-alpha1;alpha2>=0;alpha2--)
{
int alpha3 = order() - alpha1 - alpha2;
tmp[bfcur] = one;
for (int i=0;i<alpha1;i++)
{
tmp[bfcur] *= b1;
}
for (int i=0;i<alpha2;i++)
{
tmp[bfcur] *= b2;
}
for (int i=0;i<alpha3;i++)
{
tmp[bfcur] *= b3;
}
for (int i=2;i<=order();i++)
{
tmp[bfcur] *= (double) i;
}
for (int i=2;i<=alpha1;i++)
{
tmp[bfcur] /= (double) i;
}
for (int i=2;i<=alpha2;i++)
{
tmp[bfcur] /= (double) i;
}
for (int i=2;i<=alpha3;i++)
{
tmp[bfcur] /= (double) i;
}
bfcur++;
}
}
}
for (int i=0; i<tmp.length(); i++)
{
if (deriv.order()==0) result[i] = tmp[i].value();
else
result[i] = tmp[i].gradient()[deriv.firstOrderDirection()];
}
}