mirror of
https://github.com/NGSolve/netgen.git
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734 lines
19 KiB
C++
734 lines
19 KiB
C++
#ifndef FILE_AUTODIFFDIFF
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#define FILE_AUTODIFFDIFF
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/**************************************************************************/
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/* File: autodiffdiff.hpp */
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/* Author: Joachim Schoeberl */
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/* Date: 13. June. 05 */
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/**************************************************************************/
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namespace ngcore
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{
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using ngcore::IfPos;
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// Automatic second differentiation datatype
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/**
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Datatype for automatic differentiation. Contains function value,
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D first derivatives, and D*D second derivatives. Algebraic operations are
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overloaded by using product-rule etc. etc.
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**/
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template <int D, typename SCAL = double>
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class AutoDiffDiff
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{
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SCAL val;
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SCAL dval[D?D:1];
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SCAL ddval[D?D*D:1];
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public:
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typedef AutoDiffDiff<D, SCAL> TELEM;
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/// elements are undefined
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AutoDiffDiff () throw() { ; }
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/// copy constructor
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AutoDiffDiff (const AutoDiffDiff & ad2) throw()
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{
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val = ad2.val;
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for (int i = 0; i < D; i++)
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dval[i] = ad2.dval[i];
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for (int i = 0; i < D*D; i++)
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ddval[i] = ad2.ddval[i];
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}
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/// initial object with constant value
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AutoDiffDiff (SCAL aval) throw()
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{
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val = aval;
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for (int i = 0; i < D; i++)
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dval[i] = 0;
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for (int i = 0; i < D*D; i++)
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ddval[i] = 0;
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}
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/// initial object with value and derivative
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AutoDiffDiff (const AutoDiff<D, SCAL> & ad2) throw()
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{
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val = ad2.Value();
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for (int i = 0; i < D; i++)
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dval[i] = ad2.DValue(i);
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for (int i = 0; i < D*D; i++)
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ddval[i] = 0;
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}
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/// init object with (val, e_diffindex)
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AutoDiffDiff (SCAL aval, int diffindex) throw()
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{
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val = aval;
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for (int i = 0; i < D; i++)
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dval[i] = 0;
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for (int i = 0; i < D*D; i++)
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ddval[i] = 0;
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dval[diffindex] = 1;
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}
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NETGEN_INLINE AutoDiffDiff (SCAL aval, const SCAL * grad)
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{
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val = aval;
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LoadGradient (grad);
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for (int i = 0; i < D*D; i++)
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ddval[i] = 0;
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}
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NETGEN_INLINE AutoDiffDiff (SCAL aval, const SCAL * grad, const SCAL * hesse)
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{
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val = aval;
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LoadGradient (grad);
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LoadHessian (hesse);
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}
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/// assign constant value
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AutoDiffDiff & operator= (SCAL aval) throw()
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{
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val = aval;
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for (int i = 0; i < D; i++)
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dval[i] = 0;
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for (int i = 0; i < D*D; i++)
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ddval[i] = 0;
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return *this;
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}
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NETGEN_INLINE void StoreGradient (SCAL * p) const
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{
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for (int i = 0; i < D; i++)
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p[i] = dval[i];
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}
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NETGEN_INLINE void LoadGradient (const SCAL * p)
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{
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for (int i = 0; i < D; i++)
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dval[i] = p[i];
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}
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NETGEN_INLINE void StoreHessian (SCAL * p) const
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{
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for (int i = 0; i < D*D; i++)
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p[i] = ddval[i];
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}
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NETGEN_INLINE void LoadHessian (const SCAL * p)
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{
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for (int i = 0; i < D*D; i++)
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ddval[i] = p[i];
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}
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/// returns value
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SCAL Value() const throw() { return val; }
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/// returns partial derivative
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SCAL DValue (int i) const throw() { return dval[i]; }
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AutoDiff<D,SCAL> DValueAD (int i) const
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{
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AutoDiff<D,SCAL> r(dval[i]);
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for (int j = 0; j < D; j++)
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r.DValue(j) = ddval[i*D+j];
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return r;
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}
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/// returns partial derivative
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SCAL DDValue (int i) const throw() { return ddval[i]; }
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/// returns partial derivative
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SCAL DDValue (int i, int j) const throw() { return ddval[i*D+j]; }
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/// access value
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SCAL & Value() throw() { return val; }
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/// accesses partial derivative
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SCAL & DValue (int i) throw() { return dval[i]; }
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/// accesses partial derivative
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SCAL & DDValue (int i) throw() { return ddval[i]; }
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/// accesses partial derivative
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SCAL & DDValue (int i, int j) throw() { return ddval[i*D+j]; }
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explicit operator AutoDiff<D,SCAL> () const
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{ return AutoDiff<D,SCAL> (val, &dval[0]); }
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/// add autodiffdiff object
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AutoDiffDiff<D, SCAL> & operator+= (const AutoDiffDiff<D, SCAL> & y) throw()
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{
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val += y.val;
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for (int i = 0; i < D; i++)
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dval[i] += y.dval[i];
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for (int i = 0; i < D*D; i++)
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ddval[i] += y.ddval[i];
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return *this;
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}
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/// subtract autodiffdiff object
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AutoDiffDiff<D, SCAL> & operator-= (const AutoDiffDiff<D, SCAL> & y) throw()
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{
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val -= y.val;
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for (int i = 0; i < D; i++)
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dval[i] -= y.dval[i];
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for (int i = 0; i < D*D; i++)
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ddval[i] -= y.ddval[i];
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return *this;
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}
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/// multiply with autodiffdiff object
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AutoDiffDiff<D, SCAL> & operator*= (const AutoDiffDiff<D, SCAL> & y) throw()
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{
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for (int i = 0; i < D*D; i++)
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ddval[i] = val * y.ddval[i] + y.val * ddval[i];
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for (int i = 0; i < D; i++)
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for (int j = 0; j < D; j++)
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ddval[i*D+j] += dval[i] * y.dval[j] + dval[j] * y.dval[i];
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for (int i = 0; i < D; i++)
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{
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dval[i] *= y.val;
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dval[i] += val * y.dval[i];
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}
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val *= y.val;
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return *this;
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}
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/// multiply with scalar
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AutoDiffDiff<D, SCAL> & operator*= (const SCAL & y) throw()
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{
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for ( int i = 0; i < D*D; i++ )
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ddval[i] *= y;
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for (int i = 0; i < D; i++)
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dval[i] *= y;
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val *= y;
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return *this;
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}
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/// divide by scalar
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AutoDiffDiff<D, SCAL> & operator/= (const SCAL & y) throw()
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{
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SCAL iy = 1.0 / y;
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for ( int i = 0; i < D*D; i++ )
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ddval[i] *= iy;
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for (int i = 0; i < D; i++)
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dval[i] *= iy;
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val *= iy;
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return *this;
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}
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/// same value
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bool operator== (SCAL val2) throw()
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{
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return val == val2;
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}
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/// different values
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bool operator!= (SCAL val2) throw()
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{
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return val != val2;
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}
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/// less
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bool operator< (SCAL val2) throw()
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{
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return val < val2;
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}
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/// greater
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bool operator> (SCAL val2) throw()
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{
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return val > val2;
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}
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};
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//@{ AutoDiff helper functions.
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/// Prints AudoDiffDiff
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template<int D, typename SCAL>
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inline ostream & operator<< (ostream & ost, const AutoDiffDiff<D, SCAL> & x)
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{
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ost << x.Value() << ", D = ";
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for (int i = 0; i < D; i++)
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ost << x.DValue(i) << " ";
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ost << ", DD = ";
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for (int i = 0; i < D*D; i++)
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ost << x.DDValue(i) << " ";
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return ost;
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}
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///
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template<int D, typename SCAL>
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inline AutoDiffDiff<D, SCAL> operator+ (const AutoDiffDiff<D, SCAL> & x, const AutoDiffDiff<D, SCAL> & y) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value () = x.Value()+y.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = x.DValue(i) + y.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = x.DDValue(i) + y.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL>
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inline AutoDiffDiff<D, SCAL> operator- (const AutoDiffDiff<D, SCAL> & x, const AutoDiffDiff<D, SCAL> & y) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = x.Value()-y.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = x.DValue(i) - y.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = x.DDValue(i) - y.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator+ (SCAL2 x, const AutoDiffDiff<D, SCAL> & y) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = x+y.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = y.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = y.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator+ (const AutoDiffDiff<D, SCAL> & y, SCAL2 x) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = x+y.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = y.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = y.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL>
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inline AutoDiffDiff<D, SCAL> operator- (const AutoDiffDiff<D, SCAL> & x) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = -x.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = -x.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = -x.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator- (const AutoDiffDiff<D, SCAL> & x, SCAL2 y) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = x.Value()-y;
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for (int i = 0; i < D; i++)
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res.DValue(i) = x.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = x.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator- (SCAL2 x, const AutoDiffDiff<D, SCAL> & y) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = x-y.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = -y.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = -y.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator* (SCAL2 x, const AutoDiffDiff<D, SCAL> & y) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = x*y.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = x*y.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = x*y.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator* (const AutoDiffDiff<D, SCAL> & y, SCAL2 x) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = x*y.Value();
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for (int i = 0; i < D; i++)
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res.DValue(i) = x*y.DValue(i);
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for (int i = 0; i < D*D; i++)
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res.DDValue(i) = x*y.DDValue(i);
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return res;
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}
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///
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template<int D, typename SCAL>
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inline AutoDiffDiff<D, SCAL> operator* (const AutoDiffDiff<D, SCAL> & x, const AutoDiffDiff<D, SCAL> & y) throw()
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL hx = x.Value();
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SCAL hy = y.Value();
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res.Value() = hx*hy;
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for (int i = 0; i < D; i++)
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res.DValue(i) = hx*y.DValue(i) + hy*x.DValue(i);
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for (int i = 0; i < D; i++)
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for (int j = 0; j < D; j++)
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res.DDValue(i,j) = hx * y.DDValue(i,j) + hy * x.DDValue(i,j)
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+ x.DValue(i) * y.DValue(j) + x.DValue(j) * y.DValue(i);
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return res;
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}
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template<int D, typename SCAL>
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inline AutoDiffDiff<D, SCAL> Inv (const AutoDiffDiff<D, SCAL> & x)
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{
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AutoDiffDiff<D, SCAL> res(1.0 / x.Value());
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for (int i = 0; i < D; i++)
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res.DValue(i) = -x.DValue(i) / (x.Value() * x.Value());
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SCAL fac1 = 2/(x.Value()*x.Value()*x.Value());
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SCAL fac2 = 1/sqr(x.Value());
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for (int i = 0; i < D; i++)
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for (int j = 0; j < D; j++)
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res.DDValue(i,j) = fac1*x.DValue(i)*x.DValue(j) - fac2*x.DDValue(i,j);
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return res;
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}
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template<int D, typename SCAL>
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inline AutoDiffDiff<D, SCAL> operator/ (const AutoDiffDiff<D, SCAL> & x, const AutoDiffDiff<D, SCAL> & y)
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{
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return x * Inv (y);
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}
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator/ (const AutoDiffDiff<D, SCAL> & x, SCAL2 y)
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{
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return (1/y) * x;
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}
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template<int D, typename SCAL, typename SCAL2,
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typename std::enable_if<std::is_convertible<SCAL2,SCAL>::value, int>::type = 0>
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inline AutoDiffDiff<D, SCAL> operator/ (SCAL2 x, const AutoDiffDiff<D, SCAL> & y)
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{
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return x * Inv(y);
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}
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template<int D, typename SCAL>
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inline AutoDiffDiff<D, SCAL> sqrt (const AutoDiffDiff<D, SCAL> & x)
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = sqrt(x.Value());
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for (int j = 0; j < D; j++)
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res.DValue(j) = IfZero(x.DValue(j),SCAL{0.},0.5 / res.Value() * x.DValue(j));
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for (int i = 0; i < D; i++)
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for (int j = 0; j < D; j++)
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res.DDValue(i,j) = IfZero(x.DDValue(i,j)+x.DValue(i) * x.DValue(j),SCAL{0.},0.5/res.Value() * x.DDValue(i,j) - 0.25 / (x.Value()*res.Value()) * x.DValue(i) * x.DValue(j));
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return res;
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}
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// df(u)/dx = exp(x) * du/dx
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// d^2 f(u) / dx^2 = exp(x) * (du/dx)^2 + exp(x) * d^2u /dx^2
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> exp (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = exp(x.Value());
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for (int k = 0; k < D; k++)
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res.DValue(k) = x.DValue(k) * res.Value();
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = (x.DValue(k) * x.DValue(l)+x.DDValue(k,l)) * res.Value();
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return res;
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}
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using std::pow;
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D,SCAL> pow (AutoDiffDiff<D,SCAL> x, AutoDiffDiff<D,SCAL> y )
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{
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return exp(log(x)*y);
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> log (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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res.Value() = log(x.Value());
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SCAL xinv = 1.0/x.Value();
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for (int k = 0; k < D; k++)
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res.DValue(k) = x.DValue(k) * xinv;
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = -xinv*xinv*x.DValue(k) * x.DValue(l) + xinv * x.DDValue(k,l);
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return res;
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> sin (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL s = sin(x.Value());
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SCAL c = cos(x.Value());
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res.Value() = s;
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for (int k = 0; k < D; k++)
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res.DValue(k) = x.DValue(k) * c;
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = -s * x.DValue(k) * x.DValue(l) + c * x.DDValue(k,l);
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return res;
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> cos (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL s = sin(x.Value());
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SCAL c = cos(x.Value());
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res.Value() = c;
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for (int k = 0; k < D; k++)
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res.DValue(k) = -s * x.DValue(k);
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = -c * x.DValue(k) * x.DValue(l) - s * x.DDValue(k,l);
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return res;
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> tan (AutoDiffDiff<D, SCAL> x)
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{ return sin(x) / cos(x); }
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> atan (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL a = atan(x.Value());
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res.Value() = a;
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for (int k = 0; k < D; k++)
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res.DValue(k) = x.DValue(k)/(1+x.Value()*x.Value()) ;
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = -2*x.Value()/((1+x.Value()*x.Value())*(1+x.Value()*x.Value())) * x.DValue(k) * x.DValue(l) + x.DDValue(k,l)/(1+x.Value()*x.Value());
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return res;
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> atan2 (AutoDiffDiff<D, SCAL> x,AutoDiffDiff<D, SCAL> y)
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL a = atan2(x.Value(), y.Value());
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res.Value() = a;
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for (int k = 0; k < D; k++)
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res.DValue(k) = (x.Value()*y.DValue(k)-y.Value()*x.DValue(k))/(y.Value()*y.Value()+x.Value()*x.Value());
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = (x.DValue(k)*y.DValue(l)+x.Value()*y.DDValue(l,k) - y.DValue(k)*x.DValue(l) - y.Value()*x.DDValue(l,k))/(y.Value()*y.Value()+x.Value()*x.Value()) - 2 * (x.Value()*y.DValue(k)-y.Value()*x.DValue(k)) * (x.Value()*x.DValue(k) + y.Value()*y.DValue(k))/( (y.Value()*y.Value()+x.Value()*x.Value()) * (y.Value()*y.Value()+x.Value()*x.Value()) );
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return res;
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}
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using std::acos;
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D,SCAL> acos (AutoDiffDiff<D,SCAL> x)
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{
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AutoDiffDiff<D,SCAL> res;
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SCAL a = acos(x.Value());
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res.Value() = a;
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auto omaa = 1-x.Value()*x.Value();
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auto s = sqrt(omaa);
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SCAL da = -1 / s;
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SCAL dda = -x.Value() / (s*omaa);
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for (int k = 0; k < D; k++)
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res.DValue(k) = x.DValue(k)*da;
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = dda * x.DValue(k) * x.DValue(l) + da * x.DDValue(k,l);
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return res;
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}
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using std::acos;
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D,SCAL> asin (AutoDiffDiff<D,SCAL> x)
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{
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AutoDiffDiff<D,SCAL> res;
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SCAL a = asin(x.Value());
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res.Value() = a;
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auto omaa = 1-x.Value()*x.Value();
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auto s = sqrt(omaa);
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SCAL da = 1 / s;
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SCAL dda = x.Value() / (s*omaa);
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for (int k = 0; k < D; k++)
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res.DValue(k) = x.DValue(k)*da;
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = dda * x.DValue(k) * x.DValue(l) + da * x.DDValue(k,l);
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return res;
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> sinh (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL sh = sinh(x.Value());
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SCAL ch = cosh(x.Value());
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res.Value() = sh;
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for (int k = 0; k < D; k++)
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res.DValue(k) = x.DValue(k) * ch;
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = sh * x.DValue(k) * x.DValue(l) + ch * x.DDValue(k,l);
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return res;
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> cosh (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL sh = sinh(x.Value());
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SCAL ch = cosh(x.Value());
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res.Value() = ch;
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for (int k = 0; k < D; k++)
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res.DValue(k) = sh * x.DValue(k);
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = ch * x.DValue(k) * x.DValue(l) + sh * x.DDValue(k,l);
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return res;
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D, SCAL> erf (AutoDiffDiff<D, SCAL> x)
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{
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AutoDiffDiff<D, SCAL> res;
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SCAL derf = 2. / sqrt(M_PI) * exp(- x.Value() * x.Value());
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res.Value() = erf(x.Value());
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for (int k = 0; k < D; k++)
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res.DValue(k) = - derf * x.DValue(k);
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for (int k = 0; k < D; k++)
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for (int l = 0; l < D; l++)
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res.DDValue(k,l) = derf * (x.DDValue(k, l) - 2 * x.Value() * x.DValue(k) * x.DValue(l));
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return res;
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}
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using std::floor;
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template<int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D,SCAL> floor (const AutoDiffDiff<D,SCAL> & x)
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{
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return floor(x.Value());
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}
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using std::ceil;
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template<int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D,SCAL> ceil (const AutoDiffDiff<D,SCAL> & x)
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{
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return ceil(x.Value());
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}
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template <int D, typename SCAL, typename TB, typename TC>
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auto IfPos (AutoDiffDiff<D,SCAL> a, TB b, TC c) -> decltype(IfPos (a.Value(), b, c))
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{
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return IfPos (a.Value(), b, c);
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}
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template <int D, typename SCAL>
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NETGEN_INLINE AutoDiffDiff<D,SCAL> IfPos (SCAL /* SIMD<double> */ a, AutoDiffDiff<D,SCAL> b, AutoDiffDiff<D,SCAL> c)
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{
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AutoDiffDiff<D,SCAL> res;
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res.Value() = IfPos (a, b.Value(), c.Value());
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for (int j = 0; j < D; j++)
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{
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res.DValue(j) = IfPos (a, b.DValue(j), c.DValue(j));
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res.DDValue(j) = IfPos (a, b.DDValue(j), c.DDValue(j));
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}
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return res;
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}
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template <int D, typename SCAL, typename TC>
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NETGEN_INLINE AutoDiffDiff<D,SCAL> IfPos (SCAL /* SIMD<double> */ a, AutoDiffDiff<D,SCAL> b, TC c)
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{
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return IfPos (a, b, AutoDiffDiff<D,SCAL> (c));
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}
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//@}
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}
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namespace ngbla
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{
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template <typename T> struct is_scalar_type;
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template <int D, typename T>
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struct is_scalar_type<ngcore::AutoDiffDiff<D,T>> { static constexpr bool value = true; };
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// not meaningful for AutoDiff<D,Complex>, since this is
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// not (complex) differentiable anyway
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template<int D, typename SCAL>
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inline auto L2Norm2 (const ngcore::AutoDiffDiff<D,SCAL> & x)
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{
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return x*x;
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}
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}
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#endif
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