Sacado_radops2.cpp

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// @HEADER
// ***********************************************************************
//
//                           Sacado Package
//                 Copyright (2007) Sandia Corporation
//
// Under the terms of Contract DE-AC04-94AL85000 with Sandia Corporation,
// the U.S. Government retains certain rights in this software.
//
// This library is free software; you can redistribute it and/or modify
// it under the terms of the GNU Lesser General Public License as
// published by the Free Software Foundation; either version 2.1 of the
// License, or (at your option) any later version.
//
// This library is distributed in the hope that it will be useful, but
// WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301
// USA
// Questions? Contact David M. Gay (dmgay@sandia.gov) or Eric T. Phipps
// (etphipp@sandia.gov).
//
// ***********************************************************************
// @HEADER
 
// Support routines for Hessian-vector products via the RAD package
// (Reverse Automatic Differentiation) -- a package specialized for
// function and gradient evaluations, and herewith extended for Hessian-
// vector products.  This specialization is for several Hessian-vector
// products at one point, where the vectors are determined on the fly,
// e.g., by the conjugate-gradient algorithm.  Where the vectors are known
// in advance, nestings of RAD and FAD might be more efficient.
// RAD Hessian-vector product extension written in 2007 by David M. Gay at
// Sandia National Labs, Albuquerque, NM.
 
#include "Sacado_rad2.hpp"
 
#ifndef SACADO_NO_NAMESPACE
namespace Sacado {
namespace Rad2d { // "2" for 2nd derivatives, "d" for "double"
#endif
 
#ifdef RAD_AUTO_AD_Const
ADvari *ADvari::First_ADvari, **ADvari::Last_ADvari = &ADvari::First_ADvari;
#undef RAD_DEBUG_BLOCKKEEP
#else
#ifdef RAD_DEBUG_BLOCKKEEP
#if !(RAD_DEBUG_BLOCKKEEP > 0)
#undef RAD_DEBUG_BLOCKKEEP
#else
extern "C" void _uninit_f2c(void *x, int type, long len);
#define TYDREAL 5
static ADmemblock *rad_Oldcurmb;
static int rad_busy_blocks;
#endif /*RAD_DEBUG_BLOCKKEEP > 0*/
#endif /*RAD_DEBUG_BLOCKKEEP*/
#endif /*RAD_AUTO_AD_Const*/
 
Derp *Derp::LastDerp = 0;
ADcontext ADvari::adc;
CADcontext ConstADvari::cadc;
ConstADvari *ConstADvari::lastcad;
static int rad_need_reinit;
#ifdef RAD_DEBUG_BLOCKKEEP
static size_t rad_mleft_save;
#endif
 
const double CADcontext::One = 1., CADcontext::negOne = -1.;
 
ADcontext::ADcontext()
{
  First.next = 0;
  Busy = &First;
  Free = 0;
  Mbase = (char*)First.memblk;
  Mleft = sizeof(First.memblk);
  Aibusy = &AiFirst;
  Aifree = 0;
  Ainext = AiFirst.pADvari;
  AiFirst.next = AiFirst.prev = 0;
  AiFirst.limit = Ailimit = AiFirst.pADvari + ADvari_block::Gulp;
  }
 
 void*
ADcontext::new_ADmemblock(size_t len)
{
  ADmemblock *mb, *mb0, *mb1, *mbf, *x;
  ADvari_block *b;
#ifdef RAD_AUTO_AD_Const
  ADvari *a, *anext;
  IndepADvar *v;
#endif /* RAD_AUTO_AD_Const */
 
  if (rad_need_reinit && this == &ADvari::adc) {
    rad_need_reinit = 0;
    Derp::LastDerp = 0;
    Aibusy = b = &AiFirst;
    Aifree = b->next;
    b->next = b->prev = 0;
    Ailimit = b->limit = (Ainext = b->pADvari) + ADvari_block::Gulp;
#ifdef RAD_DEBUG_BLOCKKEEP
    Mleft = rad_mleft_save;
    if (Mleft < sizeof(First.memblk))
      _uninit_f2c(Mbase + Mleft, TYDREAL,
       (sizeof(First.memblk) - Mleft)/sizeof(double));
    if ((mb = Busy->next)) {
      if (!(mb0 = rad_Oldcurmb))
        mb0 = &First;
      for(;; mb = mb->next) {
        _uninit_f2c(mb->memblk, TYDREAL,
          sizeof(First.memblk)/sizeof(double));
        if (mb == mb0)
          break;
        }
      }
    rad_Oldcurmb = Busy;
    if (rad_busy_blocks >= RAD_DEBUG_BLOCKKEEP) {
      rad_busy_blocks = 0;
      rad_Oldcurmb = 0;
      mb0 = &First;
      mbf =  Free;
      for(mb = Busy; mb != mb0; mb = mb1) {
        mb1 = mb->next;
        mb->next = mbf;
        mbf = mb;
        }
      Free = mbf;
      Busy = mb;
      Mbase = (char*)First.memblk;
      Mleft = sizeof(First.memblk);
      }
 
#else /* !RAD_DEBUG_BLOCKKEEP */
 
    mb0 = &ADvari::adc.First;
    mbf =  ADvari::adc.Free;
    for(mb = ADvari::adc.Busy; mb != mb0; mb = mb1) {
      mb1 = mb->next;
      mb->next = mbf;
      mbf = mb;
      }
    ADvari::adc.Free = mbf;
    ADvari::adc.Busy = mb;
    ADvari::adc.Mbase = (char*)ADvari::adc.First.memblk;
    ADvari::adc.Mleft = sizeof(ADvari::adc.First.memblk);
#ifdef RAD_AUTO_AD_Const
    *ADvari::Last_ADvari = 0;
    ADvari::Last_ADvari = &ADvari::First_ADvari;
    if ((anext = ADvari::First_ADvari)) {
      while((a = anext)) {
        anext = a->Next;
        if ((v = a->padv))
          v->cv = new ADvari(v, a->Val);
        }
      }
#endif /*RAD_AUTO_AD_Const*/
#endif /* RAD_DEBUG_BLOCKKEEP */
    if (Mleft >= len)
      return Mbase + (Mleft -= len);
    }
 
  if ((x = Free))
    Free = x->next;
  else
    x = new ADmemblock;
#ifdef RAD_DEBUG_BLOCKKEEP
  rad_busy_blocks++;
#endif
  x->next = Busy;
  Busy = x;
  return (Mbase = (char*)x->memblk) +
    (Mleft = sizeof(First.memblk) - len);
  }
 
 void
ADcontext::new_ADvari_block()
{
  ADvari_block *ob, *nb;
  ob = Aibusy;
  ob->limit = Ailimit;  // should be unnecessary, but harmless
  if ((nb = Aifree))
    Aifree = nb->next;
  else
    nb = new ADvari_block;
  Aibusy = Aibusy->next = nb;
  nb->limit = Ailimit = (Ainext = nb->pADvari) + ADvari_block::Gulp;
  ob->next = nb;
  nb->prev = ob;
  nb->next = 0;
  }
 
 void
ADcontext::Gradcomp(int wantgrad)
{
  Derp *d;
 
  if (rad_need_reinit && wantgrad) {
    for(d = Derp::LastDerp; d; d = d->next)
      d->c->aval = 0;
    }
  else {
    rad_need_reinit = 1;
#ifdef RAD_DEBUG_BLOCKKEEP
    rad_mleft_save = ADvari::adc.Mleft;
#endif
    ADvari::adc.Mleft = 0;
    }
 
  if ((d = Derp::LastDerp) && wantgrad) {
    d->b->aval = 1;
    do d->c->aval += *d->a * d->b->aval;
    while((d = d->next));
    }
  }
 
 void
ADcontext::Weighted_Gradcomp(int n, ADvar **v, double *w)
{
  Derp *d;
  int i;
 
  if (rad_need_reinit) {
    for(d = Derp::LastDerp; d; d = d->next)
      d->c->aval = 0;
    }
  else {
    rad_need_reinit = 1;
#ifdef RAD_DEBUG_BLOCKKEEP
    rad_mleft_save = ADvari::adc.Mleft;
#endif
    ADvari::adc.Mleft = 0;
    }
  if ((d = Derp::LastDerp)) {
    for(i = 0; i < n; i++)
      v[i]->cv->aval = w[i];
    do d->c->aval += *d->a * d->b->aval;
    while((d = d->next));
    }
  }
 
#ifdef RAD_AUTO_AD_Const
 
IndepADvar::IndepADvar(double d)
{
  cv = new ADvari(this, d);
  }
 
IndepADvar::IndepADvar(int d)
{
  cv = new ADvari(this, (double)d);
  }
 
IndepADvar::IndepADvar(long d)
{
  cv = new ADvari(this, (double)d);
  }
 
#else 
IndepADvar::IndepADvar(double d)
{
  ADvari *x = new ADvari(Hv_const, d);
  cv = x;
  }
 
IndepADvar::IndepADvar(int d)
{
  ADvari *x = new ADvari(Hv_const, (double)d);
  cv = x;
  }
 
IndepADvar::IndepADvar(long d)
{
  ADvari *x = new ADvari(Hv_const, (double)d);
  cv = x;
  }
 
#endif /*RAD_AUTO_AD_Const*/
 
 void
ADvar::ADvar_ctr(double d)
{
#ifdef RAD_AUTO_AD_Const
  ADvari *x = new ADvari(this, d);
#else
  ADvari *x = ConstADvari::cadc.fpval_implies_const
    ? new ConstADvari(d)
    : new ADvari(Hv_const, d);
#endif
  cv = x;
  }
 
ConstADvar::ConstADvar()
{
  ConstADvari *x = new ConstADvari(0.);
  cv = x;
  }
 
 void
ConstADvar::ConstADvar_ctr(double d)
{
  ConstADvari *x = new ConstADvari(d);
  cv = x;
  }
ConstADvar::ConstADvar(const ADvari &x)
{
  ConstADvari *y = new ConstADvari(x.Val);
  new Derp(&CADcontext::One, y, &x); /* for side effect; value not used */
  cv = y;
  }
 
 void
IndepADvar::AD_Const(const IndepADvar &v)
{
  ConstADvari *ncv = new ConstADvari(v.val());
  ((IndepADvar*)&v)->cv = ncv;
  }
 
 void
ConstADvari::aval_reset()
{
  ConstADvari *x = ConstADvari::lastcad;
  while(x) {
    x->aval = 0;
    x = x->prevcad;
    }
  }
 
#ifdef RAD_AUTO_AD_Const
 
ADvari::ADvari(const IndepADvar *x, double d): Val(d), aval(0.)
{
  opclass = Hv_const;
  *Last_ADvari = this;
  Last_ADvari = &Next;
  padv = (IndepADvar*)x;
  }
 
ADvar1::ADvar1(const IndepADvar *x, const IndepADvar &y):
  ADvari(Hv_copy, y.cv->Val), d(&CADcontext::One, this, y.cv)
{
  *ADvari::Last_ADvari = this;
  ADvari::Last_ADvari = &Next;
  padv = (IndepADvar*)x;
  }
 
ADvar1::ADvar1(const IndepADvar *x, const ADvari &y):
  ADvari(Hv_copy, y.Val), d(&CADcontext::One, this, &y)
{
  *ADvari::Last_ADvari = this;
  ADvari::Last_ADvari = &Next;
  padv = (IndepADvar*)x;
  }
 
#else
 
 ADvar&
ADvar_operatoreq(ADvar *This, const ADvari &x)
{ This->cv = new ADvar1(Hv_copy, x.Val, &CADcontext::One, (ADvari*)&x); return *This; }
 
 IndepADvar&
ADvar_operatoreq(IndepADvar *This, const ADvari &x)
{ This->cv = new ADvar1(Hv_copy, x.Val, &CADcontext::One, (ADvari*)&x); return *This; }
 
#endif /*RAD_AUTO_AD_Const*/
 
ADvar2q::ADvar2q(double val1, double Lp, double Rp,
    double LR, double R2, const ADvari *Lcv, const ADvari *Rcv):
      ADvar2(Hv_quotLR,val1,Lcv,&pL,Rcv,&pR),
      pL(Lp), pR(Rp), pLR(LR), pR2(R2) { }
 
ADvar2g::ADvar2g(double val1, double Lp, double Rp,
    double L2, double LR, double R2, const ADvari *Lcv, const ADvari *Rcv):
      ADvar2(Hv_binary,val1,Lcv,&pL,Rcv,&pR),
      pL(Lp), pR(Rp), pL2(L2), pLR(LR), pR2(R2) { }
 
 IndepADvar&
IndepADvar::operator=(double d)
{
#ifdef RAD_AUTO_AD_Const
  if (cv)
    cv->padv = 0;
  cv = new ADvari(this,d);
#else
  cv = new ADvari(Hv_const, d);
#endif
  return *this;
  }
 
 ADvar&
ADvar::operator=(double d)
{
#ifdef RAD_AUTO_AD_Const
  if (cv)
    cv->padv = 0;
  cv = new ADvari(this, d);
#else
  cv = ConstADvari::cadc.fpval_implies_const
    ? new ConstADvari(d)
    : new ADvari(Hv_const, d);
#endif
  return *this;
  }
 
 ADvari&
operator-(const ADvari &T) {
  return *(new ADvar1(Hv_negate, -T.Val, &CADcontext::negOne, &T));
  }
 
 ADvari&
operator+(const ADvari &L, const ADvari &R) {
  return *(new ADvar2(Hv_plusLR,L.Val + R.Val, &L, &CADcontext::One, &R, &CADcontext::One));
  }
 
 ADvar&
ADvar::operator+=(const ADvari &R) {
  ADvari *Lcv = cv;
#ifdef RAD_AUTO_AD_Const
  Lcv->padv = 0;
#endif
  cv = new ADvar2(Hv_plusLR, Lcv->Val + R.Val, Lcv, &CADcontext::One, &R, &CADcontext::One);
  return *this;
  }
 
 ADvari&
operator+(const ADvari &L, double R) {
  return *(new ADvar1(Hv_copy, L.Val + R, &CADcontext::One, &L));
  }
 
 ADvar&
ADvar::operator+=(double R) {
  ADvari *tcv = cv;
#ifdef RAD_AUTO_AD_Const
  tcv->padv = 0;
#endif
  cv = new ADvar1(Hv_copy, tcv->Val + R, &CADcontext::One, tcv);
  return *this;
  }
 
 ADvari&
operator+(double L, const ADvari &R) {
  return *(new ADvar1(Hv_copy, L + R.Val, &CADcontext::One, &R));
  }
 
 ADvari&
operator-(const ADvari &L, const ADvari &R) {
  return *(new ADvar2(Hv_minusLR,L.Val - R.Val, &L, &CADcontext::One, &R, &CADcontext::negOne));
  }
 
 ADvar&
ADvar::operator-=(const ADvari &R) {
  ADvari *Lcv = cv;
#ifdef RAD_AUTO_AD_Const
  Lcv->padv = 0;
#endif
  cv = new ADvar2(Hv_minusLR, Lcv->Val - R.Val, Lcv, &CADcontext::One, &R, &CADcontext::negOne);
  return *this;
  }
 
 ADvari&
operator-(const ADvari &L, double R) {
  return *(new ADvar1(Hv_copy, L.Val - R, &CADcontext::One, &L));
  }
 
 ADvar&
ADvar::operator-=(double R) {
  ADvari *tcv = cv;
#ifdef RAD_AUTO_AD_Const
  tcv->padv = 0;
#endif
  cv = new ADvar1(Hv_copy, tcv->Val - R, &CADcontext::One, tcv);
  return *this;
  }
 
 ADvari&
operator-(double L, const ADvari &R) {
  return *(new ADvar1(Hv_negate, L - R.Val, &CADcontext::negOne, &R));
  }
 
 ADvari&
operator*(const ADvari &L, const ADvari &R) {
  return *(new ADvar2(Hv_timesLR, L.Val * R.Val, &L, &R.Val, &R, &L.Val));
  }
 
 ADvar&
ADvar::operator*=(const ADvari &R) {
  ADvari *Lcv = cv;
#ifdef RAD_AUTO_AD_Const
  Lcv->padv = 0;
#endif
  cv = new ADvar2(Hv_timesLR, Lcv->Val * R.Val, Lcv, &R.Val, &R, &Lcv->Val);
  return *this;
  }
 
 ADvari&
operator*(const ADvari &L, double R) {
  return *(new ADvar1s(L.Val * R, R, &L));
  }
 
 ADvar&
ADvar::operator*=(double R) {
  ADvari *Lcv = cv;
#ifdef RAD_AUTO_AD_Const
  Lcv->padv = 0;
#endif
  cv = new ADvar1s(Lcv->Val * R, R, Lcv);
  return *this;
  }
 
 ADvari&
operator*(double L, const ADvari &R) {
  return *(new ADvar1s(L * R.Val, L, &R));
  }
 
 ADvari&
operator/(const ADvari &L, const ADvari &R) {
  double Lv = L.Val, Rv = R.Val, pL = 1. / Rv, q = Lv/Rv, qpL = q*pL;
  return *(new ADvar2q(q, pL, -qpL, -pL*pL, 2.*pL*qpL, &L, &R));
  }
 
 ADvar&
ADvar::operator/=(const ADvari &R) {
  ADvari *Lcv = cv;
#ifdef RAD_AUTO_AD_Const
  Lcv->padv = 0;
#endif
  double Lv = Lcv->Val, Rv = R.Val, pL = 1. / Rv, q = Lv/Rv, qpL = q*pL;
  cv = new ADvar2q(q, pL, -qpL, -pL*pL, 2.*pL*qpL, Lcv, &R);
  return *this;
  }
 
 ADvari&
operator/(const ADvari &L, double R) {
  return *(new ADvar1s(L.Val / R, 1./R, &L));
  }
 
 ADvari&
operator/(double L, const ADvari &R) {
  double recip = 1. / R.Val;
  double q = L * recip;
  double d1 = -q*recip;
  return *(new ADvar1g(q, d1, -q*d1, &R));
  }
 
 ADvar&
ADvar::operator/=(double R) {
  ADvari *Lcv = cv;
#ifdef RAD_AUTO_AD_Const
  Lcv->padv = 0;
#endif
  cv = new ADvar1s(Lcv->Val / R, 1./R, Lcv);
  return *this;
  }
 
 ADvari&
acos(const ADvari &v) {
  double t = v.Val;
  double t1 = 1. - t*t;
  double d1 = -1. / sqrt(t1);
  return *(new ADvar1g(acos(t), d1, t*d1/t1, &v));
  }
 
 ADvari&
acosh(const ADvari &v) {
  double d1, t, t1, t2;
  t = v.Val;
  t1 = sqrt(t2 = t*t - 1.);
  d1 = 1./t1;
  return *(new ADvar1g(log(t + t1), d1, -t*d1/t2, &v));
  }
 
 ADvari&
asin(const ADvari &v) {
  double d1, t, t1;
  t = v.Val;
  d1 = 1. / sqrt(t1 = 1. - t*t);
  return *(new ADvar1g(asin(t), d1, t*d1/t1, &v));
  }
 
 ADvari&
asinh(const ADvari &v) {
  double d1, t, t1, t2, td;
  t = v.Val;
  t1 = sqrt(t2 = t*t + 1.);
  d1 = 1. / t1;
  td = 1.;
  if (t < 0.)
    td = -1.;
  return *(new ADvar1g(td*log(t*td + t1), d1, -(t/t2)*d1, &v));
  }
 
 ADvari&
atan(const ADvari &v) {
  double d1, t;
  t = v.Val;
  d1 = 1./(1. + t*t);
  return *(new ADvar1g(atan(t), d1, -(t+t)*d1*d1, &v));
  }
 
 ADvari&
atanh(const ADvari &v) {
  double t = v.Val;
  double d1 = 1./(1. - t*t);
  return *(new ADvar1g(0.5*log((1.+t)/(1.-t)), d1, (t+t)*d1*d1, &v));
  }
 
 ADvari&
max(const ADvari &L, const ADvari &R) {
  const ADvari &x = L.Val >= R.Val ? L : R;
  return *(new ADvar1(Hv_copy, x.Val, &CADcontext::One, &x));
  }
 
 ADvari&
max(double L, const ADvari &R) {
  if (L >= R.Val)
    return *(new ADvari(Hv_const, L));
  return *(new ADvar1(Hv_copy, R.Val, &CADcontext::One, &R));
  }
 
 ADvari&
max(const ADvari &L, double R) {
  if (L.Val >= R)
    return *(new ADvar1(Hv_copy, L.Val, &CADcontext::One, &L));
  return *(new ADvari(Hv_const, R));
  }
 
 ADvari&
min(const ADvari &L, const ADvari &R) {
  const ADvari &x = L.Val <= R.Val ? L : R;
  return *(new ADvar1(Hv_copy, x.Val, &CADcontext::One, &x));
  }
 
 ADvari&
min(double L, const ADvari &R) {
  if (L <= R.Val)
    return *(new ADvari(Hv_const, L));
  return *(new ADvar1(Hv_copy, R.Val, &CADcontext::One, &R));
  }
 
 ADvari&
min(const ADvari &L, double R) {
  if (L.Val <= R)
    return *(new ADvar1(Hv_copy, L.Val, &CADcontext::One, &L));
  return *(new ADvari(Hv_const, R));
  }
 
 ADvari&
atan2(const ADvari &L, const ADvari &R) {
  double x = L.Val, y = R.Val;
  double x2 = x*x, y2 = y*y;
  double t = 1./(x2 + y2);
  double t2 = t*t;
  double R2 = 2.*t2*x*y;
  return *(new ADvar2g(atan2(x,y), y*t, -x*t, -R2, t2*(x2 - y2), R2, &L, &R));
  }
 
 ADvari&
atan2(double x, const ADvari &R) {
  double y = R.Val;
  double x2 = x*x, y2 = y*y;
  double t = 1./(x2 + y2);
  return *(new ADvar1g(atan2(x,y), -x*t, 2.*t*t*x*y, &R));
  }
 
 ADvari&
atan2(const ADvari &L, double y) {
  double x = L.Val;
  double x2 = x*x, y2 = y*y;
  double t = 1./(x2 + y2);
  return *(new ADvar1g(atan2(x,y), y*t, -2.*t*t*x*y, &L));
  }
 
 ADvari&
cos(const ADvari &v) {
  double t = cos(v.Val);
  return *(new ADvar1g(t, -sin(v.Val), -t, &v));
  }
 
 ADvari&
cosh(const ADvari &v) {
  double t = cosh(v.Val);
  return *(new ADvar1g(t, sinh(v.Val), t, &v));
  }
 
 ADvari&
exp(const ADvari &v) {
  double t = exp(v.Val);
  return *(new ADvar1g(t, t, t, &v));
  }
 
 ADvari&
log(const ADvari &v) {
  double x = v.Val;
  double d1 = 1. / x;
  return *(new ADvar1g(log(x), d1, -d1*d1, &v));
  }
 
 ADvari&
log10(const ADvari &v) {
  static double num = 1. / log(10.);
  double x = v.Val, t = 1. / x;
  double d1 = num * t;
  return *(new ADvar1g(log10(x), d1, -d1*t, &v));
  }
 
 ADvari&
pow(const ADvari &L, const ADvari &R) {
  double x = L.Val, y = R.Val, t = pow(x,y);
  double xym1 = t/x;
  double xlog = log(x);
  double dx = y*xym1;
  double dy = t * xlog;
  return *(new ADvar2g(t, dx, dy, (y-1.)*dx/x, xym1*(1. + y*xlog), dy*xlog, &L, &R));
  }
 
 ADvari&
pow(double x, const ADvari &R) {
  double y = R.Val, t = pow(x,y);
  double xlog = log(x);
  double dy = t * xlog;
  return *(new ADvar1g(t, dy, dy*xlog, &R));
  }
 
 ADvari&
pow(const ADvari &L, double y) {
  double x = L.Val, t = pow(x,y);
  double dx = y*t/x;
  return *(new ADvar1g(t, dx, (y-1.)*dx/x, &L));
  }
 
 ADvari&
sin(const ADvari &v) {
  double t = sin(v.Val);
  return *(new ADvar1g(t, cos(v.Val), -t, &v));
  }
 
 ADvari&
sinh(const ADvari &v) {
  double t = sinh(v.Val);
  return *(new ADvar1g(t, cosh(v.Val), t, &v));
  }
 
 ADvari&
sqrt(const ADvari &v) {
  double t = sqrt(v.Val);
  double d1 = 0.5/t;
  return *(new ADvar1g(t, d1, -0.5*d1/v.Val, &v));
  }
 
 ADvari&
tan(const ADvari &v) {
  double d1, rv, t;
  rv = tan(v.Val);
  t = 1. / cos(v.Val);
  d1 = t*t;
  return *(new ADvar1g(rv, d1, (rv+rv)*d1, &v));
  }
 
 ADvari&
tanh(const ADvari &v) {
  double d1, rv, t;
  rv = tanh(v.Val);
  t = 1. / cosh(v.Val);
  d1 = t*t;
  return *(new ADvar1g(rv, d1, -(rv+rv)*d1, &v));
  }
 
 ADvari&
fabs(const ADvari &v) { // "fabs" is not the best choice of name,
      // but this name is used at Sandia.
  double t, p;
  p = 1;
  if ((t = v.Val) < 0) {
    t = -t;
    p = -p;
    }
  return *(new ADvar1g(t, p, 0., &v));
  }
 
 ADvari&
ADf1(double f, double g, double h, const ADvari &x) {
  return *(new ADvar1g(f, g, h, &x));
  }
 
 ADvari&
ADf2(double f, double gx, double gy, double hxx, double hxy, double hyy,
    const ADvari &x, const ADvari &y) {
  return *(new ADvar2g(f, gx, gy, hxx, hxy, hyy, &x, &y));
  }
 
 ADvarn::ADvarn(double val1, int n1, const ADvar *x, const double *g, const double *h):
    ADvari(Hv_nary,val1), n(n1)
{
  Derp *d1, *dlast;
  double *a1;
  int i, nh;
 
  a1 = G = (double*)ADvari::adc.Memalloc(n1*sizeof(*g));
  d1 = D =  (Derp*)ADvari::adc.Memalloc(n1*sizeof(Derp));
  dlast = Derp::LastDerp;
  for(i = 0; i < n1; i++, d1++) {
    d1->next = dlast;
    dlast = d1;
    a1[i] = g[i];
    d1->a = &a1[i];
    d1->b = this;
    d1->c = x[i].cv;
    }
  Derp::LastDerp = dlast;
  nh = (n1*(n1+1)) >> 1;
  a1 = H = (double*)ADvari::adc.Memalloc(nh * sizeof(*h));
  for(i = 0; i < nh; i++)
    a1[i] = h[i];
  }
 
 ADvari&
ADfn(double f, int n, const ADvar *x, const double *g, const double *h) {
  return *(new ADvarn(f, n, x, g, h));
  }
 
 void
ADcontext::Hvprod(int n, ADvar **x, double *v, double *hv)
{
  ADvari *a, *aL, *aR, **ap, **ape;
  ADvari_block *b, *b0;
  Derp *d;
  double aO, adO, *g, *h, *h0, t, tL, tR;
  int i, j, k, m;
  for(i = 0; i < n; i++) {
    a = x[i]->cv;
    a->dO = v[i];
    a->aO = a->adO = 0.;
    }
  ADvari::adc.Aibusy->limit = ADvari::adc.Ainext;
  a = 0;
  for(b0 = 0, b = &ADvari::adc.AiFirst; b; b0 = b, b = b->next) {
    ap = b->pADvari;
    ape = b->limit;
    while(ap < ape) {
      a = *ap++;
      a->aO = a->adO = 0.;
      switch(a->opclass) {
       case Hv_copy:
        a->dO = ((ADvar1*)a)->d.c->dO;
        break;
       case Hv_binary:
        a->dO =   ((ADvar2g*)a)->pL * ((ADvar2g*)a)->dL.c->dO
          + ((ADvar2g*)a)->pR * ((ADvar2g*)a)->dR.c->dO;
        break;
       case Hv_unary:
        a->dO = ((ADvar1g*)a)->pL * ((ADvar1g*)a)->d.c->dO;
        break;
       case Hv_negate:
        a->dO = -((ADvar1*)a)->d.c->dO;
        break;
       case Hv_plusLR:
        a->dO = ((ADvar2*)a)->dL.c->dO + ((ADvar2*)a)->dR.c->dO;
        break;
       case Hv_minusLR:
        a->dO = ((ADvar2*)a)->dL.c->dO - ((ADvar2*)a)->dR.c->dO;
        break;
       case Hv_timesL:
        a->dO = ((ADvar1s*)a)->pL * ((ADvar1s*)a)->d.c->dO;
        break;
       case Hv_timesLR:
        a->dO =   ((ADvar2*)a)->dR.c->Val * ((ADvar2*)a)->dL.c->dO
          + ((ADvar2*)a)->dL.c->Val * ((ADvar2*)a)->dR.c->dO;
        break;
       case Hv_quotLR:
        a->dO =   ((ADvar2q*)a)->pL * ((ADvar2q*)a)->dL.c->dO
          + ((ADvar2q*)a)->pR * ((ADvar2q*)a)->dR.c->dO;
       case Hv_const: // This case does not arise; the intent here
          // is to eliminate a red-herring compiler warning.
        break;
       case Hv_nary:
        d = ((ADvarn*)a)->D;
        m = ((ADvarn*)a)->n;
        g = ((ADvarn*)a)->G;
        t = 0.;
        for(i = 0; i < m; i++)
          t += g[i] * d[i].c->dO;
        a->dO = t;
       }
      }
    }
  if (a)
    a->adO = 1.;
  for(b = b0; b; b = b->prev) {
    ape = b->pADvari;
    ap = b->limit;
    while(ap > ape) {
      a = *--ap;
      aO = a->aO;
      adO = a->adO;
      switch(a->opclass) {
       case Hv_copy:
        aL = ((ADvar1*)a)->d.c;
        aL->aO += aO;
        aL->adO += adO;
        break;
       case Hv_binary:
        aL = ((ADvar2g*)a)->dL.c;
        aR = ((ADvar2g*)a)->dR.c;
        tL = adO*aL->dO;
        tR = adO*aR->dO;
        aL->aO += aO*((ADvar2g*)a)->pL
          + tL*((ADvar2g*)a)->pL2
          + tR*((ADvar2g*)a)->pLR;
        aR->aO += aO*((ADvar2g*)a)->pR
          + tL*((ADvar2g*)a)->pLR
          + tR*((ADvar2g*)a)->pR2;
        aL->adO += adO * ((ADvar2g*)a)->pL;
        aR->adO += adO * ((ADvar2g*)a)->pR;
        break;
       case Hv_unary:
        aL = ((ADvar1g*)a)->d.c;
        aL->aO += aO * ((ADvar1g*)a)->pL
          + adO * aL->dO * ((ADvar1g*)a)->pL2;
        aL->adO += adO * ((ADvar1g*)a)->pL;
        break;
       case Hv_negate:
        aL = ((ADvar1*)a)->d.c;
        aL->aO -= aO;
        aL->adO -= adO;
        break;
       case Hv_plusLR:
        aL = ((ADvar2*)a)->dL.c;
        aR = ((ADvar2*)a)->dR.c;
        aL->aO += aO;
        aL->adO += adO;
        aR->aO += aO;
        aR->adO += adO;
        break;
       case Hv_minusLR:
        aL = ((ADvar2*)a)->dL.c;
        aR = ((ADvar2*)a)->dR.c;
        aL->aO += aO;
        aL->adO += adO;
        aR->aO -= aO;
        aR->adO -= adO;
        break;
       case Hv_timesL:
        aL = ((ADvar1s*)a)->d.c;
        aL->aO += aO * (tL = ((ADvar1s*)a)->pL);
        aL->adO += adO * tL;
        break;
       case Hv_timesLR:
        aL = ((ADvar2*)a)->dL.c;
        aR = ((ADvar2*)a)->dR.c;
        aL->aO += aO * (tL = aR->Val) + adO*aR->dO;
        aR->aO += aO * (tR = aL->Val) + adO*aL->dO;
        aL->adO += adO * tL;
        aR->adO += adO * tR;
        break;
       case Hv_quotLR:
        aL = ((ADvar2q*)a)->dL.c;
        aR = ((ADvar2q*)a)->dR.c;
        tL = adO*aL->dO;
        tR = adO*aR->dO;
        aL->aO += aO*((ADvar2q*)a)->pL
          + tR*((ADvar2q*)a)->pLR;
        aR->aO += aO*((ADvar2q*)a)->pR
          + tL*((ADvar2q*)a)->pLR
          + tR*((ADvar2q*)a)->pR2;
        aL->adO += adO * ((ADvar2q*)a)->pL;
        aR->adO += adO * ((ADvar2q*)a)->pR;
       case Hv_const: // This case does not arise; the intent here
          // is to eliminate a red-herring compiler warning.
        break;
       case Hv_nary:
        d  = ((ADvarn*)a)->D;
        m  = ((ADvarn*)a)->n;
        g  = ((ADvarn*)a)->G;
        h0 = ((ADvarn*)a)->H;
        for(i = 0; i < m; i++) {
          aL = d[i].c;
          aL->adO += adO * (t = g[i]);
          aL->aO += t*aO;
          t = adO * aL->dO;
          for(h = h0, j = 0; j <= i; j++)
            d[j].c->aO += t * *h++;
          h0 = h--;
          for(k = j; j < m; j++)
            d[j].c->aO += t * *(h += k++);
          }
       }
      }
    }
  for(i = 0; i < n; i++) {
    a = x[i]->cv;
    a->dO = 0.;
    hv[i] = a->aO;
    }
  }
 
#ifndef SACADO_NO_NAMESPACE
} // namespace Rad2d
} // namespace Sacado
#endif