dynare/dynare++/kord/first_order.cc

// Copyright 2004, Ondra Kamenik

#include "kord_exception.hh"
#include "first_order.hh"

#include <dynlapack.h>

double qz_criterium = 1.000001;

/* This is a function which selects the eigenvalues pair used by
   |dgges|. See documentation to DGGES for details. Here we want
   to select (return true) the pairs for which $\alpha<\beta$. */

lapack_int
order_eigs(const double *alphar, const double *alphai, const double *beta)
{
  return (*alphar **alphar + *alphai **alphai < *beta **beta * qz_criterium * qz_criterium);
}

/* Here we solve the linear approximation. The result are the matrices
   $g_{y^*}$ and $g_u$. The method solves the first derivatives of $g$ so
   that the following equation would be true:
   $$E_t[F(y^*_{t-1},u_t,u_{t+1},\sigma)] =
   E_t[f(g^{**}(g^*(y_{t-1}^*,u_t,\sigma), u_{t+1}, \sigma), g(y_{t-1}^*,u_t,\sigma),
   y^*_{t-1},u_t)]=0$$
   where $f$ is a given system of equations.

   It is known that $g_{y^*}$ is given by $F_{y^*}=0$, $g_u$ is given by
   $F_u=0$, and $g_\sigma$ is zero. The only input to the method are the
   derivatives |fd| of the system $f$, and partitioning of the vector $y$
   (from object). */

void
FirstOrder::solve(const TwoDMatrix &fd)
{
  JournalRecordPair pa(journal);
  pa << "Recovering first order derivatives " << endrec;

  ::qz_criterium = FirstOrder::qz_criterium;

  // **********************
  // solve derivatives |gy|
  // **********************

  /* The derivatives $g_{y^*}$ are retrieved from the equation
     $F_{y^*}=0$. The calculation proceeds as follows:

     \orderedlist

     \li For each variable appearing at both $t-1$ and $t-1$ we add a dummy
     variable, so that the predetermined variables and forward looking would
     be disjoint. This is, the matrix of the first derivatives of the
     system written as:
     $$\left[\matrix{f_{y^{**}_+}&f_{ys}&f_{yp}&f_{yb}&f_{yf}&f_{y^*_-}}\right],$$
     where $f_{ys}$, $f_{yp}$, $f_{yb}$, and $f_{yf}$ are derivatives wrt
     static, predetermined, both, forward looking at time $t$, is rewritten
     to the matrix:
     $$\left[
     \matrix{f_{y^{**}_+}&f_{ys}&f_{yp}&f_{yb}&0&f_{yf}&f_{y^*_-}\cr
     0           &0     &0     &I   &-I&0    &0}
     \right],$$
     where the second line has number of rows equal to the number of both variables.

     \li Next, provided that forward looking and predetermined are
     disjoint, the equation $F_{y^*}=0$ is written as:
     $$\left[f_+{y^{**}_+}\right]\left[g_{y^*}^{**}\right]\left[g_{y^*}^*\right]
     +\left[f_{ys}\right]\left[g^s_{y^*}\right]
     +\left[f_{y^*}\right]\left[g^*_{y^*}\right]
     +\left[f_{y^{**}}\right]\left[g^{**}_{y^*}\right]+\left[f_{y^*_-}\right]=0$$
     This is rewritten as
     $$\left[\matrix{f_{y^*}&0&f_{y^{**}_+}}\right]
     \left[\matrix{I\cr g^s_{y^*}\cr g^{**}_{y^*}}\right]\left[g_{y^*}^*\right]+
     \left[\matrix{f_{y^*_-}&f_{ys}&f_{y^{**}}}\right]
     \left[\matrix{I\cr g^s_{y^*}\cr g^{**}_{y^*}}\right]=0
     $$
     Now, in the above equation, there are the auxiliary variables standing
     for copies of both variables at time $t+1$. This equation is then
     rewritten as:
     $$
     \left[\matrix{f_{yp}&f_{yb}&0&f_{y^{**}_+}\cr 0&I&0&0}\right]
     \left[\matrix{I\cr g^s_{y^*}\cr g^{**}_{y^*}}\right]\left[g_{y^*}^*\right]+
     \left[\matrix{f_{y^*_-}&f_{ys}&0&f_{yf}\cr 0&0&-I&0}\right]
     \left[\matrix{I\cr g^s_{y^*}\cr g^{**}_{y^*}}\right]=0
     $$
     The two matrices are denoted as $D$ and $-E$, so the equation takes the form:
     $$D\left[\matrix{I\cr g^s_{y^*}\cr g^{**}_{y^*}}\right]\left[g_{y^*}^*\right]=
     E\left[\matrix{I\cr g^s_{y^*}\cr g^{**}_{y^*}}\right]$$

     \li Next we solve the equation by Generalized Schur decomposition:
     $$
     \left[\matrix{T_{11}&T_{12}\cr 0&T_{22}}\right]
     \left[\matrix{Z_{11}^T&Z_{21}^T\cr Z_{12}^T&Z_{22}^T}\right]
     \left[\matrix{I\cr X}\right]\left[g_{y^*}^*\right]=
     \left[\matrix{S_{11}&S_{12}\cr 0&S_{22}}\right]
     \left[\matrix{Z_{11}^T&Z_{21}^T\cr Z_{12}^T&Z_{22}^T}\right]
     \left[\matrix{I\cr X}\right]
     $$
     We reorder the eigenvalue pair so that $S_{ii}/T_{ii}$ with modulus
     less than one would be in the left-upper part.

     \li The Blanchard--Kahn stability argument implies that the pairs
     with modulus less that one will be in and only in $S_{11}/T_{11}$.
     The exploding paths will be then eliminated when
     $$
     \left[\matrix{Z_{11}^T&Z_{21}^T\cr Z_{12}^T&Z_{22}^T}\right]
     \left[\matrix{I\cr X}\right]=
     \left[\matrix{Y\cr 0}\right]
     $$
     From this we have, $Y=Z_{11}^{-1}$, and $X=Z_{21}Y$, or equivalently
     $X=-Z_{22}^{-T}Z_{12}^T$.  From the equation, we get
     $\left[g_{y^*}^*\right]=Y^{-1}T_{11}^{-1}S_{11}Y$, which is
     $Z_{11}T_{11}^{-1}S_{11}Z_{11}^{-1}$.

     \li We then copy the derivatives to storage |gy|. Note that the
     derivatives of both variables are in $X$ and in
     $\left[g_{y^*}^*\right]$, so we check whether the two submatrices are
     the same. The difference is only numerical error.

     \endorderedlist */

  // setup submatrices of |f|
  /* Here we setup submatrices of the derivatives |fd|. */
  int off = 0;
  ConstTwoDMatrix fyplus(fd, off, ypart.nyss());
  off += ypart.nyss();
  ConstTwoDMatrix fyszero(fd, off, ypart.nstat);
  off += ypart.nstat;
  ConstTwoDMatrix fypzero(fd, off, ypart.npred);
  off += ypart.npred;
  ConstTwoDMatrix fybzero(fd, off, ypart.nboth);
  off += ypart.nboth;
  ConstTwoDMatrix fyfzero(fd, off, ypart.nforw);
  off += ypart.nforw;
  ConstTwoDMatrix fymins(fd, off, ypart.nys());
  off += ypart.nys();
  ConstTwoDMatrix fuzero(fd, off, nu);
  off += nu;

  // form matrix $D$
  lapack_int n = ypart.ny()+ypart.nboth;
  TwoDMatrix matD(n, n);
  matD.zeros();
  matD.place(fypzero, 0, 0);
  matD.place(fybzero, 0, ypart.npred);
  matD.place(fyplus, 0, ypart.nys()+ypart.nstat);
  for (int i = 0; i < ypart.nboth; i++)
    matD.get(ypart.ny()+i, ypart.npred+i) = 1.0;

  // form matrix $E$
  TwoDMatrix matE(n, n);
  matE.zeros();
  matE.place(fymins, 0, 0);
  matE.place(fyszero, 0, ypart.nys());
  matE.place(fyfzero, 0, ypart.nys()+ypart.nstat+ypart.nboth);
  for (int i = 0; i < ypart.nboth; i++)
    matE.get(ypart.ny()+i, ypart.nys()+ypart.nstat+i) = -1.0;
  matE.mult(-1.0);

  // solve generalized Schur
  TwoDMatrix vsl(n, n);
  TwoDMatrix vsr(n, n);
  lapack_int lwork = 100*n+16;
  Vector work(lwork);
  auto *bwork = new lapack_int[n];
  lapack_int info;
  lapack_int sdim2 = sdim;
  dgges("N", "V", "S", order_eigs, &n, matE.getData().base(), &n,
        matD.getData().base(), &n, &sdim2, alphar.base(), alphai.base(),
        beta.base(), vsl.getData().base(), &n, vsr.getData().base(), &n,
        work.base(), &lwork, bwork, &info);
  if (info)
    {
      throw KordException(__FILE__, __LINE__,
                          "DGGES returns an error in FirstOrder::solve");
    }
  sdim = sdim2;
  bk_cond = (sdim == ypart.nys());
  delete[] bwork;

  // make submatrices of right space
  /* Here we setup submatrices of the matrix $Z$. */
  ConstGeneralMatrix z11(vsr, 0, 0, ypart.nys(), ypart.nys());
  ConstGeneralMatrix z12(vsr, 0, ypart.nys(), ypart.nys(), n-ypart.nys());
  ConstGeneralMatrix z21(vsr, ypart.nys(), 0, n-ypart.nys(), ypart.nys());
  ConstGeneralMatrix z22(vsr, ypart.nys(), ypart.nys(), n-ypart.nys(), n-ypart.nys());

  // calculate derivatives of static and forward
  /* Here we calculate $X=-Z_{22}^{-T}Z_{12}^T$, where $X$ is |sfder| in the
     code. */
  GeneralMatrix sfder(z12, "transpose");
  z22.multInvLeftTrans(sfder);
  sfder.mult(-1);

  // calculate derivatives of predetermined
  /* Here we calculate
     $g_{y^*}^*=Z_{11}T^{-1}_{11}S_{11}Z_{11}^{-1}
     =Z_{11}T^{-1}_{11}(Z_{11}^{-T}S^T_{11})^T$. */
  ConstGeneralMatrix s11(matE, 0, 0, ypart.nys(), ypart.nys());
  ConstGeneralMatrix t11(matD, 0, 0, ypart.nys(), ypart.nys());
  GeneralMatrix dumm(s11, "transpose");
  z11.multInvLeftTrans(dumm);
  GeneralMatrix preder(dumm, "transpose");
  t11.multInvLeft(preder);
  preder.multLeft(z11);

  // copy derivatives to |gy|
  gy.place(preder, ypart.nstat, 0);
  GeneralMatrix sder(sfder, 0, 0, ypart.nstat, ypart.nys());
  gy.place(sder, 0, 0);
  GeneralMatrix fder(sfder, ypart.nstat+ypart.nboth, 0, ypart.nforw, ypart.nys());
  gy.place(fder, ypart.nstat+ypart.nys(), 0);

  // check difference for derivatives of both
  GeneralMatrix bder((const GeneralMatrix &)sfder, ypart.nstat, 0, ypart.nboth, ypart.nys());
  GeneralMatrix bder2(preder, ypart.npred, 0, ypart.nboth, ypart.nys());
  bder.add(-1, bder2);
  b_error = bder.getData().getMax();

  // **********************
  // solve derivatives |gu|
  // **********************
  /* The equation $F_u=0$ can be written as
     $$
     \left[f_{y^{**}_+}\right]\left[g^{**}_{y^*}\right]\left[g_u^*\right]+
     \left[f_y\right]\left[g_u\right]+\left[f_u\right]=0
     $$
     and rewritten as
     $$
     \left[f_y +
     \left[\matrix{0&f_{y^{**}_+}g^{**}_{y^*}&0}\right]\right]g_u=f_u
     $$
     This is exactly done here. The matrix
     $\left[f_y +\left[\matrix{0&f_{y^{**}_+}g^{**}_{y^*}&0}\right]\right]$ is |matA|
     in the code. */
  GeneralMatrix matA(ypart.ny(), ypart.ny());
  matA.zeros();
  ConstGeneralMatrix gss(gy, ypart.nstat+ypart.npred, 0, ypart.nyss(), ypart.nys());
  GeneralMatrix aux(fyplus, gss);
  matA.place(aux, 0, ypart.nstat);
  ConstGeneralMatrix fyzero(fd, 0, ypart.nyss(), ypart.ny(), ypart.ny());
  matA.add(1.0, fyzero);
  gu.zeros();
  gu.add(-1.0, fuzero);
  ConstGeneralMatrix(matA).multInvLeft(gu);

  journalEigs();

  if (!gy.isFinite() || !gu.isFinite())
    {
      throw KordException(__FILE__, __LINE__,
                          "NaN or Inf asserted in first order derivatives in FirstOrder::solve");
    }
}

void
FirstOrder::journalEigs()
{
  if (bk_cond)
    {
      JournalRecord jr(journal);
      jr << "Blanchard-Kahn conditition satisfied, model stable" << endrec;
    }
  else
    {
      JournalRecord jr(journal);
      jr << "Blanchard-Kahn condition not satisfied, model not stable: sdim=" << sdim
         << " " << "npred=" << ypart.nys() << endrec;
    }
  if (!bk_cond)
    {
      for (int i = 0; i < alphar.length(); i++)
        {
          if (i == sdim || i == ypart.nys())
            {
              JournalRecord jr(journal);
              jr << "---------------------------------------------------- ";
              if (i == sdim)
                jr << "sdim";
              else
                jr << "npred";
              jr << endrec;
            }
          JournalRecord jr(journal);
          double mod = sqrt(alphar[i]*alphar[i]+alphai[i]*alphai[i]);
          mod = mod/round(100000*std::abs(beta[i]))*100000;
          jr << i << "\t(" << alphar[i] << "," << alphai[i] << ") / " << beta[i]
             << "  \t" << mod << endrec;
        }
    }
}