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first_order.cweb

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    first_order.cweb 10.27 KiB 
    @q $Id: first_order.cweb 2351 2009-09-03 14:58:03Z kamenik $ @>
@q Copyright 2004, Ondra Kamenik @>

@ Start of {\tt first\_order.cpp} file.

@c

#include "kord_exception.h"
#include "first_order.h"

#include <dynlapack.h>

double qz_criterium = 1.000001;
@<|order_eigs| function code@>;
@<|FirstOrder::solve| code@>;
@<|FirstOrder::journalEigs| code@>;

@ 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 $\alpha<\beta$.

@<|order_eigs| function code@>=
lapack_int order_eigs(const double* alphar, const double* alphai, const double* beta)
{
	return (*alphar * *alphar + *alphai * *alphai < *beta * *beta * 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).

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

	::qz_criterium = FirstOrder::qz_criterium;

	@<solve derivatives |gy|@>;
	@<solve derivatives |gu|@>;
	journalEigs();

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

@ 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

@<solve derivatives |gy|@>=
	@<setup submatrices of |f|@>;
	@<form matrix $D$@>;
	@<form matrix $E$@>;
	@<solve generalized Schur@>;
    	@<make submatrices of right space@>;
	@<calculate derivatives of static and forward@>;
	@<calculate derivatives of predetermined@>;
	@<copy derivatives to |gy|@>;
	@<check difference for derivatives of both@>;


@ Here we setup submatrices of the derivatives |fd|.
@<setup submatrices of |f|@>=
	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);
	lapack_int *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);
	sdim = sdim2;
	bk_cond = (sdim == ypart.nys());
	delete[] bwork;


@ Here we setup submatrices of the matrix $Z$.
@<make submatrices of right space@>=
	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());
	
@ Here we calculate $X=-Z_{22}^{-T}Z_{12}^T$, where $X$ is |sfder| in the code.
@<calculate derivatives of static and forward@>=
	GeneralMatrix sfder(z12, "transpose");
	z22.multInvLeftTrans(sfder);
	sfder.mult(-1);

@ Here we calculate
$g_{y^*}^*=Z_{11}T^{-1}_{11}S_{11}Z_{11}^{-1}
=Z_{11}T^{-1}_{11}(Z_{11}^{-T}S^T_{11})^T$.

@<calculate derivatives of predetermined@>=
    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();

@ 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.

@<solve derivatives |gu|@>=
	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);

@ 
@<|FirstOrder::journalEigs| code@>=
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; 
		}
	}
}


@ End of {\tt first\_order.cpp} file.