/*
* Copyright © 2004 Ondra Kamenik
* Copyright © 2019 Dynare Team
*
* This file is part of Dynare.
*
* Dynare is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* Dynare 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 General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with Dynare. If not, see .
*/
#include "kord_exception.hh"
#include "first_order.hh"
#include
double FirstOrder::qz_criterium_global;
std::mutex FirstOrder::mut;
/* This is a function which selects the eigenvalues pair used by LAPACK’s
dgges. Here we want to select the pairs for which α<β (up to the QZ
criterium). */
lapack_int
FirstOrder::order_eigs(const double *alphar, const double *alphai, const double *beta)
{
return (*alphar **alphar + *alphai **alphai < *beta **beta * qz_criterium_global * qz_criterium_global);
}
/* Here we solve the linear approximation. The result are the matrices
g_y* and gᵤ. The method solves the first derivatives of g so
that the following equation would be true:
𝔼ₜ[F(y*ₜ₋₁,uₜ,uₜ₊₁,σ)] = 𝔼ₜ[f(g**(g*(y*ₜ₋₁,uₜ,σ), uₜ₊₁, σ), g(y*ₜ₋₁,uₜ,σ), y*ₜ₋₁,uₜ)]=0
where f is a given system of equations.
It is known that g_y* is given by F_y*=0, gᵤ is given by Fᵤ=0, and g_σ 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;
// Solve derivatives ‘gy’
/* The derivatives g_y* are retrieved from the equation F_y*=0. The
calculation proceeds as follows:
1. 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:
[ f_y**₊ f_ys f_yp f_yb f_yf f_y*₋ ]
where f_ys, f_yp, f_yb, and f_yf are derivatives w.r.t. static,
predetermined, both, forward looking at time t, is rewritten as:
⎡ f_y**₊ f_ys f_yp f_yb 0 f_yf f_y*₋ ⎤
⎣ 0 0 0 I −I 0 0 ⎦
where the second line has number of rows equal to the number of both
variables.
2. Next, provided that forward looking and predetermined are disjoint, the
equation F_y*=0 is written as:
[f_y**₊][g**_y*][g*_y*] + [f_ys][gˢ_y*] + [f_y*][g*_y*] + [f_y**][g**_y*] + [f_y*₋] = 0
This is rewritten as
⎡ I ⎤ ⎡ I ⎤
[f_y* 0 f_y**₊]⎢ gˢ_y*⎥[g*_y*] + [f_y*₋ f_ys f_y**]⎢ gˢ_y*⎥ = 0
⎣g**_y*⎦ ⎣g**_y*⎦
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:
⎡ I ⎤ ⎡ I ⎤
⎡f_yp f_yb 0 f_y**₊⎤⎢ gˢ_y*⎥[g*_y*] + ⎡f_y*₋ f_ys 0 f_yf⎤⎢ gˢ_y*⎥ = 0
⎣ 0 I 0 0 ⎦⎣g**_y*⎦ ⎣ 0 0 −I 0 ⎦⎣g**_y*⎦
The two matrices are denoted as D and −E, so the equation takes the form:
⎡ I ⎤ ⎡ I ⎤
D ⎢ gˢ_y*⎥[g*_y*] = E ⎢ gˢ_y*⎥
⎣g**_y*⎦ ⎣g**_y*⎦
3. Next we solve the equation by Generalized Schur decomposition:
⎡T₁₁ T₁₂⎤⎡Z₁₁ᵀ Z₂₁ᵀ⎤⎡I⎤ ⎡S₁₁ S₁₂⎤⎡Z₁₁ᵀ Z₂₁ᵀ⎤⎡I⎤
⎣ 0 T₂₂⎦⎣Z₁₂ᵀ Z₂₂ᵀ⎦⎣X⎦[g*_y*] = ⎣ 0 S₂₂⎦⎣Z₁₂ᵀ Z₂₂ᵀ⎦⎣X⎦
We reorder the eigenvalue pair so that Sᵢᵢ/Tᵢᵢ with modulus less than
one would be in the left-upper part.
4. The Blanchard-Kahn stability argument implies that the pairs with
modulus less that one will be in and only in S₁₁/T₁₁. The exploding
paths will be then eliminated when
⎡Z₁₁ᵀ Z₂₁ᵀ⎤⎡I⎤ ⎡Y⎤
⎣Z₁₂ᵀ Z₂₂ᵀ⎦⎣X⎦ = ⎣0⎦
From this we have, Y=Z₁₁⁻¹, and X=Z₂₁Y, or equivalently X=−Z₂₂⁻ᵀZ₁₂ᵀ.
From the equation, we get [g*_y*]=Y⁻¹T₁₁⁻¹S₁₁Y, which is
Z₁₁T₁₁⁻¹S₁₁Z₁₁⁻¹.
5. We then copy the derivatives to storage ‘gy’. Note that the derivatives
of both variables are in X and in [g*_y*], so we check whether the two
submatrices are the same. The difference is only numerical error.
*/
// 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;
lapack_int ldb = matD.getLD();
// 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);
lapack_int lda = matE.getLD();
// Solve generalized Schur decomposition
TwoDMatrix vsl(n, n);
TwoDMatrix vsr(n, n);
lapack_int ldvsl = vsl.getLD(), ldvsr = vsr.getLD();
lapack_int lwork = 100*n+16;
Vector work(lwork);
auto bwork = std::make_unique(n);
lapack_int info;
lapack_int sdim2 = sdim;
{
std::lock_guard lk{mut};
qz_criterium_global = qz_criterium;
dgges("N", "V", "S", order_eigs, &n, matE.getData().base(), &lda,
matD.getData().base(), &ldb, &sdim2, alphar.base(), alphai.base(),
beta.base(), vsl.getData().base(), &ldvsl, vsr.getData().base(), &ldvsr,
work.base(), &lwork, bwork.get(), &info);
}
if (info)
throw KordException(__FILE__, __LINE__,
"DGGES returns an error in FirstOrder::solve");
sdim = sdim2;
bk_cond = (sdim == ypart.nys());
// Setup submatrices of 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=−Z₂₂⁻ᵀZ₁₂ᵀ, where X is ‘sfder’ in the code. */
GeneralMatrix sfder(transpose(z12));
z22.multInvLeftTrans(sfder);
sfder.mult(-1);
// Calculate derivatives of predetermined
/* Here we calculate g*_y* = Z₁₁T₁₁⁻¹S₁₁Z₁₁⁻¹ = Z₁₁T₁₁⁻¹(Z₁₁⁻ᵀS₁₁ᵀ)ᵀ. */
ConstGeneralMatrix s11(matE, 0, 0, ypart.nys(), ypart.nys());
ConstGeneralMatrix t11(matD, 0, 0, ypart.nys(), ypart.nys());
GeneralMatrix dumm(transpose(s11));
z11.multInvLeftTrans(dumm);
GeneralMatrix preder(transpose(dumm));
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_cast(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 Fᵤ=0 can be written as
[f_y**₊][g**_y*][gᵤ*] + [f_y][gᵤ] + [fᵤ] = 0
and rewritten as
[f_y + [0 f_y**₊·g**_y* 0] ] gᵤ = fᵤ
This is exactly what is done here. The matrix [f_y + [0 f_y**₊·g**_y* 0] ]
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());
matA.place(fyplus * gss, 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();
KORD_RAISE_IF_X(!bk_cond,
"The model is not Blanchard-Kahn stable",
KORD_MD_NOT_STABLE);
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 << u8"──────────────────────────────────────────────────── ";
if (i == sdim)
jr << "sdim";
else
jr << "npred";
jr << endrec;
}
JournalRecord jr(journal);
double mod = std::sqrt(alphar[i]*alphar[i]+alphai[i]*alphai[i]);
mod = mod/std::round(100000*std::abs(beta[i]))*100000;
jr << i << "\t(" << alphar[i] << "," << alphai[i] << ") / " << beta[i]
<< " \t" << mod << endrec;
}
}