time_series_models/lu_tricks.ipynb

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   "source": [
    "\n",
    "<a id='lu-tricks'></a>\n",
    "<div id=\"qe-notebook-header\" style=\"text-align:right;\">\n",
    "        <a href=\"https://quantecon.org/\" title=\"quantecon.org\">\n",
    "                <img style=\"width:250px;display:inline;\" src=\"https://assets.quantecon.org/img/qe-menubar-logo.svg\" alt=\"QuantEcon\">\n",
    "        </a>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Classical Control with Linear Algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contents\n",
    "\n",
    "- [Classical Control with Linear Algebra](#Classical-Control-with-Linear-Algebra)  \n",
    "  - [Overview](#Overview)  \n",
    "  - [A Control Problem](#A-Control-Problem)  \n",
    "  - [Finite Horizon Theory](#Finite-Horizon-Theory)  \n",
    "  - [The Infinite Horizon Limit](#The-Infinite-Horizon-Limit)  \n",
    "  - [Undiscounted Problems](#Undiscounted-Problems)  \n",
    "  - [Implementation](#Implementation)  \n",
    "  - [Exercises](#Exercises)  "
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview\n",
    "\n",
    "In an earlier lecture [Linear Quadratic Dynamic Programming Problems](https://lectures.quantecon.org/dynamic_programming/lqcontrol.html) we have studied how to solve a special\n",
    "class of dynamic optimization and prediction problems by applying the method of dynamic programming. In this class of problems\n",
    "\n",
    "> - the objective function is **quadratic** in **states** and **controls**  \n",
    "- the one-step transition function is **linear**  \n",
    "- shocks are i.i.d. Gaussian or martingale differences  \n",
    "\n",
    "\n",
    "\n",
    "In this lecture and a companion lecture [Classical Filtering with Linear Algebra](https://lectures.quantecon.org/classical_filtering.html), we study the classical theory of linear-quadratic (LQ) optimal control problems.\n",
    "\n",
    "The classical approach does not use the two closely related methods – dynamic programming and  Kalman filtering – that we describe in other lectures, namely, [Linear Quadratic Dynamic Programming Problems](https://lectures.quantecon.org/dynamic_programming/lqcontrol.html) and [A First Look at the Kalman Filter](https://lectures.quantecon.org/tools_and_techniques/kalman.html)\n",
    "\n",
    "Instead they use either\n",
    "\n",
    "> - $ z $-transform and lag operator methods, or  \n",
    "- matrix decompositions applied to linear systems of first-order conditions for optimum problems.  \n",
    "\n",
    "\n",
    "\n",
    "In this lecture and the sequel [Classical Filtering with Linear Algebra](https://lectures.quantecon.org/classical_filtering.html), we mostly rely on  elementary linear algebra\n",
    "\n",
    "The main tool from linear algebra we’ll put to work here is [LU decomposition](https://en.wikipedia.org/wiki/LU_decomposition)\n",
    "\n",
    "We’ll begin with discrete horizon problems\n",
    "\n",
    "Then we’ll view infinite horizon problems as appropriate limits of these finite horizon problems\n",
    "\n",
    "Later, we will examine the close connection between LQ control and least squares prediction and filtering problems\n",
    "\n",
    "These classes of problems are connected in the sense that to solve each, essentially the same mathematics is used"
   ]
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   "source": [
    "### References\n",
    "\n",
    "Useful references include [[Whi63]](https://lectures.quantecon.org/zreferences.html#whittle1963), [[HS80]](https://lectures.quantecon.org/zreferences.html#hansar1980), [[Orf88]](https://lectures.quantecon.org/zreferences.html#orfanidisoptimum1988), [[AP91]](https://lectures.quantecon.org/zreferences.html#athanasios1991), and [[Mut60]](https://lectures.quantecon.org/zreferences.html#muth1960)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setup"
   ]
  },
  {
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   "execution_count": 1,
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     "text": [
      "\u001b[32m\u001b[1mActivated\u001b[0m /home/qebuild/repos/lecture-source-jl/_build/website/jupyter/Project.toml\u001b[39m\n",
      "\u001b[36m\u001b[1mInfo\u001b[0m quantecon-notebooks-julia 0.1.0 activated, 0.2.0 requested\u001b[39m\n"
     ]
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   ],
   "source": [
    "using InstantiateFromURL\n",
    "github_project(\"QuantEcon/quantecon-notebooks-julia\", version = \"0.2.0\")"
   ]
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   "source": [
    "using Polynomials, Plots, Random, Parameters\n",
    "using LinearAlgebra, Statistics"
   ]
  },
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   "metadata": {},
   "source": [
    "## A Control Problem\n",
    "\n",
    "Let $ L $ be the **lag operator**, so that, for sequence $ \\{x_t\\} $ we have $ L x_t = x_{t-1} $\n",
    "\n",
    "More generally, let $ L^k x_t = x_{t-k} $ with $ L^0 x_t = x_t $ and\n",
    "\n",
    "$$\n",
    "d(L) = d_0 + d_1 L+ \\ldots + d_m L^m\n",
    "$$\n",
    "\n",
    "where $ d_0, d_1, \\ldots, d_m $ is a given scalar sequence\n",
    "\n",
    "Consider the discrete time control problem\n",
    "\n",
    "\n",
    "<a id='equation-oneone'></a>\n",
    "$$\n",
    "\\max_{\\{y_t\\}}\n",
    "\\lim_{N \\to \\infty} \\sum^N_{t=0} \\beta^t\\,\n",
    "\\left\\{\n",
    "     a_t y_t - {1 \\over 2}\\, hy^2_t - {1 \\over 2} \\,\n",
    "         \\left[ d(L)y_t \\right]^2\n",
    "\\right\\}, \\tag{1}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "- $ h $ is a positive parameter and $ \\beta \\in (0,1) $ is a discount factor  \n",
    "- $ \\{a_t\\}_{t \\geq 0} $ is a sequence of exponential order less than $ \\beta^{-1/2} $, by which we mean $ \\lim_{t \\rightarrow \\infty} \\beta^{\\frac{t}{2}} a_t = 0 $  \n",
    "\n",
    "\n",
    "Maximization in [(1)](#equation-oneone) is subject to  initial conditions for $ y_{-1}, y_{-2} \\ldots, y_{-m} $\n",
    "\n",
    "Maximization is over infinite sequences $ \\{y_t\\}_{t \\geq 0} $"
   ]
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    "### Example\n",
    "\n",
    "The formulation of the LQ problem given above is broad enough to encompass\n",
    "many useful models\n",
    "\n",
    "As a simple illustration, recall that in [lqcontrol](https://lectures.quantecon.org/dynamic_programming/lqcontrol.html) we consider a monopolist facing stochastic demand\n",
    "shocks and adjustment costs\n",
    "\n",
    "Let’s consider a deterministic version of this problem, where the monopolist\n",
    "maximizes the discounted sum\n",
    "\n",
    "$$\n",
    "\\sum_{t=0}^{\\infty} \\beta^t \\pi_t\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\pi_t = p_t q_t - c q_t - \\gamma (q_{t+1} - q_t)^2\n",
    "\\quad \\text{with} \\quad\n",
    "p_t = \\alpha_0 - \\alpha_1 q_t + d_t\n",
    "$$\n",
    "\n",
    "In this expression, $ q_t $ is output, $ c $ is average cost of production, and $ d_t $ is a demand shock\n",
    "\n",
    "The term $ \\gamma (q_{t+1} - q_t)^2 $ represents adjustment costs\n",
    "\n",
    "You will be able to confirm that the objective function can be rewritten as [(1)](#equation-oneone) when\n",
    "\n",
    "- $ a_t := \\alpha_0 + d_t - c $  \n",
    "- $ h := 2 \\alpha_1 $  \n",
    "- $ d(L) := \\sqrt{2 \\gamma}(I - L) $  \n",
    "\n",
    "\n",
    "Further examples of this problem for factor demand, economic growth, and government policy problems are given in ch. IX of [[Sar87]](https://lectures.quantecon.org/zreferences.html#sargent1987)"
   ]
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    "## Finite Horizon Theory\n",
    "\n",
    "We first study a finite $ N $ version of the problem\n",
    "\n",
    "Later we will study an infinite horizon problem solution as a limiting version of a finite horizon problem\n",
    "\n",
    "(This will require being careful because the limits as $ N \\to \\infty $ of the necessary and sufficient conditions for maximizing finite $ N $ versions of [(1)](#equation-oneone)\n",
    "are not sufficient for maximizing [(1)](#equation-oneone))\n",
    "\n",
    "We begin by\n",
    "\n",
    "1. fixing $ N > m $,  \n",
    "1. differentiating the finite version of [(1)](#equation-oneone) with respect to $ y_0, y_1, \\ldots, y_N $, and  \n",
    "1. setting these derivatives to zero  \n",
    "\n",
    "\n",
    "For $ t=0, \\ldots, N-m $ these first-order necessary conditions are the\n",
    "*Euler equations*\n",
    "\n",
    "For $ t = N-m + 1, \\ldots, N $, the first-order conditions are a set of\n",
    "*terminal conditions*\n",
    "\n",
    "Consider the term\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "J\n",
    "& = \\sum^N_{t=0} \\beta^t [d(L) y_t] [d(L) y_t]\n",
    "\\\\\n",
    "& = \\sum^N_{t=0}\n",
    "    \\beta^t \\, (d_0 \\, y_t + d_1 \\, y_{t-1} + \\cdots + d_m \\, y_{t-m}) \\,\n",
    "               (d_0 \\, y_t + d_1 \\, y_{t-1} + \\cdots  + d_m\\, y_{t-m})\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Differentiating $ J $ with respect to $ y_t $ for\n",
    "$ t=0,\\ 1,\\ \\ldots,\\ N-m $ gives\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "{\\partial {J} \\over \\partial y_t}\n",
    "   & = 2 \\beta^t \\, d_0 \\, d(L)y_t +\n",
    "       2 \\beta^{t+1} \\, d_1\\, d(L)y_{t+1} + \\cdots +\n",
    "       2 \\beta^{t+m}\\, d_m\\, d(L) y_{t+m} \\\\\n",
    "   & = 2\\beta^t\\, \\bigl(d_0 + d_1 \\, \\beta L^{-1} + d_2 \\, \\beta^2\\, L^{-2} +\n",
    "       \\cdots + d_m \\, \\beta^m \\, L^{-m}\\bigr)\\, d (L) y_t\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "We can write this more succinctly as\n",
    "\n",
    "\n",
    "<a id='equation-onetwo'></a>\n",
    "$$\n",
    "{\\partial {J} \\over \\partial y_t}\n",
    "    = 2 \\beta^t \\, d(\\beta L^{-1}) \\, d (L) y_t \\tag{2}\n",
    "$$\n",
    "\n",
    "Differentiating $ J $ with respect to $ y_t $ for $ t = N-m + 1, \\ldots, N $ gives\n",
    "\n",
    "\n",
    "<a id='equation-onethree'></a>\n",
    "$$\n",
    "\\begin{aligned}\n",
    " {\\partial J \\over \\partial y_N}\n",
    " &= 2 \\beta^N\\, d_0 \\, d(L) y_N \\cr\n",
    "   {\\partial J \\over \\partial y_{N-1}}\n",
    " &= 2\\beta^{N-1} \\,\\bigl[d_0 + \\beta \\,\n",
    "   d_1\\, L^{-1}\\bigr] \\, d(L)y_{N-1} \\cr\n",
    "   \\vdots\n",
    " & \\quad \\quad \\vdots \\cr\n",
    "   {\\partial {J} \\over \\partial y_{N-m+1}}\n",
    " &= 2 \\beta^{N-m+1}\\,\\bigl[d_0 + \\beta\n",
    "   L^{-1} \\,d_1 + \\cdots + \\beta^{m-1}\\, L^{-m+1}\\, d_{m-1}\\bigr]  d(L)y_{N-m+1}\n",
    "\\end{aligned} \\tag{3}\n",
    "$$\n",
    "\n",
    "With these preliminaries under our belts, we are ready to differentiate [(1)](#equation-oneone)\n",
    "\n",
    "Differentiating [(1)](#equation-oneone) with respect to $ y_t $ for $ t=0, \\ldots, N-m $ gives the Euler equations\n",
    "\n",
    "\n",
    "<a id='equation-onefour'></a>\n",
    "$$\n",
    "\\bigl[h+d\\,(\\beta L^{-1})\\,d(L)\\bigr] y_t = a_t,\n",
    "\\quad t=0,\\, 1,\\, \\ldots, N-m \\tag{4}\n",
    "$$\n",
    "\n",
    "The system of equations [(4)](#equation-onefour) form  a  $ 2 \\times m $ order linear *difference\n",
    "equation* that must hold for the values of $ t $ indicated.\n",
    "\n",
    "Differentiating [(1)](#equation-oneone) with respect to $ y_t $ for $ t = N-m + 1, \\ldots, N $ gives the terminal conditions\n",
    "\n",
    "\n",
    "<a id='equation-onefive'></a>\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\beta^N  (a_N - hy_N - d_0\\,d(L)y_N)\n",
    "&= 0  \\cr\n",
    "  \\beta^{N-1} \\left(a_{N-1}-hy_{N-1}-\\Bigl(d_0 + \\beta \\, d_1\\,\n",
    "L^{-1}\\Bigr)\\, d(L)\\, y_{N-1}\\right)\n",
    "& = 0 \\cr\n",
    " \\vdots & \\vdots\\cr\n",
    "\\beta^{N-m+1} \\biggl(a_{N-m+1} - h y_{N-m+1} -(d_0+\\beta L^{-1}\n",
    "d_1+\\cdots\\  +\\beta^{m-1} L^{-m+1} d_{m-1}) d(L) y_{N-m+1}\\biggr)\n",
    "& = 0\n",
    "\\end{aligned} \\tag{5}\n",
    "$$\n",
    "\n",
    "In the finite $ N $ problem, we want simultaneously to solve [(4)](#equation-onefour) subject to the $ m $ initial conditions\n",
    "$ y_{-1}, \\ldots, y_{-m} $ and the $ m $ terminal conditions\n",
    "[(5)](#equation-onefive)\n",
    "\n",
    "These conditions uniquely pin down the solution of the finite $ N $ problem\n",
    "\n",
    "That is, for the finite $ N $ problem,\n",
    "conditions [(4)](#equation-onefour) and [(5)](#equation-onefive) are necessary and sufficient for a maximum,\n",
    "by concavity of the objective function\n",
    "\n",
    "Next we describe how to obtain the solution using matrix methods\n",
    "\n",
    "\n",
    "<a id='fdlq'></a>"
   ]
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    "### Matrix Methods\n",
    "\n",
    "Let’s look at how linear algebra can be used to tackle and shed light on the finite horizon LQ control problem"
   ]
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    "#### A Single Lag Term\n",
    "\n",
    "Let’s begin with the special case in which $ m=1 $\n",
    "\n",
    "We want to solve the system of $ N+1 $ linear equations\n",
    "\n",
    "\n",
    "<a id='equation-oneff'></a>\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\bigl[h & + d\\, (\\beta L^{-1})\\, d\\, (L) ] y_t = a_t, \\quad\n",
    "t = 0,\\ 1,\\ \\ldots,\\, N-1\\cr\n",
    "\\beta^N & \\bigl[a_N-h\\, y_N-d_0\\, d\\, (L) y_N\\bigr] = 0\n",
    "\\end{aligned} \\tag{6}\n",
    "$$\n",
    "\n",
    "where $ d(L) = d_0 + d_1 L $\n",
    "\n",
    "These equations are to be solved for\n",
    "$ y_0, y_1, \\ldots, y_N $ as functions of\n",
    "$ a_0, a_1, \\ldots,  a_N $ and $ y_{-1} $\n",
    "\n",
    "Let\n",
    "\n",
    "$$\n",
    "\\phi (L)\n",
    "= \\phi_0 + \\phi_1 L + \\beta \\phi_1 L^{-1}\n",
    "= h + d (\\beta L^{-1}) d(L)\n",
    "= (h + d_0^2 + d_1^2) + d_1 d_0 L+ d_1 d_0 \\beta L^{-1}\n",
    "$$\n",
    "\n",
    "Then we can represent [(6)](#equation-oneff) as the matrix equation\n",
    "\n",
    "\n",
    "<a id='equation-onefourfive'></a>\n",
    "$$\n",
    "\\left[\n",
    "    \\begin{matrix}\n",
    "        (\\phi_0-d_1^2) & \\phi_1 & 0 & 0 & \\ldots & \\ldots & 0 \\cr\n",
    "        \\beta \\phi_1 & \\phi_0 & \\phi_1 & 0 & \\ldots & \\dots & 0 \\cr\n",
    "        0 & \\beta \\phi_1 & \\phi_0 & \\phi_1 & \\ldots & \\ldots & 0 \\cr\n",
    "        \\vdots &\\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\cr\n",
    "        0 & \\ldots & \\ldots & \\ldots & \\beta \\phi_1 & \\phi_0 &\\phi_1 \\cr\n",
    "        0 & \\ldots & \\ldots & \\ldots & 0 & \\beta \\phi_1 & \\phi_0\n",
    "    \\end{matrix}\n",
    "\\right]\n",
    "\\left [\n",
    "    \\begin{matrix}\n",
    "        y_N \\cr y_{N-1} \\cr y_{N-2} \\cr \\vdots \\cr\n",
    "        y_1 \\cr y_0\n",
    "    \\end{matrix}\n",
    "\\right ] =\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "    a_N \\cr a_{N-1} \\cr a_{N-2} \\cr \\vdots \\cr a_1 \\cr\n",
    "    a_0 - \\phi_1 y_{-1}\n",
    "\\end{matrix}\n",
    "\\right] \\tag{7}\n",
    "$$\n",
    "\n",
    "or\n",
    "\n",
    "\n",
    "<a id='equation-onefoursix'></a>\n",
    "$$\n",
    "W\\bar y = \\bar a \\tag{8}\n",
    "$$\n",
    "\n",
    "Notice how we have chosen to arrange the $ y_t $’s in reverse\n",
    "time order.\n",
    "\n",
    "The matrix $ W $ on the left side of [(7)](#equation-onefourfive) is “almost” a\n",
    "[Toeplitz matrix](https://en.wikipedia.org/wiki/Toeplitz_matrix) (where each\n",
    "descending diagonal is constant)\n",
    "\n",
    "There are two sources of deviation from the  form  of a Toeplitz matrix\n",
    "\n",
    "1. The first element differs from the remaining diagonal elements, reflecting the terminal condition  \n",
    "1. The subdiagonal elements equal $ \\beta $ time the superdiagonal elements  \n",
    "\n",
    "\n",
    "The solution of [(8)](#equation-onefoursix) can be expressed in the form\n",
    "\n",
    "\n",
    "<a id='equation-onefourseven'></a>\n",
    "$$\n",
    "\\bar y = W^{-1} \\bar a \\tag{9}\n",
    "$$\n",
    "\n",
    "which represents each element $ y_t $ of $ \\bar y $ as a function of the entire vector $ \\bar a $\n",
    "\n",
    "That is, $ y_t $ is a function of past, present, and future values of $ a $’s, as well as of the initial condition $ y_{-1} $"
   ]
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    "#### An Alternative Representation\n",
    "\n",
    "An alternative way to express the solution to [(7)](#equation-onefourfive) or\n",
    "[(8)](#equation-onefoursix) is in so called **feedback-feedforward** form\n",
    "\n",
    "The idea here is to find a solution expressing $ y_t $ as a function of *past* $ y $’s and *current* and *future* $ a $’s\n",
    "\n",
    "To achieve this solution, one can use an [LU decomposition](https://en.wikipedia.org/wiki/LU_decomposition) of $ W $\n",
    "\n",
    "There always exists a decomposition of $ W $ of the form $ W= LU $\n",
    "where\n",
    "\n",
    "- $ L $ is an $ (N+1) \\times (N+1) $ lower trangular matrix  \n",
    "- $ U $ is an $ (N+1) \\times (N+1) $ upper trangular matrix.  \n",
    "\n",
    "\n",
    "The factorization can be normalized so that the diagonal elements of $ U $ are unity\n",
    "\n",
    "Using the LU representation in [(9)](#equation-onefourseven), we obtain\n",
    "\n",
    "\n",
    "<a id='equation-onefournine'></a>\n",
    "$$\n",
    "U \\bar y = L^{-1} \\bar a \\tag{10}\n",
    "$$\n",
    "\n",
    "Since $ L^{-1} $ is lower trangular, this representation expresses\n",
    "$ y_t $ as a function of\n",
    "\n",
    "- lagged $ y $’s (via the term $ U \\bar y $), and  \n",
    "- current and future $ a $’s (via the term $ L^{-1} \\bar a $)  \n",
    "\n",
    "\n",
    "Because there are zeros everywhere in the matrix\n",
    "on the left of [(7)](#equation-onefourfive) except on the diagonal, superdiagonal, and\n",
    "subdiagonal, the $ LU $ decomposition takes\n",
    "\n",
    "- $ L $ to be zero except in the diagional and the leading subdiagonal  \n",
    "- $ U $ to be zero except on the diagonal and the superdiagional  \n",
    "\n",
    "\n",
    "Thus, [(10)](#equation-onefournine) has the form\n",
    "\n",
    "$$\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "    1& U_{12} & 0 & 0 & \\ldots & 0 & 0 \\cr\n",
    "    0 & 1 & U_{23} & 0 & \\ldots & 0 & 0 \\cr\n",
    "    0 & 0 & 1 & U_{34} & \\ldots & 0 & 0 \\cr\n",
    "    0 & 0 & 0 & 1 & \\ldots & 0 & 0\\cr\n",
    "    \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\cr\n",
    "    0 & 0 & 0 & 0 & \\ldots & 1 & U_{N,N+1} \\cr\n",
    "    0 & 0 & 0 & 0 & \\ldots & 0 & 1\n",
    "\\end{matrix}\n",
    "\\right] \\ \\ \\\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "    y_N \\cr y_{N-1} \\cr y_{N-2} \\cr y_{N-3} \\cr \\vdots \\cr y_1 \\cr y_0\n",
    "\\end{matrix}\n",
    "\\right] =\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\quad\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "    L^{-1}_{11} & 0 & 0 & \\ldots & 0 \\cr\n",
    "    L^{-1}_{21} & L^{-1}_{22} & 0 & \\ldots & 0 \\cr\n",
    "    L^{-1}_{31} & L^{-1}_{32} & L^{-1}_{33}& \\ldots & 0 \\cr\n",
    "    \\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\cr\n",
    "    L^{-1}_{N,1} & L^{-1}_{N,2} & L^{-1}_{N,3} & \\ldots & 0 \\cr\n",
    "    L^{-1}_{N+1,1} & L^{-1}_{N+1,2} & L^{-1}_{N+1,3} & \\ldots &\n",
    "    L^{-1}_{N+1\\, N+1}\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "    a_N \\cr a_{N-1} \\cr a_{N-2} \\cr \\vdots \\cr a_1 \\cr a_0 -\n",
    "    \\phi_1 y_{-1}\n",
    "\\end{matrix}\n",
    "\\right ]\n",
    "$$\n",
    "\n",
    "where $ L^{-1}_{ij} $ is the $ (i,j) $ element of $ L^{-1} $ and $ U_{ij} $ is the $ (i,j) $ element of $ U $\n",
    "\n",
    "Note how the left side for a given $ t $ involves  $ y_t $ and one lagged value $ y_{t-1} $ while the right side involves all future values of the forcing process $ a_t, a_{t+1}, \\ldots, a_N $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Additional Lag Terms\n",
    "\n",
    "We briefly indicate how this approach extends to the problem with\n",
    "$ m > 1 $\n",
    "\n",
    "Assume that $ \\beta = 1 $ and  let $ D_{m+1} $ be the\n",
    "$ (m+1) \\times (m+1) $ symmetric matrix whose elements are\n",
    "determined from the following formula:\n",
    "\n",
    "$$\n",
    "D_{jk} = d_0 d_{k-j} + d_1 d_{k-j+1} + \\ldots + d_{j-1} d_{k-1}, \\qquad k\n",
    "\\geq j\n",
    "$$\n",
    "\n",
    "Let $ I_{m+1} $ be the $ (m+1) \\times (m+1) $ identity matrix\n",
    "\n",
    "Let $ \\phi_j $ be the coefficients in the expansion $ \\phi (L) = h + d (L^{-1}) d (L) $\n",
    "\n",
    "Then the first order conditions [(4)](#equation-onefour) and [(5)](#equation-onefive) can be expressed as:\n",
    "\n",
    "$$\n",
    "(D_{m+1} + hI_{m+1})\\ \\\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "    y_N \\cr y_{N-1} \\cr \\vdots \\cr y_{N-m}\n",
    "\\end{matrix}\n",
    "\\right]\\\n",
    "= \\ \\left[\n",
    "\\begin{matrix}\n",
    "     a_N \\cr a_{N-1} \\cr \\vdots \\cr a_{N-m}\n",
    " \\end{matrix}\n",
    "\\right] + M\\\n",
    "\\left[\n",
    "    \\begin{matrix}\n",
    "        y_{N-m+1}\\cr y_{N-m-2}\\cr \\vdots\\cr y_{N-2m}\n",
    "    \\end{matrix}\n",
    "\\right]\n",
    "$$\n",
    "\n",
    "where $ M $ is $ (m+1)\\times m $ and\n",
    "\n",
    "$$\n",
    "M_{ij} = \\begin{cases}\n",
    "D_{i-j,\\,m+1} \\textrm{ for } i>j  \\\\\n",
    " 0  \\textrm{ for }  i\\leq j\\end{cases}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\phi_m y_{N-1} &+ \\phi_{m-1} y_{N-2} + \\ldots + \\phi_0 y_{N-m-1} +\n",
    "\\phi_1 y_{N-m-2} +\\cr\n",
    "&\\hskip.75in \\ldots + \\phi_m y_{N-2m-1} = a_{N-m-1} \\cr\n",
    "\\phi_m y_{N-2} &+ \\phi_{m-1} y_{N-3} + \\ldots + \\phi_0 y_{N-m-2} + \\phi_1\n",
    "y_{N-m-3} +\\cr\n",
    "&\\hskip.75in \\ldots + \\phi_m y_{N-2m-2} = a_{N-m-2} \\cr\n",
    "&\\qquad \\vdots \\cr\n",
    "\\phi_m y_{m+1} &+ \\phi_{m-1} y_m + + \\ldots + \\phi_0 y_1 + \\phi_1 y_0 +\n",
    "\\phi_m y_{-m+1} = a_1 \\cr\n",
    "\\phi_m y_m + \\phi_{m-1}& y_{m-1} + \\phi_{m-2} + \\ldots + \\phi_0 y_0 + \\phi_1\n",
    "y_{-1} + \\ldots + \\phi_m y_{-m} = a_0\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "As before, we can express this equation as $ W \\bar y = \\bar a $\n",
    "\n",
    "The matrix on the left of this equation is “almost” Toeplitz, the\n",
    "exception being the leading $ m \\times m $ sub matrix in the upper\n",
    "left hand corner\n",
    "\n",
    "We can represent the solution in feedback-feedforward form by obtaining a decomposition $ LU = W $, and obtain\n",
    "\n",
    "\n",
    "<a id='equation-onefivetwo'></a>\n",
    "$$\n",
    "U \\bar y = L^{-1} \\bar a \\tag{11}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\begin{aligned} \\sum^t_{j=0}\\, U_{-t+N+1,\\,-t+N+j+1}\\,y_{t-j} &= \\sum^{N-t}_{j=0}\\,\n",
    "L_{-t+N+1,\\, -t+N+1-j}\\, \\bar a_{t+j}\\ ,\\cr\n",
    "&\\qquad t=0,1,\\ldots, N\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "where $ L^{-1}_{t,s} $ is the element in the $ (t,s) $ position\n",
    "of $ L $, and similarly for $ U $\n",
    "\n",
    "The left side of equation [(11)](#equation-onefivetwo) is the “feedback” part of the optimal\n",
    "control law for $ y_t $, while the right-hand side is the “feedforward” part\n",
    "\n",
    "We note that there is a different control law for each $ t $\n",
    "\n",
    "Thus, in the finite horizon case, the optimal control law is time dependent\n",
    "\n",
    "It is natural to suspect that as $ N \\rightarrow\\infty $, [(11)](#equation-onefivetwo)\n",
    "becomes equivalent to the solution of our infinite horizon problem,\n",
    "which below we shall show can be expressed as\n",
    "\n",
    "$$\n",
    "c(L) y_t = c (\\beta L^{-1})^{-1} a_t\\ ,\n",
    "$$\n",
    "\n",
    "so that as $ N \\rightarrow \\infty $ we expect that for each fixed\n",
    "$ t, U^{-1}_{t, t-j}\n",
    "\\rightarrow c_j $ and $ L_{t,t+j} $ approaches the coefficient on\n",
    "$ L^{-j} $ in the expansion of $ c(\\beta L^{-1})^{-1} $\n",
    "\n",
    "This suspicion is true under general conditions that we shall study later\n",
    "\n",
    "For now, we note that by creating the matrix $ W $ for large\n",
    "$ N $ and factoring it into the $ LU $ form, good approximations\n",
    "to $ c(L) $ and $ c(\\beta L^{-1})^{-1} $ can be obtained"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Infinite Horizon Limit\n",
    "\n",
    "For the infinite horizon problem, we propose to discover first-order\n",
    "necessary conditions by taking the limits of [(4)](#equation-onefour) and [(5)](#equation-onefive) as\n",
    "$ N \\to \\infty $\n",
    "\n",
    "This approach is valid, and the limits of [(4)](#equation-onefour) and [(5)](#equation-onefive) as $ N $ approaches infinity are first-order necessary conditions for a maximum\n",
    "\n",
    "However, for the infinite horizon problem with $ \\beta < 1 $, the limits of [(4)](#equation-onefour) and [(5)](#equation-onefive) are, in general, not sufficient for a maximum\n",
    "\n",
    "That is, the limits of [(5)](#equation-onefive) do not provide enough information uniquely to determine the solution of the Euler equation [(4)](#equation-onefour) that maximizes [(1)](#equation-oneone)\n",
    "\n",
    "As we shall see below, a side condition on the path of $ y_t $ that together with [(4)](#equation-onefour) is sufficient for an optimum is\n",
    "\n",
    "\n",
    "<a id='equation-onesix'></a>\n",
    "$$\n",
    "\\sum^\\infty_{t=0}\\ \\beta^t\\, hy^2_t < \\infty \\tag{12}\n",
    "$$\n",
    "\n",
    "All paths that satisfy the Euler equations, except the one that we shall\n",
    "select below, violate this condition and, therefore, evidently lead to\n",
    "(much) lower values of [(1)](#equation-oneone) than does the\n",
    "optimal path selected by the solution procedure below\n",
    "\n",
    "Consider the *characteristic equation* associated with the Euler equation\n",
    "\n",
    "\n",
    "<a id='equation-oneseven'></a>\n",
    "$$\n",
    "h+d \\, (\\beta z^{-1})\\, d \\, (z) = 0 \\tag{13}\n",
    "$$\n",
    "\n",
    "Notice that if $ \\tilde z $ is a root of equation [(13)](#equation-oneseven), then so is $ \\beta \\tilde z^{-1} $\n",
    "\n",
    "Thus, the roots of [(13)](#equation-oneseven) come in “$ \\beta $-reciprocal” pairs\n",
    "\n",
    "Assume that the roots of [(13)](#equation-oneseven) are distinct\n",
    "\n",
    "Let the roots be, in descending order according to their moduli, $ z_1, z_2, \\ldots, z_{2m} $\n",
    "\n",
    "From the reciprocal pairs property and the assumption of distinct\n",
    "roots, it follows that $ \\vert z_j \\vert > \\sqrt \\beta\\ \\hbox{ for } j\\leq m \\hbox\n",
    "{ and } \\vert z_j \\vert < \\sqrt\\beta\\ \\hbox { for } j > m $\n",
    "\n",
    "It also follows that $ z_{2m-j} = \\beta z^{-1}_{j+1}, j=0, 1, \\ldots, m-1 $\n",
    "\n",
    "Therefore, the characteristic polynomial on the left side of [(13)](#equation-oneseven) can be expressed as\n",
    "\n",
    "\n",
    "<a id='equation-oneeight'></a>\n",
    "$$\n",
    "\\begin{aligned}\n",
    "h+d(\\beta z^{-1})d(z)\n",
    "&= z^{-m} z_0(z-z_1)\\cdots\n",
    "(z-z_m)(z-z_{m+1}) \\cdots (z-z_{2m}) \\cr\n",
    "&= z^{-m} z_0 (z-z_1)(z-z_2)\\cdots (z-z_m)(z-\\beta z_m^{-1})\n",
    "\\cdots  (z-\\beta z^{-1}_2)(z-\\beta z_1^{-1})\n",
    "\\end{aligned} \\tag{14}\n",
    "$$\n",
    "\n",
    "where $ z_0 $ is a constant\n",
    "\n",
    "In [(14)](#equation-oneeight), we substitute $ (z-z_j) = -z_j (1- {1 \\over z_j}z) $ and\n",
    "$ (z-\\beta z_j^{-1}) = z(1 - {\\beta \\over z_j} z^{-1}) $ for $ j = 1, \\ldots, m $ to get\n",
    "\n",
    "$$\n",
    "h+d(\\beta z^{-1})d(z)\n",
    "= (-1)^m(z_0z_1\\cdots z_m)\n",
    "(1- {1\\over z_1} z) \\cdots (1-{1\\over z_m} z)(1- {1\\over z_1} \\beta z^{-1})\n",
    "\\cdots(1-{1\\over z_m} \\beta z^{-1})\n",
    "$$\n",
    "\n",
    "Now define $ c(z) = \\sum^m_{j=0} c_j \\, z^j $ as\n",
    "\n",
    "\n",
    "<a id='equation-onenine'></a>\n",
    "$$\n",
    "c\\,(z)=\\Bigl[(-1)^m z_0\\, z_1 \\cdots z_m\\Bigr]^{1/2} (1-{z\\over z_1}) \\,\n",
    "(1-{z\\over z_2}) \\cdots (1- {z\\over z_m}) \\tag{15}\n",
    "$$\n",
    "\n",
    "Notice that [(14)](#equation-oneeight) can be written\n",
    "\n",
    "\n",
    "<a id='equation-oneten'></a>\n",
    "$$\n",
    "h + d \\ (\\beta z^{-1})\\ d\\ (z) = c\\,(\\beta z^{-1})\\,c\\,(z) \\tag{16}\n",
    "$$\n",
    "\n",
    "It is useful to write [(15)](#equation-onenine) as\n",
    "\n",
    "\n",
    "<a id='equation-oneeleven'></a>\n",
    "$$\n",
    "c(z) = c_0(1-\\lambda_1\\, z) \\ldots (1-\\lambda_m z) \\tag{17}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\n",
    "c_0\n",
    "= \\left[(-1)^m\\, z_0\\, z_1 \\cdots z_m\\right]^{1/2};\n",
    "\\quad \\lambda_j={1 \\over z_j},\\,\\  j=1, \\ldots, m\n",
    "$$\n",
    "\n",
    "Since $ \\vert z_j \\vert > \\sqrt \\beta \\hbox { for } j = 1, \\ldots, m $ it\n",
    "follows that $ \\vert \\lambda_j \\vert < 1/\\sqrt \\beta $ for $ j = 1,\n",
    "\\ldots, m $\n",
    "\n",
    "Using [(17)](#equation-oneeleven), we can express the factorization [(16)](#equation-oneten) as\n",
    "\n",
    "$$\n",
    "h+d (\\beta z^{-1})d(z) = c^2_0 (1-\\lambda_1 z) \\cdots\n",
    "(1 - \\lambda_m z) (1-\\lambda_1 \\beta z^{-1})\n",
    "\\cdots (1 - \\lambda_m \\beta z^{-1})\n",
    "$$\n",
    "\n",
    "In sum, we have constructed a factorization [(16)](#equation-oneten) of the characteristic\n",
    "polynomial for the Euler equation in which the zeros of $ c(z) $\n",
    "exceed $ \\beta^{1/2} $ in modulus, and the zeros of\n",
    "$ c\\,(\\beta z^{-1}) $ are less than $ \\beta^{1/2} $ in modulus\n",
    "\n",
    "Using [(16)](#equation-oneten), we now write the Euler equation as\n",
    "\n",
    "$$\n",
    "c(\\beta L^{-1})\\,c\\,(L)\\, y_t = a_t\n",
    "$$\n",
    "\n",
    "The unique solution of the Euler equation that satisfies condition [(12)](#equation-onesix)\n",
    "is\n",
    "\n",
    "\n",
    "<a id='equation-onethirteen'></a>\n",
    "$$\n",
    "c(L)\\,y_t = c\\,(\\beta L^{-1})^{-1}a_t \\tag{18}\n",
    "$$\n",
    "\n",
    "This can be established by using an argument paralleling that in\n",
    "chapter IX of [[Sar87]](https://lectures.quantecon.org/zreferences.html#sargent1987)\n",
    "\n",
    "To exhibit the solution in a form\n",
    "paralleling that of [[Sar87]](https://lectures.quantecon.org/zreferences.html#sargent1987), we use [(17)](#equation-oneeleven) to write\n",
    "[(18)](#equation-onethirteen) as\n",
    "\n",
    "\n",
    "<a id='equation-junk'></a>\n",
    "$$\n",
    "(1-\\lambda_1 L) \\cdots (1 - \\lambda_mL)y_t = {c^{-2}_0 a_t \\over (1-\\beta \\lambda_1 L^{-1}) \\cdots (1 - \\beta \\lambda_m L^{-1})} \\tag{19}\n",
    "$$\n",
    "\n",
    "Using [partial fractions](https://en.wikipedia.org/wiki/Partial_fraction_decomposition), we can write the characteristic polynomial on\n",
    "the right side of [(19)](#equation-junk) as\n",
    "\n",
    "$$\n",
    "\\sum^m_{j=1} {A_j \\over 1 - \\lambda_j \\, \\beta L^{-1}}\n",
    " \\quad \\text{where} \\quad\n",
    "A_j := {c^{-2}_0 \\over \\prod_{i \\not= j}(1-{\\lambda_i \\over \\lambda_j})}\n",
    "$$\n",
    "\n",
    "Then [(19)](#equation-junk) can be written\n",
    "\n",
    "$$\n",
    "(1-\\lambda_1 L) \\cdots (1-\\lambda_m L) y_t = \\sum^m_{j=1} \\, {A_j \\over 1 -\n",
    "\\lambda_j \\, \\beta L^{-1}} a_t\n",
    "$$\n",
    "\n",
    "or\n",
    "\n",
    "\n",
    "<a id='equation-onefifteen'></a>\n",
    "$$\n",
    "(1 - \\lambda_1 L) \\cdots (1 - \\lambda_m L) y_t = \\sum^m_{j=1}\\, A_j\n",
    "\\sum^\\infty_{k=0}\\, (\\lambda_j\\beta)^k\\, a_{t+k} \\tag{20}\n",
    "$$\n",
    "\n",
    "Equation [(20)](#equation-onefifteen) expresses the optimum sequence for $ y_t $ in terms\n",
    "of $ m $ lagged $ y $’s, and $ m $ weighted infinite\n",
    "geometric sums of future $ a_t $’s\n",
    "\n",
    "Furthermore, [(20)](#equation-onefifteen) is the unique solution of the Euler equation that satisfies the initial conditions and condition [(12)](#equation-onesix)\n",
    "\n",
    "In effect, condition [(12)](#equation-onesix) compels us to\n",
    "solve the “unstable” roots of $ h+d (\\beta z^{-1})d(z) $ forward\n",
    "(see [[Sar87]](https://lectures.quantecon.org/zreferences.html#sargent1987))\n",
    "\n",
    "The step of factoring the polynomial $ h+d (\\beta z^{-1})\\, d(z) $ into\n",
    "$ c\\, (\\beta z^{-1})c\\,(z) $, where the zeros of $ c\\,(z) $ all\n",
    "have modulus exceeding $ \\sqrt\\beta $, is central to solving the problem\n",
    "\n",
    "We note two features of the solution [(20)](#equation-onefifteen)\n",
    "\n",
    "- Since $ \\vert \\lambda_j \\vert < 1/\\sqrt \\beta $ for all $ j $, it follows that $ (\\lambda_j \\ \\beta) < \\sqrt \\beta $  \n",
    "- The assumption that $ \\{ a_t \\} $ is of exponential order less than $ 1 /\\sqrt \\beta $ is sufficient to guarantee that the geometric sums of future $ a_t $’s on the right side of [(20)](#equation-onefifteen) converge  \n",
    "\n",
    "\n",
    "We immediately see that those sums will\n",
    "converge under the weaker condition that $ \\{ a_t\\} $ is of\n",
    "exponential order less than $ \\phi^{-1} $ where\n",
    "$ \\phi = \\max \\, \\{\\beta \\lambda_i, i=1,\\ldots,m\\} $\n",
    "\n",
    "Note that with $ a_t $ identically zero, [(20)](#equation-onefifteen) implies that\n",
    "in general $ \\vert y_t \\vert $ eventually grows exponentially at a\n",
    "rate given by $ \\max_i \\vert \\lambda_i \\vert $\n",
    "\n",
    "The condition\n",
    "$ \\max_i \\vert \\lambda_i \\vert <1 /\\sqrt \\beta $ guarantees that\n",
    "condition [(12)](#equation-onesix) is satisfied\n",
    "\n",
    "In fact, $ \\max_i \\vert \\lambda_i\n",
    "\\vert < 1 /\\sqrt \\beta $ is a necessary condition for [(12)](#equation-onesix) to hold\n",
    "\n",
    "Were [(12)](#equation-onesix) not satisfied, the objective function would diverge to $ - \\infty $, implying that the $ y_t $ path could not be optimal\n",
    "\n",
    "For example, with $ a_t = 0 $, for all $ t $, it is easy to describe a naive (nonoptimal) policy for $ \\{y_t, t\\geq 0\\} $ that gives a finite value of [(17)](#equation-oneeleven)\n",
    "\n",
    "We can simply let $ y_t = 0 \\hbox { for } t\\geq 0 $\n",
    "\n",
    "This policy involves at most $ m $ nonzero values of\n",
    "$ hy^2_t $ and $ [d(L)y_t]^2 $, and so yields a finite value of\n",
    "[(1)](#equation-oneone)\n",
    "\n",
    "Therefore it is easy to dominate a path that violates [(12)](#equation-onesix)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Undiscounted Problems\n",
    "\n",
    "It is worthwhile focusing on a special case of the LQ problems above:\n",
    "the undiscounted problem that emerges when $ \\beta = 1 $\n",
    "\n",
    "In this case, the Euler equation is\n",
    "\n",
    "$$\n",
    "\\Bigl( h + d(L^{-1})d(L) \\Bigr)\\, y_t = a_t\n",
    "$$\n",
    "\n",
    "The factorization of the characteristic polynomial [(16)](#equation-oneten) becomes\n",
    "\n",
    "$$\n",
    "\\Bigl(h+d \\, (z^{-1})d(z)\\Bigr) = c\\,(z^{-1})\\, c\\,(z)\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "c\\,(z) &= c_0 (1 - \\lambda_1 z) \\ldots (1 - \\lambda_m z) \\cr\n",
    "c_0 &= \\Bigl[(-1)^m z_0 z_1 \\ldots z_m\\Bigr ] \\cr\n",
    "\\vert \\lambda_j \\vert &< 1 \\, \\hbox { for } \\, j = 1, \\ldots, m\\cr\n",
    "\\lambda_j &= \\frac{1}{z_j} \\hbox{ for } j=1,\\ldots, m\\cr\n",
    "z_0 &= \\hbox{ constant}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The solution of the problem becomes\n",
    "\n",
    "$$\n",
    "(1 - \\lambda_1 L) \\cdots (1 - \\lambda_m L) y_t = \\sum^m_{j=1} A_j\n",
    "\\sum^\\infty_{k=0} \\lambda^k_j a_{t+k}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transforming discounted to undiscounted problem\n",
    "\n",
    "Discounted problems can always be converted into undiscounted problems via a simple transformation\n",
    "\n",
    "Consider problem [(1)](#equation-oneone) with $ 0 < \\beta < 1 $\n",
    "\n",
    "Define the transformed variables\n",
    "\n",
    "\n",
    "<a id='equation-onetwenty'></a>\n",
    "$$\n",
    "\\tilde a_t = \\beta^{t/2} a_t,\\ \\tilde y_t = \\beta^{t/2} y_t \\tag{21}\n",
    "$$\n",
    "\n",
    "Then notice that $ \\beta^t\\,[d\\, (L) y_t ]^2=[\\tilde d\\,(L)\\tilde y_t]^2 $ with\n",
    "$ \\tilde d \\,(L)=\\sum^m_{j=0} \\tilde d_j\\, L^j $ and $ \\tilde d_j = \\beta^{j/2}\n",
    "d_j $\n",
    "\n",
    "Then the original criterion function [(1)](#equation-oneone) is equivalent to\n",
    "\n",
    "\n",
    "<a id='equation-oneoneprime'></a>\n",
    "$$\n",
    "\\lim_{N \\rightarrow \\infty}\n",
    "\\sum^N_{t=0}\n",
    "\\{\\tilde a_t\\, \\tilde y_t - {1 \\over 2} h\\,\\tilde y^2_t - {1\\over 2}\n",
    "[ \\tilde d\\,(L)\\, \\tilde y_t]^2 \\} \\tag{22}\n",
    "$$\n",
    "\n",
    "which is to be maximized over sequences $ \\{\\tilde y_t,\\  t=0, \\ldots\\} $ subject to\n",
    "$ \\tilde y_{-1}, \\cdots, \\tilde y_{-m} $ given and $ \\{\\tilde a_t,\\  t=1, \\ldots\\} $ a known bounded sequence\n",
    "\n",
    "The Euler equation for this problem is $ [h+\\tilde d \\,(L^{-1}) \\, \\tilde d\\, (L) ]\\, \\tilde y_t = \\tilde a_t $\n",
    "\n",
    "The solution is\n",
    "\n",
    "$$\n",
    "(1 - \\tilde \\lambda_1 L) \\cdots (1 - \\tilde \\lambda_m L)\\,\\tilde y_t =\n",
    "\\sum^m_{j=1} \\tilde A_j \\sum^\\infty_{k=0} \\tilde \\lambda^k_j \\, \\tilde a_{t+k}\n",
    "$$\n",
    "\n",
    "or\n",
    "\n",
    "\n",
    "<a id='equation-onetwentyone'></a>\n",
    "$$\n",
    "\\tilde y_t = \\tilde f_1 \\, \\tilde y_{t-1} + \\cdots + \\tilde f_m\\,\n",
    "\\tilde y_{t-m} + \\sum^m_{j=1} \\tilde A_j \\sum^\\infty_{k=0} \\tilde \\lambda^k_j\n",
    "\\, \\tilde a_{t+k}, \\tag{23}\n",
    "$$\n",
    "\n",
    "where $ \\tilde c \\,(z^{-1}) \\tilde c\\,(z) = h + \\tilde d\\,(z^{-1}) \\tilde d \\,(z) $, and where\n",
    "\n",
    "$$\n",
    "\\bigl[(-1)^m\\, \\tilde z_0 \\tilde z_1 \\ldots \\tilde z_m \\bigr]^{1/2}\n",
    "(1 - \\tilde \\lambda_1\\, z) \\ldots (1 - \\tilde \\lambda_m\\, z) = \\tilde c\\,(z),\n",
    "\\hbox { where } \\ \\vert \\tilde \\lambda_j \\vert < 1\n",
    "$$\n",
    "\n",
    "We leave it to the reader to show that [(23)](#equation-onetwentyone) implies the equivalent form of the solution\n",
    "\n",
    "$$\n",
    "y_t = f_1\\, y_{t-1} + \\cdots + f_m\\, y_{t-m} + \\sum^m_{j=1} A_j\n",
    "\\sum^\\infty_{k=0} \\, (\\lambda_j\\, \\beta)^k \\, a_{t+k}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "\n",
    "<a id='equation-onetwentythree'></a>\n",
    "$$\n",
    "f_j = \\tilde f_j\\, \\beta^{-j/2},\\  A_j = \\tilde A_j,\\  \\lambda_j = \\tilde\n",
    "\\lambda_j \\, \\beta^{-1/2} \\tag{24}\n",
    "$$\n",
    "\n",
    "The transformations [(21)](#equation-onetwenty) and the inverse formulas\n",
    "[(24)](#equation-onetwentythree) allow us to solve a discounted problem by first\n",
    "solving a related undiscounted problem"
   ]
  },
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   "source": [
    "## Implementation\n",
    "\n",
    "Code that computes solutions to the LQ problem using the methods described\n",
    "above can be found in file [control_and_filter.jl](https://github.com/QuantEcon/QuantEcon.lectures.code/blob/master/lu_tricks/control_and_filter.jl)\n",
    "\n",
    "Here’s how it looks"
   ]
  },
  {
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   "metadata": {
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    {
     "data": {
      "text/plain": [
       "optimal_y (generic function with 2 methods)"
      ]
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     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function LQFilter(d, h, y_m;\n",
    "                    r = nothing,\n",
    "                    β = nothing,\n",
    "                    h_eps = nothing)\n",
    "\n",
    "    m = length(d) - 1\n",
    "    m == length(y_m) || throw(ArgumentError(\"y_m and d must be of same length = $m\"))\n",
    "\n",
    "    # define the coefficients of ϕ up front\n",
    "    ϕ = zeros(2m + 1)\n",
    "    for i in -m:m\n",
    "        ϕ[m-i+1] = sum(diag(d*d', -i))\n",
    "    end\n",
    "    ϕ[m+1] = ϕ[m+1] + h\n",
    "\n",
    "    # if r is given calculate the vector ϕ_r\n",
    "    if isnothing(r)\n",
    "        k = nothing\n",
    "        ϕ_r = nothing\n",
    "    else\n",
    "        k = size(r, 1) - 1\n",
    "        ϕ_r = zeros(2k + 1)\n",
    "\n",
    "        for i = -k:k\n",
    "            ϕ_r[k-i+1] = sum(diag(r*r', -i))\n",
    "        end\n",
    "\n",
    "        if h_eps != nothing\n",
    "            ϕ_r[k+1] = ϕ_r[k+1] + h_eps\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # if β is given, define the transformed variables\n",
    "    if isnothing(β)\n",
    "        β = 1.0\n",
    "    else\n",
    "        d = β.^(collect(0:m)/2) * d\n",
    "        y_m = y_m * β.^(- collect(1:m)/2)\n",
    "    end\n",
    "\n",
    "    return (d = d, h = h, y_m = y_m, m = m, ϕ = ϕ, β = β, ϕ_r = ϕ_r, k = k)\n",
    "end\n",
    "\n",
    "function construct_W_and_Wm(lqf, N)\n",
    "\n",
    "    @unpack d, m = lqf\n",
    "    W = zeros(N + 1, N + 1)\n",
    "    W_m = zeros(N + 1, m)\n",
    "\n",
    "    # terminal conditions\n",
    "    D_m1 = zeros(m + 1, m + 1)\n",
    "    M = zeros(m + 1, m)\n",
    "\n",
    "    # (1) Constuct the D_{m+1} matrix using the formula\n",
    "\n",
    "    for j in 1:(m+1)\n",
    "        for k in j:(m+1)\n",
    "            D_m1[j, k] = dot(d[1:j, 1], d[k-j+1:k, 1])\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # Make the matrix symmetric\n",
    "    D_m1 = D_m1 + D_m1' - Diagonal(diag(D_m1))\n",
    "\n",
    "    # (2) Construct the M matrix using the entries of D_m1\n",
    "\n",
    "    for j in 1:m\n",
    "        for i in (j + 1):(m + 1)\n",
    "            M[i, j] = D_m1[i-j, m+1]\n",
    "        end\n",
    "    end\n",
    "    M\n",
    "\n",
    "    # Euler equations for t = 0, 1, ..., N-(m+1)\n",
    "    @unpack ϕ, h = lqf\n",
    "\n",
    "    W[1:(m + 1), 1:(m + 1)] = D_m1 + h * I\n",
    "    W[1:(m + 1), (m + 2):(2m + 1)] = M\n",
    "\n",
    "    for (i, row) in enumerate((m + 2):(N + 1 - m))\n",
    "        W[row, (i + 1):(2m + 1 + i)] = ϕ'\n",
    "    end\n",
    "\n",
    "    for i in 1:m\n",
    "        W[N - m + i + 1 , end-(2m + 1 - i)+1:end] = ϕ[1:end-i]\n",
    "    end\n",
    "\n",
    "    for i in 1:m\n",
    "        W_m[N - i + 2, 1:(m - i)+1] = ϕ[(m + 1 + i):end]\n",
    "    end\n",
    "\n",
    "    return W, W_m\n",
    "end\n",
    "\n",
    "function roots_of_characteristic(lqf)\n",
    "    @unpack m, ϕ = lqf\n",
    "\n",
    "    # Calculate the roots of the 2m-polynomial\n",
    "    ϕ_poly=Poly(ϕ[end:-1:1])\n",
    "    proots = roots(ϕ_poly)\n",
    "\n",
    "    # sort the roots according to their length (in descending order)\n",
    "    roots_sorted = sort(proots, by=abs)[end:-1:1]\n",
    "    z_0 = sum(ϕ) / polyval(poly(proots), 1.0)\n",
    "    z_1_to_m = roots_sorted[1:m]     # we need only those outside the unit circle\n",
    "    λ = 1 ./ z_1_to_m\n",
    "    return z_1_to_m, z_0, λ\n",
    "end\n",
    "\n",
    "function coeffs_of_c(lqf)\n",
    "    m = lqf.m\n",
    "    z_1_to_m, z_0, λ = roots_of_characteristic(lqf)\n",
    "    c_0 = (z_0 * prod(z_1_to_m) * (-1.0)^m)^(0.5)\n",
    "    c_coeffs = coeffs(poly(z_1_to_m)) * z_0 / c_0\n",
    "    return c_coeffs\n",
    "end\n",
    "\n",
    "function solution(lqf)\n",
    "    z_1_to_m, z_0, λ = roots_of_characteristic(lqf)\n",
    "    c_0 = coeffs_of_c(lqf)[end]\n",
    "    A = zeros(lqf.m)\n",
    "    for j in 1:m\n",
    "        denom = 1 - λ/λ[j]\n",
    "        A[j] = c_0^(-2) / prod(denom[1:m .!= j])\n",
    "    end\n",
    "    return λ, A\n",
    "end\n",
    "\n",
    "function construct_V(lqf; N=nothing)\n",
    "    @unpack ϕ_r, k = lqf\n",
    "    V = zeros(N, N)\n",
    "    for i in 1:N\n",
    "        for j in 1:N\n",
    "            if abs(i-j) <= k\n",
    "                V[i, j] = ϕ_r[k + abs(i-j)+1]\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    return V\n",
    "end\n",
    "\n",
    "function simulate_a(lqf, N)\n",
    "    V = construct_V(lqf, N + 1)\n",
    "    d = MVNSampler(zeros(N + 1), V)\n",
    "    return rand(d)\n",
    "end\n",
    "\n",
    "function predict(lqf, a_hist, t)\n",
    "    N = length(a_hist) - 1\n",
    "    V = construct_V(lqf, N + 1)\n",
    "\n",
    "    aux_matrix = zeros(N + 1, N + 1)\n",
    "    aux_matrix[1:t+1 , 1:t+1 ] .= I + zeros(t+1, t+1)\n",
    "    L = chol(V)'\n",
    "    Ea_hist = inv(L) * aux_matrix * L * a_hist\n",
    "\n",
    "    return Ea_hist\n",
    "end\n",
    "\n",
    "function optimal_y(lqf, a_hist, t = nothing)\n",
    "    @unpack β, y_m, m = lqf\n",
    "\n",
    "    N = length(a_hist) - 1\n",
    "    W, W_m = construct_W_and_Wm(lqf, N)\n",
    "\n",
    "    F = lu(W)\n",
    "\n",
    "    L, U = F.L, F.U\n",
    "    D = Diagonal(1.0./diag(U))\n",
    "    U = D * U\n",
    "    L = L * Diagonal(1.0./diag(D))\n",
    "\n",
    "    J = reverse(I + zeros(N+1, N + 1), dims = 2)\n",
    "\n",
    "    if isnothing(t)                     # if the problem is deterministic\n",
    "        a_hist = J * a_hist\n",
    "\n",
    "        # transform the a sequence if β is given\n",
    "        if β != 1\n",
    "            a_hist =  reshape(a_hist * (β^(collect(N:0)/ 2)), N + 1, 1)\n",
    "        end\n",
    "\n",
    "        ā = a_hist - W_m * y_m        # ā from the lecutre\n",
    "        Uy = \\(L, ā)                  # U @ ȳ = L^{-1}ā from the lecture\n",
    "        ȳ = \\(U, Uy)                  # ȳ = U^{-1}L^{-1}ā\n",
    "        # Reverse the order of ȳ with the matrix J\n",
    "        J = reverse(I + zeros(N+m+1, N + m + 1), dims = 2)\n",
    "        y_hist = J * vcat(ȳ, y_m)     # y_hist : concatenated y_m and ȳ\n",
    "        # transform the optimal sequence back if β is given\n",
    "        if β != 1\n",
    "            y_hist = y_hist .* β.^(- collect(-m:N)/2)\n",
    "        end\n",
    "\n",
    "    else  # if the problem is stochastic and we look at it\n",
    "        Ea_hist = reshape(predict(lqf, a_hist, t), N + 1, 1)\n",
    "        Ea_hist = J * Ea_hist\n",
    "\n",
    "        ā = Ea_hist - W_m * y_m       # ā from the lecutre\n",
    "        Uy = \\(L, ā)                  # U @ ȳ = L^{-1}ā from the lecture\n",
    "        ȳ = \\(U, Uy)                  # ȳ = U^{-1}L^{-1}ā\n",
    "\n",
    "        # Reverse the order of ȳ with the matrix J\n",
    "        J = reverse(I + zeros(N + m + 1, N + m + 1), dims = 2)\n",
    "        y_hist = J * vcat(ȳ, y_m)     # y_hist : concatenated y_m and ȳ\n",
    "    end\n",
    "    return y_hist, L, U, ȳ\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "In this application we’ll have one lag, with\n",
    "\n",
    "$$\n",
    "d(L) y_t =  \\gamma(I - L) y_t = \\gamma (y_t - y_{t-1})\n",
    "$$\n",
    "\n",
    "Suppose for the moment that $ \\gamma = 0 $\n",
    "\n",
    "Then the intertemporal component of the LQ problem disappears, and the agent\n",
    "simply wants to maximize $ a_t y_t - hy^2_t / 2 $ in each period\n",
    "\n",
    "This means that the agent chooses $ y_t = a_t / h $\n",
    "\n",
    "In the following we’ll set $ h = 1 $, so that the agent just wants to\n",
    "track the $ \\{a_t\\} $ process\n",
    "\n",
    "However, as we increase $ \\gamma $, the agent gives greater weight to a smooth time path\n",
    "\n",
    "Hence $ \\{y_t\\} $ evolves as a smoothed version of $ \\{a_t\\} $\n",
    "\n",
    "The $ \\{a_t\\} $ sequence we’ll choose as a stationary cyclic process plus some white noise\n",
    "\n",
    "Here’s some code that generates a plot when $ \\gamma = 0.8 $"
   ]
  },
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    {
     "data": {
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"
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gr(fmt=:png);\n",
    "\n",
    "# set seed and generate a_t sequence\n",
    "Random.seed!(123)\n",
    "n = 100\n",
    "a_seq = sin.(range(0,  5 * pi, length = n)) .+ 2 + 0.1 * randn(n)\n",
    "\n",
    "function plot_simulation(;γ=0.8, m=1, h=1., y_m=2.)\n",
    "  d = γ * [1, -1]\n",
    "  y_m = [y_m]\n",
    "\n",
    "  testlq = LQFilter(d, h, y_m)\n",
    "  y_hist, L, U, y = optimal_y(testlq, a_seq)\n",
    "  y = y[end:-1:1]  # reverse y\n",
    "\n",
    "  # plot simulation results\n",
    "  time = 1:length(y)\n",
    "  plt = plot(time, a_seq / h, lw=2, color=:black, alpha=0.8, marker = :circle,\n",
    "             markersize = 2, label=\"a_t\")\n",
    "  plot!(plt, time, y, lw=2, color=:blue, marker = :circle,  markersize = 2, alpha=0.8,\n",
    "        label=\"y_t\")\n",
    "  plot!(plt, xlabel=\"Time\", grid=true, xlim=(0,maximum(time)), legend=:bottomleft)\n",
    "end\n",
    "\n",
    "plot_simulation()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here’s what happens when we change $ \\gamma $ to 5.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot_simulation(γ=5.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here’s $ \\gamma = 10 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot_simulation(γ=10.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "Consider solving a discounted version $ (\\beta < 1) $ of problem\n",
    "[(1)](#equation-oneone), as follows\n",
    "\n",
    "Convert [(1)](#equation-oneone) to the undiscounted problem [(22)](#equation-oneoneprime)\n",
    "\n",
    "Let the solution of [(22)](#equation-oneoneprime) in feedback form be\n",
    "\n",
    "$$\n",
    "(1 - \\tilde \\lambda_1 L)\\, \\cdots\\, (1 - \\tilde \\lambda_m L) \\tilde y_t =\n",
    "\\sum^m_{j=1} \\tilde A_j \\sum^\\infty_{k=0} \\tilde \\lambda^k_j \\tilde a_{t+k}\n",
    "$$\n",
    "\n",
    "or\n",
    "\n",
    "\n",
    "<a id='equation-estar'></a>\n",
    "$$\n",
    "\\tilde y_t = \\tilde f_1 \\tilde y_{t-1} + \\cdots + \\tilde f_m \\tilde y_{t-m} +\n",
    "\\sum^m_{j=1} \\tilde A_j \\sum^\\infty_{k=0} \\tilde \\lambda^k_j \\tilde a_{t+k} \\tag{25}\n",
    "$$\n",
    "\n",
    "Here\n",
    "\n",
    "- $ h + \\tilde d (z^{-1}) \\tilde d (z) = \\tilde c (z^{-1}) \\tilde c (z) $  \n",
    "- $ \\tilde c (z) = [(-1)^m \\tilde z_0 \\tilde z_1 \\cdots \\tilde z_m ]^{1/2} (1 - \\tilde \\lambda_1 z) \\cdots (1 - \\tilde \\lambda_m z) $  \n",
    "\n",
    "\n",
    "where the $ \\tilde z_j $ are the zeros of $ h +\\tilde d (z^{-1})\\, \\tilde d(z) $\n",
    "\n",
    "Prove that [(25)](#equation-estar) implies that the solution for $ y_t $ in feedback form is\n",
    "\n",
    "$$\n",
    "y_t = f_1 y_{t-1} + \\ldots + f_m y_{t-m} + \\sum^m_{j=1} A_j\n",
    "\\sum^\\infty_{k=0} \\beta^k \\lambda^k_j a_{t+k}\n",
    "$$\n",
    "\n",
    "where $ f_j = \\tilde f_j \\beta^{-j/2}, A_j = \\tilde A_j $, and $ \\lambda_j = \\tilde \\lambda_j \\beta^{-1/2} $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2\n",
    "\n",
    "Solve the optimal control problem, maximize\n",
    "\n",
    "$$\n",
    "\\sum^2_{t=0}\\ \\Bigl\\{a_t y_t - {1 \\over 2} [(1 - 2 L) y_t]^2\\Bigr\\}\n",
    "$$\n",
    "\n",
    "subject to $ y_{-1} $ given, and $ \\{ a_t\\} $ a known bounded sequence\n",
    "\n",
    "Express the solution in the “feedback form” [(20)](#equation-onefifteen), giving numerical values for the coefficients\n",
    "\n",
    "Make sure that the boundary conditions [(5)](#equation-onefive) are satisfied\n",
    "\n",
    "(Note: this problem differs from the problem in the text in one important way: instead of $ h > 0 $ in [(1)](#equation-oneone), $ h = 0 $. This has an important influence on the solution.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 3\n",
    "\n",
    "Solve the infinite time optimal control problem to maximize\n",
    "\n",
    "$$\n",
    "\\lim_{N \\rightarrow \\infty}\n",
    "\\sum^N_{t=0}\\, -\\, {1 \\over 2} [(1 -2 L) y_t]^2,\n",
    "$$\n",
    "\n",
    "subject to $ y_{-1} $ given. Prove that the solution is\n",
    "\n",
    "$$\n",
    "y_t = 2y_{t-1} = 2^{t+1} y_{-1} \\qquad t > 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 4\n",
    "\n",
    "Solve the infinite time problem, to maximize\n",
    "\n",
    "$$\n",
    "\\lim_{N \\rightarrow \\infty}\\ \\sum^N_{t=0}\\ (.0000001)\\, y^2_t - {1 \\over 2}\n",
    "[(1 - 2 L) y_t]^2\n",
    "$$\n",
    "\n",
    "subject to $ y_{-1} $ given. Prove that the solution $ y_t = 2y_{t-1} $  violates condition [(12)](#equation-onesix), and so\n",
    "is not optimal\n",
    "\n",
    "Prove that the optimal solution is approximately $ y_t = .5 y_{t-1} $"
   ]
  }
 ],
 "metadata": {
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   "language": "julia",
   "name": "julia-1.2"
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   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.2.0"
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  "title": "Classical Control with Linear Algebra"
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