{ "cells": [ { "cell_type": "markdown", "id": "c7c70e96-a2ad-46be-8ff6-8755b9366427", "metadata": {}, "source": [ "Submit your answers to gradescope before or at the latest at 12:50pm." ] }, { "cell_type": "code", "execution_count": 1, "id": "71baefdc-e6df-4d82-ace7-a177755492fc", "metadata": {}, "outputs": [], "source": [ "using Plots" ] }, { "cell_type": "code", "execution_count": 2, "id": "75b1368a-c89d-46db-832d-a08c5d03f7d0", "metadata": {}, "outputs": [], "source": [ "using SymPy" ] }, { "cell_type": "code", "execution_count": 3, "id": "72ad536b-743a-425a-8368-f28f74adaa34", "metadata": {}, "outputs": [], "source": [ "using FastGaussQuadrature" ] }, { "cell_type": "code", "execution_count": 4, "id": "185c4616-6338-49b2-9c9b-acff0de58d47", "metadata": {}, "outputs": [], "source": [ "using NLsolve" ] }, { "cell_type": "markdown", "id": "083feb68-cdb0-44bb-8ead-ccab2f5ab18a", "metadata": {}, "source": [ "# Question 1" ] }, { "cell_type": "markdown", "id": "8fb299a9-748f-4d66-9218-7e0bfd31ac75", "metadata": {}, "source": [ "Experience has shown that 6\\% of customers who make reservations\n", "for a restaurant with 20 tables do not show up.\n", "\n", "How many more reservations can the restaurant accept and be able\n", "to give everybody who shows up a table with 90\\% confidence?" ] }, { "cell_type": "markdown", "id": "773de547-497f-447d-9399-0614a8012c9b", "metadata": {}, "source": [ "## answer to question 1" ] }, { "cell_type": "markdown", "id": "5252f1af-f6d3-46fa-9282-d68907a1826c", "metadata": {}, "source": [ "We apply the binomial distribution." ] }, { "cell_type": "code", "execution_count": 5, "id": "b7bbddee-9328-40e4-b1e5-c2fc8ed820be", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "f (generic function with 1 method)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "c(n,k) = factorial(n)/(factorial(k)*factorial(n-k))\n", "f(n) = sum([c(big(n),big(k))*((94/100)^k)*((4/100)^(n-k)) for k=0:19])" ] }, { "cell_type": "markdown", "id": "3ec828f4-dc3f-4978-8f6e-bfe361967a38", "metadata": {}, "source": [ "The function ``f(n)`` gives the probability that **not** all 20 tables are occupied." ] }, { "cell_type": "code", "execution_count": 6, "id": "719c8a80-efe8-4232-b222-0cf88aa1c431", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "20: 0.3775017306236322695474394676061501508837123575201014606188838732137063697638269\n", "21: 0.1378667031059904123915053535900890400483630975796435726839118717882948353069729\n", "22: 0.03763367202369953607724637094980651436733237445176038187176837156724172791764408\n", "23: 0.00828812745730869555387131760092324938064864509848971778255057691875047326453888\n", "24: 0.001546004549201647255970317668307509972173611203169369596951141122311275113222475\n" ] } ], "source": [ "for n=20:24\n", " println(n, \": \", f(n))\n", "end" ] }, { "cell_type": "markdown", "id": "30a1b01e-5ae4-408a-bfb3-1aef0449892b", "metadata": {}, "source": [ "If we do not accept more than 20 reservations, then we have a 56% probability of an empty table. With 22, this probability drops to 6%. Alternatively, accepting 2 more reservations, gives a 94% probability of all 20 tables to be occupied." ] }, { "cell_type": "markdown", "id": "30e2ccfd-d59b-44a5-892e-e9b2fe9709da", "metadata": {}, "source": [ "# Question 2" ] }, { "cell_type": "markdown", "id": "1640376e-f7e6-4708-8c57-bda78dc63e02", "metadata": {}, "source": [ "Consider the following transfer function\n", "\n", "$$\n", " H(z) = \\frac{3 z - 2}{z^2 - (7/3)z + 2/3}.\n", "$$\n", "Is the filter with this transfer function $H$ stable?\n", "\n", "In your justification of your answer, refer to the definition\n", "of a stable filter." ] }, { "cell_type": "markdown", "id": "d174dd6f-0f43-480d-aa62-6a6aed0cac14", "metadata": {}, "source": [ "## answer to question 2" ] }, { "cell_type": "markdown", "id": "be67a713-9932-4da8-a69a-595efac7af2e", "metadata": {}, "source": [ "A filter with transfer function $H$ is stable if the absolute value its coefficients $h_k$ of its z-transform are all strictly less than 1: $|h_k| < 1$ for all $k$." ] }, { "cell_type": "markdown", "id": "9398a3d6-f0ca-4065-8b90-9f813f58f1ee", "metadata": {}, "source": [ "We look at the poles of $H(z)$, the roots of the denominator." ] }, { "cell_type": "code", "execution_count": 7, "id": "4179b4ef-2193-462c-8ca3-0b42e7453ab2", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$z$" ], "text/plain": [ "z" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z = Sym(\"z\")" ] }, { "cell_type": "code", "execution_count": 8, "id": "4e03e220-dce2-4310-9a0c-c91247fa4956", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "\\begin{equation*}\\begin{cases}0.333333333333333 & \\text{=>} &1\\\\2.0 & \\text{=>} &1\\\\\\end{cases}\\end{equation*}" ], "text/plain": [ "Dict{Sym{PyCall.PyObject}, Sym{PyCall.PyObject}} with 2 entries:\n", " 0.333333333333333 => 1\n", " 2.00000000000000 => 1" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "q = z^2 - (7/3)*z + 2/3\n", "roots(q)" ] }, { "cell_type": "markdown", "id": "b82be253-4b95-45df-847b-ec7799be4caf", "metadata": {}, "source": [ "We see that one of the roots is larger than 1, that is: 2.\n", "If we apply a partial fraction decomposition, then we have one term where $z-2$ appears in the denominator\n", "and then consider the expansion:\n", "$$\n", " \\frac{1}{2 - z} = \\sum_{k=0}^\\infty 2^k z^k,\n", "$$\n", "or equivalently:\n", "$$\n", " \\frac{-z}{1 - 2 z^{-1}} = -z \\sum_{k=0}^\\infty 2^k z^{-k}.\n", "$$" ] }, { "cell_type": "markdown", "id": "7ff9d15e-e02f-4386-98a8-b91b6d0ea737", "metadata": {}, "source": [ "As $2^k$ grows exponentially large, we have one component in the decomposition of $H$ where the coefficients are larger than 1 in absolute value, so the filter is not stable." ] }, { "cell_type": "markdown", "id": "51c17771-de57-42a4-bf7f-06b88c501f33", "metadata": {}, "source": [ "# Question 3" ] }, { "cell_type": "markdown", "id": "70eaec52-1c17-4cfd-9b91-05127ffb51e9", "metadata": {}, "source": [ "Let $f(x) = 7 \\exp(4 x) + 3 \\exp(8 x)$.\n", "\n", "Compute a least squares approximation for $f$ using a basis\n", "of Chebyshev polynomials.\n", "\n", "How large should the degree of the approximant be for\n", "the largest error to be less than $10^{-4}$ over $[-1,+1]$?" ] }, { "cell_type": "markdown", "id": "02be84c3-faf2-4893-9da9-96251c90a732", "metadata": {}, "source": [ "## answer to question 3" ] }, { "cell_type": "code", "execution_count": 9, "id": "bf4c7c82-1d4c-487e-bba0-51bdac1cd97c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "f (generic function with 1 method)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(x) = 7*exp(4*x) + 3*exp(8*x)" ] }, { "cell_type": "markdown", "id": "5197d349-2dff-4b3d-8a1e-c61700d3a428", "metadata": {}, "source": [ "Observe that we will need more than 16 nodes in the quadrature rule!" ] }, { "cell_type": "code", "execution_count": 10, "id": "db46f16f-e886-4f9a-aee1-09b45cbba151", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "rule (generic function with 1 method)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nodes, weights = gausschebyshev(24)\n", "rule(f) = sum([weights[k]*f(nodes[k]) for k=1:length(weights)])" ] }, { "cell_type": "code", "execution_count": 11, "id": "5ee2fb4a-b8e0-4ad7-9afc-d74d84dca163", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "innerproduct" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\"\"\"\n", " function innerproduct(f::Function, g::Function)\n", "\n", "returns the value of the inner product of the functions f and g,\n", "by the integral of f and g over [-1,+1] with weight 1/sqrt(1-x^2).\n", "\"\"\"\n", "function innerproduct(f::Function, g::Function)\n", " fun(x) = f(x)*g(x)\n", " return (2.0/pi)*rule(fun)\n", "end" ] }, { "cell_type": "code", "execution_count": 12, "id": "7ad88469-902f-4c33-9276-40e818278d58", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "basis" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\"\"\"\n", " function basis(n::Int64)\n", "\n", "returns an array of functions with the first n Chebyshev polynomials.\n", "\"\"\"\n", "function basis(n::Int64)\n", " t0(x) = 1 \n", " result = [t0]\n", " if(n > 1)\n", " t1(x) = x\n", " result = [result; t1]\n", " t1two(x) = 2*x\n", " for i=3:n\n", " newt(x) = t1two(x)*result[i-1](x) - result[i-2](x)\n", " result = [result; newt]\n", " end\n", " end\n", " return result\n", "end" ] }, { "cell_type": "code", "execution_count": 13, "id": "4de6ce56-5e74-4d0e-b762-c0c7d9ced477", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "coefficients" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\"\"\"\n", " function coefficients(f::Function,b::Array{Function,1})\n", "\n", "returns the inner products of f with the basis functions in b.\n", "\"\"\"\n", "function coefficients(f::Function,b::Array{Function,1})\n", " dim = length(b)\n", " result = zeros(dim)\n", " for i=1:dim\n", " result[i] = innerproduct(f,b[i])\n", " end\n", " return result\n", "end" ] }, { "cell_type": "code", "execution_count": 14, "id": "d7d54f4a-96ce-4710-8b3a-9f71b4c70afe", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "approximation" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\"\"\"\n", " function approximation(c::Array{Float64,1},b::Array{Function,1})\n", "\n", "Given in c are the coefficients of the Chebyshev approximation\n", "with the basis functions in f.\n", "Returns the approximation as a function.\n", "\"\"\"\n", "function approximation(c::Array{Float64,1},b::Array{Function,1})\n", " first(x) = c[1]*b[1](x)/2\n", " rest(x) = sum([c[k]*b[k](x) for k=2:length(c)])\n", " result(x) = first(x) + rest(x)\n", " return result\n", "end" ] }, { "cell_type": "markdown", "id": "44c5e5a9-2e5d-4cb5-97a2-11b06b09a094", "metadata": {}, "source": [ "Starting with 8 basis functions, we moved to 16, and then to 20." ] }, { "cell_type": "code", "execution_count": 15, "id": "09d74709-3a08-4427-946d-c394e9bca0e7", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(::var\"#result#approximation##4\"{var\"#rest#approximation##1\"{Vector{Float64}, Vector{Function}}, var\"#first#approximation##0\"{Vector{Float64}, Vector{Function}}}) (generic function with 1 method)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bN = basis(19)\n", "cN = coefficients(f, bN)\n", "aN = approximation(cN,bN)" ] }, { "cell_type": "code", "execution_count": 16, "id": "8361846e-742f-407d-b287-1197eb912a34", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xr = -1.0:0.01:1.0\n", "errors = [abs(f(t) - aN(t)) for t in xr]\n", "plot(xr, errors, label=\"|f(x)-pN(x)|\",legend=:topleft)" ] }, { "cell_type": "markdown", "id": "ae9a973d-d761-40d6-a63e-7c1886303487", "metadata": {}, "source": [ "The largest error occurs at the end." ] }, { "cell_type": "code", "execution_count": 17, "id": "76506bd1-1b70-496c-a115-9b234129471a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.675578409456648e-5" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eN = abs(f(1.0) - aN(1.0))" ] }, { "cell_type": "markdown", "id": "8ec88b67-4465-4ec0-a04d-365a76c2fa6b", "metadata": {}, "source": [ "We need a polynomial of degree 19 to get four decimal places correct in the least squares approximation." ] }, { "cell_type": "markdown", "id": "e2de5e4b-e350-4dd9-abb7-76016f567362", "metadata": {}, "source": [ "# Question 4" ] }, { "cell_type": "markdown", "id": "0b1b1d30-40a1-4896-b4f7-036acc86499f", "metadata": {}, "source": [ "Purchasing new equipment for $\\$10,000$ will give an annual savings\n", "of $\\$800$. \n", "\n", "Using annual compounding at an interest rate of 5\\%,\n", "how long should this new equipment last to justify its purchase?" ] }, { "cell_type": "markdown", "id": "f15ef9be-b1a8-4f3d-9841-aa5621e7a168", "metadata": {}, "source": [ "## answer to question 4" ] }, { "cell_type": "markdown", "id": "3910ef7d-27fa-40f6-af91-b85a0bca301e", "metadata": {}, "source": [ "The equation we want to solve is\n", "\n", "$$\n", " 10000 = 800 \\sum_{k=1}^{N} \\exp(-0.05k)\n", " = 800 \\left( \\frac{\\exp(-0.05(N+1)) - 1}{\\exp(-0.05) - 1} - 1 \\right)\n", "$$\n", "\n", "where $N$ is the number of years we want the equipment to last." ] }, { "cell_type": "code", "execution_count": 18, "id": "2c5447be-69e7-423a-98cc-61c3533db584", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "f!" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\"\"\"\n", "function f!(F, x)\n", "defines F[1] as the input to nlsolve.\n", "\"\"\"\n", "function f!(F, x)\n", " N = x[1]\n", " F[1] = 10000 - 800*( (exp(-0.05*(N+1)) - 1)/(exp(-0.05) - 1) - 1)\n", "end" ] }, { "cell_type": "code", "execution_count": 19, "id": "46d3cc5d-a806-4e67-a66d-24ace654f9a1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Results of Nonlinear Solver Algorithm\n", " * Algorithm: Trust-region with dogleg and autoscaling\n", " * Starting Point: [10.0]\n", " * Zero: [20.48245847552222]\n", " * Inf-norm of residuals: 0.000000\n", " * Iterations: 5\n", " * Convergence: true\n", " * |x - x'| < 0.0e+00: false\n", " * |f(x)| < 1.0e-08: true\n", " * Function Calls (f): 6\n", " * Jacobian Calls (df/dx): 6" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sol = nlsolve(f!, [10.0])" ] }, { "cell_type": "code", "execution_count": 20, "id": "70b6990d-e779-4a59-b4ad-8588625cb0ff", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "10143.137897754556" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sum([exp(-0.05*k)*800 for k=1:21])" ] }, { "cell_type": "markdown", "id": "f33efa30-c474-43c9-81cf-554aa23d8483", "metadata": {}, "source": [ "Indeed, if we compute the present values of the annual savings, then it takes 21 years till the sum of those present values exceeded the first costs." ] }, { "cell_type": "markdown", "id": "dcd78529-d5a3-4bef-8b86-ea2bf0abc236", "metadata": {}, "source": [ "# Question 5" ] }, { "cell_type": "markdown", "id": "62ac40f8-687a-4b35-b5f0-0c69786e64ab", "metadata": {}, "source": [ "Describe in one paragraph what you learned \n", "from *another* project presentation." ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.12", "language": "julia", "name": "julia-1.12" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.12.4" } }, "nbformat": 4, "nbformat_minor": 5 }