{
"cells": [
{
"cell_type": "markdown",
"id": "cb1d5eb9-cf28-46a8-9495-4e5346fcf67e",
"metadata": {},
"source": [
"# 2025-09-12 Formulation\n",
"\n",
"* Goal sharing\n",
"\n",
"* Exploration\n",
"\n",
"* Multiple roots\n",
"\n",
"* Conditioning of the rootfinding problem\n",
"\n",
"* Forward and backward stability"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "1b71447f-9300-4a9e-89d6-d09175fd7df6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"newton_hist (generic function with 1 method)"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"default(lw=4, ms=5, legendfontsize=12, xtickfontsize=12, ytickfontsize=12)\n",
"\n",
"# Here's the Newton's method with history we've been using\n",
"function newton_hist(f, fp, x0; tol=1e-12)\n",
" x = x0\n",
" hist = []\n",
" for k in 1:100 # max number of iterations\n",
" fx = f(x)\n",
" fpx = fp(x)\n",
" push!(hist, [x fx fpx])\n",
" if abs(fx) < tol\n",
" return vcat(hist...)\n",
" end\n",
" x = x - fx / fpx\n",
" end\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "f8e0f515-7d2f-4f56-8b23-4fad5e91f8dc",
"metadata": {},
"source": [
"## Exploration\n",
"\n",
"* Find a function $f \\left( x \\right)$ that models something you're interested in.\n",
"You could consider nonlinear physical models (aerodynamic drag, nonlinear elasticity), behavioral models, probability distributions, or anything else that that catches your interest.\n",
"Implement the function in Julia or another language.\n",
"\n",
"* Consider how you might know the output of such functions, but not an input.\n",
"Think from the position of different stakeholders: is the equation used differently by an experimentalist collecting data versus by someone making predictions through simulation?\n",
"How about a company or government reasoning about people versus the people their decisions may impact?\n",
"\n",
"* Formulate the map from known to desired data as a rootfinding problem and try one or more methods (Newton, bisection, etc., or use a rootfinding library).\n",
"\n",
"* Plot the inverse function (output versus input) from the standpoint of one or more stakeholder.\n",
"Are there interesting inflection points?\n",
"Are the methods reliable?\n",
"\n",
"* If there are a hierarchy of models for the application you're interested in, consider using a simpler model to provide an initial guess to a more complicated model."
]
},
{
"cell_type": "markdown",
"id": "643c9fe9-b6ca-44dd-9b2e-a779165385ae",
"metadata": {},
"source": [
"## Roots with multiplicity\n",
"\n",
"There are multiple ways to represent (monic) polynomials\n",
"\n",
"$$ f \\left( x \\right) = \\left( x - a_1 \\right) \\left( x - a_2 \\right) $$\n",
"$$ g \\left( x \\right) = x^2 - \\left( a_1 + a_2 \\right) x + a_1 a_2 $$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "635a0f6a-94a6-47a4-8808-6c0dfcf09e22",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"g (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Here is the f(x) style above\n",
"function poly_eval_prod(x, a) \n",
" product = 1\n",
" for c in a\n",
" product *= (x - c)\n",
" end\n",
" product\n",
"end\n",
"# Note, compactly poly_eval_prod(x, a) = prod(x .- a)\n",
"\n",
"# And here is the g(x) style\n",
"function poly_eval_sum(x, b)\n",
" sum = 1\n",
" for c in b\n",
" # This is known as Horner's rule\n",
" sum = x * sum + c\n",
" end\n",
" sum\n",
"end\n",
"\n",
"# Let's make a couple of functions and compare the effect of perturbations ot each\n",
"eps = 1e-10\n",
"a = [1e5, 1e5*(1+eps)] # tiny perturbation to root\n",
"b = [-(a[1] + a[2])*(1+eps), # tiny perturbation to monomial coefficent\n",
" a[1]*a[2]]\n",
"f(x) = poly_eval_prod(x, a)\n",
"g(x) = poly_eval_sum(x, b)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "5dcb0ddb-c340-492e-958e-9ac7c0037e97",
"metadata": {},
"outputs": [
{
"data": {
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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# And let's plot these two and look at their roots\n",
"plot([f, g], xlim=(a[1]-2, a[2]+2), ylim=(-3, 3), label=[\"\\$f(x)\\$\" \"\\$g(x)\\$\"])\n",
"plot!(zero, color=:black, label=:none)"
]
},
{
"cell_type": "markdown",
"id": "85df8582-b03b-495e-b066-6d06270572ff",
"metadata": {},
"source": [
"* Perturbing the coefficient in the 10th digit made the difference between one root or two\n",
"\n",
"* The distance between the roots was large when $b_1$ was perturbed"
]
},
{
"cell_type": "markdown",
"id": "40647b04-8d46-4db6-a3ca-f0db5486b6a2",
"metadata": {},
"source": [
"## How did this happen?\n",
"\n",
"Take a polynomial with two roots at $x = a$ and perturb the middle coefficient.\n",
"\n",
"$$ x^2 + b_1 x + b_2 = 0 $$\n",
"$$ x^2 - 2a \\left( 1 + \\epsilon \\right) x + a^2 = 0 $$\n",
"\n",
"Using the quadratic formula, we know that the roots are at\n",
"\n",
"$$ \\begin{align} x_* = \\frac{b_1}{2} \\pm \\sqrt{\\frac{b_1^2}{4} - b_2} &= a \\left(1 + \\epsilon \\right) \\pm \\sqrt{a^2 \\left( 1 + \\epsilon \\right)^2 - a^2}\\\\\n",
"&= a \\left( 1 + \\epsilon \\right) \\pm a \\sqrt{2 \\epsilon + \\epsilon^2}\\\\\n",
"&\\approx a \\left( 1 + \\epsilon \\pm \\sqrt{2 \\epsilon} \\right)\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "4c5790dc-cc5f-4d9a-8cbc-3eab19d89b43",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# What happens to small values when we take their square root?\n",
"plot([x -> x, sqrt], xlims=(0, .001), label=[\"\\$x\\$\" \"\\$\\\\sqrt{x}\\$\"])"
]
},
{
"cell_type": "markdown",
"id": "a6f0bc7b-2194-481c-9bc0-3cfc2806c87f",
"metadata": {},
"source": [
"Note that the condition number is well behaved.\n",
"\n",
"$$ \\kappa \\left( \\sqrt{\\epsilon} \\right) = \\frac{1}{2 \\sqrt{\\epsilon}} \\frac{\\epsilon}{\\sqrt{\\epsilon}} = \\frac{1}{2} $$"
]
},
{
"cell_type": "markdown",
"id": "18eb6f76-50c4-4788-a7c2-e051676f13e8",
"metadata": {},
"source": [
"## Which root does Newton find?"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "36c80819-6142-47d7-8fa3-bc8fa07d1c7e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x_k, g(x_k), g'(x_k)\n"
]
},
{
"data": {
"text/plain": [
"17×3 Matrix{Float64}:\n",
" 110000.0 1.0e8 20000.0\n",
" 105000.0 2.5e7 10000.0\n",
" 102500.0 6.25e6 5000.0\n",
" 101250.0 1.5625e6 2500.0\n",
" 100625.0 3.90624e5 1250.0\n",
" 1.00313e5 97655.6 625.004\n",
" 1.00156e5 24413.4 312.509\n",
" 1.00078e5 6102.85 156.267\n",
" 1.00039e5 1525.21 78.1591\n",
" 1.0002e5 380.804 39.1307\n",
" 1.0001e5 94.7036 19.6676\n",
" 100005.0 23.1862 10.0372\n",
" 1.00003e5 5.33626 5.41711\n",
" 1.00002e5 0.970375 3.44696\n",
" 1.00001e5 0.0792503 2.88392\n",
" 1.00001e5 0.00075531 2.82896\n",
" 1.00001e5 0.0 2.82843"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Note the rather slow convergence\n",
"println(\"x_k, g(x_k), g'(x_k)\")\n",
"hist = newton_hist(g, x -> 2*x + b[1], 1.1e5; tol=1e-14)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "dded298a-ede8-4568-9b8f-28333cd1a0ea",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's plot the newton iterations\n",
"plot(hist[:,1], hist[:,2], seriestype=:path, marker=:auto, label=\"\\$g(x_k)\\$\")\n",
"plot!(g, title=\"Root $(hist[end,1])\", alpha=.5, label=\"\\$g(x)\\$\")\n",
"plot!(zero, color=:black, label=:none)"
]
},
{
"cell_type": "markdown",
"id": "e9abb500-a0b6-417b-ace8-cc17bb4f1926",
"metadata": {},
"source": [
"* Convergence is slower than we've seen before\n",
"\n",
"* The solution is not nearly as accurate as machine precision"
]
},
{
"cell_type": "markdown",
"id": "e6254f75-2442-4c74-a86f-7788e863409c",
"metadata": {},
"source": [
"## Using [Polynomials](https://juliamath.github.io/Polynomials.jl/stable/)\n",
"\n",
"Julia has a package for evaluating and manipulating polynomials.\n",
"Lets, use it instead of our own hand-rolled approach.\n",
"\n",
"
\n",
"\n",
"Note that the coefficients are given in the order $b_0 + b_1 x + b_2 x^2 + \\cdots$."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "6670d275-3e51-4ea6-9e03-de87d62c7c92",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Defining roots (x - a_0)(x - a_1)...\n",
"fpoly = fromroots([1, 2] .+ 100000.0) = Polynomial(1.0000300002e10 - 200003.0*x + 1.0*x^2)\n",
"derivative(fpoly) = Polynomial(-200003.0 + 2.0*x)\n",
"\n",
"Defining coefficients b_0, b_1, ...\n",
"gpoly = Polynomial([2, -3, 1]) = Polynomial(2 - 3*x + x^2)\n",
"derivative(gpoly) = Polynomial(-3 + 2*x)\n"
]
}
],
"source": [
"using Polynomials\n",
"\n",
"# Here's how we'd call the equivelent of our two functions above\n",
"println(\"Defining roots (x - a_0)(x - a_1)...\")\n",
"@show fpoly = fromroots([1, 2] .+ 1e5)\n",
"@show derivative(fpoly);\n",
"\n",
"println()\n",
"println(\"Defining coefficients b_0, b_1, ...\")\n",
"@show gpoly = Polynomial([2, -3, 1])\n",
"@show derivative(gpoly);"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "6cd78520-aa1f-4cc7-bfb3-08b0cdd823d9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"24×3 Matrix{Float64}:\n",
" 300000.0 3.99994e10 399997.0\n",
" 2.00001e5 9.99985e9 1.99999e5\n",
" 1.50001e5 2.49996e9 99999.3\n",
" 1.25001e5 6.24991e8 49999.6\n",
" 1.12501e5 1.56248e8 24999.8\n",
" 1.06251e5 3.90619e7 12499.9\n",
" 1.03126e5 9.76548e6 6249.95\n",
" 101564.0 2.44137e6 3124.98\n",
" 1.00783e5 6.10342e5 1562.49\n",
" 1.00392e5 1.52586e5 781.245\n",
" 1.00197e5 38146.3 390.623\n",
" 1.00099e5 9536.52 195.313\n",
" 1.0005e5 2384.07 97.6589\n",
" 1.00026e5 595.954 48.8346\n",
" 1.00014e5 148.926 24.4275\n",
" 1.00008e5 37.1691 12.2342\n",
" 1.00005e5 9.2302 6.15799\n",
" 1.00003e5 2.2467 3.16019\n",
" 1.00002e5 0.505433 1.73831\n",
" 1.00002e5 0.0845417 1.15679\n",
" 100002.0 0.00534111 1.01063\n",
" 100002.0 2.79307e-5 1.00006\n",
" 100002.0 7.85803e-10 1.0\n",
" 100002.0 0.0 1.0"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# And we can feed this directly into our Newton function\n",
"newton_hist(fpoly, derivative(fpoly), 3e5)"
]
},
{
"cell_type": "markdown",
"id": "0def07da-2fc9-453e-9b98-14fdefbc68d1",
"metadata": {},
"source": [
"## Finding *all* roots of polynomials"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "baf1ead0-7af1-4eab-ac51-00d787b85754",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Vector{Float64}:\n",
" 99998.58579643762\n",
" 100001.41422356237"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# There's a convenient function to call!\n",
"roots(Polynomial([1e10, -2e5*(1 + 1e-10), 1]))"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "9bd1083a-dc25-4650-abb0-3d7c2270322c",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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