{ "cells": [ { "cell_type": "markdown", "id": "29409e29-a7b5-4d20-9b41-07c8faa431d5", "metadata": {}, "source": [ "# 2025-09-08 Newton Methods\n", "\n", "* Newton's method via Taylor series\n", "\n", "* Convergence theory for fixed point methods\n", "\n", "* Derive Newton's method via convergence theory\n", "\n", "* Newton methods in computing culture\n", "\n", "* Breaking Newton's method\n", "\n", "See also [FNC](https://tobydriscoll.net/fnc-julia/nonlineqn/newton.html)" ] }, { "cell_type": "markdown", "id": "77026b22-cc90-4e07-aa35-b2366ac19312", "metadata": {}, "source": [ "## Bisection" ] }, { "cell_type": "code", "execution_count": 1, "id": "77b281da-7807-436f-a551-bc4a5979431c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "f (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", "# Bisection function\n", "hasroot(f, a, b) = f(a) * f(b) < 0\n", "function bisect_iter(f, a, b, tol)\n", " hist = Float64[]\n", " while abs(b - a) > tol\n", " mid = (a + b) / 2\n", " push!(hist, mid)\n", " if hasroot(f, a, mid)\n", " b = mid\n", " else\n", " a = mid\n", " end\n", " end\n", " hist\n", "end\n", "\n", "# And a target function\n", "f(x) = cos(x) - x" ] }, { "cell_type": "markdown", "id": "9d7f508b-ff22-47a2-9814-bc58145a72e7", "metadata": {}, "source": [ "## Convergence classes\n", "\n", "A convergent rootfinding algorithm produces a sequence of approximations $x_k$ such that\n", "\n", "$$ \\lim_{k \\rightarrow \\infty} x_k \\rightarrow x_* $$\n", "\n", "where $f \\left( x_* \\right) = 0$.\n", "For analysis, it is convenient to define the errors $e_k = x_k - x_*$. We say that an iterative algorithm is [**$q$-linearly convergent**](https://en.wikipedia.org/wiki/Rate_of_convergence#Q-convergence) if\n", "\n", "$$ \\lim_{k \\rightarrow \\infty} \\lvert e_{k + 1} \\rvert / \\lvert e_k \\rvert = \\rho < 1 $$\n", "\n", "(**$q$** stands for \"quotient\")\n", "A smaller convergence factor $\\rho$ represents faster convergence.\n", "\n", "A slightly weaker condition, [$r$-linear convergence](https://en.wikipedia.org/wiki/Rate_of_convergence#R-convergence) or **linear convergence**, is that\n", "\n", "\n", "$$ \\lvert e_{k + 1} \\rvert \\leq \\lvert e_k \\rvert $$\n", "\n", "for all sufficiently large $k$ when the sequence $\\left\\lbrace e_k \\right\\rbrace$ converges $r$-linearly to 0." ] }, { "cell_type": "code", "execution_count": 2, "id": "7757ebea-0ebe-451c-a688-0a32899f4bba", "metadata": {}, "outputs": [ { "data": { "image/png": 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"markdown", "id": "f27b0db3-0a0a-4809-a222-4b8f1bba0d40", "metadata": {}, "source": [ "## Newton-Raphson method\n", "\n", "Much of the numerical analysis reduces to [Taylor series](https://en.wikipedia.org/wiki/Taylor_series), the approximation\n", "\n", "$$ f \\left( x \\right) = f \\left( x_0 \\right) + f' \\left( x_0 \\right) \\left( x - x_0 \\right) + f'' \\left( x_0 \\right) \\left( x - x_0 \\right)^2 / 2 + \\cdots $$\n", "\n", "centered on some reference point $x_0$.\n", "\n", "In numerical computation, it is exceedingly rare to look beyond the first-order approximation\n", "\n", "$$ \\tilde{f}_{x_0} \\left( x \\right) = f \\left( x_0 \\right) + f' \\left( x_0 \\right) \\left( x - x_0 \\right) $$\n", "\n", "Since $\\tilde{f}_{x_0} \\left( x \\right)$ is a linear function, we can explicitly compute the unique solution of $\\tilde{f}_{x_0} \\left( x \\right) = 0$ as\n", "\n", "$$ x = x_0 - \\frac{f \\left( x_0 \\right)}{f' \\left( x \\right)} $$\n", "\n", "This is Newton's Method (aka Newton-Raphson or Newton-Raphson-Simpson) for finding the roots of differentiable functions." ] }, { "cell_type": "markdown", "id": "3610d236-8c09-4fa8-b600-8745a6162e6a", "metadata": {}, "source": [ "### An implementation" ] }, { "cell_type": "code", "execution_count": 3, "id": "15bbc468-8dd5-4f1b-8af6-ff0996c571b8", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "k=1, x_k=1.0000000000000000, f_k=-4.5969769413186023e-01\n", "k=2, x_k=0.7503638678402439, f_k=-1.8923073822117442e-02\n", "k=3, x_k=0.7391128909113617, f_k=-4.6455898990771516e-05\n", "k=4, x_k=0.7390851333852840, f_k=-2.8472058044570758e-10\n", "k=5, x_k=0.7390851332151607, f_k=0.0000000000000000e+00\n" ] }, { "data": { "text/plain": [ "(0.7390851332151607, 0.0, 5)" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using Printf\n", "\n", "# Newton's method\n", "function newton(f, fp, x0; tol=1e-8, verbose=false)\n", " x = x0\n", " for k in 1:100 # max number of iterations\n", " fx = f(x)\n", " fpx = fp(x)\n", " if verbose\n", " println(\"k=$k, x_k=$(@sprintf(\"%.16f\", x)), f_k=$(@sprintf(\"%.16e\", fx))\")\n", " end\n", " if abs(fx) < tol\n", " return x, fx, k\n", " end\n", " x = x - fx / fpx\n", " end \n", "end\n", "\n", "# And a target function\n", "f(x) = cos(x) - x\n", "fp(x) = -sin(x) - 1\n", "\n", "# Let's plot our target function\n", "plot(f, xlims=[-2, 2], label=\"\\$cos(x) - x\\$\")\n", "\n", "# And find the root\n", "newton(f, fp, 1; tol=1e-15, verbose=true)" ] }, { "cell_type": "markdown", "id": "74f84119-055f-42b3-9853-0b95369f85e4", "metadata": {}, "source": [ "That's fast convergence!\n", "\n", "* 10 digits of accuracy in 4 iterations\n", "\n", "* How is this convergence test different than the one for bisection?\n", "\n", "* How can Newton's method break down?\n", "\n", "$$ x_{k + 1} = x_k - \\frac{f \\left( x_k \\right)}{f' \\left( x_k \\right)} $$" ] }, { "cell_type": "code", "execution_count": 4, "id": "61b05a9d-e85f-4191-a367-50b4a5f7af9f", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "k=1, x_k=-1.4707963267948965, f_k=1.5706297434417247e+00\n", "k=2, x_k=312.9170549232223948, f_k=-3.1259435002533314e+02\n", "k=3, x_k=-5529.9275427528937144, f_k=5.5306763919178247e+03\n", "k=4, x_k=10868.9459369702435652, f_k=-1.0868376227850798e+04\n", "k=5, x_k=-50136.7073225235581049, f_k=5.0135707777419018e+04\n", "k=6, x_k=-1468.7903453577164328, f_k=1.4688859787856973e+03\n", "k=7, x_k=-732.6603742863298976, f_k=7.3187610943622951e+02\n", "k=8, x_k=-281.0038172368655864, f_k=2.8083589138981239e+02\n", "k=9, x_k=-139.5817486699348819, f_k=1.3979912372811944e+02\n", "k=10, x_k=5706.8561562109989609, f_k=-5.7070086597647050e+03\n", "k=11, x_k=2836.5648158265976235, f_k=-2.8375220962814674e+03\n", "k=12, x_k=635.5038791777569713, f_k=-6.3488396511814790e+02\n", "k=13, x_k=279.7607629875441830, f_k=-2.8074814643252921e+02\n", "k=14, x_k=-53.8070079658355667, f_k=5.2885920826340048e+01\n", "k=15, x_k=-15.7419606182276794, f_k=1.4742538472479499e+01\n", "k=16, x_k=-1.4840596284124175, f_k=1.5706876106501757e+00\n", "k=17, x_k=416.3331582875110257, f_k=-4.1640522744068772e+02\n", "k=18, x_k=207.8594910282616013, f_k=-2.0698889108021109e+02\n", "k=19, x_k=69.1262088526432592, f_k=-6.8126271241735495e+01\n", "k=20, x_k=1.7525179991632598, f_k=-1.9332411629172330e+00\n", "k=21, x_k=0.7778731690296721, f_k=-6.5465483571587102e-02\n", "k=22, x_k=0.7394040200105153, f_k=-5.3373035134818281e-04\n", "k=23, x_k=0.7390851556610822, f_k=-3.7565764610114627e-08\n", "k=24, x_k=0.7390851332151608, f_k=-2.2204460492503131e-16\n" ] }, { "data": { "text/plain": [ "(0.7390851332151608, -2.220446049250313e-16, 24)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# What if we start with a poor initial guess?\n", "# Note that our convergence from before starts at k=20\n", "newton(f, fp, -pi/2+.1; verbose=true)" ] }, { 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" } }, "cell_type": "markdown", "id": "3bb765e6-9780-48ca-9b28-a185e1b593e8", "metadata": {}, "source": [ "## Convergence of fixed-point methods (mean value theorem)\n", "\n", "Consider the iteration\n", "\n", "$$ x_{k + 1} = g \\left( x_k \\right) $$\n", "\n", "where $g$ is a continuously differentiable function.\n", "Suppose that there exists a fixed point $x_* = g \\left( x_* \\right)$.\n", "By the [mean value theorem](https://en.wikipedia.org/wiki/Mean_value_theorem), we have that\n", "\n", "$$ x_{k + 1} - x_* = g \\left( x_k \\right) - g \\left( x_* \\right) = g' \\left( c_k \\right) \\left( x_k - x_* \\right) $$\n", "\n", "for some $c_i$ between $x_k$ and $x_*$.\n", "\n", "Taking absolute values,\n", "\n", "$$ \\left\\lvert e_{k + 1} \\right\\rvert = \\left\\lvert g' \\left( c_k \\right) \\right\\rvert \\left\\lvert e_k \\right\\rvert $$\n", "\n", "which converges to zero if $\\left\\lvert g' \\left( c_k \\right) \\right\\rvert < 1$.\n", "\n", "![image.png](attachment:94013db6-ba0e-41fc-9b16-73c226d89c83.png)" ] }, { "cell_type": "markdown", "id": "d2042eba-7748-4fe5-afed-f6d2b588a3c5", "metadata": {}, "source": [ "### Convergence of fixed-point methods (Taylor series)\n", "\n", "Consider the iteration\n", "\n", "$$ x_{k + 1} = g \\left( x_k \\right) $$\n", "\n", "where $g$ is a continuously differentiable function.\n", "Suppose there exists a fixed point $x_* = g \\left( x_* \\right)$.\n", "there exists a Taylor series at $x_*$,\n", "\n", "$$ g \\left( x_k \\right) = g \\left( x_* \\right) + g' \\left( x_* \\right) \\left( x_k - x_* \\right) + \\mathcal{O} \\left( \\left( x_k - x_* \\right)^2 \\right) $$\n", "\n", "and thus\n", "\n", "$$ x_{k + 1} - x_* = g \\left( x_k \\right) - g \\left( x_* \\right) = g' \\left( x_* \\right) \\left( x_k - x_* \\right) + \\mathcal{O} \\left( \\left( x_k - x_* \\right)^2 \\right) $$\n", "\n", "In terms of the error $e_k = x_k - x_*$,\n", "\n", "$$ \\left\\lvert \\frac{e_{k + 1}}{e_k} \\right\\rvert = \\left\\lvert g' \\left( x_* \\right) \\right\\rvert + \\mathcal{O} \\left( e_k^2 \\right) $$" ] }, { "cell_type": "markdown", "id": "d406ac47-55b9-4ecb-88e9-28d14c695dd5", "metadata": {}, "source": [ "## [Big $\\mathcal{O}$ notation](https://en.wikipedia.org/wiki/Big_O_notation)\n", "\n", "### Limit $n \\rightarrow \\infty$\n", "\n", "We’d say an algorithm costs $\\mathcal{O} \\left( n^2 \\right)$ if its running time on input of size $n$ is less than $c n^2$ for some constant $c$ and sufficiently large $n$.\n", "\n", "Sometimes we write $\\text{cost} \\left( \\text{algorithm} \\right), n = \\mathcal{O} \\left( n^2 \\right)$ or (preferably) $\\text{cost} \\left( \\text{algorithm} \\right) \\in \\mathcal{O} \\left( n^2 \\right)$.\n", "\n", "Note that $\\mathcal{O} \\left( \\log \\left( n \\right) \\right) \\subset \\mathcal{O} \\left( n \\right) \\subset \\mathcal{O} \\left( n \\log \\left( n \\right) \\right) \\subset \\mathcal{O} \\left( n^2 \\right) \\subset \\cdots$ so it’s correct to say \"binary search is in $\\mathcal{O} \\left( n^2 \\right)$\", even though a sharper statement is also true.\n", "\n", "We say the algorithm is in $\\mathcal{\\Omega} \\left( n^2 \\right)$ (“big theta”) if\n", "\n", "$$ c_1 n^2 < \\text{cost} \\left( \\text{algorithm} \\right) < c_2 n^2 $$\n", "\n", "for some positive constants $c_1, c_2$ and sufficiently large $n$.\n", "\n", "### Limit $h \\rightarrow 0$\n", "\n", "In numerical analysis, we often have a small real number, and now the definitions take the limit as the small number goes to zero.\n", "So we say a term in an expression is in $\\mathcal{O} \\left( h^2 \\right)$ if\n", "\n", "$$ \\lim_{h \\rightarrow \\infty} \\frac{\\text{term} \\left( h \\right)}{h^2} < \\infty $$\n", "\n", "Big $\\mathcal{O}$ terms can be manipulated as\n", "\n", "$$ h \\mathcal{O} \\left( h^k \\right) = \\mathcal{O} \\left( h^{k + 1} \\right) $$\n", "$$ \\mathcal{O} \\left( h^k \\right) / h = \\mathcal{O} \\left( h^{k - 1} \\right) $$\n", "$$ c \\mathcal{O} \\left( h^k \\right) = \\mathcal{O} \\left( h^k \\right) $$\n", "$$ \\mathcal{O} \\left( h^k \\right) - \\mathcal{O} \\left( h^k \\right) = ? $$" ] }, { "cell_type": "markdown", "id": "a7bc987a-cc33-460c-b011-3500ae54ad41", "metadata": {}, "source": [ "## Fixed point iteration example\n", "\n", "We wanted to solve $\\cos \\left( x \\right) - x = 0$, which occurs when $g \\left( k \\right) = \\cos \\left( x \\right)$ is a fixed point." ] }, { "cell_type": "code", "execution_count": 5, "id": "967f8b9d-da6f-4050-ac70-9b115b5cafad", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "xstar = 0.739085133385284\n", "gp(xstar) = -0.6736120293089505\n" ] }, { "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", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Let's find the root\n", "xstar, _ = newton(f, fp, 1.)\n", "\n", "# And reformulate our target function as a fixed point iteration\n", "# Note: 0 = cos(x) - x => x = cos(x)\n", "g(x) = cos(x)\n", "gp(x) = -sin(x)\n", "\n", "# What is our root and the slope of g there?\n", "@show xstar\n", "@show gp(xstar)\n", "\n", "plot([x->x, g], xlims=(-2, 3), label=[\"\\$x\\$\" \"\\$cos (x)\\$\"])\n", "scatter!([xstar], [xstar], label=\"\\$x_*\\$\")" ] }, { "cell_type": "code", "execution_count": 6, "id": "c5b21017-7e58-47be-8dc2-552c0d30a579", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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"source": [ "## Verifying fixed point convergence theory\n", "\n", "$$ \\left\\lvert \\frac{e_{k + 1}}{e_k} \\right\\rvert \\rightarrow \\left\\lvert g' \\left( x_* \\right) \\right\\rvert $$" ] }, { "cell_type": "code", "execution_count": 7, "id": "ee612363-3675-4317-8e0d-94725db34ecd", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "gp(xstar) = -0.6736120293089505\n", "\n", "ratios e_k+1 / e_k =\n" ] }, { "data": { "text/plain": [ "15-element Vector{Float64}:\n", " -0.9161855415615605\n", " -0.15197657010596488\n", " -0.734870205299266\n", " -0.624132525531327\n", " -0.7026257933893496\n", " -0.6523498121376077\n", " -0.6870971782336925\n", " -0.664168570025122\n", " -0.6798044680427148\n", " -0.6693659427636027\n", " -0.6764378047956165\n", " -0.6716930541785153\n", " -0.6748976495459512\n", " -0.6727427617641084\n", " -0.6741962236114177" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# The ratio of errors should approach g'(x_*)\n", "# First, what is g'(x_*)\n", "@show gp(xstar)\n", "\n", "# And then look at the ratio\n", "ek = xk .- xstar\n", "println(\"\\nratios e_k+1 / e_k =\")\n", "(ek[2:end] ./ ek[1:end-1])" ] }, { "cell_type": "code", "execution_count": 8, "id": "c8565fc4-6155-48c8-9a2b-1746f77f9bfe", "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", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# And lets put it on a log-linear plot to visualize it\n", "scatter(abs.(ek), yscale=:log10, xlabel=\"\\$k\\$\", ylabel=\"\\$e_k\\$\", label=\"\\$e_k\\$\")\n", "plot!(k -> abs(gp(xstar))^k, label=\"\\$|g'|^k\\$\")" ] }, { "cell_type": "markdown", "id": "327ea506-e5df-47af-89e6-e3abe97167b7", "metadata": {}, "source": [ "## Plotting Newton convergence" ] }, { "cell_type": "code", "execution_count": 9, "id": "399c1ebc-6a76-4547-be4c-92a10edfda30", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "newton_hist (generic function with 1 method)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Ok, this is for fixed points, but what about Newton?!?\n", "# Let's look at the history of our iterates\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": "code", "execution_count": 10, "id": "8d2d438a-e173-4bed-964c-b4be9fd56157", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "x_star = xk[end, 1] = 0.7390851332151607\n" ] }, { "data": { "image/png": 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"cell_type": "markdown", "id": "e41d9233-7ae1-47b7-99ec-b09d3fd9d445", "metadata": {}, "source": [ "## Bringing fixed point and Newton together\n", "\n", "Let's try to bridge the gap and bring these two ideas together.\n", "\n", "### Formulations are not unique (constants)\n", "\n", "If $x_* = g \\left( x_* \\right)$ then $g \\left( x_* \\right) - x_* = 0$.\n", "\n", "Consider the fixed point iteration\n", "\n", "$$ x = x + c \\left( g \\left( x \\right) - x \\right) $$\n", "\n", "for any constant $c \\neq 0$.\n", "Let $g_2 \\left( x \\right) = x + c \\left( g \\left( x \\right) - x \\right)$.\n", "Can we choose $c$ to make $\\left\\lvert g'_2 \\left( x_* \\right) \\right\\rvert$ small?\n", "\n", "

\n", "\n", "$$ g'_3 \\left( x \\right) = 1 + c \\left( g' \\left( x \\right) - 1 \\right) $$" ] }, { "cell_type": "code", "execution_count": 11, "id": "fb922b7a-3e96-459b-8ee5-b8a9b0e5a6d3", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "g2p(xstar) = 0.16319398534552476\n" ] }, { "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", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# This looks like Newton, but with c instead of f'(x)...\n", "c = .5\n", "g2(x) = x + c * (cos(x) - x)\n", "g2p(x) = 1 + c * (-sin(x) - 1)\n", "\n", "@show g2p(xstar)\n", "plot([x -> x, g, g2], ylims=(-5, 5), label=[\"\\$x\\$\" \"\\$g(x)\\$\" \"\\$g_2(x)\\$\"])" ] }, { "cell_type": "code", "execution_count": 12, "id": "241f5d5b-83a9-408b-853c-33dc9e722fdd", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "11-element Vector{Float64}:\n", " 0.26091486661471597\n", " 0.03106601954878585\n", " 0.004893162344945079\n", " 0.0007941171212053622\n", " 0.00012947850276123773\n", " 2.112687301181193e-5\n", " 3.4475537732392425e-6\n", " 5.62475483634195e-7\n", " 9.16501970982253e-8\n", " 1.4814399151852342e-8\n", " 2.2752605355336186e-9" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Let's check the convergence\n", "xk = fixed_point(g2, 1., 10)\n", "xk .- xstar" ] }, { "cell_type": "code", "execution_count": 13, "id": "b47361c9-3efa-4838-bb9a-77fb46c5a1cd", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "g2p(xstar) = 0.16319398534552476\n", "\n", "ratios e_k+1 / e_k =\n" ] }, { "data": { "text/plain": [ "10-element Vector{Float64}:\n", " 0.11906573186824182\n", " 0.15750850659386512\n", " 0.1622911861949014\n", " 0.16304711144460268\n", " 0.16316896288776617\n", " 0.1631833433803352\n", " 0.1631520552341395\n", " 0.16294078544733492\n", " 0.16164066876992242\n", " 0.15358439530428933" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# The ratio of errors should approach g'(x_*)\n", "# First, what is g'(x_*)\n", "@show g2p(xstar)\n", "\n", "# And then look at the ratio\n", "ek = xk .- xstar\n", "println(\"\\nratios e_k+1 / e_k =\")\n", "(ek[2:end] ./ ek[1:end-1])" ] }, { "cell_type": "markdown", "id": "25fc8ef4-e65e-4b38-a4c4-b1bc8ff0bd17", "metadata": {}, "source": [ "### Formulations are not unique (functions)\n", "\n", "If $x_* = g \\left( x_* \\right)$ then $g \\left( x_* \\right) - x_* = 0$.\n", "\n", "Consider the fixed point iteration\n", "\n", "$$ x = x + h \\left( x \\right) \\left( g \\left( x \\right) - x \\right) $$\n", "\n", "for any smooth $h \\left( x \\right) \\neq 0$.\n", "Let $g_3 \\left( x \\right) = x - h \\left( x \\right) \\left( g \\left( x \\right) - x \\right)$.\n", "Can we choose $h \\left( x \\right)$ to make $\\left\\lvert g'_3 \\left( x_* \\right) \\right\\rvert$ small?\n", "\n", "

\n", "\n", "$$ g'_3 \\left( x \\right) = 1 + h' \\left( x \\right) \\left( g \\left( x \\right) - x \\right) + h \\left( x \\right) \\left( g' \\left( x \\right) - 1 \\right) $$" ] }, { "cell_type": "code", "execution_count": 14, "id": "ffc9ece9-687d-4c53-9144-aceb9aa4e39c", "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", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Let's try something that looks more like Newton\n", "h(x) = -1 / (gp(x) - 1)\n", "g3(x) = x + h(x) * (g(x) - x)\n", "plot([x -> x, cos, g2, g3], ylims=(-5, 5), label=[\"\\$x\\$\" \"\\$cos (x)\\$\" \"\\$g_2 (x)\\$\" \"\\$g_3 (x)\\$\"])" ] }, { "cell_type": "markdown", "id": "ddc45b39-8274-4dc7-8e6d-0d6ce72ec42e", "metadata": {}, "source": [ "* We don't know $g' \\left( x_* \\right)$ in advance because we don't know $x_*$\n", "\n", "* This method converges fast\n", "\n", "* We just re-derived Newton's method" ] }, { "cell_type": "markdown", "id": "a7cc46c3-09ae-4aa7-ba93-6392f43bc32e", "metadata": {}, "source": [ "## Newton's method - a new derivation\n", "\n", "A rootfinding problem $f \\left( x \\right) = 0$ can be converted to a fixed point problem\n", "\n", "$$ x = x + f \\left( x \\right) := g \\left( x \\right) $$\n", "\n", "but there is no guarantee that $g' \\left( x_* \\right) = 1 + f' \\left( x_* \\right)$ will have magnitude less than 1.\n", "\n", "Problem-specific algebraic manipulation can be used to make $\\left\\lvert g' \\left( x_* \\right) \\right\\rvert$ small.\n", "\n", "

\n", "\n", "$x = x + h \\left( x \\right) f \\left( x \\right)$ is a valid formulation for any $h \\left( x \\right)$ bounded away from $0$.\n", "\n", "Can we choose $h \\left( x \\right)$ such that\n", "\n", "$$ g' \\left( x \\right) = 1 + h' \\left( x \\right) f \\left( x \\right) + h \\left( x \\right) f' \\left( x \\right) = 0$$\n", "\n", "when $f \\left( x \\right) = 0$?\n", "\n", "Then, we have\n", "\n", "$$ g' \\left( x \\right) = 1 + h \\left( x \\right) f' \\left( x \\right) = 0$$\n", "\n", "or\n", "\n", "$$ h \\left( x \\right) = \\frac{-1}{f' \\left( x \\right)} $$\n", "\n", "In other words,\n", "\n", "$$ x_{k + 1} = x_k + \\frac{-1}{f' \\left( x_k \\right)} f \\left( x_k \\right) = x_k + h \\left( x_k \\right) f \\left( x_k \\right) $$\n", "\n", "which is our Newton method." ] }, { "cell_type": "markdown", "id": "8fb8733f-cb3b-440c-858d-d5b2e9a5f818", "metadata": {}, "source": [ "### Quadratic convergence\n", "\n", "$$ \\left\\lvert \\frac{e_{k + 1}}{e_k} \\right\\rvert \\rightarrow \\left\\lvert g' \\left( x_* \\right) \\right\\rvert $$\n", "\n", "* What does it mean that $g' \\left( x_* \\right) = 0$?\n", "\n", "* It turns out that Newton's method has **locally quadratic** convergence to simple roots,\n", "\n", "$$ \\lim_{k \\rightarrow \\infty} \\frac{\\left\\lvert e_{k + 1} \\right\\rvert}{\\left\\lvert e_k \\right\\rvert^2} < \\infty $$\n", "\n", "* \"The number of correct digits doubles each iteration\"\n", "\n", "* Now that we know how to make a good guess accurate, the hard part is getting a good guess" ] }, { "cell_type": "markdown", "id": "2211a76f-9cdb-4e88-b9ee-402d38df2b9a", "metadata": {}, "source": [ "## Sidebar - Fast inverse square root\n", "\n", "The following code appeared literally (with these comments) in the Quake III Arena source code (late 1990s).\n", "\n", "```C\n", "float Q_rsqrt( float number )\n", "{\n", "\tlong i;\n", "\tfloat x2, y;\n", "\tconst float threehalfs = 1.5F;\n", "\n", "\tx2 = number * 0.5F;\n", "\ty = number;\n", "\ti = * ( long * ) &y; // evil floating point bit level hacking\n", "\ti = 0x5f3759df - ( i >> 1 ); // what the fuck? \n", "\ty = * ( float * ) &i;\n", " y = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration\n", "// y = y * ( threehalfs - ( x2 * y * y ) ); // 2nd iteration, this can be removed\n", "\n", "\treturn y;\n", "}\n", "```\n", "\n", "We now have [vector instructions](https://software.intel.com/sites/landingpage/IntrinsicsGuide/#text=rsqrt&expand=2989,1224,4470) for approximate inverse square root.\n", "See also [Wikipedia](https://en.wikipedia.org/wiki/Fast_inverse_square_root).\n", "For details about the magic number, read this [section on Wikipedia](https://en.wikipedia.org/wiki/Fast_inverse_square_root#Aliasing_to_an_integer_as_an_approximate_logarithm)." ] }, { "cell_type": "markdown", "id": "6d5ddf23-0c65-4156-b9e0-680ac41d23d8", "metadata": {}, "source": [ "### How does it work?\n", "\n", "Let's look at the last line.\n", "\n", "```C\n", "y = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration\n", "```\n", "\n", "We want a function $f \\left( y \\right)$ such that $f \\left( 1 / \\sqrt{x} \\right) = 0$.\n", "One such function is\n", "\n", "$$ f \\left( y \\right) = 1 / y^2 - x, f' \\left( y \\right) = -2 / y^3 $$\n", "\n", "There are others (see the homework activity) that require a division.\n", "\n", "Newton's method is\n", "\n", "$$ \\begin{align} y_{k + 1} &= y_k - \\frac{f \\left( y_k \\right)}{f' \\left( y_k \\right)}\\\\\n", " &= y_k - \\frac{ 1 / y_k^2 - x}{-2 / y_k^3}\\\\\n", " &= y_k + \\frac{1}{2} \\left( y_k - x y_k^2 \\right)\\\\\n", " &= y_k \\left( \\frac{3}{2} - \\frac{1}{2} x f_k^2 \\right)\n", "\\end{align} $$" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.11.6", "language": "julia", "name": "julia-1.11" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }