343 lines
14 KiB
Plaintext
343 lines
14 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "3c51341a",
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"metadata": {},
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"source": [
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"# Tutorial 4: Root-Finding Methods\n",
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"---\n",
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"In this tutorial, we study classical **root-finding algorithms** for nonlinear equations. We will:\n",
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"\n",
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"- Define the root-finding problem mathematically\n",
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"- Derive several algorithms (bisection, fixed-point, Newton, secant)\n",
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"- Discuss convergence conditions and error behavior\n",
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"- Compare methods with worked examples using the `numethods` package.\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2999aa4f",
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"metadata": {},
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"source": [
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"## 1. Problem Setup and Notation\n",
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"\n",
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"We seek to solve a nonlinear scalar equation:\n",
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"$$\n",
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"f(x) = 0, \\quad f: \\mathbb{R} \\to \\mathbb{R}.\n",
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"$$\n",
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"with a continuously differentiable function.\n",
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"\n",
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"\n",
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"### Root, residual, and error\n",
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"- A **root** $r$ satisfies $f(r)=0$.\n",
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"- **Absolute error:** $(e_k = |x_k - x^\\star|)$.\n",
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"- **Residual:** $(r_k = |f(x_k)|)$. \n",
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"Note that small residual does not always imply small error.\n",
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"\n",
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"### Multiplicity\n",
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"A root $r$ has **multiplicity** $m$ if\n",
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"$$\n",
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"f(r) = f'(r) = \\dots = f^{(m-1)}(r) = 0, \\quad f^{(m)}(r) \\neq 0.\n",
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"$$\n",
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"If $x^\\star$ satisfies $f(x^\\star)=0$ and $f'(x^\\star)\\ne 0$, we say the root is **simple** (multiplicity 1).\n",
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"\n",
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"If $f'(x^\\star)=\\cdots=f^{(m-1)}(x^\\star)=0$ and $f^{(m)}(x^\\star)\\ne 0$, we say the root has **multiplicity** (m).\n",
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"\n",
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"- **Simple roots** ($m=1$): most methods converge rapidly.\n",
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"- **Multiple roots** ($m>1$): convergence often slows.\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "b610cea2",
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"metadata": {},
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"source": [
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"## 2. Bisection Method\n",
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"\n",
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"**Assumption (Intermediate Value Theorem):** If f is continuous on ([a,b]) and (f(a),f(b) < 0),\n",
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"then there exists $x^\\star$ in (a,b) with $f(x^\\star)=0$.\n",
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"\n",
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"- Assumes $f$ is continuous on $[a,b]$ with $f(a)f(b)<0$.\n",
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"- Repeatedly bisect interval and select subinterval containing the root.\n",
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"\n",
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"**Iteration:**\n",
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"$$\n",
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"c_k = \\frac{a_k+b_k}{2}, \\quad f(c_k).\n",
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"$$\n",
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"\n",
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"**Error bound:** interval length halves each step:\n",
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"$$\n",
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"|c_k-r| \\le \\frac{b-a}{2^k}.\n",
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"$$\n",
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"- Convergence: **linear**, guaranteed.\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3be4a510",
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"metadata": {},
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"source": [
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"## 3. Fixed-Point Iteration\n",
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"- Rewrite equation as $x=g(x)$.\n",
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"- Iterate:\n",
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"$$\n",
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"x_{k+1} = g(x_k).\n",
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"$$\n",
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"\n",
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"**Convergence theorem (Banach fixed-point):** If g is continuously differentiable near $(x^\\star)$ and\n",
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"$$\n",
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"|g'(x_\\star)| < 1,\n",
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"$$\n",
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"then for initial guesses $x_0$ sufficiently close to $x^\\star$, the iterates converge **linearly** to $x^\\star$ with asymptotic rate $|g'(x^\\star)|$.\n",
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"\n",
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"**Choice of g.** Different rearrangements yield different g's with different convergence properties.\n",
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"A poor choice (with $(|g'|\\ge 1))$ can diverge.\n",
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"\n",
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"- Rate: linear with factor $|g'(r)|$.\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "40e66a30",
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"metadata": {},
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"source": [
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"## 4. Newton’s Method\n",
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"From Taylor expansion:\n",
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"$$\n",
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"f(x) \\approx f(x_k) + f'(x_k)(x-x_k).\n",
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"$$\n",
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"Set $f(x)=0$ to solve for next iterate:\n",
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"$$\n",
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"x_{k+1} = x_k - \\frac{f(x_k)}{f'(x_k)}.\n",
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"$$\n",
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"\n",
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"- **Quadratic convergence** for simple roots.\n",
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"- For multiple roots: drops to linear.\n",
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"- Requires derivative, sensitive to initial guess.\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "84888305",
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"metadata": {},
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"source": [
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"## 5. Secant Method\n",
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"- Avoids derivative by approximating slope with finite difference:\n",
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"$$\n",
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"x_{k+1} = x_k - f(x_k) \\frac{x_k - x_{k-1}}{f(x_k)-f(x_{k-1})}.\n",
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"$$\n",
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"\n",
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"- Convergence order: $\\approx 1.618$ (superlinear).\n",
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"- More efficient than Newton if derivative expensive.\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7ed8a0b5",
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"metadata": {},
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"source": [
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"## 6. Stopping Criteria\n",
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"We stop iteration when:\n",
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"- $|f(x_k)| \\le \\varepsilon$ (residual small), or\n",
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"- $|x_{k+1}-x_k| \\le \\varepsilon(1+|x_{k+1}|)$ (relative step small).\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "4c9f93d9",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Bisection root: 1.4142135605216026\n",
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"Newton root: 1.4142135623730951\n",
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"Secant root: 1.414213562373095\n",
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"Fixed-point failed: Fixed-point iteration did not converge\n"
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]
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}
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],
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"source": [
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"from numethods import Bisection, FixedPoint, NewtonRoot, Secant, print_trace\n",
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"import math\n",
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"\n",
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"# Example function: f(x) = x^2 - 2\n",
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"f = lambda x: x**2 - 2\n",
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"df = lambda x: 2*x\n",
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"\n",
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"# Bisection\n",
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"bisect = Bisection(f, 0, 2, tol=1e-8)\n",
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"root_b = bisect.solve()\n",
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"print('Bisection root:', root_b)\n",
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"\n",
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"# Newton\n",
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"newton = NewtonRoot(f, df, 1.0, tol=1e-12)\n",
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"root_n = newton.solve()\n",
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"print('Newton root:', root_n)\n",
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"\n",
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"# Secant\n",
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"sec = Secant(f, 0, 2, tol=1e-12)\n",
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"root_s = sec.solve()\n",
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"print('Secant root:', root_s)\n",
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"\n",
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"# Fixed point: g(x)=sqrt(2) ~ rewriting\n",
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"g = lambda x: (2/x) # not always convergent, but demonstrates\n",
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"try:\n",
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" fp = FixedPoint(g, 1.0, tol=1e-8)\n",
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" root_fp = fp.solve()\n",
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" print('Fixed-point root:', root_fp)\n",
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"except Exception as e:\n",
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" print('Fixed-point failed:', e)\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "7f8fe13e",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Newton Method Trace (x^2 - 2):\n",
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" iter | x | f(x) | df(x) | x_new | error\n",
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"------------------------------------------------------------------------------\n",
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" 0 | 1 | -1 | 2 | 1.5 | 0.5\n",
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" 1 | 1.5 | 0.25 | 3 | 1.41667 | 0.0833333\n",
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" 2 | 1.41667 | 0.00694444 | 2.83333 | 1.41422 | 0.00245098\n",
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" 3 | 1.41422 | 6.0073e-06 | 2.82843 | 1.41421 | 2.1239e-06\n",
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" 4 | 1.41421 | 4.51061e-12 | 2.82843 | 1.41421 | 1.59472e-12\n",
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"\n",
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"Secant Method Trace (x^2 - 2):\n",
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" iter | x0 | x1 | x2 | f(x0) | f(x1) | error\n",
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"-------------------------------------------------------------------------------------------\n",
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" 0 | 0 | 2 | 1 | -2 | 2 | 1\n",
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" 1 | 2 | 1 | 1.33333 | 2 | -1 | 0.333333\n",
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" 2 | 1 | 1.33333 | 1.42857 | -1 | -0.222222 | 0.0952381\n",
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" 3 | 1.33333 | 1.42857 | 1.41379 | -0.222222 | 0.0408163 | 0.0147783\n",
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" 4 | 1.42857 | 1.41379 | 1.41421 | 0.0408163 | -0.00118906 | 0.000418335\n",
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" 5 | 1.41379 | 1.41421 | 1.41421 | -0.00118906 | -6.00729e-06 | 2.12421e-06\n",
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" 6 | 1.41421 | 1.41421 | 1.41421 | -6.00729e-06 | 8.93146e-10 | 3.15775e-10\n",
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" 7 | 1.41421 | 1.41421 | 1.41421 | 8.93146e-10 | -8.88178e-16 | 2.22045e-16\n",
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"\n",
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"Bisection Method Trace (x^2 - 2):\n",
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" iter | a | b | c | f(a) | f(b) | f(c) | interval\n",
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"--------------------------------------------------------------------------------------------------------\n",
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" 0 | 0 | 2 | 1 | -2 | 2 | -1 | 2\n",
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" 1 | 1 | 2 | 1.5 | -1 | 2 | 0.25 | 1\n",
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" 2 | 1 | 1.5 | 1.25 | -1 | 0.25 | -0.4375 | 0.5\n",
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" 3 | 1.25 | 1.5 | 1.375 | -0.4375 | 0.25 | -0.109375 | 0.25\n",
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" 4 | 1.375 | 1.5 | 1.4375 | -0.109375 | 0.25 | 0.0664062 | 0.125\n",
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" 5 | 1.375 | 1.4375 | 1.40625 | -0.109375 | 0.0664062 | -0.0224609 | 0.0625\n",
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" 6 | 1.40625 | 1.4375 | 1.42188 | -0.0224609 | 0.0664062 | 0.0217285 | 0.03125\n",
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" 7 | 1.40625 | 1.42188 | 1.41406 | -0.0224609 | 0.0217285 | -0.000427246 | 0.015625\n",
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" 8 | 1.41406 | 1.42188 | 1.41797 | -0.000427246 | 0.0217285 | 0.0106354 | 0.0078125\n",
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" 9 | 1.41406 | 1.41797 | 1.41602 | -0.000427246 | 0.0106354 | 0.00510025 | 0.00390625\n",
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" 10 | 1.41406 | 1.41602 | 1.41504 | -0.000427246 | 0.00510025 | 0.00233555 | 0.00195312\n",
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" 11 | 1.41406 | 1.41504 | 1.41455 | -0.000427246 | 0.00233555 | 0.000953913 | 0.000976562\n",
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" 12 | 1.41406 | 1.41455 | 1.41431 | -0.000427246 | 0.000953913 | 0.000263274 | 0.000488281\n",
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" 13 | 1.41406 | 1.41431 | 1.41418 | -0.000427246 | 0.000263274 | -8.20011e-05 | 0.000244141\n",
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" 14 | 1.41418 | 1.41431 | 1.41425 | -8.20011e-05 | 0.000263274 | 9.06326e-05 | 0.00012207\n",
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" 15 | 1.41418 | 1.41425 | 1.41422 | -8.20011e-05 | 9.06326e-05 | 4.31482e-06 | 6.10352e-05\n",
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" 16 | 1.41418 | 1.41422 | 1.4142 | -8.20011e-05 | 4.31482e-06 | -3.88434e-05 | 3.05176e-05\n",
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" 17 | 1.4142 | 1.41422 | 1.41421 | -3.88434e-05 | 4.31482e-06 | -1.72643e-05 | 1.52588e-05\n",
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" 18 | 1.41421 | 1.41422 | 1.41421 | -1.72643e-05 | 4.31482e-06 | -6.47477e-06 | 7.62939e-06\n",
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" 19 | 1.41421 | 1.41422 | 1.41421 | -6.47477e-06 | 4.31482e-06 | -1.07998e-06 | 3.8147e-06\n",
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" 20 | 1.41421 | 1.41422 | 1.41421 | -1.07998e-06 | 4.31482e-06 | 1.61742e-06 | 1.90735e-06\n",
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" 21 | 1.41421 | 1.41421 | 1.41421 | -1.07998e-06 | 1.61742e-06 | 2.68718e-07 | 9.53674e-07\n",
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" 22 | 1.41421 | 1.41421 | 1.41421 | -1.07998e-06 | 2.68718e-07 | -4.05632e-07 | 4.76837e-07\n",
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" 23 | 1.41421 | 1.41421 | 1.41421 | -4.05632e-07 | 2.68718e-07 | -6.84571e-08 | 2.38419e-07\n",
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" 24 | 1.41421 | 1.41421 | 1.41421 | -6.84571e-08 | 2.68718e-07 | 1.0013e-07 | 1.19209e-07\n",
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" 25 | 1.41421 | 1.41421 | 1.41421 | -6.84571e-08 | 1.0013e-07 | 1.58366e-08 | 5.96046e-08\n",
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" 26 | 1.41421 | 1.41421 | 1.41421 | -6.84571e-08 | 1.58366e-08 | -2.63102e-08 | 2.98023e-08\n",
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" 27 | 1.41421 | 1.41421 | 1.41421 | -2.63102e-08 | 1.58366e-08 | -5.23681e-09 | 1.49012e-08\n",
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" 28 | 1.41421 | 1.41421 | 1.41421 | -5.23681e-09 | 1.58366e-08 | 5.2999e-09 | 7.45058e-09\n",
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" 29 | 1.41421 | 1.41421 | 1.41421 | -5.23681e-09 | 5.2999e-09 | 3.15445e-11 | 3.72529e-09\n",
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"\n",
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"Fixed-Point Iteration Trace (x^2 - 2):\n",
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" iter | x | x_new | error\n",
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"----------------------------------------------------\n",
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" 0 | 1 | 1.5 | 0.5\n",
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" 1 | 1.5 | 1.41667 | 0.0833333\n",
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" 2 | 1.41667 | 1.41422 | 0.00245098\n",
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" 3 | 1.41422 | 1.41421 | 2.1239e-06\n",
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" 4 | 1.41421 | 1.41421 | 1.59495e-12\n"
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]
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}
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],
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"source": [
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"# Newton\n",
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"steps = NewtonRoot(f, df, x0=1.0).trace()\n",
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"print(\"Newton Method Trace (x^2 - 2):\")\n",
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"print_trace(steps)\n",
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"\n",
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"# Secant\n",
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"steps = Secant(f, 0, 2).trace()\n",
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"print(\"\\nSecant Method Trace (x^2 - 2):\")\n",
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"print_trace(steps)\n",
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"\n",
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"# Bisection\n",
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"steps = Bisection(f, 0, 2).trace()\n",
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"print(\"\\nBisection Method Trace (x^2 - 2):\")\n",
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"print_trace(steps)\n",
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"\n",
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"# Fixed-point: solve\n",
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"g = lambda x: 0.5 * (x + 2 / x)\n",
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"steps = FixedPoint(g, 1.0).trace()\n",
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"print(\"\\nFixed-Point Iteration Trace (x^2 - 2):\")\n",
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"print_trace(steps)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "833b538c",
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"metadata": {},
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"source": [
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"## 7. Comparison of Methods\n",
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"| Method | Requires derivative | Convergence rate | Guarantee? |\n",
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"|--------|---------------------|------------------|------------|\n",
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"| Bisection | No | Linear | Yes (if sign change) |\n",
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"| Fixed-Point | No | Linear | Not always |\n",
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"| Newton | Yes | Quadratic | Locally (good guess) |\n",
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"| Secant | No | ~1.618 (superlinear) | Locally (good guess) |\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0d372aa0",
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||
"metadata": {},
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"source": [
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"## 8. Exercises\n",
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"1. Apply all four methods to $f(x)=\\cos x - x$.\n",
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"2. Try Newton’s method on $f(x)=(x-1)^2$ and compare convergence rate with $f(x)=x^2-2$.\n",
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"3. Modify Secant to stop if denominator too small.\n"
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]
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}
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],
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"metadata": {
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||
"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.13.7"
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