physics-simulation/examples/Newton_minimization.html at main · InteractiveComputerGraphics/physics-simulation · GitHub

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 <!doctype html>
<html class="no-js" lang="en">
<head>
	<meta charset="utf-8">
	<style>
		body {font-family: Helvetica, sans-serif;}
		table {background-color:#CCDDEE;text-align:left}
	</style>
	<script type="text/x-mathjax-config">
	  MathJax.Hub.Config({
		extensions: ["tex2jax.js"],
		jax: ["input/TeX", "output/HTML-CSS"],
		tex2jax: {
		  inlineMath: [ ['$','$'], ["\\(","\\)"] ],
		  displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
		  processEscapes: true
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	<script type="text/javascript" aync src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js"></script>
	<script src="https://cdn.plot.ly/plotly-2.5.1.min.js"></script>
	<title>Newton's method for minimization</title>
</head>
<body>
<main>
	<h1 style="text-align:center">Newton's method for minimization</h1>
	<table style="align_center;border-radius: 20px;padding: 20px;margin:auto">
		<col width="1000">
		<tr>
			<td>
				<div id="plotOutput" style="width: 1000px; height: 600px;border:2px solid #000000;border-radius: 0px;background-color:#EEEEEE"></div>
			</td>
		</tr>
		<tr>
			<td><table style="margin:20px">
				<col width="200" style="padding-right:10px">
				<col width="100">
				<tr>
					<td><label for="newton_steps">Newton steps</label></td>
					<td><input type="text" id="textInput" value="1" readonly></td>
				</tr>
				<tr>
					<td></td>
					<td><input onchange="document.getElementById('textInput').value=this.value;plot.reset()" id="newton_steps" value="1" type="range"  min="1" max="50" step="1"></td>
				</tr>
				<tr>
					<td><label for="fct">Function</label></td>
					<td><select onchange="plot.reset()" id="fct" size="1">
						  <option selected="selected">Quartic function</option>
						  <option>Sinusoidal function</option>
						  <option>Exponential function</option>
						  <option>Logarithmic function</option>
						</select>
					</td>
				</tr>
			</table></td>
		</tr>

		<tr><td>
			<h2>Newton's method for minimization</h2>

			<p>Newton's method can be used to find the minimum of a function $f(x)$. At a minimum the derivative vanishes, $f'(x^*) = 0$, so minimization reduces to finding the root of $f'(x)$.</p>

			<p>At each iterate $x_n$ a quadratic (second-order Taylor) approximation is formed:</p>
			$$q(x) = f(x_n) + f'(x_n)(x-x_n) + \frac{1}{2}f''(x_n)(x-x_n)^2$$
			<p>The next iterate $x_{n+1}$ is the minimizer of $q(x)$, giving the update rule:</p>
			$$\begin{equation*}
			x_{n+1} = x_{n} - \frac{f'(x_n)}{f''(x_n)}
		   \end{equation*}$$
		   <p>The green curves show the quadratic approximation at each step. Provided $f''(x_n) > 0$ and $x_0$ is close enough to a local minimum, the method converges rapidly.</p>
		</td></tr>
	</table>

</main>

<script id="simulation_code" type="text/javascript">
	class Plot
	{
		constructor()
		{
			this.reset();
			this.num_newton_steps = 1;
		}

		reset()
		{
			this.num_newton_steps = parseInt(document.getElementById('newton_steps').value);
			this.fct = document.getElementById('fct').value;
			this.plotFunctions();
		}

		// f(x) = x^4 - 4x^2 + 2  (two minima at x = ±√2 ≈ ±1.414)
		quartic_function(x)
		{
			return x*x*x*x - 4*x*x + 2;
		}
		grad_quartic_function(x)
		{
			return 4*x*x*x - 8*x;
		}
		hess_quartic_function(x)
		{
			return 12*x*x - 8;
		}

		// f(x) = 0.5x^2 + 2sin(x)  (local minimum near x ≈ -1.03)
		sinusoidal_function(x)
		{
			return 0.5*x*x + 2*Math.sin(x);
		}
		grad_sinusoidal_function(x)
		{
			return x + 2*Math.cos(x);
		}
		hess_sinusoidal_function(x)
		{
			return 1 - 2*Math.sin(x);
		}

		// f(x) = exp(x) - 3x  (minimum at x = ln 3 ≈ 1.099)
		exponential_function(x)
		{
			return Math.exp(x) - 3*x;
		}
		grad_exponential_function(x)
		{
			return Math.exp(x) - 3;
		}
		hess_exponential_function(x)
		{
			return Math.exp(x);
		}

		// f(x) = x²/2 - 3·ln(x+4)  (minimum at x = -2+√7 ≈ 0.646)
		// Strictly convex; the log term makes the quadratic approximations
		// visibly different from the true curve.
		logarithmic_function(x)
		{
			return 0.5*x*x - 3*Math.log(x + 4);
		}
		grad_logarithmic_function(x)
		{
			return x - 3/(x + 4);
		}
		hess_logarithmic_function(x)
		{
			return 1 + 3/((x + 4)*(x + 4));
		}

		newton_step(grad_f, hess_f, x)
		{
			const h = hess_f(x);
			if (Math.abs(h) < 1e-10)
				return x;
			return x - grad_f(x) / h;
		}

		computeData(fct, grad_fct, hess_fct, x0, data)
		{
			let xValues = [];
			let yValues = [];

			let x = -3;
			let num_steps = 5000;
			for (let i = 0; i <= num_steps; i++)
			{
				xValues.push(x);
				yValues.push(fct(x));
				x += 6 / num_steps;
			}

			// Collect Newton iterates: x_0, x_1, ..., x_{num_newton_steps}
			let iterates = [x0];
			let current_x = x0;
			for (let i = 0; i < this.num_newton_steps; i++)
			{
				current_x = this.newton_step(grad_fct, hess_fct, current_x);
				iterates.push(current_x);
			}

			// Main function trace
			data.push({
				x: xValues,
				y: yValues,
				name: "f(x)",
				showlegend: true,
				line: { color: 'rgb(31, 119, 180)', width: 2 }
			});

			// For each Newton step: draw the local quadratic approximation and
			// a vertical dashed line marking the current iterate x_n
			for (let i = 0; i < this.num_newton_steps; i++)
			{
				const xn  = iterates[i];
				const xn1 = iterates[i + 1];
				const fn  = fct(xn);
				const gn  = grad_fct(xn);
				const hn  = hess_fct(xn);

				// q(x) = f(xn) + f'(xn)(x-xn) + 0.5*f''(xn)(x-xn)^2
				// drawn over a range that covers both xn and xn1
				const lo = Math.min(xn, xn1) - 0.4;
				const hi = Math.max(xn, xn1) + 0.4;
				const qx = [], qy = [];
				for (let j = 0; j <= 200; j++)
				{
					const xq = lo + j * (hi - lo) / 200;
					const dx = xq - xn;
					qx.push(xq);
					qy.push(fn + gn * dx + 0.5 * hn * dx * dx);
				}
				data.push({
					x: qx,
					y: qy,
					line: { color: 'rgb(0, 150, 0)', width: 2 },
					name: "quadratic approx.",
					showlegend: i == 0
				});

				// Vertical dashed line from x-axis up to f(x_n)
				data.push({
					type: 'line',
					x: [xn, xn],
					y: [0, fn],
					line: { color: 'rgb(0, 0, 0)', width: 2, dash: 'dash' },
					text: ["x_" + i.toString(), ""],
					textposition: "bottom center",
					mode: 'lines+markers+text',
					name: "x_n",
					showlegend: i == 0
				});
			}

			// Vertical dashed line for the final iterate
			const xFinal = iterates[this.num_newton_steps];
			const fFinal = fct(xFinal);
			data.push({
				type: 'line',
				x: [xFinal, xFinal],
				y: [0, fFinal],
				line: { color: 'rgb(0, 0, 0)', width: 2, dash: 'dash' },
				text: ["x_" + this.num_newton_steps.toString(), ""],
				textposition: "bottom center",
				mode: 'lines+markers+text',
				name: "x_n",
				showlegend: false
			});
		}

		plotFunctions()
		{
			var data = [];
			if (this.fct == "Quartic function")
			{
				this.computeData(
					this.quartic_function,
					this.grad_quartic_function,
					this.hess_quartic_function,
					2.5, data)
			}

			if (this.fct == "Sinusoidal function")
			{
				this.computeData(
					this.sinusoidal_function,
					this.grad_sinusoidal_function,
					this.hess_sinusoidal_function,
					-2.5, data)
			}

			if (this.fct == "Exponential function")
			{
				this.computeData(
					this.exponential_function,
					this.grad_exponential_function,
					this.hess_exponential_function,
					2.0, data)
			}

			if (this.fct == "Logarithmic function")
			{
				this.computeData(
					this.logarithmic_function,
					this.grad_logarithmic_function,
					this.hess_logarithmic_function,
					-2.5, data)
			}

			var layout = {
			  title: "Minimization with Newton's method",
			  width: 1000,
			  height: 600
			};

			Plotly.newPlot('plotOutput', data, layout);
		}

	}

	plot = new Plot();
	plot.reset();
</script>

</body>
</html>