{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "nm-0",
   "metadata": {},
   "source": "# Numerical methods\n\nPolynomials, interpolation, the FFT, statistics, and quadrature."
  },
  {
   "cell_type": "markdown",
   "id": "nm-1",
   "metadata": {},
   "source": "## Polynomials: fit, evaluate, roots"
  },
  {
   "cell_type": "code",
   "id": "nm-2",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "x = 0:0.5:5;\ny = 2 * x.^2 - 3 * x + 1;\np = polyfit(x, y, 2)\npolyval(p, 10)\nroots([1 -3 2])'"
  },
  {
   "cell_type": "markdown",
   "id": "nm-3",
   "metadata": {},
   "source": "## Interpolation"
  },
  {
   "cell_type": "code",
   "id": "nm-4",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "xs = 0:10;\nys = xs.^2;\ninterp1(xs, ys, 3.5)\ninterp1(xs, ys, [1.5 5.5 8.5])"
  },
  {
   "cell_type": "markdown",
   "id": "nm-5",
   "metadata": {},
   "source": "## The FFT"
  },
  {
   "cell_type": "code",
   "id": "nm-6",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "t = 0:0.01:1 - 0.01;\nsig = sin(2 * pi * 5 * t) + 0.5 * sin(2 * pi * 12 * t);\nF = abs(fft(sig));\n[~, idx] = max(F(1:50));\nfprintf('dominant frequency near %d Hz\\n', idx - 1);"
  },
  {
   "cell_type": "markdown",
   "id": "nm-7",
   "metadata": {},
   "source": "## Basic statistics"
  },
  {
   "cell_type": "code",
   "id": "nm-8",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "data = [4 8 15 16 23 42];\n[mean(data), median(data), std(data)]\n[min(data), max(data)]"
  },
  {
   "cell_type": "markdown",
   "id": "nm-9",
   "metadata": {},
   "source": "## Sorting, differences, cumulative sums"
  },
  {
   "cell_type": "code",
   "id": "nm-10",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "v = [3 1 4 1 5 9 2 6];\nsort(v)\nunique(v)\ncumsum(v)\ndiff(v)"
  },
  {
   "cell_type": "markdown",
   "id": "nm-11",
   "metadata": {},
   "source": "## Numerical integration with trapz"
  },
  {
   "cell_type": "code",
   "id": "nm-12",
   "metadata": {},
   "execution_count": null,
   "outputs": [],
   "source": "x = linspace(0, pi, 200);\napprox = trapz(x, sin(x))    % exact value is 2"
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "numbl (MATLAB syntax)",
   "language": "numbl",
   "name": "numbl"
  },
  "language_info": {
   "codemirror_mode": "octave",
   "file_extension": ".m",
   "mimetype": "text/x-octave",
   "name": "numbl",
   "nbconvert_exporter": "script",
   "pygments_lexer": "matlab"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}