{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Trapezoid rule"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "# n intervals, n+1 points\n",
    "def trapz(a,b,n,f):\n",
    "    h = (b-a)/n\n",
    "    x = np.linspace(a,b,n+1)\n",
    "    y = f(x)\n",
    "    res = np.sum(y[1:n]) + 0.5*(y[0] + y[n])\n",
    "    return h*res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example\n",
    "$$\n",
    "f(x) = x, \\qquad x \\in [0,1]\n",
    "$$\n",
    "Exact integral is 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Integral =  0.5\n"
     ]
    }
   ],
   "source": [
    "f = lambda x: x\n",
    "print(\"Integral = \", trapz(0.0,1.0,10,f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example\n",
    "$$\n",
    "f(x) = x^2, \\qquad x \\in [0,1]\n",
    "$$\n",
    "Exact integral is $1/3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Integral =  0.3350000000000001\n"
     ]
    }
   ],
   "source": [
    "f = lambda x: x**2\n",
    "print(\"Integral = \", trapz(0.0,1.0,10,f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example\n",
    "$$\n",
    "f(x) = \\exp(x)\\cos(x), \\qquad x \\in [0,\\pi]\n",
    "$$\n",
    "The exact integral is $-\\frac{1}{2}(1+\\exp(\\pi))$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     4    -1.26567653098185e+00\n",
      "     8    -3.11816113365945e-01    4.05905e+00\n",
      "    16    -7.76577835071954e-02    4.01526e+00\n",
      "    32    -1.93958006245669e-02    4.00385e+00\n",
      "    64    -4.84778281250620e-03    4.00096e+00\n",
      "   128    -1.21187271271594e-03    4.00024e+00\n",
      "   256    -3.02963615787633e-04    4.00006e+00\n",
      "   512    -7.57406187865683e-05    4.00002e+00\n",
      "  1024    -1.89351368735657e-05    4.00000e+00\n",
      "  2048    -4.73378310594796e-06    4.00000e+00\n"
     ]
    }
   ],
   "source": [
    "f = lambda x: np.exp(x)*np.cos(x)\n",
    "qe = -0.5*(1.0 + np.exp(np.pi)) # Exact integral\n",
    "\n",
    "n,N = 4,10\n",
    "e = np.zeros(N)\n",
    "for i in range(N):\n",
    "    e[i] = trapz(0.0,np.pi,n,f) - qe\n",
    "    if i > 0:\n",
    "        print('%6d %24.14e %14.5e'%(n,e[i],e[i-1]/e[i]))\n",
    "    else:\n",
    "        print('%6d %24.14e'%(n,e[i]))\n",
    "    n = 2*n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example\n",
    "\n",
    "An example of a periodic function\n",
    "$$\n",
    "f(x) = \\exp(\\sin(x)), \\qquad x \\in [0,2\\pi]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     2    -1.67174121383321e+00\n",
      "     3    -2.82600848168890e-04\n",
      "     4     3.43969188092368e-02\n",
      "     5    -3.45941231216784e-09\n",
      "     6    -2.82600848168890e-04\n",
      "     7     3.46389583683049e-14\n",
      "     8     1.25168897735506e-06\n",
      "     9     4.52970994047064e-14\n",
      "    10    -3.45941053581100e-09\n",
      "    11     4.44089209850063e-14\n",
      "    12     6.57429666262033e-12\n",
      "    13     4.35207425653061e-14\n",
      "    14     3.46389583683049e-14\n",
      "    15     4.44089209850063e-14\n",
      "    16     4.61852778244065e-14\n"
     ]
    }
   ],
   "source": [
    "f = lambda x: np.exp(np.sin(x))\n",
    "qe = 7.9549265210128 # Exact integral\n",
    "\n",
    "n,N = 2,15\n",
    "e = np.zeros(N)\n",
    "for i in range(N):\n",
    "    e[i] = trapz(0.0,2*np.pi,n,f) - qe\n",
    "    print('%6d %24.14e'%(n,e[i]))\n",
    "    n = n + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The error converges rapidly to machine zero. It is very small for even number of points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     3    -2.82600848168890e-04\n",
      "     5    -3.45941231216784e-09\n",
      "     7     3.46389583683049e-14\n",
      "     9     4.52970994047064e-14\n",
      "    11     4.44089209850063e-14\n",
      "    13     4.35207425653061e-14\n",
      "    15     4.44089209850063e-14\n",
      "    17     4.52970994047064e-14\n",
      "    19     4.52970994047064e-14\n",
      "    21     4.52970994047064e-14\n"
     ]
    }
   ],
   "source": [
    "n,N = 3,10\n",
    "e = np.zeros(N)\n",
    "for i in range(N):\n",
    "    e[i] = trapz(0.0,2*np.pi,n,f) - qe\n",
    "    print('%6d %24.14e'%(n,e[i]))\n",
    "    n = n + 2"
   ]
  }
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