Lec09-accurate-integral.ipynb

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    "### More accurate integration is possible using \"Simpson's\" rule\n",
    "\n",
    "https://en.wikipedia.org/wiki/Simpson%27s_rule\n",
    "\n",
    "this fits quadratic polynomial (instead of quadrilaterals from earlier) to consecutive sets of 3 points \n",
    "\n",
    "$\n",
    "f(x_{j})\n",
    "$\n",
    ".\n",
    "\n",
    "\n",
    "For a function divided into an even number of subintervals, N, then Simpsons rule for the approximate integral, I, of a function f can be written\n",
    "\n",
    "\n",
    "$$\n",
    "I = \\sum_{i=0}^{i=\\frac{N}{2}-1}\\frac{h}{3}\\left[ f( x_{2i} ) + 4f(x_{2i+1}) + f(x_{2i+2} ) \\right]\n",
    "$$\n",
    "\n",
    "where $h$ is the subinterval width."
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    "import numpy as np\n",
    "\n",
    "def trap(f, a, b, N ):\n",
    "    total_a=0\n",
    "    dx=(b-a)/(N)\n",
    "    xval=a\n",
    "    for i in [*range(0,N)]:\n",
    "        midy=0.5*(f(xval)+f(xval+dx))\n",
    "        area=midy*dx\n",
    "        total_a=total_a+area\n",
    "        xval=xval+dx\n",
    "    # print(xval)\n",
    "    return total_a\n",
    "\n",
    "def simpson(f, a, b, N):\n",
    "    h=(b-a)/(N)\n",
    "    x0=a\n",
    "    ival=0\n",
    "    for i in [*range(0, int(N/2) )]:\n",
    "        x1=x0+h\n",
    "        x2=x0+2*h\n",
    "        ival=ival+h/3*(f(x0) + 4*f(x1) + f(x2))\n",
    "        x0=x0+2*h\n",
    "    # print(x0)\n",
    "    return ival\n",
    "\n",
    "f=np.cos\n",
    "\n",
    "a=0\n",
    "b=2*np.pi/4\n",
    "N=10\n",
    "tans=1.\n",
    "print('simpson = ', simpson(f,a,b,N)-tans )\n",
    "print('   trap = ', trap(f,a,b,N)-tans    )"
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    "import scipy.integrate as si\n",
    "N=10\n",
    "dx=(b-a)/(N)\n",
    "x=np.linspace(a,b,N+1)\n",
    "y=f(x)\n",
    "si.simpson(y,x)-1."
   ]
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