Homework2.ipynb

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                "# Problem Set 2\n",
                "## Due Tuesday 18 October 2022\n",
                "\n",
                "Answer each of the following questions. **Read each question carefully to see what you need to do.** Each question has multiple parts. Don't miss any of the questions."
            ]
        },
        {
            "attachments": {},
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            "source": [
                "\u003cfont color = 'red'\u003e\n",
                "29/30\n",
                "\u003cfont\u003e"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 21,
            "metadata": {},
            "outputs": [],
            "source": [
                "# importing neccessary packages\n",
                "import numpy as np\n",
                "import pandas as pd\n",
                "import matplotlib.pyplot as plt"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Question 1\n",
                "\n",
                "We want to compute the derivative of $f(x) = \\log x$ at $x=3$. In doing so, you will learn about a powerful technique called Richardson extrapolation.\n",
                "\n",
                "a) first, find the exact expression for $f'(x)$ using calculus by hand. Evaluate that function at $x = 3$, and store that value in a variable `dfdx_exact`."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 22,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": "dfdx_exact = 0.3333333333333333\n"
                }
            ],
            "source": [
                "def dfdx_log(x):\n",
                "    if x\u003c=0:\n",
                "        raise ValueError(\"log(x) is only defined for x\u003e0: x must be greater than 0.\")\n",
                "    return 1/x\n",
                "\n",
                "dfdx_exact = dfdx_log(3)\n",
                "print(\"dfdx_exact =\",dfdx_exact)"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "b) Now compute the second order central difference approximation at $x = 3$ using a stepsize of $h_1=0.4$.  Save that in `dfdx_h1`. Calculate the relative error $\\epsilon_{h_1}$ with the exact solution."
            ]
        },
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            "attachments": {},
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            "source": [
                "\n",
                "$$f'_{cd}(x) \\equiv \\frac{f(x+h/2) - f(x-h/2)}{h}$$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 23,
            "metadata": {},
            "outputs": [],
            "source": [
                "def cd(f,x,h):\n",
                "    return (f(x+h/2)-f(x-h/2))/h"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 24,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": "dfdx_h1=  0.33382848156130684 \n\nThe relative error in the calculations of derivitive of log(x) at x=3 using the \ncentral difference approximation and a step size of 0.4 is about 0.00149\n"
                }
            ],
            "source": [
                "dfdx_h1 = cd(np.log, 3, .4)\n",
                "err_h1 = abs(dfdx_h1-dfdx_exact)/dfdx_exact\n",
                "\n",
                "print(\"dfdx_h1= \", dfdx_h1, '\\n')\n",
                "print(\"The relative error in the calculations of derivitive of log(x) at x=3 using the \\n\" + \\\n",
                "      \"central difference approximation and a step size of 0.4 is about\", round(err_h1, 5))"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "c) Do the same thing again, but use $h_2 = h_1/2 = 0.2$. Save this as `dfdx_h2`. Calculate the relative error $\\epsilon_{h_2}$ with the exact solution."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 25,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": "dfdx_h2=  0.33345687249336176 \n\nThe relative error in the calculations of derivitive of log(x) at x=3 using the \ncentral difference approximation and a step size of 0.2 is about 0.00037\n"
                }
            ],
            "source": [
                "dfdx_h2 = cd(np.log, 3, .2)\n",
                "err_h2 = abs(dfdx_h2-dfdx_exact)/dfdx_exact\n",
                "\n",
                "print(\"dfdx_h2= \", dfdx_h2, '\\n')\n",
                "print(\"The relative error in the calculations of derivitive of log(x) at x=3 using the \\n\" + \\\n",
                "      \"central difference approximation and a step size of 0.2 is about\", round(err_h2, 5))"
            ]
        },
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            "attachments": {},
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            "metadata": {},
            "source": [
                "Remember that if we continue to decrease $h$ arbitrarily, we will run into round off errors. We would like to further improve accuracy *without* decreasing the stepsize. This leads us to the extremely powerful technique of Richardson Extrapolation. Here, we compute $f'(x)$ twice using the same method with two different stepsizes and combine them together to get a *higher order* solution that we started with:\n",
                "\n",
                "$$f'_r(x) = f'_{h_2}(x) + \\frac{1}{2^2 - 1} ( f'_{h_2}(x) - f'_{h_1}(x))$$\n",
                "\n",
                "Compute this $f'_r(x)$ and compare its value at $x = 3$ to the analytic value. How does it compare? Make a table showing the absolute (e.g. $f'(3) - f'_r(3)$) and relative error ($(f'(3) - f'_r(3))/f'_r(3)$) for all three approximations."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 26,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": "dfdx_r= 0.3333330028040467\nThe relative error in the calculations of derivitive of log(x) at x=3 using the \nRichardson Extrapolation approximation and an original step size of 0.4 is about 9.916e-07\n"
                }
            ],
            "source": [
                "def rich_extra(f, x, h):\n",
                "    \"\"\"\n",
                "    Here we assume h is h_1 and h_2 is (h_1)/2.\n",
                "\n",
                "    We also use the Central Difference Approximation to get the derivitives using these different\n",
                "    step sizes\n",
                "    \"\"\"\n",
                "\n",
                "    h_1 = h\n",
                "    h_2 = h_1 / 2\n",
                "\n",
                "    approx = cd(f,x,h_2) +1/(2**2-1)*( cd(f,x,h_2)-cd(f,x,h_1) )\n",
                "    return approx\n",
                "\n",
                "dfdx_r = rich_extra(np.log,3,.4)\n",
                "err_r = abs(dfdx_r - dfdx_exact) / dfdx_exact\n",
                "\n",
                "print(\"dfdx_r=\", dfdx_r)\n",
                "print(\"The relative error in the calculations of derivitive of log(x) at x=3 using the \\n\" + \\\n",
                "      \"Richardson Extrapolation approximation and an original step size of 0.4 is about\", round(err_r, 10))"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 27,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": "\u003cdiv\u003e\n\u003cstyle scoped\u003e\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n\u003c/style\u003e\n\u003ctable border=\"1\" class=\"dataframe\"\u003e\n  \u003cthead\u003e\n    \u003ctr style=\"text-align: right;\"\u003e\n      \u003cth\u003e\u003c/th\u003e\n      \u003cth\u003eCD Approximation using h=0.4\u003c/th\u003e\n      \u003cth\u003eCD Approximation using h=0.2\u003c/th\u003e\n      \u003cth\u003eRE Approximation using h_1=0.4 and h_1=0.2\u003c/th\u003e\n    \u003c/tr\u003e\n  \u003c/thead\u003e\n  \u003ctbody\u003e\n    \u003ctr\u003e\n      \u003cth\u003eValue of Derivitive\u003c/th\u003e\n      \u003ctd\u003e0.3338284815613068\u003c/td\u003e\n      \u003ctd\u003e0.3334568724933618\u003c/td\u003e\n      \u003ctd\u003e0.3333330028040467\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003eRelative Error\u003c/th\u003e\n      \u003ctd\u003e0.0014854446839206\u003c/td\u003e\n      \u003ctd\u003e0.0003706174800853\u003c/td\u003e\n      \u003ctd\u003e0.0000009915878598\u003c/td\u003e\n    \u003c/tr\u003e\n  \u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e",
                        "text/plain": "                     CD Approximation using h=0.4  \\\nValue of Derivitive            0.3338284815613068   \nRelative Error                 0.0014854446839206   \n\n                     CD Approximation using h=0.2  \\\nValue of Derivitive            0.3334568724933618   \nRelative Error                 0.0003706174800853   \n\n                     RE Approximation using h_1=0.4 and h_1=0.2  \nValue of Derivitive                          0.3333330028040467  \nRelative Error                               0.0000009915878598  "
                    },
                    "execution_count": 27,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "data = [[dfdx_h1,dfdx_h2,dfdx_r],[err_h1,err_h2,err_r]]\n",
                "df = pd.DataFrame(data, index = [\"Value of Derivitive\", \"Relative Error\"], \\\n",
                "                  columns = [\"CD Approximation using h=0.4\", \"CD Approximation using h=0.2\", \\\n",
                "                  r\"RE Approximation using h_1=0.4 and h_1=0.2\"])\n",
                "pd.options.display.float_format = \"{:,.16f}\".format\n",
                "df"
            ]
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            "attachments": {},
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            "metadata": {},
            "source": [
                "\u003cfont color = 'red'\u003e\n",
                "9/10\n",
                "\n",
                "You were asked for relative and **Absolute** Error calculations. \n",
                "\u003cfont\u003e"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\n",
                "\n",
                "---\n",
                "\n",
                ""
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Question 2\n",
                "\n",
                "Debye's theory of solids gives the heat capacity of a solid at temperature $T$ to be \n",
                "\n",
                "$$C_V = 9 V \\rho k_B \\left(\\frac{T}{\\theta_D} \\right)^3 \\int_0^{\\theta_D/T} \\frac{x^4 e^x}{(e^x - 1)^2} dx,$$\n",
                "\n",
                "where $V$ is the volume of the solid, $\\rho$ is the number density of atoms, $k_B$ is Boltzmann's constant, and $\\theta_D$ is the *Debye temperature*, a property of solids dependent on their density and sound speed.\n",
                "\n",
                "a) write a function `cv(T)` that calculates $C_V$ for a given temperature `T` for the specific case of a $1000\\ \\mathrm{cm}^3$ block of solid aluminum, with number density $\\rho = 6.022 \\times 10^{28}\\ \\mathrm{m^{-3}}$ and a Debye temperature $\\theta_D = 428\\ \\mathrm{K}$. Use Gaussian quadruture to compute the integral with $N=50$ points\n",
                ""
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 28,
            "metadata": {},
            "outputs": [],
            "source": [
                "def gauss(f,a,b,N):\n",
                "    y,wprime = np.polynomial.legendre.leggauss(N)\n",
                "    x = 0.5*(y + 1)*(b - a) + a\n",
                "    w = wprime* 0.5*(b-a)\n",
                "    return (f(x)*w).sum()"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 29,
            "metadata": {},
            "outputs": [],
            "source": [
                "def cv(T):\n",
                "    V = 0.001 # converting to m\n",
                "    rho = 6.022 * 10**28\n",
                "    theta_D = 428\n",
                "    k_B = 1.380649 * 10**(-23) #SI units\n",
                "    integrand = lambda x: x**4 * np.exp(x)/(np.exp(x)-1)**2\n",
                "    if type(T) == int:\n",
                "        integral = gauss(integrand,0,theta_D/T,50)\n",
                "    else: #assume its a numpy array\n",
                "        integral = np.array([gauss(integrand,0,theta_D/T_ele,50) for T_ele in T])\n",
                "    return 9*V*rho*k_B*(T/theta_D)**3 * integral"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "b) Use that function to make a plot of the heat capacity as a function of temperature from $T=5\\ \\mathrm{K}$ K to $T=500\\ \\mathrm{K}$.\n",
                ""
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 30,
            "metadata": {},
            "outputs": [
                {
                    "data": {
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\n",
                        "text/plain": "\u003cFigure size 640x480 with 1 Axes\u003e"
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "T_array = np.linspace(5,500,496)\n",
                "cv_array = cv(T_array)\n",
                "\n",
                "plt.plot(T_array, cv_array)\n",
                "plt.xlabel(\"Tempature (K)\")\n",
                "plt.ylabel(r\"Heat Capacity $JK^{-1}$\")\n",
                "plt.title(\"Heat Capacity of the Block of Aluminum Using Debye's Theory of Solids \\n\")\n",
                "plt.show()"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "c) Could you have used the Simpson's rule to do this integral? Why or why not?"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\u003cfont color = 'blue'\u003e\n",
                "No, we could not use Simpson's rule to do this integral as the integral had 50 bins and Simpson's rule requires that the number of bins be odd and greater than 3. \n",
                "\n",
                "\u003c/font\u003e"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\u003cfont color = 'red'\u003e\n",
                "10/10\n",
                "\n",
                "Even if you use an odd number of bins Simposon wont work, why?\n",
                "\u003cfont\u003e"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\n",
                "\n",
                "---\n",
                "\n",
                ""
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Question 3\n",
                "Make a plot of the semi-circle fucntion \n",
                "\n",
                "$$\n",
                "f(x) \\ = \\ \\sqrt{1-x^2}\n",
                "$$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 31,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": "\u003cFigure size 640x480 with 1 Axes\u003e"
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "f_x_ = lambda x: np.sqrt(1-x**2) # only defined form x = -1 to x = 1(positive values)\n",
                "\n",
                "x_array = np.linspace(-1,1,201) # this function is only defined from x=-1 to x=1\n",
                "f_x_array = f_x_(x_array)\n",
                "\n",
                "plt.plot(x_array, f_x_array)\n",
                "plt.xlabel(\"x\")\n",
                "plt.ylabel(\"f(x)\")\n",
                "plt.title(\"Plot of this Unit Semi-Circle\")\n",
                "plt.show()"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Use your implementation of Simpson's rule to find $\\pi$.\n",
                "\n",
                "$$\n",
                "\\pi \\ = \\ 4 \\int_{0}^{1} \\sqrt{1-x^{2}}\\, \\mathrm{d} x\n",
                "$$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 32,
            "metadata": {},
            "outputs": [],
            "source": [
                "def qsimp(f,a,b,N):\n",
                "    \"\"\"qsimp uses Simpson's rule to evaluate an approximation to the integral of f(x) on the interval [a, b] using N points.\n",
                "    \n",
                "    inputs\n",
                "    ------\n",
                "    f: function that can operate on numpy arrays\n",
                "    a: float representing the lower limit of the integral\n",
                "    b: float representing the upper limit of the integral\n",
                "    N: int representing the number of samples\n",
                "    \n",
                "    \"\"\"\n",
                "    if N%2!=1 and N\u003e=3:\n",
                "        raise ValueError(\"Simpson's rule requires that N, the number of points be odd and at least 3.\")\n",
                "    h = (b-a)/(N-1) # first we define the width between each x_i\n",
                "    x = a+np.arange(N)*h # np.arange(N) returns a np array; we can multiply it and add numbers to each element in it\n",
                "    \n",
                "    w = np.zeros(N) #initializing a weight array\n",
                "\n",
                "    for i in range(len(x)):\n",
                "        if i%2==1:\n",
                "            w[i] = h *4/3 # i is even\n",
                "        else:\n",
                "            w[i] = h *2/3 # i is odd\n",
                "    w[0] = w[-1] = h/3 # fixing end points\n",
                "        \n",
                "    data = f(x) # f is a function that we \"pass\" into the qtrap; here we have it evaluate at every point in x, \n",
                "                # storing the results in a new numpy array I called \"data\"\n",
                "    \n",
                "    # this is the actual integral: data*w is a new numpy array \n",
                "    # of the same length as data and w, which I call \"integrand\"\n",
                "    integrand = data*w\n",
                "    \n",
                "    # numpy arrays can sum themeselves by calling their .sum() method:\n",
                "    integral = integrand.sum()\n",
                "\n",
                "    return integral"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 33,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": "pi= 3.141592639067041\n"
                }
            ],
            "source": [
                "pi = 4*qsimp(f_x_,0,1,100001) # lets try and get it super accurate but not take forever\n",
                "\n",
                "print(\"pi=\",pi)"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Make a log-log plot of the convergence as a function of the number of sample points. What is the convergence exponent? Why is it not -4? "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 34,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": "\u003cFigure size 640x480 with 1 Axes\u003e"
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "exp_arr = np.linspace(0,5,100)\n",
                "N_ls = []\n",
                "for exp in exp_arr:\n",
                "    N = 3+(int(10**exp)) + (int(10**exp))%2 #encures N in odd and greater than 3 while also scaling logarithmically\n",
                "    N_ls.append(N)\n",
                "N_ls = np.unique(N_ls) #makes sure we don't repeat elements\n",
                "\n",
                "pi_approx_arr = np.array([4*qsimp(f_x_,0,1,N) for N in N_ls])\n",
                "\n",
                "plt.semilogx(N_ls, pi_approx_arr, label = r\"Approximation of $\\pi$\")\n",
                "plt.semilogx(N_ls, [np.pi for i in range(len(N_ls))], label = r\"$\\pi$\")\n",
                "plt.xlabel(\"Number of Sample Points\")\n",
                "plt.ylabel(\"Value\")\n",
                "plt.legend()\n",
                "plt.title(r\"The Convergence of $\\pi$ using our f(x) and Simpson's Rule\")\n",
                "plt.show()"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 35,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": "\u003cFigure size 640x480 with 1 Axes\u003e"
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "err_1 = abs(pi_approx_arr-np.pi)/np.pi\n",
                "\n",
                "\n",
                "plt.loglog(N_ls, err_1)\n",
                "plt.xlabel(\"Number of Sample Points\")\n",
                "plt.ylabel(\"Fractional Error\")\n",
                "plt.title(r\"The Fractional Error in Approximations of $\\pi$ using our f(x) and Simpson's Rule\")\n",
                "plt.show()"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\u003cfont color = 'blue'\u003e\n",
                "The convergence exponent can be estimated by the slope of this curve. It is clear that the slope of this curve is not -4. I would estimate the convergence exponent is $1.6\\pm0.5$. While someone might expect it to be -4 form the factor of 4 outside our integral: $\\pi \\ = \\ 4 \\int_{0}^{1} \\sqrt{1-x^{2}}\\, \\mathrm{d} x$. However, this is not how we go about calculating the convergence exponent. \n",
                "\n",
                "\u003c/font\u003e"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "There are other ways to do an integral and get the same result. Define the *sphere function*\n",
                "\n",
                "$$\n",
                "S(x,y) \\ = \\ \\begin{cases} 1  \u0026 \\mathrm{if} \\quad x^{2} + y^{2} \u003c 1 \\\\ \\\\\n",
                "                         0  \u0026 \\mathrm{if} \\quad x^{2} + y^{2} \\ge 1\n",
                "           \\end{cases}\n",
                "$$\n",
                "\n",
                "Therefore we can also compute $\\pi$ with the two dimensional integral\n",
                "\n",
                "$$\n",
                "\\pi \\ = \\ \\int_{-1}^{1} \\int_{-1}^{1}  S(x,y) \\, \\mathrm{d} x \\, \\mathrm{d} y \\ = \\ \\mathrm{Area \\ of \\ a\\  unit \\ circle.}\n",
                "$$\n",
                "\n",
                "\n",
                "Make an integration routine that uses your Simpson's integrator to compute two-dimensional integrals.  Compute $\\pi$ using this integrator.\n",
                "\n",
                "\n",
                "Note: You can save yourself a factor of $4 = 2^{2}$ in cost by just integrating over the positive values. "
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\u003cfont color = 'blue'\u003e\n",
                "\n",
                "Due to symmetry we can says a factor of 4 in this calculation by integrating:\n",
                "$$\n",
                "\\pi \\ = 4\\ \\int_{0}^{1} \\int_{0}^{1}  S(x,y) \\, \\mathrm{d} x \\, \\mathrm{d} y \\ = \\ \\mathrm{Area \\ of \\ a\\  unit \\ circle.}\n",
                "$$\n",
                "\n",
                "\n",
                "\u003c/font\u003e"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 36,
            "metadata": {},
            "outputs": [],
            "source": [
                "def S(x,y):\n",
                "    if x**2 + y**2 \u003c 1:\n",
                "        return 1\n",
                "    else:\n",
                "        return 0"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 37,
            "metadata": {},
            "outputs": [],
            "source": [
                "def qsimp_2_dim(f,x_i,x_f,y_i,y_f,N):\n",
                "    \"\"\"qsimp uses Simpson's rule to evaluate an approximation to the integral of f(x) on the interval [a, b] using N points.\n",
                "    \n",
                "    inputs\n",
                "    ------\n",
                "    f: function that can operate on numpy arrays\n",
                "    x_0: float representing the lower limit of the integral with respect to the x variable \n",
                "    x_f: float representing the upper limit of the integral with respect to the x variable\n",
                "    y_0: float representing the lower limit of the integral with respect to the y variable \n",
                "    y_f: float representing the upper limit of the integral with respect to the y variable\n",
                "    N: int representing the number of samples which is the same for both variables\n",
                "    \n",
                "    \"\"\"\n",
                "    if N%2!=1 and N\u003e=3:\n",
                "        raise ValueError(\"Simpson's rule requires that N, the number of points be odd and at least 3.\")\n",
                "    h = (x_f-x_i)/(N-1) # first we define the width between each x_i\n",
                "    x_array = x_i+np.arange(N)*h # np.arange(N) returns a np array; we can multiply it and add numbers to each element in it\n",
                "    y_array = y_i+np.arange(N)*h \n",
                "\n",
                "    w_x = np.zeros(N) #initializing a weight array\n",
                "\n",
                "    for i in range(len(x_array)):\n",
                "        if i%2==1:\n",
                "            w_x[i] = h *4/3 \n",
                "        else:\n",
                "            w_x[i] = h *2/3 \n",
                "    w_x[0] = w_x[-1] = h/3 # fixing end points\n",
                "\n",
                "    w_x_y = np.array([w_x for i in range(len(w_x))]) # and array of arrays where we must get each\n",
                "                                                     # individual weight using 2 dimensions\n",
                "    \n",
                "    for i in range(1,len(y_array)-1):   # doesnt fix endpint weights\n",
                "        if i%2==1:\n",
                "            w_x_y[i] = w_x_y[i] * h *4/3 \n",
                "        else:\n",
                "            w_x_y[i] = w_x_y[i] * h *2/3 \n",
                "    w_x_y[0] = w_x_y[-1] = w_x_y[0] *h/3 # fixing end points\n",
                "        \n",
                "    integral = 0\n",
                "\n",
                "    # summing each value by their weights\n",
                "    for i in range(len(x_array)):\n",
                "        for j in range(len(y_array)):\n",
                "            x = x_array[i]\n",
                "            y = y_array[j]\n",
                "            val = f(x,y)\n",
                "            integral += w_x_y[i][j] * val\n",
                "\n",
                "    return integral"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 38,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": "pi_2= 3.1415844443064866\n"
                }
            ],
            "source": [
                "pi_2= 4*qsimp_2_dim(S,0,1,0,1,3001)\n",
                "\n",
                "print(\"pi_2=\",pi_2)"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Make a log-log plot of the error in your calculation *as a function of the of the total number of calls to your function $S$*."
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\u003cfont color = 'blue'\u003e\n",
                "It is clear that we call S, $N^2$ times.\n",
                "\u003c/font\u003e"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 39,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "image/png": 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\n",
                        "text/plain": "\u003cFigure size 640x480 with 1 Axes\u003e"
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "exp_arr = np.linspace(0,3,120)\n",
                "N_ls = []\n",
                "for exp in exp_arr:\n",
                "    N = 1+(int(10**exp)) + (int(10**exp))%2 #encures N in odd and greater than 3 while also scaling logarithmically\n",
                "    N_ls.append(N)\n",
                "N_ls = np.unique(N_ls) #makes sure we don't repeat elements\n",
                "S_calls = N_ls**2\n",
                "\n",
                "pi_approx_arr = np.array([4*qsimp_2_dim(S,0,1,0,1,N) for N in N_ls])\n",
                "err = abs(pi_approx_arr - np.pi)/np.pi\n",
                "\n",
                "plt.loglog(S_calls, err)\n",
                "plt.xlabel(\"Number of calls to S function\")\n",
                "plt.ylabel(\"Fractional Error\")\n",
                "plt.title(r\"The Fractional Error in Approximations of $\\pi$ using our S(x,y) and Simpson's Rule\")\n",
                "plt.show()"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 40,
            "metadata": {},
            "outputs": [],
            "source": [
                ""
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\u003cfont color = 'red'\u003e\n",
                "10/10\n",
                "\u003cfont\u003e"
            ]
        }
    ]
}