hw02_approximation.ipynb

{
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   "source": [
    "## ASTR 2100, Winter 2020\n",
    "\n",
    "## Homework Assignment 2 (70 points + 10 extra-credit points)\n",
    "\n",
    "### Distributed: Monday, January 27\n",
    "\n",
    "### Due: Wednesday,  Feb 5 at 11:59pm \n",
    "\n",
    "#### Exercises you need to do for the assignment grade are tasks in Exercises 1 and 2 along with relative difficulty of each task in the exercise denoted by associated points\n",
    "#### Optional exercises are marked as extra-credit\n",
    "\n",
    "#### Please submit you notebook (i.e. its *.ipynb file) as an attachment via Canvas. You can also attach other files, if needed to your submission. Make sure that all of the cells in the notebook have output that you want, when you run all cells just before the submission\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
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   "source": [
    "#### Jupyter notebooks with class notes and assignments: https://github.com/a-kravtsov/a211w20\n",
    "\n",
    "#### Slack workspace for this class: https://a211w20.slack.com\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Instructor: Andrey Kravtsov (kravtsov@uchicago.edu)\n",
    "\n",
    "#### TAs: Dimitrios Tanoglidis (dtanoglidis@uchicaago.edu), Georgios Zakharegkas (gzakharegkas@uchicago.edu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "import packages needed by the codes below. Run this cell first before using these codes. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "# use jupyter \"magic\" command to tell it to embed plot into the notebook \n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "from codes.plot_utils import plot_pretty\n",
    "plot_pretty(fontsize=12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1: Using splines for interpolation and estimates of  derivatives and integrals\n",
    "### (40 points)\n",
    "\n",
    "**Background.** I introduced splines as a way to construct piecewise polynomial approximation that is continuous and smooth and also has smooth first derivative (see the <a href=\"https://github.com/a-kravtsov/a211w20/blob/master/03_approximation.ipynb\"><tt>03_approximation.ipynb</tt></a> notebook). Splines not only provide a general flexible and accurate way to approximate a function represented by some tabulated values at the training points, but can also automatically provides approximations for the function first three derivatives and its integral. Moreover, splines can be used as a tool for numerical solution (finding roots) of equations and inverting functions. In this sense splines are one of the \"Swiss army knife\"-type numerical methods that are very useful to have in one's toolbox. \n",
    "\n",
    "In this exercise you will practice using splines for these tasks. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### What you are learning in this exercise\n",
    "\n",
    "* To use polynomials and splines to approximate a function using SciPy routines\n",
    "* Use splines to obtain accurate and smooth estimate of the function derivative\n",
    "* Use spline to obtain accurate estimate of the function integral for integration intervals that fall within the interval, in which the approximating spline was constructed.  \n",
    "* Use splines to invert a complicated function (which is equivalent to finding a root of a function). Inversion is when we instead of estimating $y=f(x)$, find $x$ that satisfies such equation given a $y$ value. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data used in this exercise\n",
    "\n",
    "For this exercise we will use a function $P(k)$ that is tabulated in a data file. \n",
    "\n",
    "For the purposes of this exercise it is not needed to know what this function is. This is something called \"power spectrum\" - the concept we will come back to when we will discuss Fourier transforms (also see below task 2). This particular power spectrum is often used in astrophysical calculations related to galaxies and the spatial structures in galaxy distribution. The tabulated values in the file were computed using a sophisticated code <a href=\"https://camb.info/\">CAMB</a> that solves a complicated system of differential equations. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# read in function that will be used for this exercise\n",
    "k, Pk = np.loadtxt('data/test_matterpower_logintk1000.dat', usecols=(0, 1),  unpack=True)\n",
    "\n",
    "# their log10 values that will be used \n",
    "lk, lPk = np.log10(k), np.log10(Pk)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 525x525 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 525x525 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from codes.plot_utils import plot_line_points\n",
    "plot_pretty(fontsize=12)\n",
    "\n",
    "plot_line_points(k, Pk, xlabel=r'$k$', ylabel=r'$P(k)$', figsize=3)\n",
    "plot_line_points(lk, lPk, xlabel=r'$\\log_{10} k$', ylabel=r'$\\log_{10} P(k)$', figsize=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plots above show that $P(k)$ is changing very rapidly for small values of $k$ and slowly for large values. Functions that change rapidly are more difficult to approximate accurately because more training points are needed in the areas of rapid change. So it is often useful to examine how $\\log_{10}$ of the function behaves as a function of $\\log_{10}$ of the parameter (i.e., examine function in logarithmic space). This is what is done in the second plot above. We can see that in this case, $\\log_{10} P(k)$ changes relatively slowly and so approximating $\\log_{10} P(k)$ as a function of $\\log_{10} k$ is preferrable. \n",
    "\n",
    "Overall, $\\log_{10} P(k)$ looks like two power law functions of different exponent smoothly joined in the middle, but notice little \"wiggles\" around $\\log_{10} k\\sim -1$. These are real features of this function and make it not entirely trivial to approximate with, say, a polynomial of low order. \n",
    "\n",
    "Next, let's see how <tt>k</tt> and <tt>lk</tt> are spaced. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.04304196 0.04303239 0.04305963 0.04303769 0.04302909 0.04304367\n",
      " 0.04303872 0.04302599 0.04304582 0.04303598]\n",
      "0.0003967464299742818 0.04306143598772039\n"
     ]
    }
   ],
   "source": [
    "# np.diff computes difference of 2 subsequent elements of lk\n",
    "dlk = np.diff(lk) \n",
    "# if we look at the first 10 elements, it may seem that l (and lk) are evenly spaced\n",
    "print(dlk[:10])\n",
    "# however, this is not the case\n",
    "print(dlk.min(), dlk.max())\n",
    "# spacing of data points is not uniform overall"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polynomial interpolation\n",
    "\n",
    "In class on Friday I showed that straitforward solution of the linear system of equations for the polynomial coefficients by computing the inverse of the Vandermonde matrix as in the code below (this is similar to the code for approximating $d_L$ in <tt>03_approximation.ipynb</tt> notebook that I showed in class on Friday). \n",
    "\n",
    "For small systems of equations this works fine. For example, consider a system of two equations\n",
    "\n",
    "\\begin{eqnarray}\n",
    "f_0 &=& a_0 + a_1 x_0\\\\\n",
    "f_1 &=& a_0 + a_1 x_1\n",
    "\\end{eqnarray}\n",
    "Suppose we have 2 training points, $x_0=1$, $x_1=2$, and function values at these points $f_0=3$, $f_1=5$. We can easily solve this system ourselves for the coefficients: $a_0=2$, $a_1=1$. \n",
    "\n",
    "The code I used on Friday in class can do this as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2. 1.]\n"
     ]
    }
   ],
   "source": [
    "xtr = np.array([1., 2.])\n",
    "ftr = np.array([3., 5.])\n",
    "\n",
    "# construct the Vandermonde matrix for vector xtr\n",
    "V = np.vander(xtr, N=np.size(xtr), increasing=True) \n",
    "# solve for the polynomial (linear order in this case)\n",
    "a = np.dot(np.linalg.inv(V), ftr) \n",
    "# the coefficients are in reverse order in a, so we can reverse it for printing using ::-1 slicing\n",
    "print(a[::-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you saw on Friday and as the code below shows again, this way of solving for coefficients is numerically unstable, and fails in interpolation using as few as 20 training points. Example below will produce reasonably accurate interpolating polynomial for the number of training points <tt>ntr<20</tt>, but will fail dramatically for any larger number of the training points. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# set up vector of training points\n",
    "xmin, xmax, ntr = 0., 2.*np.pi, 21\n",
    "xtr = np.linspace(xmin, xmax, ntr)\n",
    "# for testing points use sin(x) function\n",
    "# compute function values at training points\n",
    "ftr = np.sin(xtr)\n",
    "\n",
    "# set up a vector of test points\n",
    "nt = 100 # should be different from nt\n",
    "xt = np.linspace(xmin, xmax, nt)\n",
    "\n",
    "# construct Vandermonde matrix for xtr vector\n",
    "V = np.vander(xtr, N=ntr, increasing=True) \n",
    "\n",
    "# solve for polynomial coefficients vector a, as a=V^{-1}f\n",
    "a = np.dot(np.linalg.inv(V), ftr) \n",
    "\n",
    "# np.poly1d initializes a 1d polynomial class with input coefficients\n",
    "# [::-1] reverses order of the coefficients, as required by this routine\n",
    "pfit = np.poly1d(a[::-1]) \n",
    "# now compute interpolating polynomial approximation at test values xt\n",
    "poly_int = pfit(xt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 525x525 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_line_points(xt, poly_int, xp=xtr, yp=ftr, points=True,\n",
    "                 xlabel=r'$x$', ylabel=r'$f(x)$', figsize=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is because, matrix $V$ involves a wide range of powers of $x$ and thus its elements have a vast range of values. Such matrices have large *condition* value (see <tt>03_approximaton.ipynb</tt> for the definition of condition) and are called \"ill- or poorly conditioned\". \n",
    "\n",
    "However, if we instead use <tt>numpy.linalg.solve</tt> routine to solve the same system of equations, namely if we replace the following line in the <tt>03_approximation.ipynb</tt>\n",
    "\n",
    "    a = np.dot(np.linalg.inv(V), ftr) # a = V^{-1} f\n",
    "\n",
    "with \n",
    "\n",
    "    a = np.linalg.solve(V, ftr)\n",
    "    \n",
    "this fixes the problem at least until the number of training of points is much larger than 20. \n",
    "This is because <tt>numpy.linalg.solve</tt> uses a more numerically stable algorithm with LU decomposition of the Vandermonde matrix to solve the system of equations (the routine uses fortran-based LAPACK library, see <a href=\"https://software.intel.com/en-us/node/468876\">here</a>). \n",
    "\n",
    "Likewise, if we replace that line with \n",
    "\n",
    "    a = np.linalg.lstsq(V, ftr, rcond=None)[0] \n",
    "   \n",
    "this solves the problem as well because this routine uses numerically robust Singular Value Decomposition (SVD) method to solve the system of normal equations. Note that we can control the order of approximating polynomial in this case, by limiting the size of the Vandermonde matrix to the required order <tt>poly_order</tt> as shown here: \n",
    "    V = np.vander(xtr, N=poly_order, increasing=True)\n",
    "\n",
    "Let's see if this will also work for approximating function $P(k)$ that was read above. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**Task 1a. (10 points)** Based on the code in the 03 notebook reproduced above, write a function that constructs interpolating (using <tt>np.linalg.solve</tt>) or approximating (using <tt>np.linalg.lstsq</tt>) polynomial for input vector of test points and corresponding tabulated function values with the following interface:\n",
    "\n",
    "    def poly_approx(xtr, ftr, xt, method='interpolation', poly_order=1):\n",
    "        \"\"\"\n",
    "        function constructing an approximating polynomial for the input \n",
    "        vector of training values of the approximated function ftr, \n",
    "        tabulated at the vector of training points xtr and compute the values\n",
    "        of constructed polynomial at the input vector of test values xt\n",
    "    \n",
    "        Parameters:\n",
    "        -----------\n",
    "        xtr, ftr: array like\n",
    "                    vectors of training points and tabulated function values\n",
    "        xt: array like\n",
    "                    vector of test points\n",
    "        method: string\n",
    "                defining the method to be used in construction of the polynomial\n",
    "                'interpolation' - construct interpolating polynomial using np.linalg.solve\n",
    "                'regression' - construct approximating polynomial np.linalg.lstsq\n",
    "        poly_order: integer\n",
    "                    order of the approximating polynomial. This parameter is ignored\n",
    "                    for method='interpolation', but is used for 'regression'\n",
    "                    must be 0<=poly_order<=np.size(xtr)\n",
    "        Returns:\n",
    "        --------\n",
    "        ft: array like\n",
    "            values of the polynomial at xt \n",
    "        a:  array like\n",
    "            polynomial coefficients\n",
    "        \"\"\"\n",
    "\n",
    "Test this function using $\\sin(x)$ as a  test function as shown below and demonstrate that your routine works (i.e., produces sensible results). Try changing the number of training points used <tt>ntr</tt> from 20 to smaller values. Identify and report the smallest <tt>ntr</tt> for which you get *interpolating* polynomial that still looks qualitatively as a sine (for example, it has a clear minimum and maximum in the $[0,2\\pi]$ interval). Try to come with your interpretation of the meaning of the minimum value that you find and briefly describe it. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'poly_approx' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-8-407d4146c318>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m     10\u001b[0m \u001b[0mxt\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlinspace\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mxmin\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mxmax\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mnt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     11\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 12\u001b[1;33m \u001b[0mpoly_int\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ma_int\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mpoly_approx\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mxtr\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mftr_int\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mxt\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'interpolation'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     13\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     14\u001b[0m \u001b[0mpoly_order\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m17\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNameError\u001b[0m: name 'poly_approx' is not defined"
     ]
    }
   ],
   "source": [
    "# test your routine using code below\n",
    "\n",
    "xmin, xmax, ntr = 0., 2.*np.pi, 20\n",
    "xtr = np.linspace(xmin, xmax, ntr)\n",
    "ftr_int = np.sin(xtr)\n",
    "# this vector is sine function + a little bit of Gaussian random noise (so mimics approximating noisy data)\n",
    "ftr_fit = np.sin(xtr) + np.random.normal(scale=0.05, size=np.size(xtr))\n",
    "\n",
    "nt = 1000 # should be different from nt\n",
    "xt = np.linspace(xmin, xmax, nt)\n",
    "\n",
    "poly_int, a_int = poly_approx(xtr, ftr_int, xt, method='interpolation')\n",
    "\n",
    "poly_order=17\n",
    "poly_fit, a_fit = poly_approx(xtr, ftr_fit, xt, method='regression', poly_order=poly_order)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot results\n",
    "plot_line_points(xt, poly_int, xp=xtr, yp=ftr_int, points=True,\n",
    "                 xlabel=r'$x$', ylabel=r'$f(x)$', figsize=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_line_points(xt, poly_fit, xp=xtr, yp=ftr_fit, points=True,\n",
    "                 xlabel=r'$x$', ylabel=r'$f(x)$', figsize=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 1b. (5 points)** Now apply the routine you implemented to interpolate $P(k)$ read in below. If you don't manage to get the working implementation in 1a, you can use <a href=\"https://docs.scipy.org/doc/numpy/reference/generated/numpy.polyfit.html\"><tt>np.polyfit</tt></a> routine for this exercise. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nkt = 1000\n",
    "kt = np.linspace(k.min(), k.max(), nkt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly_int, a_int = poly_approx(lk, lPk, kt, method='regression', poly_order=np.size(kt))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(a_int, np.size(a_int))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_line_points(kt, poly_int, xlabel=r'$x$', ylabel=r'$f(x)$', figsize=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happens? Try doing this for $\\log_{10} k$ and $\\log_{10} P(k)$ instead of $P(k)$. Then try with regression. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "lkt = np.linspace(lk.min(), lk.max(), nkt)\n",
    "poly_intl, a_intl = poly_approx(lk, lPk, lkt, method='interpolation')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly_int, a_int = poly_approx(lk, lPk, lkt, method='regression', poly_order=20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_line_points(lkt, poly_int, xlabel=r'$x$', ylabel=r'$f(x)$', figsize=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do you get a sensible result? Think about what the routine is doing and try to come up with an explanation for the outcome of this exercise. (7 points)\n",
    "\n",
    "Try regression with smaller poly_order values and find polynomial order for which at least somewhat sensible approximation can be obtained. What is that order? Think about a possible explanation for why this works for small order (3 points). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### SciPy routines for spline interpolation and approximation\n",
    "\n",
    "Spline interpolation is implemented in the SciPy library <a href=\"http://scipy.github.io/devdocs/interpolate.html\"><tt>scipy.interpolate</tt></a> in routines <a href=\"https://docs.scipy.org/doc/scipy-0.19.1/reference/generated/scipy.interpolate.interp1d.html\"><tt>interp1d</tt></a> and <a href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.UnivariateSpline.html\"><tt>UnivariateSpline</tt></a>. Scipy   has more spline routines for more specialized tasks (see the list <a href=\"http://scipy.github.io/devdocs/interpolate.html\">here</a> and examples of their use <a href=\"https://docs.scipy.org/doc/scipy-1.0.0/reference/tutorial/interpolate.html\">here</a>), but the routines listed here are most frequently used for 1-dimensional function interpolation/approximation. \n",
    "\n",
    "<a href=\"https://docs.scipy.org/doc/scipy-0.19.1/reference/generated/scipy.interpolate.interp1d.html\"><tt>interp1d</tt></a> is a routine written in python, although parts of it are written in <a href=\"https://cython.org/\">Cython</a> and use a couple of <a href=\"\">Fitpack library routines</a> and can perform only spline interpolation (i.e. approximation matching the function values at the training values to machine precision). \n",
    "\n",
    "<a href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.UnivariateSpline.html\"><tt>UnivariateSpline</tt></a> is a class-based python wrapper around routines <a href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.splev.html\"><tt>splev</tt></a>, <a href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.splrep.html#scipy.interpolate.splrep\"><tt>splrep</tt></a>, and <a href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.splprep.html#scipy.interpolate.splprep\"><tt>splprep</tt></a> that use old, efficient fortran-based Fitpack library of interpolation routines. This function allows construction of the spline that is not interpolating data, but approximating it using the least squares regression discussed in class on Friday. The effective weights of tabulated function data points are set by the input parameter <tt>s</tt>.   \n",
    "\n",
    "For <tt>s=0</tt> <tt>UnivariateSpline</tt> is equivalent to routine\n",
    "SciPy <a href=\"http://scipy.github.io/devdocs/generated/scipy.interpolate.InterpolatedUnivariateSpline.html#scipy.interpolate.InterpolatedUnivariateSpline\"><tt>InterpolatedUnivariateSpline</tt></a> that constructs spline that matches the function values at the training points to machine precision.\n",
    "\n",
    "For <tt>s>0</tt> <tt>UnivariateSpline</tt> constructs spline that satisfies condition for the $N+1$ training points $x_i$:\n",
    "\n",
    "$$\\sum\\limits_{i=0}^N w_i [f_i-{\\rm spline}(x_i)]^2 \\leq s$$\n",
    "\n",
    "by solving the system of normal equations for the spline coefficients. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below are my routines that compute coefficients of the interpolating \n",
    "natural cubic spline and the spline itself, given the vector of these coefficients.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cubic_spline_coefficients(xi, fi):\n",
    "    \"\"\"\n",
    "    compute coefficients of the interpolating natural cubic spline\n",
    "    see Appendix in the note for the derivation and details of the algorithm\n",
    "    \n",
    "    Parameters: \n",
    "        xi, fi: numpy float vectors\n",
    "                tabulated points and function values\n",
    "            \n",
    "    Returns:\n",
    "        a, b, c, d: numpy float vectors\n",
    "                cubic spline coefficients \n",
    "                dx = x - xi[i]\n",
    "                fx = a[i] + dx*(b[i] + c[i]*dx + d[i]*dx*dx)\n",
    "\n",
    "    \"\"\"\n",
    "    delx = np.diff(xi); delf = np.diff(fi)\n",
    "    # form matrices to solve for spline coefficients\n",
    "    vx = np.zeros_like(xi)\n",
    "    # form rhs vector using python's array slicing \n",
    "    vx[1:-1:] = 3.*(delf[1::]/delx[1::] - delf[:-1:]/delx[:-1:])\n",
    "    # construct 3 diagonals\n",
    "    nx = np.size(xi)\n",
    "    diags = np.zeros((3, nx))\n",
    "    diags[1,0] = 1.; diags[1,-1] = 1.\n",
    "    diags[1,1:-1:] = 2. * (delx[1::] + delx[:-1:])\n",
    "    diags[0,1:] = delx[:]\n",
    "    diags[2,1:-1] = delx[1:]\n",
    "    # solve for coefficients c using Thomas algorithm for tri-diagonal matrices\n",
    "    # see https://www.cfd-online.com/Wiki/Tridiagonal_matrix_algorithm_-_TDMA_(Thomas_algorithm)\n",
    "    ac, bc, cc, dc = map(np.array, (diags[0,:], diags[1,:], diags[2,:], vx)) # copy arrays\n",
    "    for k in range(1, nx):\n",
    "        mk = ac[k] / bc[k-1]\n",
    "        bc[k] = bc[k] - mk * cc[k-1] \n",
    "        dc[k] = dc[k] - mk * dc[k-1]\n",
    "    \n",
    "    c = np.zeros_like(bc)\n",
    "    c[-1] = dc[-1] / bc[-1]\n",
    "\n",
    "    for k in range(nx-2, -1, -1):\n",
    "        c[k] = (dc[k]-cc[k]*c[k+1])/bc[k]\n",
    "\n",
    "    # now get the rest of the coefficients\n",
    "    b = delf[::]/delx[::] - (c[1::] + 2.*c[:-1:])*delx[::]/3.\n",
    "    d = (c[1::] - c[:-1:])/(3.*delx[::]) \n",
    "    a = fi\n",
    "    return a, b, c, d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cubic_spline(xt, xi, a, b, c, d):\n",
    "    \"\"\"\n",
    "    piecewise linear approximation of f(x) given input of tabulated values of xi and fi\n",
    "    note that xi are expected in ascending order\n",
    "    \n",
    "    Returns:\n",
    "        vector of spline values at test points x \n",
    "\n",
    "    \"\"\"\n",
    "    n = np.size(xi) - 1\n",
    "    fxt = np.empty_like(xt)\n",
    "    for j, x in enumerate(xt):\n",
    "        for i in range(n):\n",
    "            if (x >= xi[i]) and (x <= xi[i+1]):\n",
    "                # reusing computations is always a good idea, but here we also can return dfdx\n",
    "                dx = x - xi[i]\n",
    "                fxt[j] = a[i] + dx*(b[i] + c[i]*dx + d[i]*dx*dx)\n",
    "        \n",
    "    return fxt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is the test that the spline interpolation produces good results for interpolation of <tt>Plk</tt> as a function of <tt>lk</tt>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c, d = cubic_spline_coefficients(lk, lPk)\n",
    "\n",
    "spline_int = cubic_spline(lkt, lk, a, b, c, d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_line_points(lkt, spline_int, \n",
    "                 xlabel=r'$\\log_{10} k$', ylabel=r'$\\log_{10} P(k)$', figsize=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that once we have the spline coefficients for a given interval $a_i$, $b_i$, $c_i$, and $d_i$, we can compute derivatives of the interpolating spline as well as its integral. For interpolating spline \n",
    "\n",
    "$$s_i(x) = a_i + b_i(x-x_i) + c_i(x-x_i)^2 + d_i(x-x_i)^3,$$\n",
    "\n",
    "we have for derivatives\n",
    "\n",
    "\\begin{eqnarray}\n",
    "f^\\prime(x\\in[x_i,x_{i+1}])\\approx s^\\prime_i(x)&=&b_i + 2c_i(x-x_i)+3d_i(x-x_i)^2,\\\\\n",
    "f^{\\prime\\prime}(x\\in[x_i,x_{i+1}])\\approx s^{\\prime\\prime}_i(x)&=&2c_i+6d_i(x-x_i),\\\\\n",
    "f^{\\prime\\prime\\prime}(x\\in[x_i,x_{i+1}])\\approx s^{\\prime\\prime\\prime}_i(x)&=&6d_i\n",
    "\\end{eqnarray}\n",
    "\n",
    "It is also straightforward to use interpolating spline to estimate the integral of the approximated function. For example, the integral in the interval $[x_i,x_{i+1}]$ (using change of variables $x\\rightarrow\\xi=x-x_i$:\n",
    "\n",
    "$$\\int\\limits_{x_i}^{x_{i+1}}f(x)dx\\approx \\int\\limits_{x_i}^{x_{i+1}}s_i(x)dx = \\int\\limits_{0}^{x_{i+1}-x_i}s_i(\\xi)d\\xi=a_i(x_{i+1}-x_i)+\\frac{1}{2}b_i(x_{i+1}-x_i)^2+\\frac{1}{3}c_i(x_{i+1}-x_i)^3+\\frac{1}{4}d_i(x_{i+1}-x_i)^4$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 1c. (10 points)** Add calculation of the first, second, and third derivatives based on the spline interpolation to the routine <tt>cubic_spline</tt> above (3 points). Write a routine to estimate integral of $f(x)$ in the interval $[a,b]$ using interpolating cubic spline for $f(x)$ and its coefficient vectors $a$, $b$, $c$, $d$. (4 points).\n",
    "\n",
    "Plot the first three derivatives for some function (for example, the sine) and comment on their behavior (smoothness, continuity). Test the accuracy of the integral for a function for which you know the exact answer (e.g., $\\sin x$ or $e^x$.) (3 points). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 1d. (5 points)** Suppose we want to compute the *logarithmic slope* of $P(k)$:\n",
    "\n",
    "$$\\frac{d\\log_{10}P}{d\\log_{10} k}=\\frac{d\\ln P}{d\\ln k}=\\frac{k}{P}\\,\\frac{dP}{dk}.$$\n",
    "\n",
    "Compute this derivative using your routine implemented in 1c using <tt>lk</tt> and <tt>lPk</tt> vectors and plot this derivative as a function of <tt>lk</tt>. Compare your plot to the left panel of Figure 4 in <a href=\"https://arxiv.org/pdf/1407.4730\">this paper</a>. It should look roughly similar, but with some differences. Describe the differences. Briefly describe what causes them. \n",
    "\n",
    "Note that if you don't complete 1c, you can construct spline interpolation of <tt>lk</tt> and <tt>lPk</tt> using SciPy <tt>UnivariateSpline</tt> with <tt>s=0</tt> and use its <tt>derivative()</tt> method to compute the derivative (see Task 1e below). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 1e (5 points)** Use SciPy <a href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.UnivariateSpline.html\"><tt>UnivariateSpline</tt></a> routine to approximate <tt>lk</tt> and <tt>lPk</tt> and this class'  <tt>derivative()</tt> method to construct approximation for the derivative, as illustrated below, to compute the derivative $$d\\log_{10}P/d\\log_{10} k$$. Compute and plot the derivative for <tt>s=0.</tt>, <tt>s=0.0001</tt>, <tt>s=0.001</tt>, and <tt>s=0.01</tt> and comment on how the derivative changes for different values of the smoothing parameter <tt>s</tt>. Try to explain the behavior you observe. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# example of use of derivative method of UnivariateSpline \n",
    "from scipy.interpolate import UnivariateSpline\n",
    "\n",
    "ntr = 4\n",
    "xtr = np.linspace(0.,2.*np.pi, ntr)\n",
    "spl = UnivariateSpline(xtr, np.sin(xtr), s=0.) # spline approximating sine\n",
    "dspl = spl.derivative() # spline approximating dsin(x)/dx=cos(x)\n",
    "\n",
    "nt = 100\n",
    "xt = np.linspace(0.,2.*np.pi, nt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_line_points(xt, dspl(xt), xlabel=r'$x$', ylabel=r'$f^\\prime(x)$', \n",
    "                 figsize=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "spl = UnivariateSpline(lk, lPk, s=0.001)\n",
    "dspl = spl.derivative()\n",
    "\n",
    "plot_line_points(lkt, dspl(lkt), xlabel=r'$\\log_{10} k$', ylabel=r'$\\log_{10}P(k)$', \n",
    "                 figsize=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 1f. (5 points)** Using spline to invert function. Using spline we can invert a function or, for $y=f(x)$ compute $x$ for a given $y$. For this we need to generate vectors of $x$ and $y$ and construct interpolating spline for $y$ as the argument and $x$ as a function value. The spline can then be used for input $y$ values to estimate corresponding $x$. ***This, of course, assumes that function is monotonic and that a unique value of $x$ can be obtained for a given $y$.*** So these assumptions must be satisfied for this approach to work.  \n",
    "\n",
    "Construct interpolating spline for $y=\\sin(x)$ on the interval $[0,\\pi/2]$ and use it to compute $x$ for $y=1$, $y=0$, $y=0.5$. Compare the values you get to the output of <tt>np.arcsin</tt> for these values of $y$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2 (30 points)\n",
    "\n",
    "**Background.** During the last week we saw that computation of the luminosity distance $d_L(z,H_0,\\Omega_{\\rm m0},\\Omega_\\Lambda)$ is quite expensive when we need to compute it for many parameter combinations, as is required, for example, when we want to find the combination that describe observational measurements of supernovae type Ia. \n",
    "\n",
    "This motivates constructing accurate approximations for $d_L(z,H_0,\\Omega_{\\rm m0},\\Omega_\\Lambda)$ using methods that we've been discussing in the last couple of classes (interpolation or regression). The nonlinear dependence of $d_L$ on $z$, $\\Omega_{\\rm m0}$ and $\\Omega_\\Lambda$ via the integral (in general case) is particularly important to approximate well, as it cannot be simply obtained by rescaling from results for other model parameters, as can be done for $H_0$.\n",
    "\n",
    "Indeed, $d_L$ is proportional to $c/H_0$ and this is the only place where $H_0$ appears in its expression. Thus, if we tabulate $\\tilde{d}_L$ without  $c/H_0$ factor, it can always be added later to compute the actual value of the luminosity distance $d_L=c/H_0\\,\\tilde{d}_L(z,\\Omega_{\\rm m0}, \\Omega_\\Lambda)$. Given this simple dependence of $d_L$ on $H_0$, there is no need to construct approximation of $d_L$ as a function of $H_0$. We can simply scale it. \n",
    "\n",
    "**Goal of the exercise.** The goal of this exercise is for you to find an optimal approximation of $\\tilde{d}_L$ (i.e. $d_L$ without $c/H_0$ factor) as a function of $\\Omega_{\\rm m0}$ and $\\Omega_\\Lambda$ (that is approximation should be two-dimensional, not two separate 1d approximations as a function of $\\Omega_{\\rm m0}$ and $\\Omega_\\Lambda$) for a given single value of $z$. \n",
    "\n",
    "\"Optimal\" here means that provides target accuracy of the approximation with the smallest training set of tabulated function values. Suppose our target fractional accuracy for $\\tilde{d}_L$ is $10^{-4}$. Experiment with 2D polynomial and piecewise spline interpolation for $\\tilde{d}_L$ with different size of training point set and their spacing along $x$ and $y$ (i.e. along $\\Omega_{\\rm m0}$ and $\\Omega_\\Lambda$ in this exercise) using codes and examples below and identify the approach that reaches this target accuracy with the smallest training sample. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### What you are learning in this exercise\n",
    "\n",
    "* Approximating functions of two parameters (i.e. approximation in 2D) \n",
    "\n",
    "* Different methods of approximating smooth functions\n",
    "\n",
    "* That method that is best for one problem may not be the best for another problem (this is why multiple methods exist and why we study more than one method of approximating functions). \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Group work\n",
    "\n",
    "This is \"open-ended\" exercise of the kind that is often encountered in research. Open-ended means that the results are not obvious from the beginning and your route to conlcusion is not prescribed, but requires thinking and experimentation. \n",
    "\n",
    "You are expected to work on this exercise in a group. Group assignment is announced when this homework is distributed. \n",
    "Try to work collaboratively. You can split the various parts of the task among group members and then combine your work, or do this independently and then discuss your findings among your group and reach consensus conclusion. Use in-person meetings or slack messaging (you can create your own private or public channel for these discussions).  Each member of the group should submit results of their and group work in their own notebook via Canvas."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 2a. (15 points)** Write a routine that constructs 2D *spline* approximation of $\\tilde{d}_L(z,\\Omega_{\\rm m0}, \\Omega_\\Lambda)$ for a given input single value of redshift $z$ and for ranges of the $\\Omega_{\\rm m0}$ and $\\Omega_\\Lambda)$ parameters of $\\Omega_{\\rm m0}\\in [0.1]$ and $\\Omega_\\Lambda\\in[0.,1]$. Experiment with different training point spacings (for example, evenly spaced vs Chebyshev nodes) and try to find the spacing that reaches the target fractional accuracy of $10^{-4}$ with the smallest size of the training set for any $z$ in the interval $z\\in [0,2]$. \n",
    "\n",
    "There are several different SciPy routines that can be used for this.  I recommend using routine <tt>scipy.interpolate.RectBivariateSpline(x, y, z, s=0, kx=3, ky=3)</tt>, where $z$ is the array of function values tabulated at training points in vectors $x$ and $y$, $s=0$ indicates interpolation (no smoothing), parameters <tt>kx=3, ky=3</tt> specify that cubic splines should be used in $x$ and $y$ variables. Example of using this function is shown below (you can read about other available options and see examples of how they are used <a href=\"https://mmas.github.io/interpolation-scipy\">here</a>).\n",
    "\n",
    "Example below uses a numpy function <a href=\"https://docs.scipy.org/doc/numpy/reference/generated/numpy.meshgrid.html\"><tt>np.meshgrid</tt></a> to generate 2D grids of values of $x$ and $y$ from 1D vectors defined using <tt>np.linspace</tt>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# example of how to use it: 2D sine function\n",
    "def sin2d(x, y):\n",
    "    return 0.5*(np.sin(x) + np.sin(y))\n",
    "\n",
    "# training vectors\n",
    "xmin, xmax, ntr = 0., 2.*np.pi, 10\n",
    "xtr = np.linspace(xmin, xmax, ntr)\n",
    "ytr = np.linspace(xmin, xmax, ntr)\n",
    "# construct grids for x and y using xtr and ytr\n",
    "xgrid, ygrid = np.meshgrid(xtr, ytr, sparse=False, indexing='ij')\n",
    "# tabulate test function on this grid\n",
    "ftr = sin2d(xgrid, ygrid)\n",
    "\n",
    "# test vectors\n",
    "nt = 200\n",
    "xt = np.linspace(xmin, xmax, nt)\n",
    "yt = np.linspace(xmin, xmax, nt)\n",
    "# test grid\n",
    "xgt, ygt = np.meshgrid(xt, yt, sparse=False, indexing='ij')\n",
    "\n",
    "# construct 2d interpolating cubic spline given training values xtr, ytr, and ftr \n",
    "from scipy.interpolate import RectBivariateSpline\n",
    "\n",
    "spl2d = RectBivariateSpline(xtr, ytr, ftr, s=0, kx=3, ky=3)\n",
    "\n",
    "ft = spl2d(xt, yt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(10, 10) (200, 200)\n"
     ]
    }
   ],
   "source": [
    "print(np.shape(ftr), np.shape(ft))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note the dimensions of <tt>ftr</tt> and <tt>ft</tt> generated in the example above. When you work on this exercise, if you use the above example, make sure your function for computing $\\tilde{d}_L$ can handle input of 2d grids like <tt>xgrid</tt> and <tt>ygrid</tt>. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 525x525 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from codes.plot_utils import plot_color_map\n",
    "\n",
    "plot_color_map(xgt, ygt, ft, xlim=[xmin,xmax], ylim=[xmin,xmax], \n",
    "                   xlabel = '$x$', ylabel = '$y$', cmap='winter', colorbar=True, \n",
    "                   plot_title=r'$0.5\\,(\\sin x+\\sin y)$', \n",
    "                   figsize=3.0)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 2b (15 points).** Although there is no convenient Scipy routine for polynomial interpolation in 2d, we can use the same approach as was used in Task 1a to construct interpolating polynomial in 2d, as is done in the following routine (a modified version of the routine provided <a href=\"https://stackoverflow.com/questions/33964913/equivalent-of-polyfit-for-a-2d-polynomial-in-python\">here</a>). \n",
    "\n",
    "Write a routine that constructs 2D *polynomial* approximation of $\\tilde{d}_L(z,\\Omega_{\\rm m0}, \\Omega_\\Lambda)$ for a given input single value of redshift $z$ and for ranges of the $\\Omega_{\\rm m0}$ and $\\Omega_\\Lambda)$ parameters of $\\Omega_{\\rm m0}\\in [0.1]$ and $\\Omega_\\Lambda\\in[0.,1]$. Just like for the spline in 2a, experiment with different training point spacings (for example, evenly spaced vs Chebyshev nodes) and try to find the spacing that reaches the target fractional accuracy of $10^{-4}$ with the smallest size of the training set for any $z$ in the interval $z\\in [0,2]$. \n",
    "\n",
    "State your conclusions about the optimal method for approximating $\\tilde{d}_L(z,\\Omega_{\\rm m0}, \\Omega_\\Lambda)$ supporting it with tests of fractional accuracy and plots, as needed. Each of the groups will make a brief presentation of their results in class Friday of next week."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def polyfit2d(xtr, ytr, ftr, kx=3, ky=3, order=None):\n",
    "    '''\n",
    "    Two dimensional polynomial fitting by least squares.\n",
    "    Fits the functional form f(x,y) = z.\n",
    "\n",
    "    Notes\n",
    "    -----\n",
    "    Resulting fit can be plotted with:\n",
    "    np.polynomial.polynomial.polygrid2d(x, y, soln.reshape((kx+1, ky+1)))\n",
    "\n",
    "    Parameters:\n",
    "    ----------\n",
    "    xtr, ytr: array-like, 1d\n",
    "        xtr and ytr coordinates.\n",
    "    ftr: 2d numpy array\n",
    "        f(xgtr, ygtr) values evaluated on meshgrid of xtr and ytr vectors to fit by polynomial\n",
    "    kx, ky: int, default is 3\n",
    "        Polynomial order in x and y, respectively.\n",
    "    order: int or None, default is None\n",
    "        If None, all coefficients up to maxiumum kx, ky, ie. up to and including x^kx*y^ky, are considered.\n",
    "        If int, coefficients up to a maximum of kx+ky <= order are considered.\n",
    "\n",
    "    Returns:\n",
    "    -------\n",
    "    Return paramters from np.linalg.lstsq.\n",
    "\n",
    "    soln: np.ndarray\n",
    "        Array of polynomial coefficients.\n",
    "    residuals: np.ndarray\n",
    "    rank: int\n",
    "    s: np.ndarray\n",
    "\n",
    "    '''\n",
    "\n",
    "    # grid coords\n",
    "    x, y = np.meshgrid(xtr, ytr)\n",
    "    # coefficient array, up to x^kx, y^ky\n",
    "    coeffs = np.ones((kx+1, ky+1))\n",
    "\n",
    "    # solve array\n",
    "    V = np.zeros((coeffs.size, x.size))\n",
    "\n",
    "    # construct Vandermonde matrix: for each coefficient produce array x^i, y^j\n",
    "    for index, (j, i) in enumerate(np.ndindex(coeffs.shape)):\n",
    "        # do not include powers greater than order\n",
    "        if order is not None and i + j > order:\n",
    "            arr = np.zeros_like(x)\n",
    "        else:\n",
    "            arr = coeffs[i, j] * x**i * y**j\n",
    "        V[index] = arr.flatten()\n",
    "        \n",
    "    # do leastsq fitting and return leastsq result\n",
    "    return np.linalg.lstsq(V.T, np.ravel(ftr), rcond=None)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# be careful with polynomial order, results quickly become unstable as order increases\n",
    "px, py = 7, 7 # polynomial order in x and y direction\n",
    "\n",
    "# get coefficients of best 2d polynomial\n",
    "a = polyfit2d(xtr, ytr, ftr, kx=px, ky=py, order=None)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# compute polynomial values on the grid of test points\n",
    "ff = np.polynomial.polynomial.polyval2d(xgt, ygt, a.reshape((px+1,py+1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 525x525 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot approximating polynomial\n",
    "plot_color_map(xgt, ygt, ff, xlim=[xmin,xmax], ylim=[xmin,xmax], \n",
    "                   xlabel = '$x$', ylabel = '$y$', cmap='winter', colorbar=True, \n",
    "                   plot_title=r'$0.5\\,(\\sin x+\\sin y)$', \n",
    "                   figsize=3.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 3 (extra-credit, 10 points): implementing spline interpolation as a Python class\n",
    "\n",
    "Implementing a spline routine as a python class. In this exercise you would learn (or practice) object-oriented programming in python, which is particularly suited for this task. To learn how classes are defined and used in Python, see <a href=\"https://www.python-course.eu/python3_object_oriented_programming.php\">this tutorial</a> or ask instructor or TAs.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task.** Using the routines for calculation of spline coefficient vectors <tt>a, b, c, d</tt> and then the calculation of the actual spline values for an arbitrary input vector $x$  provided in this notebook above (routines <tt>cubic_spline_coefficients</tt> and <tt>cubic_spline</tt>) implement a class for spline interpolation that can be initialized and then used for an arbitrary input vector, similar to the way SciPy routines <a href=\"https://docs.scipy.org/doc/scipy-0.19.1/reference/generated/scipy.interpolate.interp1d.html\"><tt>interp1d</tt></a> and <a href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.UnivariateSpline.html\"><tt>UnivariateSpline</tt></a> are used.  "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
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 },
 "nbformat": 4,
 "nbformat_minor": 1
}