{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Lecture 5" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "%run set_env.py\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Universal Functions (UFuncs)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A universal function (ufunc) is:\n", "* a function which operates on an ndarray object in an element-by-element fashion\n", "* an instance of the numpy.ufunc class\n", "* a function of which many are implemented in compiled C code\n", "* to which broadcasting rules are applied. \n", "\n", "The concept is similar to the map function in standard Python." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Some ufuncs within NumPy: " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Math operations:\n", " * add(x1,x2) (called when invoked a+b)\n", " * power(x1,x2) (same as '**')\n", " * mod(x1,x2)\n", " * exp(x)\n", " * sqrt(x)\n", " * log(x) (Napierian/natural logarithm)\n", " * ...\n", "* Trig operations:\n", " * sin(x)\n", " * sinh(x)\n", " * arcsinh(x)\n", " * deg2rad(x)\n", " * rad2deg(x)\n", " * ..\n", "* Bit-twiddling operations:\n", " * bitwise_and(x1,x2)\n", " * ...\n", "* Comparison functions:\n", " * greater(x1,x2) (called when x1>x2 is invoked)\n", " * not_equal(x1,x2) (called when x1!=x2 is invoked)\n", " * maximum(x1,x2) (el.-wise max.)\n", " * isfinite(x) (el. test for finiteness i.e. neither Infinity nor Not a Number)\n", " * isinf(x)\n", " * isnan(x)\n", " * ...\n", " \n", "To see all the available ufuncs, see:
\n", "https://docs.scipy.org/doc/numpy-1.13.0/reference/ufuncs.html#available-ufuncs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note:\n", "* One can write its own UFunc -> C-API" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Examples/Applications of UFuncs:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Example 1: no BC\n", "np.set_printoptions(precision=5)\n", "import numpy as np\n", "x = np.random.random((2,3,7))\n", "y = np.exp(x)\n", "print(f\" x:\\n{x}\\n\")\n", "print(f\" y:\\n{y}\\n\")\n", "import math\n", "z=0.5\n", "print(np.exp(z))" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Example 2: with BC\n", "x=np.arange(90,103,dtype=int)\n", "y=np.arange(2,7,dtype=int).reshape((5,1))\n", "print(f\" x:{x.shape}\\n{x}\\n\")\n", "print(f\" y:{y.shape}\\n{y}\\n\")\n", "z=np.mod(x,y)\n", "print(f\" z:{z.shape}\\n{z}\\n\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Reductions on ndarrays" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Besides Numpy functions which operate on ndarrays element-wise (UFuncs, vide supra),
\n", " there are also Numpy functions which perform reductions on ndarrays. \n", "\n", "* By default, the reductions operate on the whole ndarray.\n", " \n", "* However, we can specify a particular axis/dimension on which to perform the reduction. \n", "\n", "* The functions all have a similar syntax:
\n", " numpy.func_name(a,[axis=None],[dtype=None],[out=None])
\n", " The function func_name can be called in 2 different ways:\n", " * a.func_name() # Object-Oriented way i.e. method associated to an object\n", " * np.func_name(a) # Procedural way i.e. array is an argument of the function" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Mathematical Operations:\n", "* numpy.sum(), numpy.cumsum() : sum vs. cumulative sum\n", "* numpy.prod(), numpy.cumprod() : prod vs. cumulative product\n", "* numpy.min(), numpy.max() : min, max of a vector\n", "* numpy.argmin(), numpy.argmax() : return indices of the min./max. values" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Statistical Operations:\n", "* numpy.mean, numpy.median : average, median\n", "* numpy.std, numpy.var : standard deviation, variance" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Logical Operations:\n", "* numpy.any(): Test whether ANY el. along a given axis evaluates to True\n", "* numpy.all(): Test whether ALL el. along a given axis evaluate to True" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Examples" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###### Example 1:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Example 1: \n", "# Invoke sum over the complete ndarray\n", "a = np.arange(1,25).reshape((2,3,4))\n", "print(f\" a:\\n{a}\\n\")\n", "print(f\" a.shape:{a.shape}\\n\")\n", "print(f\" a.sum() (Object-oriented syntax): {a.sum()}\\n\")\n", "print(f\" np.sum(a) (Procedural syntax) : {np.sum(a)}\\n\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Invoke sums over certain axes\n", "a = np.arange(1,25).reshape((2,3,4))\n", "red0 = a.sum(axis=0)\n", "print(f\" a.sum(axis=0) shape:{red0.shape}:\\n{red0}\\n\")\n", "red1 = a.sum(axis=1)\n", "print(f\" a.sum(axis=1) shape:{red1.shape}:\\n{red1}\\n\")\n", "red2 = a.sum(axis=2)\n", "print(f\" a.sum(axis=2) shape:{red2.shape}:\\n{red2}\\n\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###### Example 2:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "np.set_printoptions(precision=4)\n", "b = rnd.random((3,7))\n", "print(f\" b:\\n{b}\\n\")\n", "print(f\" b.shape:{b.shape}\\n\")\n", "\n", "av = b.mean(axis=0)\n", "print(f\" b.mean(axis=0):\\n{av}\\n\")\n", "\n", "bool_matrix = b < 0.05\n", "print(f\" bool_matrix:\\n{bool_matrix}\\n\")\n", "print(f\" Are they any values < 0.01? {bool_matrix.any()}\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercises:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Generate the following vector [ 1, 3, 9, 27, ... , 729] using a UFunc.\n", " \n", "* Generate a 5x10 array A with random numbers $x$ $\\in$ $[0,1[$.\n", " * What is the maximum value for all $x$ in A?\n", " * What is the minimum value in each column?\n", " * What is the minimum value in the fourth row?\n", " * Are there any random numbers $x<\\alpha$ or $x>\\beta$?
You can set $\\alpha:=0.02$ and $\\beta:=0.98$\n", " \n", "* Write the function *calc_sn(n)* (**without the use of for loops!**): \n", " * The function *calc_sn(n)* returns an array of partial sums $S_n$ ($n>0$) given by:
\n", " $\\begin{equation*}\n", " S_n := \\sum_{k=1}^{k=n} \\frac{sin(k)}{k^2} \n", " \\end{equation*}\n", " $ \n", " * Generate the plot $S_n$ where $n$ $\\in$ $\\{1,\\ldots,100\\}$ to visualize the absolute convergency of the series.
\n", " You can use the following code to create the matplotlib plot:
\n", " *import matplotlib.pyplot as plt*
\n", " *plt.plot(calc_sn(100))*
\n", " *plt.show()*
\n", " \n", " \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solutions:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# %load ../solutions/ex5.py" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.9.7" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": false, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }