{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Math examples\n",
    "With attempts to use Unicode operators where possible to make the Raku code look like the math."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Cauchy-Schwarz Inequality\n",
    " from the [jupyter docs](http://jupyter-notebook.readthedocs.io/en/latest/examples/Notebook/Typesetting%20Equations.html)\n",
    "\\begin{equation*}\n",
    "\\left( \\sum_{k=1}^n a_k b_k \\right)^2 \\leq \\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right)\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sub cauchy-schwarz(@a,@b) {\n",
    "   ( [+] @a Z× @b )² ≤ ( [+] @a»² ) × ( [+] @b»² )\n",
    "}\n",
    "\n",
    "cauchy-schwarz( ( ^100 ).pick(5), ( ^100 ).pick(5) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Cubic formula\n",
    "see this [discussion](http://www.perlmonks.org/?node_id=1189383) and [wikipedia](https://en.wikipedia.org/wiki/Cubic_function#Algebraic_solution)\n",
    "\\begin{equation*}\n",
    "\\Delta_0 = b^2 - 3ac\n",
    "\\\\\n",
    "\\Delta_1 = 2b^3 - 9abc + 27a^2d\n",
    "\\\\\n",
    "C = \\sqrt[3]{ \\frac{ \\Delta_1 \\pm \\sqrt{\\Delta_1^2 - 4 \\Delta_0^3 } }{2} }\n",
    "\\\\\n",
    "x_k = - \\frac{1}{3a}( b + ς^k C + \\frac{\\Delta_0}{ς^k C}), k ∊ {0,1,2}\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[True True True]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sub cubic(\\a,\\b,\\c,\\d) {\n",
    "    my \\Δ0 = b²\t- 3 × a × c;\n",
    "    # note: special case when Δ0 == 0\n",
    "    my \\Δ1 = 2 * b³ - 9 × a × b × c + 27 × a² × d;\n",
    "    my \\C = ( ( Δ1 + sqrt( Δ1² - 4 × Δ0³ + 0i) ) / 2 ).roots(3)[0];\n",
    "    my \\ς = 1.roots(3);  # cubic roots of unity\n",
    "    return [0,1,2].map: -> \\k {\n",
    "        ( -1 / ( 3 × a ) ) × ( b + ς[k] × C + Δ0 / ( C × ς[k] ) )\n",
    "    }\n",
    "}\n",
    "\n",
    "my @vals = cubic(1,10,10,-10);\n",
    "my $f = -> \\x { x³ + 10 * x² + 10 * x - 10 };\n",
    "\n",
    "my $*TOLERANCE = 1e-10;\n",
    "\n",
    "[ $f( $_ ) ≅ 0 for @vals ]"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Raku",
   "language": "raku",
   "name": "raku"
  },
  "language_info": {
   "file_extension": ".raku",
   "mimetype": "text/x-raku",
   "name": "raku",
   "version": "6.d"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}