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   "source": [
    "# Chebyshev polynomials"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a7535578-9bde-4d0b-bdc4-11bb1903a767",
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   "source": [
    "**Chebyshev polynomials** is a particular form of *special functions* that are important in\n",
    "numerical analysis.\n",
    "\n",
    "Chebyshev polynomials (of the first kind) are defined as follows:\n",
    "\n",
    "$$T_{n}\\left( x \\right) \\equiv \\cos \\left(n \\arccos x \\right), \\qquad n = 0, 1, 2, \\ldots \\quad -1 \\le x \\le 1.$$\n",
    "\n",
    "The polynomials are named after the mathematician Pafnuty Chebyshev. The\n",
    "letter `T` is used because of the alternative transliterations of\n",
    "the name Chebyshev as Tchebycheff or Tchebyshev.\n",
    "\n",
    "The first few Chebyshev polynomials are as follows:\n",
    "  \n",
    "$$T_{0}(x) = 1, \\quad T_{1}(x) = x , \\quad T_2(x) = 2x^2 - 1,\n",
    "    \\quad T_3(x) = 4x^3 - 3x .$$\n",
    "\n",
    "Chebyshev polynomial $T_n(x)$ satisfy the following recurrence relations:\n",
    "\n",
    "$$T_{n+1}(x) = 2 \\, x \\,T_{n}(x) - T_{n-1}(x), \\qquad n = 1, 2, \\ldots$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2c4ef57d",
   "metadata": {},
   "outputs": [],
   "source": [
    "using PyPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1aa3aab0",
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "    mycheb(n, x)\n",
    "\n",
    "Calculate Chebyshev polynomials using the recurrence relation T_{n+1} = 2*x*T_nc - T_(n-1}\n",
    "\"\"\"\n",
    "function mycheb(n, x)\n",
    "    \n",
    "    tp = 1.0\n",
    "    if n == 0\n",
    "        return tp\n",
    "    end\n",
    "    \n",
    "    tc = x\n",
    "    if n == 1\n",
    "        return tc\n",
    "    end\n",
    "\n",
    "    for i = 1:(n-1)\n",
    "        tn = 2*x*tc - tp\n",
    "        tp = tc\n",
    "        tc = tn\n",
    "    end\n",
    "    \n",
    "    return tc\n",
    "        \n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3fc4b76e",
   "metadata": {},
   "outputs": [],
   "source": [
    "np = 100\n",
    "x = range(-1.0, 1.0, np);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8d64dc9f",
   "metadata": {},
   "outputs": [],
   "source": [
    "for n = [2, 3, 5]\n",
    "    plot(x, mycheb.(n, x), label=L\"T_%$n(x)\")\n",
    "end\n",
    "grid(true)\n",
    "title(\"Chebyshev polynomials\")\n",
    "xlabel(\"x\")\n",
    "ylabel(\"y(x)\")\n",
    "legend()\n",
    "ylim(-1.1, 1.1); # optional in this example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1bb63038",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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