{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ea873303-4ac4-4aad-8a85-c9fee4f6cbea",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\"\"\"\n",
    "    mylegendre(m, x)\n",
    "\n",
    "Calculate Legendre polynomial P_n(x) using the recurrence relation    \n",
    "(n+1)P_{n+1}(x) = (2*n+1)*x*P_n(x) - n*P_{n-1}(x), and P_0(x) = 1, P_1(x) = x\n",
    "\"\"\"\n",
    "function mylegendre(m, x)\n",
    "\n",
    "    pp = 1\n",
    "    if m == 0\n",
    "        return pp\n",
    "    end\n",
    "\n",
    "    pc = x\n",
    "    if m == 1\n",
    "        return pc\n",
    "    end\n",
    "\n",
    "    for n = 1:(m-1)\n",
    "        # pn = (2*n + 1)/(n + 1) * x * pc - n/(n + 1) * pp\n",
    "        # pp = pc\n",
    "        # pc = pn\n",
    "        pc, pp = (2*n + 1)/(n + 1) * x * pc - n/(n + 1) * pp, pc\n",
    "    end\n",
    "\n",
    "    return pc\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2d62ab24-fe9c-41d3-8c92-70f69cdc3e7d",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "@show mylegendre(3, 0.0);   # must be 0\n",
    "@show mylegendre(3, -1.0);  # must be -1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "21724284-6f4b-4932-ae47-77513f13236a",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "@time mylegendre(1, 1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "779f73fc-be4b-4d35-bc4f-edcb60f75aa8",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "@time mylegendre(51, -1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a37b5850-fd54-4c43-b82b-defdca4a0b61",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "np = 100\n",
    "x = range(-1.0, 1.0, np);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b2605dbc-fb47-4552-a614-3f27ddac0b9c",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "using PyPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "80298906-317d-4276-8e7f-67465812c82d",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "for n = [2, 3, 5, 7]\n",
    "    plot(x, mylegendre.(n, x), label=L\"P_{%$n}(x)\")\n",
    "end\n",
    "\n",
    "grid(true)\n",
    "title(\"Legendre polynomials\")\n",
    "xlabel(\"x\")\n",
    "ylabel(L\"P_n(x)\")\n",
    "legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f884dcee-fd9a-4088-bd7d-6271bf353a08",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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