Discrete Fourier analysis.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "np.set_printoptions(suppress=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3-point centered"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implicit Euler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = 9\n",
    "F = np.zeros((m,m),dtype=complex)\n",
    "for j in range(m):\n",
    "    for k in range(m):\n",
    "        F[j,k] = np.exp(2j*np.pi*j*k/m)/np.sqrt(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "nu = 10000.1\n",
    "xi = 2*np.pi*np.arange(m)/m\n",
    "d = 1./(1- nu*1j*np.sin(xi))\n",
    "D = np.diag(d)\n",
    "L = np.dot(F,np.dot(np.diag(1j*np.sin(xi)),F.conj()))\n",
    "M = np.dot(F,np.dot(D,F.conj()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0. ,  0.5, -0. ,  0. , -0. , -0. ,  0. , -0. , -0.5],\n",
       "       [-0.5,  0. ,  0.5,  0. ,  0. , -0. , -0. , -0. , -0. ],\n",
       "       [ 0. , -0.5, -0. ,  0.5, -0. ,  0. , -0. ,  0. , -0. ],\n",
       "       [-0. , -0. , -0.5, -0. ,  0.5, -0. ,  0. ,  0. ,  0. ],\n",
       "       [ 0. , -0. ,  0. , -0.5,  0. ,  0.5,  0. , -0. , -0. ],\n",
       "       [ 0. ,  0. , -0. ,  0. , -0.5, -0. ,  0.5, -0. , -0. ],\n",
       "       [-0. ,  0. ,  0. , -0. , -0. , -0.5, -0. ,  0.5, -0. ],\n",
       "       [ 0. ,  0. , -0. , -0. ,  0. ,  0. , -0.5, -0. ,  0.5],\n",
       "       [ 0.5,  0. ,  0. , -0. ,  0. ,  0. ,  0. , -0.5,  0. ]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.real(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.11111114, 0.11119998, 0.1110889 , 0.11117777, 0.11106667,\n",
       "        0.11115555, 0.11104443, 0.11113334, 0.11102221],\n",
       "       [0.11102221, 0.11111114, 0.11119998, 0.1110889 , 0.11117777,\n",
       "        0.11106667, 0.11115555, 0.11104443, 0.11113334],\n",
       "       [0.11113334, 0.11102221, 0.11111114, 0.11119998, 0.1110889 ,\n",
       "        0.11117777, 0.11106667, 0.11115555, 0.11104443],\n",
       "       [0.11104443, 0.11113334, 0.11102221, 0.11111114, 0.11119998,\n",
       "        0.1110889 , 0.11117777, 0.11106667, 0.11115555],\n",
       "       [0.11115555, 0.11104443, 0.11113334, 0.11102221, 0.11111114,\n",
       "        0.11119998, 0.1110889 , 0.11117777, 0.11106667],\n",
       "       [0.11106667, 0.11115555, 0.11104443, 0.11113334, 0.11102221,\n",
       "        0.11111114, 0.11119998, 0.1110889 , 0.11117777],\n",
       "       [0.11117777, 0.11106667, 0.11115555, 0.11104443, 0.11113334,\n",
       "        0.11102221, 0.11111114, 0.11119998, 0.1110889 ],\n",
       "       [0.1110889 , 0.11117777, 0.11106667, 0.11115555, 0.11104443,\n",
       "        0.11113334, 0.11102221, 0.11111114, 0.11119998],\n",
       "       [0.11119998, 0.1110889 , 0.11117777, 0.11106667, 0.11115555,\n",
       "        0.11104443, 0.11113334, 0.11102221, 0.11111114]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.real(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.1111111111111111\n"
     ]
    }
   ],
   "source": [
    "print(1./m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.11119998429364004+5.722178132562385e-20j)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = 2\n",
    "np.sum(np.exp(-1j*(j-1)*xi)/(1.-nu*1j*np.sin(xi)))/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.11102220829954801+5.421010862427522e-20j)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(np.exp(1j*xi)/(1./1-nu*1j*np.sin(xi)))/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.        -0.j        , 0.00000009-0.00029238j,\n",
       "       0.00000009+0.00029238j, 0.00000002-0.00015557j,\n",
       "       0.00000001-0.00011547j, 0.00000001-0.00010154j,\n",
       "       0.00000002+0.00015557j, 0.00000001+0.00010154j,\n",
       "       0.00000001+0.00011547j])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eigvals(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.        +0.j        , 0.00000002+0.00015557j,\n",
       "       0.00000001+0.00010154j, 0.00000001+0.00011547j,\n",
       "       0.00000009+0.00029238j, 0.00000009-0.00029238j,\n",
       "       0.00000001-0.00011547j, 0.00000001-0.00010154j,\n",
       "       0.00000002-0.00015557j])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.64278761,  0.98480775,  0.8660254 ,  0.34202014,\n",
       "       -0.34202014, -0.8660254 , -0.98480775, -0.64278761])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sin(xi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "       3.4906585 , 4.1887902 , 4.88692191, 5.58505361])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.9999999999999974"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 9\n",
    "xi = 2*np.pi*np.arange(1,m)/m\n",
    "np.sum(np.cos(5*xi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $\\theta$ method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = 7\n",
    "F = np.zeros((m,m),dtype=complex)\n",
    "for j in range(m):\n",
    "    for k in range(m):\n",
    "        F[j,k] = np.exp(2j*np.pi*j*k/m)/np.sqrt(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.88057   ,  0.47773371,  0.05870032,  0.00813111, -0.00634858,\n",
       "         0.05891971, -0.47770629],\n",
       "       [-0.47770629,  0.88057   ,  0.47773371,  0.05870032,  0.00813111,\n",
       "        -0.00634858,  0.05891971],\n",
       "       [ 0.05891971, -0.47770629,  0.88057   ,  0.47773371,  0.05870032,\n",
       "         0.00813111, -0.00634858],\n",
       "       [-0.00634858,  0.05891971, -0.47770629,  0.88057   ,  0.47773371,\n",
       "         0.05870032,  0.00813111],\n",
       "       [ 0.00813111, -0.00634858,  0.05891971, -0.47770629,  0.88057   ,\n",
       "         0.47773371,  0.05870032],\n",
       "       [ 0.05870032,  0.00813111, -0.00634858,  0.05891971, -0.47770629,\n",
       "         0.88057   ,  0.47773371],\n",
       "       [ 0.47773371,  0.05870032,  0.00813111, -0.00634858,  0.05891971,\n",
       "        -0.47770629,  0.88057   ]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nu = 1\n",
    "theta = 0.25\n",
    "xi = 2*np.pi*np.arange(m)/m\n",
    "d = (1+(1-theta)*nu*1j*np.sin(xi))/(1- theta*nu*1j*np.sin(xi))\n",
    "D = np.diag(d)\n",
    "M = np.dot(F,np.dot(D,F.conj().T))\n",
    "np.real(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.4777062858435684-5.075305255429287e-16j)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = m\n",
    "np.sum(np.exp(-1j*(j-1)*xi)*(1+(1-theta)*nu*1j*np.sin(xi))/(1.-theta*nu*1j*np.sin(xi)))/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.47771999766898965"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(-nu*(np.sin(xi))**2/(1+theta**2*nu**2*(np.sin(xi))**2))/m"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Spectral collocation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Two ways of constructing the differentiation matrix\n",
    "For some reason this formula is not right when $m$ is even (it gives complex entries)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.          0.52359878  1.04719755  1.57079633  2.0943951   2.61799388\n",
      "  0.         -2.61799388 -2.0943951  -1.57079633 -1.04719755 -0.52359878]\n",
      "[[ 0.        +0.j  0.97704862-0.j -0.45344984+0.j  0.26179939-0.j\n",
      "  -0.15114995-0.j  0.07014893+0.j -0.        -0.j -0.07014893+0.j\n",
      "   0.15114995-0.j -0.26179939+0.j  0.45344984-0.j -0.97704862-0.j]\n",
      " [-0.97704862-0.j -0.        -0.j  0.97704862+0.j -0.45344984-0.j\n",
      "   0.26179939-0.j -0.15114995+0.j  0.07014893+0.j -0.        +0.j\n",
      "  -0.07014893+0.j  0.15114995-0.j -0.26179939-0.j  0.45344984+0.j]\n",
      " [ 0.45344984+0.j -0.97704862+0.j  0.        -0.j  0.97704862-0.j\n",
      "  -0.45344984+0.j  0.26179939+0.j -0.15114995-0.j  0.07014893+0.j\n",
      "   0.        -0.j -0.07014893+0.j  0.15114995+0.j -0.26179939-0.j]\n",
      " [-0.26179939+0.j  0.45344984+0.j -0.97704862-0.j -0.        +0.j\n",
      "   0.97704862+0.j -0.45344984-0.j  0.26179939+0.j -0.15114995-0.j\n",
      "   0.07014893-0.j -0.        +0.j -0.07014893+0.j  0.15114995-0.j]\n",
      " [ 0.15114995+0.j -0.26179939-0.j  0.45344984-0.j -0.97704862+0.j\n",
      "   0.        -0.j  0.97704862-0.j -0.45344984+0.j  0.26179939-0.j\n",
      "  -0.15114995-0.j  0.07014893+0.j -0.        -0.j -0.07014893-0.j]\n",
      " [-0.07014893-0.j  0.15114995+0.j -0.26179939-0.j  0.45344984+0.j\n",
      "  -0.97704862+0.j -0.        -0.j  0.97704862+0.j -0.45344984-0.j\n",
      "   0.26179939+0.j -0.15114995+0.j  0.07014893+0.j -0.        -0.j]\n",
      " [ 0.        -0.j -0.07014893+0.j  0.15114995-0.j -0.26179939+0.j\n",
      "   0.45344984-0.j -0.97704862-0.j  0.        -0.j  0.97704862-0.j\n",
      "  -0.45344984+0.j  0.26179939-0.j -0.15114995+0.j  0.07014893+0.j]\n",
      " [ 0.07014893+0.j  0.        -0.j -0.07014893-0.j  0.15114995+0.j\n",
      "  -0.26179939+0.j  0.45344984-0.j -0.97704862+0.j -0.        -0.j\n",
      "   0.97704862-0.j -0.45344984+0.j  0.26179939-0.j -0.15114995-0.j]\n",
      " [-0.15114995+0.j  0.07014893-0.j -0.        +0.j -0.07014893-0.j\n",
      "   0.15114995-0.j -0.26179939+0.j  0.45344984-0.j -0.97704862+0.j\n",
      "  -0.        +0.j  0.97704862-0.j -0.45344984-0.j  0.26179939+0.j]\n",
      " [ 0.26179939+0.j -0.15114995-0.j  0.07014893+0.j  0.        +0.j\n",
      "  -0.07014893-0.j  0.15114995+0.j -0.26179939+0.j  0.45344984-0.j\n",
      "  -0.97704862+0.j  0.        +0.j  0.97704862-0.j -0.45344984-0.j]\n",
      " [-0.45344984-0.j  0.26179939+0.j -0.15114995+0.j  0.07014893-0.j\n",
      "  -0.        -0.j -0.07014893+0.j  0.15114995-0.j -0.26179939+0.j\n",
      "   0.45344984+0.j -0.97704862-0.j  0.        +0.j  0.97704862+0.j]\n",
      " [ 0.97704862-0.j -0.45344984+0.j  0.26179939+0.j -0.15114995-0.j\n",
      "   0.07014893+0.j  0.        +0.j -0.07014893-0.j  0.15114995-0.j\n",
      "  -0.26179939-0.j  0.45344984-0.j -0.97704862+0.j  0.        +0.j]]\n"
     ]
    }
   ],
   "source": [
    "def F_matrix(m):\n",
    "    F = np.zeros((m,m),dtype=complex)\n",
    "    for j in range(m):\n",
    "        v = np.zeros(m)\n",
    "        v[j] = 1.\n",
    "        F[:,j] = np.fft.fft(v)\n",
    "    return F\n",
    "\n",
    "m = 12\n",
    "L = 1#2*np.pi\n",
    "F = F_matrix(m)\n",
    "x = np.arange(m)*L/m   # Grid points\n",
    "xi = np.fft.fftfreq(m)*2*np.pi  # Wavenumber \"grid\"\n",
    "if m % 2 == 0:  # Handle Nyquist frequency\n",
    "    xi[m//2]=0\n",
    "print(xi)\n",
    "\n",
    "D = np.diag(1j*xi)\n",
    "DD = np.dot(np.linalg.inv(F),np.dot(D,F))\n",
    "print(DD)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.        +0.j  0.97704862+0.j -0.45344984-0.j  0.26179939+0.j\n",
      "  -0.15114995+0.j  0.07014893+0.j  0.        -0.j -0.07014893+0.j\n",
      "   0.15114995-0.j -0.26179939+0.j  0.45344984-0.j -0.97704862-0.j]\n",
      " [-0.97704862+0.j -0.        -0.j  0.97704862+0.j -0.45344984-0.j\n",
      "   0.26179939+0.j -0.15114995+0.j  0.07014893+0.j -0.        +0.j\n",
      "  -0.07014893+0.j  0.15114995-0.j -0.26179939-0.j  0.45344984+0.j]\n",
      " [ 0.45344984-0.j -0.97704862+0.j -0.        -0.j  0.97704862+0.j\n",
      "  -0.45344984-0.j  0.26179939-0.j -0.15114995-0.j  0.07014893-0.j\n",
      "  -0.        +0.j -0.07014893-0.j  0.15114995-0.j -0.26179939-0.j]\n",
      " [-0.26179939+0.j  0.45344984-0.j -0.97704862+0.j  0.        +0.j\n",
      "   0.97704862+0.j -0.45344984+0.j  0.26179939+0.j -0.15114995+0.j\n",
      "   0.07014893-0.j -0.        -0.j -0.07014893-0.j  0.15114995-0.j]\n",
      " [ 0.15114995+0.j -0.26179939+0.j  0.45344984-0.j -0.97704862+0.j\n",
      "  -0.        -0.j  0.97704862-0.j -0.45344984+0.j  0.26179939+0.j\n",
      "  -0.15114995+0.j  0.07014893+0.j -0.        +0.j -0.07014893+0.j]\n",
      " [-0.07014893+0.j  0.15114995-0.j -0.26179939-0.j  0.45344984+0.j\n",
      "  -0.97704862-0.j  0.        -0.j  0.97704862+0.j -0.45344984-0.j\n",
      "   0.26179939+0.j -0.15114995+0.j  0.07014893+0.j -0.        -0.j]\n",
      " [-0.        -0.j -0.07014893-0.j  0.15114995-0.j -0.26179939+0.j\n",
      "   0.45344984-0.j -0.97704862+0.j  0.        +0.j  0.97704862-0.j\n",
      "  -0.45344984-0.j  0.26179939-0.j -0.15114995+0.j  0.07014893-0.j]\n",
      " [ 0.07014893+0.j -0.        +0.j -0.07014893-0.j  0.15114995+0.j\n",
      "  -0.26179939+0.j  0.45344984-0.j -0.97704862-0.j  0.        -0.j\n",
      "   0.97704862+0.j -0.45344984+0.j  0.26179939-0.j -0.15114995+0.j]\n",
      " [-0.15114995-0.j  0.07014893+0.j  0.        +0.j -0.07014893-0.j\n",
      "   0.15114995+0.j -0.26179939+0.j  0.45344984-0.j -0.97704862+0.j\n",
      "  -0.        -0.j  0.97704862-0.j -0.45344984+0.j  0.26179939-0.j]\n",
      " [ 0.26179939+0.j -0.15114995-0.j  0.07014893-0.j  0.        -0.j\n",
      "  -0.07014893+0.j  0.15114995+0.j -0.26179939-0.j  0.45344984+0.j\n",
      "  -0.97704862-0.j  0.        +0.j  0.97704862-0.j -0.45344984-0.j]\n",
      " [-0.45344984-0.j  0.26179939+0.j -0.15114995+0.j  0.07014893-0.j\n",
      "  -0.        +0.j -0.07014893-0.j  0.15114995+0.j -0.26179939-0.j\n",
      "   0.45344984+0.j -0.97704862-0.j  0.        -0.j  0.97704862+0.j]\n",
      " [ 0.97704862-0.j -0.45344984+0.j  0.26179939-0.j -0.15114995+0.j\n",
      "   0.07014893-0.j  0.        +0.j -0.07014893+0.j  0.15114995-0.j\n",
      "  -0.26179939+0.j  0.45344984-0.j -0.97704862+0.j -0.        -0.j]]\n"
     ]
    }
   ],
   "source": [
    "F = np.zeros((m,m),dtype=complex)\n",
    "for j in range(m):\n",
    "    for k in range(m):\n",
    "        F[j,k] = np.exp(2j*np.pi*j*k/m)/np.sqrt(m) # Note the minus sign\n",
    "        \n",
    "L=np.dot(F,np.dot(np.diag(1j*xi),F.conj()))\n",
    "print(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.-0.j  0.+0.j  0.-0.j -0.+0.j -0.+0.j -0.+0.j  0.+0.j -0.-0.j  0.-0.j\n",
      "   0.+0.j  0.-0.j -0.-0.j]\n",
      " [-0.+0.j  0.-0.j  0.+0.j -0.+0.j  0.+0.j -0.+0.j -0.+0.j -0.+0.j  0.+0.j\n",
      "   0.-0.j  0.-0.j  0.+0.j]\n",
      " [ 0.-0.j -0.+0.j -0.-0.j  0.+0.j  0.-0.j -0.-0.j  0.-0.j -0.-0.j -0.+0.j\n",
      "  -0.-0.j -0.-0.j  0.+0.j]\n",
      " [ 0.+0.j  0.-0.j -0.+0.j  0.-0.j -0.-0.j  0.+0.j  0.+0.j -0.+0.j  0.-0.j\n",
      "  -0.-0.j -0.-0.j -0.-0.j]\n",
      " [ 0.+0.j -0.+0.j  0.-0.j  0.-0.j -0.-0.j  0.-0.j -0.+0.j  0.+0.j -0.+0.j\n",
      "  -0.+0.j -0.+0.j -0.+0.j]\n",
      " [-0.+0.j  0.-0.j -0.-0.j -0.-0.j -0.-0.j  0.+0.j  0.+0.j -0.-0.j  0.+0.j\n",
      "   0.+0.j -0.+0.j -0.+0.j]\n",
      " [-0.+0.j  0.-0.j -0.-0.j  0.-0.j  0.-0.j -0.+0.j  0.+0.j  0.-0.j  0.-0.j\n",
      "   0.-0.j -0.+0.j -0.-0.j]\n",
      " [-0.+0.j -0.+0.j  0.-0.j  0.+0.j  0.+0.j  0.-0.j  0.-0.j  0.+0.j  0.+0.j\n",
      "  -0.+0.j -0.-0.j  0.+0.j]\n",
      " [ 0.-0.j -0.+0.j  0.-0.j -0.-0.j  0.+0.j -0.+0.j -0.-0.j -0.+0.j  0.-0.j\n",
      "   0.-0.j  0.+0.j  0.-0.j]\n",
      " [-0.+0.j -0.-0.j  0.-0.j  0.-0.j  0.+0.j -0.+0.j -0.-0.j  0.+0.j -0.-0.j\n",
      "   0.-0.j  0.-0.j  0.-0.j]\n",
      " [-0.+0.j  0.+0.j  0.+0.j -0.-0.j -0.+0.j -0.-0.j  0.+0.j  0.-0.j -0.+0.j\n",
      "   0.-0.j  0.-0.j  0.+0.j]\n",
      " [ 0.-0.j  0.+0.j  0.-0.j  0.+0.j -0.-0.j -0.-0.j -0.+0.j  0.-0.j  0.+0.j\n",
      "   0.-0.j -0.+0.j -0.-0.j]]\n"
     ]
    }
   ],
   "source": [
    "print(L-DD)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Backward Euler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.47719441,  0.38491871,  0.16656684, -0.00000855,  0.07873336,\n",
       "        -0.06131545,  0.07700798, -0.07993946,  0.08827245, -0.09709409,\n",
       "         0.11222496, -0.14656117],\n",
       "       [-0.14656117,  0.47719441,  0.38491871,  0.16656684, -0.00000855,\n",
       "         0.07873336, -0.06131545,  0.07700798, -0.07993946,  0.08827245,\n",
       "        -0.09709409,  0.11222496],\n",
       "       [ 0.11222496, -0.14656117,  0.47719441,  0.38491871,  0.16656684,\n",
       "        -0.00000855,  0.07873336, -0.06131545,  0.07700798, -0.07993946,\n",
       "         0.08827245, -0.09709409],\n",
       "       [-0.09709409,  0.11222496, -0.14656117,  0.47719441,  0.38491871,\n",
       "         0.16656684, -0.00000855,  0.07873336, -0.06131545,  0.07700798,\n",
       "        -0.07993946,  0.08827245],\n",
       "       [ 0.08827245, -0.09709409,  0.11222496, -0.14656117,  0.47719441,\n",
       "         0.38491871,  0.16656684, -0.00000855,  0.07873336, -0.06131545,\n",
       "         0.07700798, -0.07993946],\n",
       "       [-0.07993946,  0.08827245, -0.09709409,  0.11222496, -0.14656117,\n",
       "         0.47719441,  0.38491871,  0.16656684, -0.00000855,  0.07873336,\n",
       "        -0.06131545,  0.07700798],\n",
       "       [ 0.07700798, -0.07993946,  0.08827245, -0.09709409,  0.11222496,\n",
       "        -0.14656117,  0.47719441,  0.38491871,  0.16656684, -0.00000855,\n",
       "         0.07873336, -0.06131545],\n",
       "       [-0.06131545,  0.07700798, -0.07993946,  0.08827245, -0.09709409,\n",
       "         0.11222496, -0.14656117,  0.47719441,  0.38491871,  0.16656684,\n",
       "        -0.00000855,  0.07873336],\n",
       "       [ 0.07873336, -0.06131545,  0.07700798, -0.07993946,  0.08827245,\n",
       "        -0.09709409,  0.11222496, -0.14656117,  0.47719441,  0.38491871,\n",
       "         0.16656684, -0.00000855],\n",
       "       [-0.00000855,  0.07873336, -0.06131545,  0.07700798, -0.07993946,\n",
       "         0.08827245, -0.09709409,  0.11222496, -0.14656117,  0.47719441,\n",
       "         0.38491871,  0.16656684],\n",
       "       [ 0.16656684, -0.00000855,  0.07873336, -0.06131545,  0.07700798,\n",
       "        -0.07993946,  0.08827245, -0.09709409,  0.11222496, -0.14656117,\n",
       "         0.47719441,  0.38491871],\n",
       "       [ 0.38491871,  0.16656684, -0.00000855,  0.07873336, -0.06131545,\n",
       "         0.07700798, -0.07993946,  0.08827245, -0.09709409,  0.11222496,\n",
       "        -0.14656117,  0.47719441]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nu = 1.\n",
    "d = 1 - nu*1j*xi\n",
    "D = np.diag(d)\n",
    "Dinv = np.diag(1./d)\n",
    "M = np.dot(F,np.dot(Dinv,F.conj()))\n",
    "M = np.real(M)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.40337615,  0.46517643,  0.08114055,  0.09234154, -0.02278769,\n",
       "        0.05541291, -0.07465989])"
      ]
     },
     "execution_count": 190,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[0,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.40337614763053153+1.586032892321652e-17j),\n",
       " (0.4651764319067382+3.7668281192639235e-17j),\n",
       " (0.08114055416726605-3.172065784643304e-17j),\n",
       " (0.09234154169973044+0j),\n",
       " (-0.022787693089044048-6.344131569286608e-17j),\n",
       " (0.055412907794449004-7.137148015447434e-17j),\n",
       " (-0.07465989010967117-1.586032892321652e-16j)]"
      ]
     },
     "execution_count": 191,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j=3\n",
    "#lamda = 2*np.pi*xi/m\n",
    "xxi = 2*np.pi*np.arange(m)/m\n",
    "[np.sum(np.exp(-1j*j*xxi)*1./(1-nu*1j*xi))/m for j in range(m)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.       ,  0.8975979,  1.7951958,  2.6927937, -2.6927937,\n",
       "       -1.7951958, -0.8975979])"
      ]
     },
     "execution_count": 192,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.8975979 , 1.7951958 , 2.6927937 , 3.5903916 ,\n",
       "       4.48798951, 5.38558741])"
      ]
     },
     "execution_count": 193,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xxi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.4033761476305315,\n",
       " 0.46517643190673824,\n",
       " 0.08114055416726607,\n",
       " 0.09234154169973044,\n",
       " -0.022787693089044016,\n",
       " 0.055412907794449025,\n",
       " -0.07465989010967118]"
      ]
     },
     "execution_count": 194,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[np.sum((np.cos(j*xxi)+nu*xi*np.sin(j*xxi))/(1+nu**2*xi**2))/m for j in range(m)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.022787693089044016"
      ]
     },
     "execution_count": 205,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = 4\n",
    "thet = 2*np.pi * np.arange(1,(m+1)/2)/m  # here should divide by domain length?\n",
    "thetj = 2*np.pi*j*np.arange(1,(m+1)/2)/m\n",
    "#thet = thet[1:(m+1)/2]\n",
    "#thetj = thetj[1:(m+1)/2]\n",
    "(1+2*np.sum((np.cos(thetj)+nu*thet*np.sin(thetj))/(1+nu**2*thet**2)))/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.8975979, 1.7951958, 2.6927937])"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "thet"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.8975979 , 1.7951958 , 2.6927937 , 3.5903916 ,\n",
       "       4.48798951, 5.38558741])"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xxi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  1.,  2.,  3., -3., -2., -1.])"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "#np.imag(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.37142857142857144-1.586032892321652e-17j)"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = 1\n",
    "np.sum(np.exp(-1j*(1-j)*xi)/d)/m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
       "        -0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j,\n",
       "        -0.16245985-0.5j,  0.22261434+0.5j, -0.28867513-0.5j,\n",
       "         0.36327126+0.5j, -0.45020202-0.5j,  0.55530626+0.5j,\n",
       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j],\n",
       "       [ 4.75718223+0.5j, -0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
       "         0.05255212-0.5j,  0.        +0.5j, -0.05255212-0.5j,\n",
       "         0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j,\n",
       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j],\n",
       "       [-2.35231505-0.5j,  4.75718223+0.5j,  0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j, -0.        +0.5j,\n",
       "        -0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j,\n",
       "         0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j,\n",
       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j],\n",
       "       [ 1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "        -0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
       "        -0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j,\n",
       "        -0.16245985-0.5j,  0.22261434+0.5j, -0.28867513-0.5j,\n",
       "         0.36327126+0.5j, -0.45020202-0.5j,  0.55530626+0.5j,\n",
       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j],\n",
       "       [-1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j,  0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
       "         0.05255212-0.5j,  0.        +0.5j, -0.05255212-0.5j,\n",
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       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j],\n",
       "       [ 0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
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       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j, -0.        +0.5j,\n",
       "        -0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j,\n",
       "         0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j,\n",
       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j],\n",
       "       [-0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "        -0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
       "         0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j,\n",
       "        -0.16245985-0.5j,  0.22261434+0.5j, -0.28867513-0.5j,\n",
       "         0.36327126+0.5j, -0.45020202-0.5j,  0.55530626+0.5j],\n",
       "       [ 0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j,  0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
       "         0.05255212-0.5j,  0.        +0.5j, -0.05255212-0.5j,\n",
       "         0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j,\n",
       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j],\n",
       "       [-0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j,  0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j, -0.        +0.5j,\n",
       "        -0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j,\n",
       "         0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j],\n",
       "       [ 0.36327126+0.5j, -0.45020202-0.5j,  0.55530626+0.5j,\n",
       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "         0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
       "        -0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j,\n",
       "        -0.16245985-0.5j,  0.22261434+0.5j, -0.28867513-0.5j],\n",
       "       [-0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j, -0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
       "         0.05255212-0.5j, -0.        +0.5j, -0.05255212-0.5j,\n",
       "         0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j],\n",
       "       [ 0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j,\n",
       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j,  0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j, -0.        +0.5j,\n",
       "        -0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j],\n",
       "       [-0.16245985-0.5j,  0.22261434+0.5j, -0.28867513-0.5j,\n",
       "         0.36327126+0.5j, -0.45020202-0.5j,  0.55530626+0.5j,\n",
       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "         0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
       "         0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j],\n",
       "       [ 0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j,\n",
       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j, -0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
       "         0.05255212-0.5j,  0.        +0.5j, -0.05255212-0.5j],\n",
       "       [-0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j,\n",
       "         0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j,\n",
       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j, -0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j, -0.        +0.5j],\n",
       "       [ 0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j,\n",
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       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "         0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j],\n",
       "       [ 0.05255212-0.5j, -0.        +0.5j, -0.05255212-0.5j,\n",
       "         0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j,\n",
       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j,  0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j],\n",
       "       [-0.10627828+0.5j,  0.05255212-0.5j,  0.        +0.5j,\n",
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       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j,  0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j],\n",
       "       [ 0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
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       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "         0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j],\n",
       "       [-0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
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       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
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       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j],\n",
       "       [ 0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
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       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j, -0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j],\n",
       "       [-0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
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       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
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       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j],\n",
       "       [ 0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
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       "         0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j,\n",
       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j,  0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j],\n",
       "       [-0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j,  0.        +0.5j,\n",
       "        -0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j,\n",
       "         0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j,\n",
       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j, -0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j],\n",
       "       [ 0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
       "         0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j,\n",
       "        -0.16245985-0.5j,  0.22261434+0.5j, -0.28867513-0.5j,\n",
       "         0.36327126+0.5j, -0.45020202-0.5j,  0.55530626+0.5j,\n",
       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "         0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j,\n",
       "        -1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j],\n",
       "       [-0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
       "         0.05255212-0.5j,  0.        +0.5j, -0.05255212-0.5j,\n",
       "         0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j,\n",
       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j, -0.        -0.5j, -4.75718223+0.5j,\n",
       "         2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j],\n",
       "       [ 1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j,  0.        +0.5j,\n",
       "        -0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j,\n",
       "         0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j,\n",
       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j, -0.        -0.5j,\n",
       "        -4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j],\n",
       "       [-1.53884177+0.5j,  1.12301839-0.5j, -0.8660254 +0.5j,\n",
       "         0.68819096-0.5j, -0.55530626+0.5j,  0.45020202-0.5j,\n",
       "        -0.36327126+0.5j,  0.28867513-0.5j, -0.22261434+0.5j,\n",
       "         0.16245985-0.5j, -0.10627828+0.5j,  0.05255212-0.5j,\n",
       "        -0.        +0.5j, -0.05255212-0.5j,  0.10627828+0.5j,\n",
       "        -0.16245985-0.5j,  0.22261434+0.5j, -0.28867513-0.5j,\n",
       "         0.36327126+0.5j, -0.45020202-0.5j,  0.55530626+0.5j,\n",
       "        -0.68819096-0.5j,  0.8660254 +0.5j, -1.12301839-0.5j,\n",
       "         1.53884177+0.5j, -2.35231505-0.5j,  4.75718223+0.5j,\n",
       "         0.        -0.5j, -4.75718223+0.5j,  2.35231505-0.5j],\n",
       "       [ 2.35231505-0.5j, -1.53884177+0.5j,  1.12301839-0.5j,\n",
       "        -0.8660254 +0.5j,  0.68819096-0.5j, -0.55530626+0.5j,\n",
       "         0.45020202-0.5j, -0.36327126+0.5j,  0.28867513-0.5j,\n",
       "        -0.22261434+0.5j,  0.16245985-0.5j, -0.10627828+0.5j,\n",
       "         0.05255212-0.5j, -0.        +0.5j, -0.05255212-0.5j,\n",
       "         0.10627828+0.5j, -0.16245985-0.5j,  0.22261434+0.5j,\n",
       "        -0.28867513-0.5j,  0.36327126+0.5j, -0.45020202-0.5j,\n",
       "         0.55530626+0.5j, -0.68819096-0.5j,  0.8660254 +0.5j,\n",
       "        -1.12301839-0.5j,  1.53884177+0.5j, -2.35231505-0.5j,\n",
       "         4.75718223+0.5j, -0.        -0.5j, -4.75718223+0.5j],\n",
       "       [-4.75718223+0.5j,  2.35231505-0.5j, -1.53884177+0.5j,\n",
       "         1.12301839-0.5j, -0.8660254 +0.5j,  0.68819096-0.5j,\n",
       "        -0.55530626+0.5j,  0.45020202-0.5j, -0.36327126+0.5j,\n",
       "         0.28867513-0.5j, -0.22261434+0.5j,  0.16245985-0.5j,\n",
       "        -0.10627828+0.5j,  0.05255212-0.5j,  0.        +0.5j,\n",
       "        -0.05255212-0.5j,  0.10627828+0.5j, -0.16245985-0.5j,\n",
       "         0.22261434+0.5j, -0.28867513-0.5j,  0.36327126+0.5j,\n",
       "        -0.45020202-0.5j,  0.55530626+0.5j, -0.68819096-0.5j,\n",
       "         0.8660254 +0.5j, -1.12301839-0.5j,  1.53884177+0.5j,\n",
       "        -2.35231505-0.5j,  4.75718223+0.5j,  0.        -0.5j]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = np.dot(F.conj().T,np.dot(np.diag(1j*xi),F))\n",
    "L"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0. +0.j,  2.5-0.j,  0. +0.j,  0. +0.j, -2.5-0.j],\n",
       "       [-2.5-0.j, -0. +0.j,  2.5-0.j, -0. +0.j, -0. -0.j],\n",
       "       [-0. +0.j, -2.5-0.j, -0. +0.j,  2.5-0.j,  0. +0.j],\n",
       "       [ 0. +0.j,  0. +0.j, -2.5-0.j,  0. +0.j,  2.5-0.j],\n",
       "       [ 2.5-0.j,  0. -0.j,  0. +0.j, -2.5-0.j,  0. +0.j]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mm = 5\n",
    "FF = F_matrix(mm)\n",
    "xi = 2*np.pi*np.arange(mm)/mm\n",
    "np.dot(FF.conj().T,np.dot(np.diag(1j*np.sin(xi)),FF))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        +12.56637061j, -4.32403133 -3.14159265j,\n",
       "        -1.02076533 -3.14159265j,  1.02076533 -3.14159265j,\n",
       "         4.32403133 -3.14159265j],\n",
       "       [ 4.32403133 -3.14159265j,  0.        +12.56637061j,\n",
       "        -4.32403133 -3.14159265j, -1.02076533 -3.14159265j,\n",
       "         1.02076533 -3.14159265j],\n",
       "       [ 1.02076533 -3.14159265j,  4.32403133 -3.14159265j,\n",
       "        -0.        +12.56637061j, -4.32403133 -3.14159265j,\n",
       "        -1.02076533 -3.14159265j],\n",
       "       [-1.02076533 -3.14159265j,  1.02076533 -3.14159265j,\n",
       "         4.32403133 -3.14159265j,  0.        +12.56637061j,\n",
       "        -4.32403133 -3.14159265j],\n",
       "       [-4.32403133 -3.14159265j, -1.02076533 -3.14159265j,\n",
       "         1.02076533 -3.14159265j,  4.32403133 -3.14159265j,\n",
       "        -0.        +12.56637061j]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(FF.conj().T,np.dot(np.diag(1j*xi),FF))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        -0.j,  0.85065081+0.j, -0.52573111-0.j,\n",
       "         0.52573111+0.j, -0.85065081-0.j],\n",
       "       [-0.85065081-0.j,  0.        +0.j,  0.85065081+0.j,\n",
       "        -0.52573111-0.j,  0.52573111+0.j],\n",
       "       [ 0.52573111+0.j, -0.85065081-0.j, -0.        +0.j,\n",
       "         0.85065081-0.j, -0.52573111+0.j],\n",
       "       [-0.52573111+0.j,  0.52573111-0.j, -0.85065081-0.j,\n",
       "        -0.        -0.j,  0.85065081-0.j],\n",
       "       [ 0.85065081-0.j, -0.52573111+0.j,  0.52573111+0.j,\n",
       "        -0.85065081-0.j, -0.        +0.j]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 5\n",
    "L = 2*np.pi\n",
    "F = F_matrix(m)\n",
    "x = np.arange(0,m)*(2*np.pi/(m))   # Grid points\n",
    "xi = np.fft.fftfreq(m)*m*2*np.pi/L  # Wavenumber \"grid\"\n",
    "\n",
    "D = np.diag(1j*xi)\n",
    "DD = np.dot(np.linalg.inv(F),np.dot(D,F))\n",
    "DD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11a05bf28>]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u = np.sin(x)\n",
    "plt.plot(x,np.dot(np.real(DD),u))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.        -0.j,  0.30901699+0.j, -0.80901699+0.j, -0.80901699-0.j,\n",
       "        0.30901699+0.j])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(DD,u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "FF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = 4\n",
    "F = np.zeros((m,m),dtype=complex)\n",
    "for j in range(m):\n",
    "    for k in range(m):\n",
    "        F[j,k] = np.exp(2j*np.pi*j*k/m)/np.sqrt(m)\n",
    "np.sqrt(m)*F.conj()-FF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.        +0.j          1.        +0.j          1.        +0.j\n",
      "   1.        +0.j          1.        +0.j        ]\n",
      " [ 1.        +0.j          0.30901699-0.95105652j -0.80901699-0.58778525j\n",
      "  -0.80901699+0.58778525j  0.30901699+0.95105652j]\n",
      " [ 1.        +0.j         -0.80901699-0.58778525j  0.30901699+0.95105652j\n",
      "   0.30901699-0.95105652j -0.80901699+0.58778525j]\n",
      " [ 1.        +0.j         -0.80901699+0.58778525j  0.30901699-0.95105652j\n",
      "   0.30901699+0.95105652j -0.80901699-0.58778525j]\n",
      " [ 1.        +0.j          0.30901699+0.95105652j -0.80901699+0.58778525j\n",
      "  -0.80901699-0.58778525j  0.30901699-0.95105652j]]\n"
     ]
    }
   ],
   "source": [
    "m = 5\n",
    "F = np.zeros((m,m),dtype=complex)\n",
    "for j in range(m):\n",
    "    for k in range(m):\n",
    "        F[j,k] = np.exp(-2j*np.pi*j*k/m)/np.sqrt(m)\n",
    "print(F/F[0,0])\n",
    "xi = np.fft.fftfreq(m)*m*2*np.pi/L"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        +0.j,  4.25325404+0.j, -2.62865556+0.j,\n",
       "         2.62865556+0.j, -4.25325404+0.j],\n",
       "       [-4.25325404+0.j,  0.        +0.j,  4.25325404+0.j,\n",
       "        -2.62865556-0.j,  2.62865556-0.j],\n",
       "       [ 2.62865556+0.j, -4.25325404+0.j,  0.        +0.j,\n",
       "         4.25325404-0.j, -2.62865556-0.j],\n",
       "       [-2.62865556+0.j,  2.62865556-0.j, -4.25325404-0.j,\n",
       "         0.        +0.j,  4.25325404+0.j],\n",
       "       [ 4.25325404+0.j, -2.62865556-0.j,  2.62865556-0.j,\n",
       "        -4.25325404+0.j,  0.        +0.j]])"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1 = F_matrix(m)\n",
    "xi = np.fft.fftfreq(m)*m*2*np.pi/L\n",
    "np.dot(F1.conj().T,np.dot(np.diag(1j*xi),F1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.-0.j],\n",
       "       [ 0.-0.j,  5.+0.j, -0.-0.j, -0.+0.j,  0.-0.j],\n",
       "       [ 0.+0.j, -0.+0.j,  5.+0.j,  0.-0.j, -0.+0.j],\n",
       "       [ 0.+0.j, -0.-0.j,  0.+0.j,  5.+0.j, -0.-0.j],\n",
       "       [ 0.+0.j,  0.+0.j, -0.-0.j, -0.+0.j,  5.+0.j]])"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(F1.conj().T,F1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [],
   "source": [
    "L=np.dot(F.conj().T,np.dot(np.diag(1j*xi),F))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.-2.j, -0.+2.j, -0.-1.j,  0.-0.j,  0.+1.j])"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eigvals(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.+0.j,  0.+1.j,  0.+2.j, -0.-2.j, -0.-1.j])"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi*1j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.+0.j        ,  0.+0.j        ,  0.+0.j        ,\n",
       "         0.+0.j        ,  0.+0.j        ,  0.+0.j        ],\n",
       "       [ 0.+0.j        ,  0.-1.73205081j,  0.-1.73205081j,\n",
       "         0.+0.j        ,  0.+1.73205081j, -0.+1.73205081j],\n",
       "       [ 0.+0.j        , -0.-1.73205081j,  0.+1.73205081j,\n",
       "         0.+0.j        ,  0.-1.73205081j,  0.+1.73205081j],\n",
       "       [ 0.+0.j        ,  0.+0.j        ,  0.+0.j        ,\n",
       "         0.+0.j        ,  0.-0.j        ,  0.-0.j        ],\n",
       "       [ 0.+0.j        , -0.+1.73205081j,  0.-1.73205081j,\n",
       "         0.-0.j        ,  0.+1.73205081j,  0.-1.73205081j],\n",
       "       [ 0.+0.j        ,  0.+1.73205081j,  0.+1.73205081j,\n",
       "         0.-0.j        ,  0.-1.73205081j,  0.-1.73205081j]])"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1-F1.conj().T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. +0.j       ,  1. +0.j       ,  1. +0.j       ,\n",
       "         1. +0.j       ,  1. +0.j       ,  1. +0.j       ],\n",
       "       [ 1. +0.j       ,  0.5-0.8660254j, -0.5-0.8660254j,\n",
       "        -1. +0.j       , -0.5+0.8660254j,  0.5+0.8660254j],\n",
       "       [ 1. +0.j       , -0.5-0.8660254j, -0.5+0.8660254j,\n",
       "         1. +0.j       , -0.5-0.8660254j, -0.5+0.8660254j],\n",
       "       [ 1. +0.j       , -1. +0.j       ,  1. +0.j       ,\n",
       "        -1. +0.j       ,  1. -0.j       , -1. -0.j       ],\n",
       "       [ 1. +0.j       , -0.5+0.8660254j, -0.5-0.8660254j,\n",
       "         1. +0.j       , -0.5+0.8660254j, -0.5-0.8660254j],\n",
       "       [ 1. +0.j       ,  0.5+0.8660254j, -0.5+0.8660254j,\n",
       "        -1. +0.j       , -0.5-0.8660254j,  0.5-0.8660254j]])"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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