神奇計算

god.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy as sci"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 3\n",
    "A = np.random.random((N,N))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.37075245, 0.6739229 , 0.20091591],\n",
       "       [0.39703536, 0.0495583 , 0.49734277],\n",
       "       [0.58168862, 0.67693551, 0.8502318 ]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "A1 = A+A.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.7415049 , 1.07095825, 0.78260453],\n",
       "       [1.07095825, 0.0991166 , 1.17427829],\n",
       "       [0.78260453, 1.17427829, 1.70046361]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "w,v = np.linalg.eig(A1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.98360474,  0.38211522, -0.82463486])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.48415423, -0.74705032,  0.4555332 ],\n",
       "       [-0.47808952, -0.21016968, -0.85279489],\n",
       "       [-0.73281996,  0.6306699 ,  0.25540239]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "U = np.zeros((N,N))\n",
    "for i in range(N):\n",
    "    lambda_D = w.copy()\n",
    "    lambda_D[i] = np.nan\n",
    "    lambda_D = lambda_D[~np.isnan(lambda_D)]\n",
    "    denominator = np.prod(w[i]-lambda_D)\n",
    "    for j in range(N):\n",
    "        M = A1.copy()\n",
    "        M[:,j] = np.nan\n",
    "        M[j,:] = np.nan\n",
    "        M = M[~np.isnan(M)]\n",
    "        M = M.reshape(N-1,N-1)\n",
    "        w2,v2 = np.linalg.eig(M)\n",
    "        fraction = np.prod(w[i]-w2)\n",
    "        U[j,i] = fraction/denominator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.63404690107061e-15"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.norm(U-v*v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.23440532, 0.55808418, 0.2075105 ],\n",
       "       [0.22856958, 0.04417129, 0.72725912],\n",
       "       [0.53702509, 0.39774453, 0.06523038]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.23440532, 0.55808418, 0.2075105 ],\n",
       "       [0.22856958, 0.04417129, 0.72725912],\n",
       "       [0.53702509, 0.39774453, 0.06523038]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v*v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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要移除指定的 index
可以使用 np.delete

U = np.zeros((N,N))
for i in range(N):
    lambda_D = np.delete(w, i)
    denominator = np.prod(w[i]-lambda_D)
    for j in range(N):
        M = np.delete(A1, j, 0)
        M = np.delete(M, j, 1)
        w2, v2 = np.linalg.eig(M)
        fraction = np.prod(w[i]-w2)
        U[j,i] = fraction/denominator

要移除指定的 index
可以使用 np.delete

U = np.zeros((N,N))
for i in range(N):
    lambda_D = np.delete(w, i)
    denominator = np.prod(w[i]-lambda_D)
    for j in range(N):
        M = np.delete(A1, j, 0)
        M = np.delete(M, j, 1)
        w2, v2 = np.linalg.eig(M)
        fraction = np.prod(w[i]-w2)
        U[j,i] = fraction/denominator

感謝！這個更加簡潔好讀