Final
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commit
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@ -1,6 +1,7 @@
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{
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"cells": [
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{
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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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@ -8,45 +9,7 @@
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"ename": "NameError",
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"evalue": "name 'matrix' is not defined",
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"output_type": "error",
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"traceback": [
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"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
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"\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)",
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"Cell \u001b[1;32mIn[1], line 1\u001b[0m\n\u001b[1;32m----> 1\u001b[0m A\u001b[38;5;241m=\u001b[39m\u001b[43mmatrix\u001b[49m(QQ,\u001b[38;5;241m5\u001b[39m,\u001b[38;5;241m3\u001b[39m,[\u001b[38;5;241m2\u001b[39m, \u001b[38;5;241m4\u001b[39m, \u001b[38;5;241m6\u001b[39m, \u001b[38;5;241m8\u001b[39m, \u001b[38;5;241m10\u001b[39m, \u001b[38;5;241m12\u001b[39m, \u001b[38;5;241m14\u001b[39m, \u001b[38;5;241m16\u001b[39m, \u001b[38;5;241m18\u001b[39m, \u001b[38;5;241m20\u001b[39m, \u001b[38;5;241m22\u001b[39m, \u001b[38;5;241m24\u001b[39m, \u001b[38;5;241m26\u001b[39m, \u001b[38;5;241m28\u001b[39m, \u001b[38;5;241m31\u001b[39m])\n\u001b[0;32m 2\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mA =\u001b[39m\u001b[38;5;124m'\u001b[39m)\n\u001b[0;32m 3\u001b[0m \u001b[38;5;28mprint\u001b[39m(A, \u001b[38;5;124m'\u001b[39m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124m'\u001b[39m)\n",
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"\u001b[1;31mNameError\u001b[0m: name 'matrix' is not defined"
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]
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}
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],
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"source": [
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"A=matrix(QQ,5,3,[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 31])\n",
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"print('A =')\n",
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"print(A, '\\n')\n",
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"\n",
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"print('b =')\n",
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"b=vector(QQ,[-1,0,1,0,1])\n",
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"print(b, '\\n')\n",
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"\n",
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"print('A^T * A =')\n",
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"print(A.transpose()*A, '\\n')\n",
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"print('Macierz A^T*A jest kwadratowa, więc rozwiązanie istnieje\\n')\n",
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"\n",
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"u=(A.transpose()*A)^(-1)*A.transpose()*b\n",
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"print('u = (A^T * A)^-1 * A^T * b =')\n",
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"print(u, '\\n')\n",
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"\n",
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"\n",
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"print('b - A * u = ')\n",
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"print(b - A * u, '\\n')"
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]
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},
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{
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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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@ -66,13 +29,97 @@
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]
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},
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{
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"cell_type": "markdown",
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"cell_type": "code",
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"execution_count": 126,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"A =\n",
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"[[ 2 4 6]\n",
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" [ 8 10 12]\n",
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" [14 16 18]\n",
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" [20 22 24]\n",
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" [26 28 31]] \n",
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"\n",
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"b =\n",
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"[-1 0 1 0 1] \n",
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"\n",
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"A^T * A =\n",
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"Matrix([[1340, 1480, 1646], [1480, 1640, 1828], [1646, 1828, 2041]]) \n",
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"\n",
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"Macierz A^T*A jest kwadratowa, więc rozwiązanie istnieje\n",
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"\n",
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"(A^T * A)^(-1)*A.transpose() =\n",
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"[[0.050000 -0.233333 -0.516667 -0.800000 1.000000]\n",
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" [-0.600000 0.216667 1.033333 1.850000 -2.000000]\n",
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" [0.500000 0.000000 -0.500000 -1.000000 1.000000]] \n",
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"\n",
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"Rozwiązanie:\n",
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"\n",
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"x = (A^T * A)^(-1) * A^T * b =\n",
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"[0.433333 -0.366667 -0.000000] \n",
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"\n",
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"\n"
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]
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}
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],
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"source": [
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"Zadanie 4.7\n"
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"import numpy as np\n",
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"from sympy import symbols, Matrix, vector\n",
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"from numpy.linalg import eig\n",
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"\n",
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"m = np.array([[2, 4, 6],\n",
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" [8, 10, 12],\n",
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" [14, 16, 18],\n",
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" [20, 22, 24],\n",
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" [26, 28, 31]])\n",
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"\n",
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"A=Matrix(m)\n",
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"\n",
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"np.set_printoptions(formatter={'float_kind':'{:f}'.format})\n",
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"\n",
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"print('A =')\n",
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"print(m, '\\n')\n",
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"\n",
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"print('b =')\n",
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"b=np.array([-1,0,1,0,1])\n",
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"print(b, '\\n')\n",
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"\n",
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"print('A^T * A =')\n",
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"print(A.transpose()*A, '\\n')\n",
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"print('Macierz A^T*A jest kwadratowa, więc rozwiązanie istnieje\\n')\n",
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"\n",
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"#print('(A^T * A)^(-1) =')\n",
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"#print(np.linalg.inv(m.transpose() @ m), '\\n')\n",
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"\n",
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"print('(A^T * A)^(-1)*A.transpose() =')\n",
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"print(np.linalg.inv(m.transpose() @ m) @ m.transpose(), '\\n')\n",
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"\n",
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"print('Rozwiązanie:\\n')\n",
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"#u=(A.transpose()*A)^(-1)*A.transpose()*b\n",
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"x = np.linalg.inv(m.transpose() @ m) @ m.transpose() @ b\n",
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"print('x = (A^T * A)^(-1) * A^T * b =')\n",
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"print(x, '\\n\\n')\n",
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"\n",
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"# SPRAWDZENIE\n",
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"x, residuals, _, _ = np.linalg.lstsq(m, b, rcond=None)\n",
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"#print(\"Przybliżone rozwiązanie:\")\n",
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"#print(x)"
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]
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},
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{
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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Zadanie 4.7"
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]
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},
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{
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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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@ -83,29 +130,84 @@
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"execution_count": 127,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"zbior punktów = [(1, 1), (2, 3), (4, 5)]\n"
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]
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},
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{
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"ename": "NameError",
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"evalue": "name 'matrix' is not defined",
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"output_type": "error",
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"traceback": [
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"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
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"\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)",
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"Cell \u001b[1;32mIn[2], line 3\u001b[0m\n\u001b[0;32m 1\u001b[0m zbior\u001b[38;5;241m=\u001b[39m[(\u001b[38;5;241m1\u001b[39m,\u001b[38;5;241m1\u001b[39m),(\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m3\u001b[39m),(\u001b[38;5;241m4\u001b[39m,\u001b[38;5;241m5\u001b[39m)]\n\u001b[0;32m 2\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mzbior punktów = \u001b[39m\u001b[38;5;124m'\u001b[39m, zbior)\n\u001b[1;32m----> 3\u001b[0m m\u001b[38;5;241m=\u001b[39m\u001b[43mmatrix\u001b[49m(\u001b[38;5;241m3\u001b[39m,\u001b[38;5;241m2\u001b[39m,[\u001b[38;5;241m1\u001b[39m,exp(\u001b[38;5;241m1.0\u001b[39m),\u001b[38;5;241m1\u001b[39m,exp(\u001b[38;5;241m2.0\u001b[39m),\u001b[38;5;241m1\u001b[39m,exp(\u001b[38;5;241m4.0\u001b[39m)])\n\u001b[0;32m 5\u001b[0m a,b,t\u001b[38;5;241m=\u001b[39mvar(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124ma,b,t\u001b[39m\u001b[38;5;124m'\u001b[39m)\n\u001b[0;32m 7\u001b[0m m\u001b[38;5;241m*\u001b[39mvector([a,b])\u001b[38;5;241m-\u001b[39mvector([\u001b[38;5;241m1\u001b[39m,\u001b[38;5;241m3\u001b[39m,\u001b[38;5;241m5\u001b[39m])\n",
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"\u001b[1;31mNameError\u001b[0m: name 'matrix' is not defined"
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"Przybliżone parametry:\n",
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"a = 1.641485981693081\n",
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"b = 0.06298603382678646\n"
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]
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}
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],
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"source": [
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"import numpy as np\n",
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"from scipy.optimize import curve_fit\n",
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"\n",
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"# Definicja funkcji, której chcemy dokonać przybliżenia\n",
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"def func(t, a, b):\n",
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" return a + b * np.exp(t)\n",
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"\n",
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"# Dane punktów\n",
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"t_data = np.array([1, 2, 4])\n",
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"y_data = np.array([1, 3, 5])\n",
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"\n",
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"# Przybliżanie funkcji do danych punktów\n",
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"params, _ = curve_fit(func, t_data, y_data)\n",
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"\n",
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"# Rozwiązania parametrów a i b\n",
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"a = params[0]\n",
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"b = params[1]\n",
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"\n",
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"# Wyświetlenie wyników\n",
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"print(\"Przybliżone parametry:\")\n",
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"print(\"a =\", a)\n",
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"print(\"b =\", b)\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 128,
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"metadata": {},
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"outputs": [
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{
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"ename": "TypeError",
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"evalue": "'module' object is not callable",
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"output_type": "error",
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"traceback": [
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"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
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"\u001b[1;31mTypeError\u001b[0m Traceback (most recent call last)",
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"Cell \u001b[1;32mIn[128], line 9\u001b[0m\n\u001b[0;32m 5\u001b[0m m1\u001b[39m=\u001b[39mnp\u001b[39m.\u001b[39marray([[\u001b[39m1\u001b[39m,np\u001b[39m.\u001b[39mexp(\u001b[39m1.0\u001b[39m)],[\u001b[39m1\u001b[39m,np\u001b[39m.\u001b[39mexp(\u001b[39m2.0\u001b[39m)],[\u001b[39m1\u001b[39m,np\u001b[39m.\u001b[39mexp(\u001b[39m4.0\u001b[39m)]])\n\u001b[0;32m 7\u001b[0m \u001b[39m#a,b,t=var('a,b,t')\u001b[39;00m\n\u001b[1;32m----> 9\u001b[0m m1\u001b[39m*\u001b[39mvector([a,b])\u001b[39m-\u001b[39mvector([\u001b[39m1\u001b[39m,\u001b[39m3\u001b[39m,\u001b[39m5\u001b[39m])\n\u001b[0;32m 11\u001b[0m M1\u001b[39m=\u001b[39mm1\u001b[39m.\u001b[39mtranspose()\u001b[39m*\u001b[39mm1\n\u001b[0;32m 12\u001b[0m M1\u001b[39m.\u001b[39mdet()\n",
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"\u001b[1;31mTypeError\u001b[0m: 'module' object is not callable"
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]
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}
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],
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"source": [
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"import numpy as np\n",
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"\n",
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"zb1=np.array([[1,1],[2,3],[4,5]])\n",
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"\n",
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"m1=np.array([[1,np.exp(1.0)],[1,np.exp(2.0)],[1,np.exp(4.0)]])\n",
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"\n",
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"#a,b,t=var('a,b,t')\n",
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"\n",
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"m1*vector([a,b])-vector([1,3,5])\n",
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"\n",
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"M1=m1.transpose()*m1\n",
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"M1.det()\n",
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"\n",
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"M1^(-1)*m1.transpose()*vector([1,3,5])\n",
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"\n",
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"##########################################################\n",
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"\n",
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"def func(t, a, b):\n",
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" return a + b * np.exp(t)\n",
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"\n",
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"x = np.linalg.inv(m.transpose() @ m) @ m.transpose() @ b\n",
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"\n",
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"zbior=[(1,1),(2,3),(4,5)]\n",
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"print('zbior punktów = ', zbior)\n",
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"m=matrix(3,2,[1,exp(1.0),1,exp(2.0),1,exp(4.0)])\n",
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@ -122,6 +224,7 @@
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]
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},
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{
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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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@ -129,6 +232,7 @@
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]
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},
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{
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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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@ -145,48 +249,207 @@
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": null,
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"metadata": {},
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"outputs": [
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{
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"ename": "NameError",
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"evalue": "name 'matrix' is not defined",
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"output_type": "error",
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"traceback": [
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"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
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"\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)",
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"Cell \u001b[1;32mIn[3], line 1\u001b[0m\n\u001b[1;32m----> 1\u001b[0m m\u001b[38;5;241m=\u001b[39m\u001b[43mmatrix\u001b[49m(\u001b[38;5;241m3\u001b[39m,\u001b[38;5;241m3\u001b[39m,[\u001b[38;5;241m1\u001b[39m,\u001b[38;5;241m1\u001b[39m,\u001b[38;5;241m0\u001b[39m,\u001b[38;5;241m1\u001b[39m,\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m0\u001b[39m,\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m3\u001b[39m])\n\u001b[0;32m 3\u001b[0m \u001b[38;5;66;03m#wartosci wlasne\u001b[39;00m\n\u001b[0;32m 4\u001b[0m eigenvalues \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mm\u001b[38;5;241m.\u001b[39meigvals(matrix)\n",
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"\u001b[1;31mNameError\u001b[0m: name 'matrix' is not defined"
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Macierz A=\n",
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"[[1 1 0]\n",
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" [1 2 2]\n",
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" [0 2 3]]\n",
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"Baza ortonormalnych wektorów własnych:\n",
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"[[-0.593233 -0.786436 0.172027]\n",
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" [0.679313 -0.374362 0.631179]\n",
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" [-0.431981 0.491296 0.756320]]\n"
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]
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}
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],
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"source": [
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"m=matrix(3,3,[1,1,0,1,2,2,0,2,3])\n",
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"import numpy as np\n",
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"from sympy import symbols, Matrix\n",
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"from numpy.linalg import eig\n",
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"\n",
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"#wartosci wlasne\n",
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"eigenvalues = np.m.eigvals(matrix)\n",
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"#Definicja macierzy A\n",
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"A = np.array([[1,1,0],[1,2,2],[0,2,3]])\n",
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"#A = Matrix([[1,1,0],[1,2,2],[0,2,3]])\n",
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"print(\"Macierz A=\")\n",
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"print(A)\n",
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"\n",
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"eigen=m.right_eigenvectors()\n",
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"e1=eigen[0][1][0]\n",
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"e2=eigen[1][1][0]\n",
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"print(e1.dot_product(e2))\n",
|
||||
"e3=eigen[2][1][0]\n",
|
||||
"print(e3.dot_product(e1))\n",
|
||||
"print(e2.dot_product(e3))\n",
|
||||
"def qr_eigval(matrix, epsilon=1e-10, max_iterations=1000):\n",
|
||||
"\n",
|
||||
"#znormalizuj wektor wlasny\n",
|
||||
" eigenv = np.diag(matrix)\n",
|
||||
"\n",
|
||||
"#czy wektory wlasne sa ortogonalne?\n",
|
||||
" for _ in range(max_iterations):\n",
|
||||
"\n",
|
||||
"#postac ortonormalna i normalizacja"
|
||||
" q, r = np.linalg.qr(matrix)\n",
|
||||
" matrix = np.dot(r, q)\n",
|
||||
" new_eigenv = np.diag(matrix)\n",
|
||||
"\n",
|
||||
" if np.allclose(eigenv, new_eigenv, atol=epsilon):\n",
|
||||
" break\n",
|
||||
" eigenv = new_eigenv\n",
|
||||
"\n",
|
||||
" return eigenv\n",
|
||||
"\n",
|
||||
"#Obliczenie wektorów własnych i wartości własnych\n",
|
||||
"#eigenvalues = qr_eigval(matrix)\n",
|
||||
"eigenvalues, eigenvectors = np.linalg.eig(A)\n",
|
||||
"\n",
|
||||
"#Dekompozycja\n",
|
||||
"Q, R = np.linalg.qr(eigenvectors)\n",
|
||||
"\n",
|
||||
"#Normalizacja wektora własnego\n",
|
||||
"orthonormal_basis = Q\n",
|
||||
"\n",
|
||||
"print(\"Baza ortonormalnych wektorów własnych:\")\n",
|
||||
"print(orthonormal_basis)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {},
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Zadanie 3.9"
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {},
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Zadanie 9\n",
|
||||
"\n",
|
||||
"Oblicz metodą ''power iteration'' wartości własne macierzy\n",
|
||||
"\n",
|
||||
"$$\\left(\\begin{array}{rrr}\n",
|
||||
"1 & 2 & 3 \\\\\n",
|
||||
"4 & 5 & 6 \\\\\n",
|
||||
"7 & 8 & 9\n",
|
||||
"\\end{array}\\right)$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Dominująca wartość własna (power iteration):\n",
|
||||
"16.116843969807043\n",
|
||||
"Dominujący wektor własny:\n",
|
||||
"[[0.107761 0.132408 0.157054]\n",
|
||||
" [0.244037 0.299852 0.355666]\n",
|
||||
" [0.380312 0.467295 0.554278]]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"\n",
|
||||
"A = np.array([[1,2,3],\n",
|
||||
" [4,5,6],\n",
|
||||
" [7,8,9]])\n",
|
||||
"\n",
|
||||
"S = np.array([[1,1,1]])\n",
|
||||
"\n",
|
||||
"def power_iteration(m, n, s):\n",
|
||||
" #punkt startowy\n",
|
||||
" st = s\n",
|
||||
" eigv = 0\n",
|
||||
" for i in range(n):\n",
|
||||
" st = np.dot(A, st)\n",
|
||||
" eigv = np.max(st)\n",
|
||||
" st = st/eigv\n",
|
||||
" return eigv\n",
|
||||
"\n",
|
||||
"def power_iteration_vec(m, n, s):\n",
|
||||
" tolerance = 1e-6\n",
|
||||
" #punkt startowy\n",
|
||||
" x = s\n",
|
||||
" for i in range(n): \n",
|
||||
" y = np.dot(m, x)\n",
|
||||
" #Normalizacja wektora\n",
|
||||
" x_new = y / np.linalg.norm(y)\n",
|
||||
" #Sprawdzenie warunku zbieżności\n",
|
||||
" if np.linalg.norm(x - x_new) < tolerance:\n",
|
||||
" break\n",
|
||||
" #Aktualizacja\n",
|
||||
" x = x_new\n",
|
||||
" return x\n",
|
||||
"\n",
|
||||
"y1 = power_iteration(A, 1000, A)\n",
|
||||
"print(\"Dominująca wartość własna (power iteration):\")\n",
|
||||
"print(y1)\n",
|
||||
"\n",
|
||||
"y2 = power_iteration_vec(A, 1000, A)\n",
|
||||
"print(\"Dominujący wektor własny:\")\n",
|
||||
"print(y2)\n",
|
||||
"\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Dominująca wartość własna:\n",
|
||||
"16.116840810027913\n",
|
||||
"Dominujący wektor własny:\n",
|
||||
"[0.231970 0.525322 0.818674]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"\n",
|
||||
"# Definicja macierzy A\n",
|
||||
"A = np.array([[1,2,3],\n",
|
||||
" [4,5,6],\n",
|
||||
" [7,8,9]])\n",
|
||||
" \n",
|
||||
"# Definicja wektora startowego\n",
|
||||
"x = np.array([1, 1, 1])\n",
|
||||
"\n",
|
||||
"# Parametry iteracji\n",
|
||||
"max_iterations = 1000\n",
|
||||
"tolerance = 1e-6\n",
|
||||
"\n",
|
||||
"# Iteracyjne obliczanie dominującej wartości własnej i wektora własnego\n",
|
||||
"for _ in range(max_iterations):\n",
|
||||
" # Mnożenie macierzy przez wektor\n",
|
||||
" y = np.dot(A, x)\n",
|
||||
"\n",
|
||||
" # Normalizacja wektora\n",
|
||||
" x_new = y / np.linalg.norm(y)\n",
|
||||
"\n",
|
||||
" # Sprawdzenie warunku zbieżności\n",
|
||||
" if np.linalg.norm(x - x_new) < tolerance:\n",
|
||||
" break\n",
|
||||
"\n",
|
||||
" # Aktualizacja wektora\n",
|
||||
" x = x_new\n",
|
||||
"\n",
|
||||
"# Obliczenie dominującej wartości własnej\n",
|
||||
"eigenvalue = np.dot(np.dot(A, x), x) / np.dot(x, x)\n",
|
||||
"\n",
|
||||
"# Wyświetlenie wyników\n",
|
||||
"print(\"Dominująca wartość własna:\")\n",
|
||||
"print(eigenvalue)\n",
|
||||
"print(\"Dominujący wektor własny:\")\n",
|
||||
"print(x)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
|
||||
Loading…
Reference in New Issue
Block a user