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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "a0564571-faf2-47f7-80b8-6b93da370de6",
"metadata": {},
"outputs": [],
"source": [
"# 12.3.1c"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "97f1a9eb-89f3-493e-a564-9ceed016b7ba",
"metadata": {},
"outputs": [],
"source": [
"N=6\n",
"alpha = vector(QQbar,[0 for _ in range(N)])\n",
"a = vector(QQbar,[0 for _ in range(N)])\n",
"t = vector(QQbar,[0 for _ in range(N)])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "43236aea-32a2-4b2e-ad87-674ea0443341",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 13,
"id": "4c58f2d2-829b-4e8e-8b41-e18241a04c39",
"metadata": {},
"outputs": [],
"source": [
"alpha[0] = sqrt(5)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "d239123d-8f1d-4c2a-8723-e488035f4882",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0 2.236067977499790? 2 0.2360679774997897?\n",
"1 4.236067977499789? 4 0.2360679774997897?\n",
"2 4.236067977499789? 4 0.2360679774997897?\n",
"3 4.23606797749979? 4 0.23606797749979?\n",
"4 4.2360679774998? 4 0.2360679774998?\n"
]
}
],
"source": [
"for k in range(N-1):\n",
" a[k], t[k] = floor(alpha[k]), alpha[k] - floor(alpha[k])\n",
" alpha[k+1] = 1 / t[k]\n",
" print(k,alpha[k], a[k], t[k])"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "91e64dfc-7d82-41da-97a3-907b70fb04cf",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[2; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(alpha[0])"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "4805e461-de3a-4ad1-b68b-0ac1254a9fa2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[4; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(alpha[1])"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "74e5c6b5-4141-48a2-9ed0-5145ec962206",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[0; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(t[1])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ce40fb53-525d-4778-ab65-d3b9de12fd0a",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 28,
"id": "5f179ca2-6fe2-4acf-8ee5-efc16c9f92d7",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0 3.358898943540674? 3 0.3588989435406736?\n",
"1 2.786299647846891? 2 0.7862996478468912?\n",
"2 1.271779788708135? 1 0.2717797887081347?\n",
"3 3.679449471770337? 3 0.679449471770337?\n",
"4 1.471779788708135? 1 0.471779788708135?\n"
]
}
],
"source": [
"alpha[0] = sqrt(19)-1\n",
"for k in range(N-1):\n",
" a[k], t[k] = floor(alpha[k]), alpha[k] - floor(alpha[k])\n",
" alpha[k+1] = 1 / t[k]\n",
" print(k,alpha[k], a[k], t[k])"
]
},
{
"cell_type": "code",
"execution_count": 29,
"id": "befb67c1-8720-4d6f-88da-1920b91edfab",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(alpha[0])"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "73ca37dd-732c-4614-87e5-f546b69ff166",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[0; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(t[0])"
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "3e32ecf1-bb17-4e8c-88cc-ab86cb26c262",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[2; 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, ...]"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(1/t[0])"
]
},
{
"cell_type": "code",
"execution_count": 34,
"id": "ad4f7e53-2f82-4592-b1b1-ce52cc96373f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(a[0] + t[0])"
]
},
{
"cell_type": "code",
"execution_count": 35,
"id": "2a3daa68-23ea-4292-b642-b9b293468494",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(a[0] + 1/(a[1] +t[1]))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e33f1445-ebf7-4ad9-8441-093613207188",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "1c92946d-cb2e-458f-967e-c10cc12be1aa",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "e5b2db5f-d125-43eb-837f-27896ceabd02",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"id": "44991d26-ebed-44ad-9883-e49b90842e8e",
"metadata": {},
"outputs": [],
"source": [
"#12.3.2.a"
]
},
{
"cell_type": "code",
"id": "a05c12c9-9df9-464d-b932-62a81a1d4429",
"metadata": {},
"outputs": [],
"source": [
"R.<x> = QQbar[]\n",
"f0 = x^3-2\n",
"d=3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7ffc1d57-0784-4119-825f-32f5459cc945",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 5,
"id": "1f050106-4454-441f-ab17-869bb1e7deb9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1.259921049894873?, 1),\n",
" (-0.6299605249474365? - 1.091123635971722?*I, 1),\n",
" (-0.6299605249474365? + 1.091123635971722?*I, 1)]"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f0.roots()"
]
},
{
"cell_type": "code",
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"execution_count": 7,
"id": "3327b8b5-8e8e-4f34-97d0-8b24e06410e2",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"1.259921049894873?"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alfa0=f0.roots()[0][0]\n",
"alfa0"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "4ddb6330-657c-4387-b902-6e055b0097b0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, ...]"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(alfa0)"
]
},
{
"cell_type": "raw",
"id": "d800b558-13db-47be-be2c-a1a878594129",
"metadata": {},
"source": []
},
{
"cell_type": "code",
"execution_count": 9,
"id": "78e5e15c-0cc0-4dff-a8a2-5693f4b4a339",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x^3 - 3*x^2 - 3*x - 1"
]
},
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"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a0 = 1\n",
"f1 = (-x)^d*f0(a0 + 1/x)\n",
"f1 =R(f1)\n",
"f1"
]
},
{
"cell_type": "code",
"execution_count": 47,
"id": "5d9d0f98-5ecc-4e6a-b074-813c329fdbf1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(3.847322101863073?, 1),\n",
" (-0.4236610509315363? - 0.2836060010268813?*I, 1),\n",
" (-0.4236610509315363? + 0.2836060010268813?*I, 1)]"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f1.roots()"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "c54abfa7-4930-4a3f-9c2c-3bf05d5b32a0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[3; 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, ...]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alfa1=f1.roots()[0][0]\n",
"continued_fraction(alfa1)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "5801d40e-a2a8-46e9-bdd9-39b33d52f83f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10*x^3 - 6*x^2 - 6*x - 1"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a1 = 3\n",
"f2 = (-x)^d*f1(a1 + 1/x)\n",
"f2 =R(f2)\n",
"f2"
]
},
{
"cell_type": "code",
"id": "c4c0bc9d-9bbf-4d8d-b063-d1dc78245ff6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1.180188735548410?, 1),\n",
" (-0.2900943677742046? - 0.02403056328910778?*I, 1),\n",
" (-0.2900943677742046? + 0.02403056328910778?*I, 1)]"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f2.roots()"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "c049c466-ebf9-4853-9ebf-2be2a169b069",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1; 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, ...]"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alfa2=f2.roots()[0][0]\n",
"continued_fraction(alfa2)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "9953eae5-e5d1-48f6-bda3-e464b1a6c6c1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3*x^3 - 12*x^2 - 24*x - 10"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a2 = 1\n",
"f3 = (-x)^d*f2(a2 + 1/x)\n",
"f3 =R(f3)\n",
"f3"
]
},
{
"cell_type": "code",
"id": "0a66ff0f-dc17-42ae-b8ee-6535c8541aac",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(5.549736485782388?, 1),\n",
" (-0.7748682428911940? - 0.01443345604526783?*I, 1),\n",
" (-0.7748682428911940? + 0.01443345604526783?*I, 1)]"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f3.roots()"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "b4fc68d2-c006-43ce-945d-f809360adb11",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[5; 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...]"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alfa3=f3.roots()[0][0]\n",
"continued_fraction(alfa3)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "0c50aa3c-5f4c-4d34-bc2f-a5de911952b9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"55*x^3 - 81*x^2 - 33*x - 3"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a3 = 5\n",
"f4 = (-x)^d*f3(a3 + 1/x)\n",
"f4 =R(f4)\n",
"f4"
]
},
{
"cell_type": "code",
"id": "d99ecd1d-0a85-4ebb-a69b-546456037c87",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1.819053357131272?, 1),\n",
" (-0.1731630422019992? - 0.00043279622203746?*I, 1),\n",
" (-0.1731630422019992? + 0.00043279622203746?*I, 1)]"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f4.roots()"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "c72bd901-5e62-4abf-ab21-758fc8a6d91d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1; 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, ...]"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alfa4=f4.roots()[0][0]\n",
"continued_fraction(alfa4)"
]
},
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{
"cell_type": "code",
"execution_count": 54,
"id": "468aa920-5d96-4aba-af0d-906c8823f6c4",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, ...]"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"continued_fraction(2^(1/3))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "73ee5ba0-8a57-4afc-a637-3ef24d89ad58",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "c9930f4b-758a-4edd-b462-827914d8f867",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 55,
"id": "d742e1d1-7935-46ee-bd91-3037e9791149",
"metadata": {},
"outputs": [],
"source": [
"#12.3.4"
]
},
{
"cell_type": "code",
"execution_count": 67,
"id": "cd9a0eb9-823a-46cf-a3ce-ce344d55ceb4",
"metadata": {},
"outputs": [],
"source": [
"a=vector([2,1,2,1,1,4])\n",
"N=len(a)\n",
"v=[]\n",
"v.append(vector([1,0]))\n",
"v.append(vector([0,1]))"
]
},
{
"cell_type": "code",
"execution_count": 68,
"id": "22b9d3bd-d905-4ecc-a841-0469335e7707",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1, 0), (0, 1)]"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v"
]
},
{
"cell_type": "code",
"execution_count": 72,
"id": "a89b77b0-93bf-4c2a-82db-f1f1a7d99938",
"metadata": {},
"outputs": [],
"source": [
"for k in range(N):\n",
" v.append(a[k]*v[-1] + v[-2])"
]
},
{
"cell_type": "code",
"execution_count": 73,
"id": "5f5c0c05-47f0-4538-8046-1a75fe629b55",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1, 0), (0, 1), (1, 2), (1, 3), (3, 8), (4, 11), (7, 19), (32, 87)]"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v"
]
},
{
"cell_type": "code",
"execution_count": 77,
"id": "b8feab8b-f3ed-477c-8ee2-9f88a8728ea2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[2, 3, 8/3, 11/4, 19/7, 87/32]"
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"conve =[c[1]/c[0] for c in v[2:]]\n",
"conve"
]
},
{
"cell_type": "code",
"execution_count": 79,
"id": "a560f987-04f8-40a1-8b1e-366949e8464e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/224"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"conve[5] - conve[4]"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "dcbebe47-9e94-4540-8f4d-99f1217ca1b0",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 10.2.beta9",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.1"
}
},
"nbformat": 4,
"nbformat_minor": 5
}