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Jan Snellman
TATA54-kurshemsida
Commits
5313d4b0
Commit
5313d4b0
authored
11 months ago
by
Jan Snellman
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rosen notebook
parent
8b0daf03
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#126365
passed
11 months ago
Stage: deploy
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.ipynb_checkpoints/Rosen12-3-checkpoint.ipynb
+165
-18
165 additions, 18 deletions
.ipynb_checkpoints/Rosen12-3-checkpoint.ipynb
Rosen12-3.ipynb
+165
-18
165 additions, 18 deletions
Rosen12-3.ipynb
with
330 additions
and
36 deletions
.ipynb_checkpoints/Rosen12-3-checkpoint.ipynb
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165
−
18
View file @
5313d4b0
...
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@@ -294,7 +294,7 @@
},
{
"cell_type": "code",
"execution_count":
36
,
"execution_count":
1
,
"id": "44991d26-ebed-44ad-9883-e49b90842e8e",
"metadata": {},
"outputs": [],
...
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@@ -304,7 +304,7 @@
},
{
"cell_type": "code",
"execution_count":
41
,
"execution_count":
2
,
"id": "a05c12c9-9df9-464d-b932-62a81a1d4429",
"metadata": {},
"outputs": [],
...
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@@ -316,7 +316,15 @@
},
{
"cell_type": "code",
"execution_count": 42,
"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": [
...
...
@@ -328,7 +336,7 @@
" (-0.6299605249474365? + 1.091123635971722?*I, 1)]"
]
},
"execution_count":
42
,
"execution_count":
5
,
"metadata": {},
"output_type": "execute_result"
}
...
...
@@ -339,7 +347,58 @@
},
{
"cell_type": "code",
"execution_count": 46,
"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": [
...
...
@@ -349,7 +408,7 @@
"x^3 - 3*x^2 - 3*x - 1"
]
},
"execution_count":
46
,
"execution_count":
9
,
"metadata": {},
"output_type": "execute_result"
}
...
...
@@ -386,7 +445,29 @@
},
{
"cell_type": "code",
"execution_count": 48,
"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": [
...
...
@@ -396,7 +477,7 @@
"10*x^3 - 6*x^2 - 6*x - 1"
]
},
"execution_count":
48
,
"execution_count":
12
,
"metadata": {},
"output_type": "execute_result"
}
...
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@@ -410,7 +491,7 @@
},
{
"cell_type": "code",
"execution_count":
49
,
"execution_count":
13
,
"id": "c4c0bc9d-9bbf-4d8d-b063-d1dc78245ff6",
"metadata": {},
"outputs": [
...
...
@@ -422,7 +503,7 @@
" (-0.2900943677742046? + 0.02403056328910778?*I, 1)]"
]
},
"execution_count":
49
,
"execution_count":
13
,
"metadata": {},
"output_type": "execute_result"
}
...
...
@@ -433,7 +514,29 @@
},
{
"cell_type": "code",
"execution_count": 50,
"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": [
...
...
@@ -443,7 +546,7 @@
"3*x^3 - 12*x^2 - 24*x - 10"
]
},
"execution_count":
50
,
"execution_count":
16
,
"metadata": {},
"output_type": "execute_result"
}
...
...
@@ -457,7 +560,7 @@
},
{
"cell_type": "code",
"execution_count":
5
1,
"execution_count": 1
7
,
"id": "0a66ff0f-dc17-42ae-b8ee-6535c8541aac",
"metadata": {},
"outputs": [
...
...
@@ -469,7 +572,7 @@
" (-0.7748682428911940? + 0.01443345604526783?*I, 1)]"
]
},
"execution_count":
5
1,
"execution_count": 1
7
,
"metadata": {},
"output_type": "execute_result"
}
...
...
@@ -480,7 +583,29 @@
},
{
"cell_type": "code",
"execution_count": 52,
"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": [
...
...
@@ -490,7 +615,7 @@
"55*x^3 - 81*x^2 - 33*x - 3"
]
},
"execution_count":
5
2,
"execution_count": 2
0
,
"metadata": {},
"output_type": "execute_result"
}
...
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@@ -504,7 +629,7 @@
},
{
"cell_type": "code",
"execution_count":
53
,
"execution_count":
21
,
"id": "d99ecd1d-0a85-4ebb-a69b-546456037c87",
"metadata": {},
"outputs": [
...
...
@@ -516,7 +641,7 @@
" (-0.1731630422019992? + 0.00043279622203746?*I, 1)]"
]
},
"execution_count":
53
,
"execution_count":
21
,
"metadata": {},
"output_type": "execute_result"
}
...
...
@@ -525,6 +650,28 @@
"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)"
]
},
{
"cell_type": "code",
"execution_count": 54,
...
...
%% Cell type:code id:a0564571-faf2-47f7-80b8-6b93da370de6 tags:
```
sage
# 12.3.1c
```
%% Cell type:code id:97f1a9eb-89f3-493e-a564-9ceed016b7ba tags:
```
sage
N=6
alpha = vector(QQbar,[0 for _ in range(N)])
a = vector(QQbar,[0 for _ in range(N)])
t = vector(QQbar,[0 for _ in range(N)])
```
%% Cell type:code id:43236aea-32a2-4b2e-ad87-674ea0443341 tags:
```
sage
```
%% Cell type:code id:4c58f2d2-829b-4e8e-8b41-e18241a04c39 tags:
```
sage
alpha[0] = sqrt(5)
```
%% Cell type:code id:d239123d-8f1d-4c2a-8723-e488035f4882 tags:
```
sage
for k in range(N-1):
a[k], t[k] = floor(alpha[k]), alpha[k] - floor(alpha[k])
alpha[k+1] = 1 / t[k]
print(k,alpha[k], a[k], t[k])
```
%% Output
0 2.236067977499790? 2 0.2360679774997897?
1 4.236067977499789? 4 0.2360679774997897?
2 4.236067977499789? 4 0.2360679774997897?
3 4.23606797749979? 4 0.23606797749979?
4 4.2360679774998? 4 0.2360679774998?
%% Cell type:code id:91e64dfc-7d82-41da-97a3-907b70fb04cf tags:
```
sage
continued_fraction(alpha[0])
```
%% Output
[2; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]
%% Cell type:code id:4805e461-de3a-4ad1-b68b-0ac1254a9fa2 tags:
```
sage
continued_fraction(alpha[1])
```
%% Output
[4; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]
%% Cell type:code id:74e5c6b5-4141-48a2-9ed0-5145ec962206 tags:
```
sage
continued_fraction(t[1])
```
%% Output
[0; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]
%% Cell type:code id:ce40fb53-525d-4778-ab65-d3b9de12fd0a tags:
```
sage
```
%% Cell type:code id:5f179ca2-6fe2-4acf-8ee5-efc16c9f92d7 tags:
```
sage
alpha[0] = sqrt(19)-1
for k in range(N-1):
a[k], t[k] = floor(alpha[k]), alpha[k] - floor(alpha[k])
alpha[k+1] = 1 / t[k]
print(k,alpha[k], a[k], t[k])
```
%% Output
0 3.358898943540674? 3 0.3588989435406736?
1 2.786299647846891? 2 0.7862996478468912?
2 1.271779788708135? 1 0.2717797887081347?
3 3.679449471770337? 3 0.679449471770337?
4 1.471779788708135? 1 0.471779788708135?
%% Cell type:code id:befb67c1-8720-4d6f-88da-1920b91edfab tags:
```
sage
continued_fraction(alpha[0])
```
%% Output
[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:73ca37dd-732c-4614-87e5-f546b69ff166 tags:
```
sage
continued_fraction(t[0])
```
%% Output
[0; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:3e32ecf1-bb17-4e8c-88cc-ab86cb26c262 tags:
```
sage
continued_fraction(1/t[0])
```
%% Output
[2; 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, ...]
%% Cell type:code id:ad4f7e53-2f82-4592-b1b1-ce52cc96373f tags:
```
sage
continued_fraction(a[0] + t[0])
```
%% Output
[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:2a3daa68-23ea-4292-b642-b9b293468494 tags:
```
sage
continued_fraction(a[0] + 1/(a[1] +t[1]))
```
%% Output
[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:e33f1445-ebf7-4ad9-8441-093613207188 tags:
```
sage
```
%% Cell type:code id:1c92946d-cb2e-458f-967e-c10cc12be1aa tags:
```
sage
```
%% Cell type:code id:e5b2db5f-d125-43eb-837f-27896ceabd02 tags:
```
sage
```
%% Cell type:code id:44991d26-ebed-44ad-9883-e49b90842e8e tags:
```
sage
#12.3.2.a
```
%% Cell type:code id:a05c12c9-9df9-464d-b932-62a81a1d4429 tags:
```
sage
R.
<x>
= QQbar[]
f0 = x^3-2
d=3
```
%% Cell type:code id:7ffc1d57-0784-4119-825f-32f5459cc945 tags:
```
sage
```
%% Cell type:code id:1f050106-4454-441f-ab17-869bb1e7deb9 tags:
```
sage
f0.roots()
```
%% Output
[(1.259921049894873?, 1),
(-0.6299605249474365? - 1.091123635971722?*I, 1),
(-0.6299605249474365? + 1.091123635971722?*I, 1)]
%% Cell type:code id:3327b8b5-8e8e-4f34-97d0-8b24e06410e2 tags:
```
sage
alfa0=f0.roots()
[
0
][
0
]
alfa0
```
%% Output
1.259921049894873?
%% Cell type:code id:4ddb6330-657c-4387-b902-6e055b0097b0 tags:
```
sage
continued_fraction(alfa0)
```
%% Output
[1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, ...]
%% Cell type:raw id:d800b558-13db-47be-be2c-a1a878594129 tags:
%% Cell type:code id:78e5e15c-0cc0-4dff-a8a2-5693f4b4a339 tags:
```
sage
a0 = 1
f1 = (-x)^d
*
f0(a0 + 1/x)
f1 =R(f1)
f1
```
%% Output
x^3 - 3*x^2 - 3*x - 1
%% Cell type:code id:5d9d0f98-5ecc-4e6a-b074-813c329fdbf1 tags:
```
sage
f1.roots()
```
%% Output
[(3.847322101863073?, 1),
(-0.4236610509315363? - 0.2836060010268813?*I, 1),
(-0.4236610509315363? + 0.2836060010268813?*I, 1)]
%% Cell type:code id:c54abfa7-4930-4a3f-9c2c-3bf05d5b32a0 tags:
```
sage
alfa1=f1.roots()
[
0
][
0
]
continued_fraction(alfa1)
```
%% Output
[3; 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, ...]
%% Cell type:code id:5801d40e-a2a8-46e9-bdd9-39b33d52f83f tags:
```
sage
a1 = 3
f2 = (-x)^d
*
f1(a1 + 1/x)
f2 =R(f2)
f2
```
%% Output
10*x^3 - 6*x^2 - 6*x - 1
%% Cell type:code id:c4c0bc9d-9bbf-4d8d-b063-d1dc78245ff6 tags:
```
sage
f2.roots()
```
%% Output
[(1.180188735548410?, 1),
(-0.2900943677742046? - 0.02403056328910778?*I, 1),
(-0.2900943677742046? + 0.02403056328910778?*I, 1)]
%% Cell type:code id:c049c466-ebf9-4853-9ebf-2be2a169b069 tags:
```
sage
alfa2=f2.roots()
[
0
][
0
]
continued_fraction(alfa2)
```
%% Output
[1; 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, ...]
%% Cell type:code id:9953eae5-e5d1-48f6-bda3-e464b1a6c6c1 tags:
```
sage
a2 = 1
f3 = (-x)^d
*
f2(a2 + 1/x)
f3 =R(f3)
f3
```
%% Output
3*x^3 - 12*x^2 - 24*x - 10
%% Cell type:code id:0a66ff0f-dc17-42ae-b8ee-6535c8541aac tags:
```
sage
f3.roots()
```
%% Output
[(5.549736485782388?, 1),
(-0.7748682428911940? - 0.01443345604526783?*I, 1),
(-0.7748682428911940? + 0.01443345604526783?*I, 1)]
%% Cell type:code id:b4fc68d2-c006-43ce-945d-f809360adb11 tags:
```
sage
alfa3=f3.roots()
[
0
][
0
]
continued_fraction(alfa3)
```
%% Output
[5; 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...]
%% Cell type:code id:0c50aa3c-5f4c-4d34-bc2f-a5de911952b9 tags:
```
sage
a3 = 5
f4 = (-x)^d
*
f3(a3 + 1/x)
f4 =R(f4)
f4
```
%% Output
55*x^3 - 81*x^2 - 33*x - 3
%% Cell type:code id:d99ecd1d-0a85-4ebb-a69b-546456037c87 tags:
```
sage
f4.roots()
```
%% Output
[(1.819053357131272?, 1),
(-0.1731630422019992? - 0.00043279622203746?*I, 1),
(-0.1731630422019992? + 0.00043279622203746?*I, 1)]
%% Cell type:code id:c72bd901-5e62-4abf-ab21-758fc8a6d91d tags:
```
sage
alfa4=f4.roots()
[
0
][
0
]
continued_fraction(alfa4)
```
%% Output
[1; 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, ...]
%% Cell type:code id:468aa920-5d96-4aba-af0d-906c8823f6c4 tags:
```
sage
continued_fraction(2^(1/3))
```
%% Output
[1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, ...]
%% Cell type:code id:73ee5ba0-8a57-4afc-a637-3ef24d89ad58 tags:
```
sage
```
%% Cell type:code id:c9930f4b-758a-4edd-b462-827914d8f867 tags:
```
sage
```
%% Cell type:code id:d742e1d1-7935-46ee-bd91-3037e9791149 tags:
```
sage
#12.3.4
```
%% Cell type:code id:cd9a0eb9-823a-46cf-a3ce-ce344d55ceb4 tags:
```
sage
a=vector([2,1,2,1,1,4])
N=len(a)
v=[]
v.append(vector([1,0]))
v.append(vector([0,1]))
```
%% Cell type:code id:22b9d3bd-d905-4ecc-a841-0469335e7707 tags:
```
sage
v
```
%% Output
[(1, 0), (0, 1)]
%% Cell type:code id:a89b77b0-93bf-4c2a-82db-f1f1a7d99938 tags:
```
sage
for k in range(N):
v.append(a[k]
*
v[-1] + v[-2])
```
%% Cell type:code id:5f5c0c05-47f0-4538-8046-1a75fe629b55 tags:
```
sage
v
```
%% Output
[(1, 0), (0, 1), (1, 2), (1, 3), (3, 8), (4, 11), (7, 19), (32, 87)]
%% Cell type:code id:b8feab8b-f3ed-477c-8ee2-9f88a8728ea2 tags:
```
sage
conve =[c[1]/c[0] for c in v[2:]]
conve
```
%% Output
[2, 3, 8/3, 11/4, 19/7, 87/32]
%% Cell type:code id:a560f987-04f8-40a1-8b1e-366949e8464e tags:
```
sage
conve[5] - conve[4]
```
%% Output
1/224
%% Cell type:code id:dcbebe47-9e94-4540-8f4d-99f1217ca1b0 tags:
```
sage
```
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},
{
"cell_type": "code",
"execution_count":
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"execution_count":
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"id": "44991d26-ebed-44ad-9883-e49b90842e8e",
"metadata": {},
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},
{
"cell_type": "code",
"execution_count":
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"metadata": {},
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},
{
"cell_type": "code",
"execution_count": 42,
"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": [
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" (-0.6299605249474365? + 1.091123635971722?*I, 1)]"
]
},
"execution_count":
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,
"execution_count":
5
,
"metadata": {},
"output_type": "execute_result"
}
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},
{
"cell_type": "code",
"execution_count": 46,
"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": [
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"x^3 - 3*x^2 - 3*x - 1"
]
},
"execution_count":
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"execution_count":
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"metadata": {},
"output_type": "execute_result"
}
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},
{
"cell_type": "code",
"execution_count": 48,
"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": [
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"10*x^3 - 6*x^2 - 6*x - 1"
]
},
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"metadata": {},
"output_type": "execute_result"
}
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},
{
"cell_type": "code",
"execution_count":
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"execution_count":
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"id": "c4c0bc9d-9bbf-4d8d-b063-d1dc78245ff6",
"metadata": {},
"outputs": [
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" (-0.2900943677742046? + 0.02403056328910778?*I, 1)]"
]
},
"execution_count":
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"execution_count":
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"metadata": {},
"output_type": "execute_result"
}
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},
{
"cell_type": "code",
"execution_count": 50,
"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": [
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"3*x^3 - 12*x^2 - 24*x - 10"
]
},
"execution_count":
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"execution_count":
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"metadata": {},
"output_type": "execute_result"
}
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},
{
"cell_type": "code",
"execution_count":
5
1,
"execution_count": 1
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"id": "0a66ff0f-dc17-42ae-b8ee-6535c8541aac",
"metadata": {},
"outputs": [
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" (-0.7748682428911940? + 0.01443345604526783?*I, 1)]"
]
},
"execution_count":
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1,
"execution_count": 1
7
,
"metadata": {},
"output_type": "execute_result"
}
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},
{
"cell_type": "code",
"execution_count": 52,
"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": [
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"55*x^3 - 81*x^2 - 33*x - 3"
]
},
"execution_count":
5
2,
"execution_count": 2
0
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"metadata": {},
"output_type": "execute_result"
}
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},
{
"cell_type": "code",
"execution_count":
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"execution_count":
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"id": "d99ecd1d-0a85-4ebb-a69b-546456037c87",
"metadata": {},
"outputs": [
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" (-0.1731630422019992? + 0.00043279622203746?*I, 1)]"
]
},
"execution_count":
53
,
"execution_count":
21
,
"metadata": {},
"output_type": "execute_result"
}
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"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)"
]
},
{
"cell_type": "code",
"execution_count": 54,
...
...
%% Cell type:code id:a0564571-faf2-47f7-80b8-6b93da370de6 tags:
```
sage
# 12.3.1c
```
%% Cell type:code id:97f1a9eb-89f3-493e-a564-9ceed016b7ba tags:
```
sage
N=6
alpha = vector(QQbar,[0 for _ in range(N)])
a = vector(QQbar,[0 for _ in range(N)])
t = vector(QQbar,[0 for _ in range(N)])
```
%% Cell type:code id:43236aea-32a2-4b2e-ad87-674ea0443341 tags:
```
sage
```
%% Cell type:code id:4c58f2d2-829b-4e8e-8b41-e18241a04c39 tags:
```
sage
alpha[0] = sqrt(5)
```
%% Cell type:code id:d239123d-8f1d-4c2a-8723-e488035f4882 tags:
```
sage
for k in range(N-1):
a[k], t[k] = floor(alpha[k]), alpha[k] - floor(alpha[k])
alpha[k+1] = 1 / t[k]
print(k,alpha[k], a[k], t[k])
```
%% Output
0 2.236067977499790? 2 0.2360679774997897?
1 4.236067977499789? 4 0.2360679774997897?
2 4.236067977499789? 4 0.2360679774997897?
3 4.23606797749979? 4 0.23606797749979?
4 4.2360679774998? 4 0.2360679774998?
%% Cell type:code id:91e64dfc-7d82-41da-97a3-907b70fb04cf tags:
```
sage
continued_fraction(alpha[0])
```
%% Output
[2; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]
%% Cell type:code id:4805e461-de3a-4ad1-b68b-0ac1254a9fa2 tags:
```
sage
continued_fraction(alpha[1])
```
%% Output
[4; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]
%% Cell type:code id:74e5c6b5-4141-48a2-9ed0-5145ec962206 tags:
```
sage
continued_fraction(t[1])
```
%% Output
[0; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...]
%% Cell type:code id:ce40fb53-525d-4778-ab65-d3b9de12fd0a tags:
```
sage
```
%% Cell type:code id:5f179ca2-6fe2-4acf-8ee5-efc16c9f92d7 tags:
```
sage
alpha[0] = sqrt(19)-1
for k in range(N-1):
a[k], t[k] = floor(alpha[k]), alpha[k] - floor(alpha[k])
alpha[k+1] = 1 / t[k]
print(k,alpha[k], a[k], t[k])
```
%% Output
0 3.358898943540674? 3 0.3588989435406736?
1 2.786299647846891? 2 0.7862996478468912?
2 1.271779788708135? 1 0.2717797887081347?
3 3.679449471770337? 3 0.679449471770337?
4 1.471779788708135? 1 0.471779788708135?
%% Cell type:code id:befb67c1-8720-4d6f-88da-1920b91edfab tags:
```
sage
continued_fraction(alpha[0])
```
%% Output
[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:73ca37dd-732c-4614-87e5-f546b69ff166 tags:
```
sage
continued_fraction(t[0])
```
%% Output
[0; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:3e32ecf1-bb17-4e8c-88cc-ab86cb26c262 tags:
```
sage
continued_fraction(1/t[0])
```
%% Output
[2; 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, ...]
%% Cell type:code id:ad4f7e53-2f82-4592-b1b1-ce52cc96373f tags:
```
sage
continued_fraction(a[0] + t[0])
```
%% Output
[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:2a3daa68-23ea-4292-b642-b9b293468494 tags:
```
sage
continued_fraction(a[0] + 1/(a[1] +t[1]))
```
%% Output
[3; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, ...]
%% Cell type:code id:e33f1445-ebf7-4ad9-8441-093613207188 tags:
```
sage
```
%% Cell type:code id:1c92946d-cb2e-458f-967e-c10cc12be1aa tags:
```
sage
```
%% Cell type:code id:e5b2db5f-d125-43eb-837f-27896ceabd02 tags:
```
sage
```
%% Cell type:code id:44991d26-ebed-44ad-9883-e49b90842e8e tags:
```
sage
#12.3.2.a
```
%% Cell type:code id:a05c12c9-9df9-464d-b932-62a81a1d4429 tags:
```
sage
R.
<x>
= QQbar[]
f0 = x^3-2
d=3
```
%% Cell type:code id:7ffc1d57-0784-4119-825f-32f5459cc945 tags:
```
sage
```
%% Cell type:code id:1f050106-4454-441f-ab17-869bb1e7deb9 tags:
```
sage
f0.roots()
```
%% Output
[(1.259921049894873?, 1),
(-0.6299605249474365? - 1.091123635971722?*I, 1),
(-0.6299605249474365? + 1.091123635971722?*I, 1)]
%% Cell type:code id:3327b8b5-8e8e-4f34-97d0-8b24e06410e2 tags:
```
sage
alfa0=f0.roots()
[
0
][
0
]
alfa0
```
%% Output
1.259921049894873?
%% Cell type:code id:4ddb6330-657c-4387-b902-6e055b0097b0 tags:
```
sage
continued_fraction(alfa0)
```
%% Output
[1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, ...]
%% Cell type:raw id:d800b558-13db-47be-be2c-a1a878594129 tags:
%% Cell type:code id:78e5e15c-0cc0-4dff-a8a2-5693f4b4a339 tags:
```
sage
a0 = 1
f1 = (-x)^d
*
f0(a0 + 1/x)
f1 =R(f1)
f1
```
%% Output
x^3 - 3*x^2 - 3*x - 1
%% Cell type:code id:5d9d0f98-5ecc-4e6a-b074-813c329fdbf1 tags:
```
sage
f1.roots()
```
%% Output
[(3.847322101863073?, 1),
(-0.4236610509315363? - 0.2836060010268813?*I, 1),
(-0.4236610509315363? + 0.2836060010268813?*I, 1)]
%% Cell type:code id:c54abfa7-4930-4a3f-9c2c-3bf05d5b32a0 tags:
```
sage
alfa1=f1.roots()
[
0
][
0
]
continued_fraction(alfa1)
```
%% Output
[3; 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, ...]
%% Cell type:code id:5801d40e-a2a8-46e9-bdd9-39b33d52f83f tags:
```
sage
a1 = 3
f2 = (-x)^d
*
f1(a1 + 1/x)
f2 =R(f2)
f2
```
%% Output
10*x^3 - 6*x^2 - 6*x - 1
%% Cell type:code id:c4c0bc9d-9bbf-4d8d-b063-d1dc78245ff6 tags:
```
sage
f2.roots()
```
%% Output
[(1.180188735548410?, 1),
(-0.2900943677742046? - 0.02403056328910778?*I, 1),
(-0.2900943677742046? + 0.02403056328910778?*I, 1)]
%% Cell type:code id:c049c466-ebf9-4853-9ebf-2be2a169b069 tags:
```
sage
alfa2=f2.roots()
[
0
][
0
]
continued_fraction(alfa2)
```
%% Output
[1; 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, ...]
%% Cell type:code id:9953eae5-e5d1-48f6-bda3-e464b1a6c6c1 tags:
```
sage
a2 = 1
f3 = (-x)^d
*
f2(a2 + 1/x)
f3 =R(f3)
f3
```
%% Output
3*x^3 - 12*x^2 - 24*x - 10
%% Cell type:code id:0a66ff0f-dc17-42ae-b8ee-6535c8541aac tags:
```
sage
f3.roots()
```
%% Output
[(5.549736485782388?, 1),
(-0.7748682428911940? - 0.01443345604526783?*I, 1),
(-0.7748682428911940? + 0.01443345604526783?*I, 1)]
%% Cell type:code id:b4fc68d2-c006-43ce-945d-f809360adb11 tags:
```
sage
alfa3=f3.roots()
[
0
][
0
]
continued_fraction(alfa3)
```
%% Output
[5; 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...]
%% Cell type:code id:0c50aa3c-5f4c-4d34-bc2f-a5de911952b9 tags:
```
sage
a3 = 5
f4 = (-x)^d
*
f3(a3 + 1/x)
f4 =R(f4)
f4
```
%% Output
55*x^3 - 81*x^2 - 33*x - 3
%% Cell type:code id:d99ecd1d-0a85-4ebb-a69b-546456037c87 tags:
```
sage
f4.roots()
```
%% Output
[(1.819053357131272?, 1),
(-0.1731630422019992? - 0.00043279622203746?*I, 1),
(-0.1731630422019992? + 0.00043279622203746?*I, 1)]
%% Cell type:code id:c72bd901-5e62-4abf-ab21-758fc8a6d91d tags:
```
sage
alfa4=f4.roots()
[
0
][
0
]
continued_fraction(alfa4)
```
%% Output
[1; 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, ...]
%% Cell type:code id:468aa920-5d96-4aba-af0d-906c8823f6c4 tags:
```
sage
continued_fraction(2^(1/3))
```
%% Output
[1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, ...]
%% Cell type:code id:73ee5ba0-8a57-4afc-a637-3ef24d89ad58 tags:
```
sage
```
%% Cell type:code id:c9930f4b-758a-4edd-b462-827914d8f867 tags:
```
sage
```
%% Cell type:code id:d742e1d1-7935-46ee-bd91-3037e9791149 tags:
```
sage
#12.3.4
```
%% Cell type:code id:cd9a0eb9-823a-46cf-a3ce-ce344d55ceb4 tags:
```
sage
a=vector([2,1,2,1,1,4])
N=len(a)
v=[]
v.append(vector([1,0]))
v.append(vector([0,1]))
```
%% Cell type:code id:22b9d3bd-d905-4ecc-a841-0469335e7707 tags:
```
sage
v
```
%% Output
[(1, 0), (0, 1)]
%% Cell type:code id:a89b77b0-93bf-4c2a-82db-f1f1a7d99938 tags:
```
sage
for k in range(N):
v.append(a[k]
*
v[-1] + v[-2])
```
%% Cell type:code id:5f5c0c05-47f0-4538-8046-1a75fe629b55 tags:
```
sage
v
```
%% Output
[(1, 0), (0, 1), (1, 2), (1, 3), (3, 8), (4, 11), (7, 19), (32, 87)]
%% Cell type:code id:b8feab8b-f3ed-477c-8ee2-9f88a8728ea2 tags:
```
sage
conve =[c[1]/c[0] for c in v[2:]]
conve
```
%% Output
[2, 3, 8/3, 11/4, 19/7, 87/32]
%% Cell type:code id:a560f987-04f8-40a1-8b1e-366949e8464e tags:
```
sage
conve[5] - conve[4]
```
%% Output
1/224
%% Cell type:code id:dcbebe47-9e94-4540-8f4d-99f1217ca1b0 tags:
```
sage
```
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