Pivot de gauss avec numpy#
Etape par étape, le pivote de Gauss implémenté en python puis avec numpy.
[7]:
%matplotlib inline
Python#
[8]:
import numpy
[9]:
def pivot_gauss(m):
n = m.copy()
for i in range(1, m.shape[0]):
j0 = i
while j0 < m.shape[0] and m[j0, i - 1] == 0:
j0 += 1
for j in range(j0, m.shape[0]):
coef = -m[j, i - 1] / m[i - 1, i - 1]
for k in range(i - 1, m.shape[1]):
m[j, k] += coef * m[i - 1, k]
return m
m = numpy.random.rand(4, 4)
piv = pivot_gauss(m)
piv * (numpy.abs(piv) > 1e-10)
[9]:
array([[ 0.58650968, 0.72315516, 0.9137217 , 0.04526746],
[ 0. , 0.70012278, 0.87604076, 0.92402115],
[ 0. , 0. , -0.36865515, 1.00362559],
[ 0. , 0. , 0. , 0.26361505]])
* (numpy.abs(piv) > 1e-10)
sert à simplifier l’affichage des valeurs quasi nulles.
Numpy 1#
[10]:
def pivot_gauss2(m):
n = m.copy()
for i in range(1, m.shape[0]):
j0 = i
while j0 < m.shape[0] and m[j0, i - 1] == 0:
j0 += 1
for j in range(j0, m.shape[0]):
coef = -m[j, i - 1] / m[i - 1, i - 1]
m[j, i - 1 :] += m[i - 1, i - 1 :] * coef
return m
piv = pivot_gauss2(m)
piv * (numpy.abs(piv) > 1e-10)
[10]:
array([[ 0.58650968, 0.72315516, 0.9137217 , 0.04526746],
[ 0. , 0.70012278, 0.87604076, 0.92402115],
[ 0. , 0. , -0.36865515, 1.00362559],
[ 0. , 0. , 0. , 0.26361505]])
Numpy 2#
[11]:
def pivot_gauss3(m):
n = m.copy()
for i in range(1, m.shape[0]):
j0 = i
while j0 < m.shape[0] and m[j0, i - 1] == 0:
j0 += 1
coef = -m[j0:, i - 1] / m[i - 1, i - 1]
m[j0:, i - 1 :] += coef.reshape((-1, 1)) * m[i - 1, i - 1 :].reshape((1, -1))
return m
piv = pivot_gauss3(m)
piv * (numpy.abs(piv) > 1e-10)
[11]:
array([[ 0.58650968, 0.72315516, 0.9137217 , 0.04526746],
[ 0. , 0.70012278, 0.87604076, 0.92402115],
[ 0. , 0. , -0.36865515, 1.00362559],
[ 0. , 0. , 0. , 0.26361505]])
Vitesse#
[12]:
from tqdm import tqdm
import pandas
from teachpyx.ext_test_case import measure_time
data = []
for n in tqdm([10, 20, 30, 40, 50, 60, 70, 80, 100]):
m = numpy.random.rand(n, n)
if n < 50:
res = measure_time(lambda: pivot_gauss(m), number=10, repeat=10)
res.update(dict(name="python", n=n))
data.append(res)
res = measure_time(lambda: pivot_gauss2(m), number=10, repeat=10)
res.update(dict(name="numpy1", n=n))
data.append(res)
res = measure_time(lambda: pivot_gauss3(m), number=10, repeat=10)
res.update(dict(name="numpy2", n=n))
data.append(res)
df = pandas.DataFrame(data)
df
100%|██████████| 9/9 [00:07<00:00, 1.17it/s]
[12]:
average | deviation | min_exec | max_exec | repeat | number | ttime | context_size | warmup_time | name | n | |
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.000051 | 0.000030 | 0.000028 | 0.000125 | 10 | 10 | 0.000512 | 64 | 0.000728 | python | 10 |
1 | 0.000045 | 0.000021 | 0.000028 | 0.000094 | 10 | 10 | 0.000447 | 64 | 0.000046 | numpy1 | 10 |
2 | 0.000137 | 0.000024 | 0.000110 | 0.000197 | 10 | 10 | 0.001367 | 64 | 0.000253 | numpy2 | 10 |
3 | 0.001134 | 0.000540 | 0.000623 | 0.002258 | 10 | 10 | 0.011340 | 64 | 0.003184 | python | 20 |
4 | 0.001312 | 0.000678 | 0.000639 | 0.002858 | 10 | 10 | 0.013121 | 64 | 0.001035 | numpy1 | 20 |
5 | 0.000375 | 0.000184 | 0.000233 | 0.000818 | 10 | 10 | 0.003753 | 64 | 0.001292 | numpy2 | 20 |
6 | 0.001557 | 0.001094 | 0.000686 | 0.003803 | 10 | 10 | 0.015571 | 64 | 0.016613 | python | 30 |
7 | 0.000404 | 0.000062 | 0.000331 | 0.000553 | 10 | 10 | 0.004045 | 64 | 0.000364 | numpy1 | 30 |
8 | 0.000471 | 0.000037 | 0.000440 | 0.000545 | 10 | 10 | 0.004713 | 64 | 0.000482 | numpy2 | 30 |
9 | 0.003798 | 0.001491 | 0.002594 | 0.007715 | 10 | 10 | 0.037978 | 64 | 0.018972 | python | 40 |
10 | 0.001324 | 0.000265 | 0.001054 | 0.002030 | 10 | 10 | 0.013239 | 64 | 0.001341 | numpy1 | 40 |
11 | 0.000820 | 0.000190 | 0.000602 | 0.001140 | 10 | 10 | 0.008200 | 64 | 0.000651 | numpy2 | 40 |
12 | 0.004260 | 0.001177 | 0.002724 | 0.006471 | 10 | 10 | 0.042603 | 64 | 0.005198 | numpy1 | 50 |
13 | 0.001690 | 0.000268 | 0.001100 | 0.002116 | 10 | 10 | 0.016902 | 64 | 0.001080 | numpy2 | 50 |
14 | 0.009320 | 0.002581 | 0.005762 | 0.013509 | 10 | 10 | 0.093197 | 64 | 0.015497 | numpy1 | 60 |
15 | 0.001664 | 0.000133 | 0.001458 | 0.001827 | 10 | 10 | 0.016641 | 64 | 0.001309 | numpy2 | 60 |
16 | 0.005314 | 0.001348 | 0.003848 | 0.008252 | 10 | 10 | 0.053141 | 64 | 0.016670 | numpy1 | 70 |
17 | 0.001629 | 0.000216 | 0.001459 | 0.002045 | 10 | 10 | 0.016286 | 64 | 0.001499 | numpy2 | 70 |
18 | 0.011036 | 0.003890 | 0.007070 | 0.018254 | 10 | 10 | 0.110356 | 64 | 0.017467 | numpy1 | 80 |
19 | 0.002637 | 0.000582 | 0.001846 | 0.003606 | 10 | 10 | 0.026375 | 64 | 0.002417 | numpy2 | 80 |
20 | 0.021563 | 0.003232 | 0.017314 | 0.029488 | 10 | 10 | 0.215629 | 64 | 0.028747 | numpy1 | 100 |
21 | 0.004673 | 0.001109 | 0.003391 | 0.005935 | 10 | 10 | 0.046730 | 64 | 0.005792 | numpy2 | 100 |
[13]:
piv = df.pivot(index="n", columns="name", values="average")
piv
[13]:
name | numpy1 | numpy2 | python |
---|---|---|---|
n | |||
10 | 0.000045 | 0.000137 | 0.000051 |
20 | 0.001312 | 0.000375 | 0.001134 |
30 | 0.000404 | 0.000471 | 0.001557 |
40 | 0.001324 | 0.000820 | 0.003798 |
50 | 0.004260 | 0.001690 | NaN |
60 | 0.009320 | 0.001664 | NaN |
70 | 0.005314 | 0.001629 | NaN |
80 | 0.011036 | 0.002637 | NaN |
100 | 0.021563 | 0.004673 | NaN |
[14]:
piv.plot(title="pivot de gauss");
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