diff --git a/Librairie/AL_Fct.py b/Librairie/AL_Fct.py
index a9fea0a..cd4fff5 100644
--- a/Librairie/AL_Fct.py
+++ b/Librairie/AL_Fct.py
@@ -1,1049 +1,1099 @@
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Mar 13 16:42:29 2019
@author: jecker
"""
from __future__ import division
import numpy as np
from IPython.display import display, Latex
import matplotlib.pyplot as plt
import math
import plotly
# import chart_studio.plotly as py
import plotly.graph_objs as go
from plotly.graph_objs import *
plotly.offline.init_notebook_mode(connected=True)
from IPython.core.magic import register_cell_magic
from IPython.display import HTML
import ipywidgets as widgets
import random
from ipywidgets import interact, interactive, fixed, interact_manual
@register_cell_magic
def bgc(color):
script = (
"var cell = this.closest('.jp-CodeCell');"
"var editor = cell.querySelector('.jp-Editor');"
"editor.style.background='{}';"
"this.parentNode.removeChild(this)"
).format(color)
display(HTML('<img src onerror="{}">'.format(script)))
###############################################################################
## PRINTS Equations, systems, matrix
def printMonomial(coeff, index=None, include_zeros=False):
"""Prints the monomial coeff*x_{index} in optimal way
:param coeff: value of the coefficient
:type coeff: float
:param index: index of the monomial. If None, only the numerical value of the coefficient is displayed
:type index: int or NoneType
:param include_zeros: if True, monomials of type 0x_n are printed. Defaults to False
:type include_zeros: bool
:return: string representative of the monomial
:rtype: str
"""
+ if index is not None:
+ coeff = abs(coeff)
+
if coeff % 1:
- return str(round(abs(coeff), 3)) + ('x_' + str(index) if index is not None else "")
+ return str(round(coeff, 3)) + ('x_' + str(index) if index is not None else "")
elif not coeff:
if index is None:
return str(0)
else:
return str(0) + 'x_' + str(index) if include_zeros else ""
elif coeff == 1:
- return 'x_' + str(index) if index is not None else str(coeff)
+ return 'x_' + str(index) if index is not None else str(int(coeff))
elif coeff == -1:
- return 'x_' + str(index) if index is not None else str(coeff)
+ return 'x_' + str(index) if index is not None else str(int(coeff))
else:
- return str(int(abs(coeff))) + ('x_' + str(index) if index is not None else "")
+ return str(int(coeff)) + ('x_' + str(index) if index is not None else "")
def printPlusMinus(coeff, include_zeros=False):
- """Prints a plus or minus sign, depending on the sign of te coefficient
+ """Prints a plus or minus sign, depending on the sign of the coefficient
:param coeff: value of the coefficient
:type coeff: float
:param include_zeros: if True, 0-coefficients are assigned a "+" sign
:type include_zeros: bool
:return: "+" if the coefficient is positive, "-" if it is negative, "" if it is 0
:rtype: str
"""
if coeff > 0:
return "+"
elif coeff < 0:
return "-"
else:
return "+" if include_zeros else ""
def strEq(n, coeff):
"""Method that provides the Latex string of a linear equation, given the number of unknowns and the values
of the coefficients. If no coefficient value is provided, then a symbolic equation with `n` unknowns is plotted.
In particular:
* **SYMBOLIC EQUATION**: if the number of unknowns is either 1 or 2, then all the equation is
displayed while, if the number of unknowns is higher than 2, only the first and last term of the equation
are displayed
* **NUMERICAL EQUATION**: whichever the number of unknowns, the whole equation is plotted. Numerical values
of the coefficients are rounded to the third digit
:param n: number of unknowns of the equation
:type n: int
:param coeff: coefficients of the linear equation. It must be [] if a symbolic equation is desired
:type: list[float]
:return: Latex string representing the equation
:rtype: str
"""
Eq = ''
if not len(coeff):
if n is 1:
Eq = Eq + 'a_1x_1 = b'
elif n is 2:
Eq = Eq + 'a_1x_1 + a_2x_2 = b'
else:
Eq = Eq + 'a_1x_1 + \ldots + ' + 'a_' + str(n) + 'x_' + str(n) + '= b'
else:
all_zeros = len(set(coeff[:-1])) == 1 and not coeff[0] # check if all lhs coefficients are 0
start_put_sign = all_zeros
if n is 1:
Eq += "-" if coeff[0] < 0 else ""
Eq += printMonomial(coeff[0], index=1, include_zeros=all_zeros) + "=" + printMonomial(coeff[-1])
else:
Eq += "-" if coeff[0] < 0 else ""
Eq += printMonomial(coeff[0], index=1, include_zeros=all_zeros)
start_put_sign = start_put_sign or coeff[0] is not 0
for i in range(1, n):
Eq += printPlusMinus(coeff[i], include_zeros=all_zeros) if start_put_sign \
else "-" if coeff[i] < 0 else ""
Eq += printMonomial(coeff[i], index=i+1, include_zeros=all_zeros)
start_put_sign = start_put_sign or coeff[i] is not 0
Eq += "=" + printMonomial(coeff[-1])
return Eq
def printEq(coeff, b, *args):
"""Method that prints the Latex string of a linear equation, given the values of the coefficients. If no coefficient
value is provided, then a symbolic equation with `n` unknowns is plotted. In particular:
* **SYMBOLIC EQUATION**: if the number of unknowns is either 1 or 2, then all the equation is
displayed while, if the number of unknowns is higher than 2, only the first and last term of the equation
are displayed
* **NUMERICAL EQUATION**: whichever the number of unknowns, the whole equation is plotted. Numerical values
of the coefficients are rounded to the third digit
:param coeff: coefficients of the left-hand side of the linear equation
:type: list[float]
:param b: right-hand side coefficient of the linear equation
:type b: float
:param *args: optional; if passed, it contains the number of unknowns to be considered. If not passed, all the
unknowns are considered, i.e. n equals the length of the coefficients list
:type: *args: list
"""
if len(args) == 1:
n = args[0]
else:
n = len(coeff)
coeff = coeff + b
texEq = '$'
texEq = texEq + strEq(n, coeff)
texEq = texEq + '$'
display(Latex(texEq))
return
def printSyst(A, b, *args):
"""Method that prints a linear system of `n` unknowns and `m` equations. If `A` and `b` are empty, then a symbolic
system is printed; otherwise a system containing the values of the coefficients stored in `A` and `b`, approximated
up to their third digit is printed.
:param A: left-hand side matrix. It must be [] if a symbolic system is desired
:type: list[list[float]]
:param b: right-hand side vector. It must be [] if a symbolic system is desired
:type b: list[float]
:param args: optional; if not empty, it is a list of two integers representing the number of equations of the
linear system (i.e. `m`) and the number of unknowns of the system (i.e. `n`)
:type: list
"""
if (len(args) == 2) or (len(A) == len(b)): # ensures that MatCoeff has proper dimensions
if len(args) == 2:
m = args[0]
n = args[1]
else:
m = len(A)
n = len(A[0])
texSyst = '$\\begin{cases}'
Eq_list = []
if len(A) and len(b):
if type(b[0]) is list:
b = np.array(b).astype(float)
A = np.concatenate((A, b), axis=1)
else:
A = [A[i] + [b[i]] for i in range(0, m)] # becomes augmented matrix
A = np.array(A) # just in case it's not
for i in range(m):
if not len(A) or not len(b):
Eq_i = ''
if n is 1:
Eq_i = Eq_i + 'a_{' + str(i + 1) + '1}' + 'x_1 = b_' + str(i + 1)
elif n is 2:
Eq_i = Eq_i + 'a_{' + str(i + 1) + '1}' + 'x_1 + ' + 'a_{' + str(i + 1) + '2}' + 'x_2 = b_' + str(
i + 1)
else:
Eq_i = Eq_i + 'a_{' + str(i + 1) + '1}' + 'x_1 + \ldots +' + 'a_{' + str(i + 1) + str(
n) + '}' + 'x_' + str(n) + '= b_' + str(i + 1)
else:
Eq_i = strEq(n, A[i, :]) # attention A is (A|b)
Eq_list.append(Eq_i)
texSyst = texSyst + Eq_list[i] + '\\\\'
texSyst = texSyst + '\\end{cases}$'
display(Latex(texSyst))
else:
print("La matrice des coefficients n'a pas les bonnes dimensions")
return
def texMatrix(*args):
"""Method which produces the Latex string corresponding to the input matrix.
.. note:: if two inputs are passed, they represent A and b respectively; as a result the augmented matrix A|B is
plotted. Otherwise, if the input is unique, just the matrix A is plotted
:param args: input arguments; they could be either a matrix and a vector or a single matrix
:type args: list
:return: Latex string representing the input matrix or the input matrix augmented by the input vector
:rtype: str
"""
if len(args) == 2: # matrice augmentée
A = np.array(args[0]).astype(float)
m = A.shape[1]
b = np.array(args[1]).astype(float)
A = np.concatenate((A, b), axis=1)
texApre = '\\left(\\begin{array}{'
texA = ''
for i in np.asarray(A):
texALigne = ''
texALigne = texALigne + str(round(i[0], 3) if i[0] % 1 else int(i[0]))
if texA == '':
texApre = texApre + 'c'
for j in i[1:m]:
if texA == '':
texApre = texApre + 'c'
texALigne = texALigne + ' & ' + str(round(j, 3) if j % 1 else int(j))
if texA == '':
texApre = texApre + '| c'
for j in i[m:]:
if texA == '':
texApre = texApre + 'c'
texALigne = texALigne + ' & ' + str(round(j, 3) if j % 1 else int(j))
texALigne = texALigne + ' \\\\'
texA = texA + texALigne
texA = texApre + '} ' + texA[:-2] + ' \\end{array}\\right)'
elif len(args) == 1: # matrice des coefficients
A = np.array(args[0]).astype(float)
texApre = '\\left(\\begin{array}{'
texA = ''
for i in np.asarray(A):
texALigne = ''
texALigne = texALigne + str(round(i[0], 3) if i[0] % 1 else int(i[0]))
if texA == '':
texApre = texApre + 'c'
for j in i[1:]:
if texA == '':
texApre = texApre + 'c'
texALigne = texALigne + ' & ' + str(round(j, 3) if j % 1 else int(j))
texALigne = texALigne + ' \\\\'
texA = texA + texALigne
texA = texApre + '} ' + texA[:-2] + ' \\end{array}\\right)'
else:
print("Ce n'est pas une matrice des coefficients ni une matrice augmentée")
+ texA = ''
return texA
def printA(*args): # Print matrix
"""Method which prints the input matrix.
.. note:: if two inputs are passed, they represent A and b respectively; as a result the augmented matrix A|B is
plotted. Otherwise, if the input is unique, just the matrix A is plotted
:param args: input arguments; they could be either a matrix and a vector or a single matrix
:type args: list
"""
texA = '$' + texMatrix(*args) + '$'
display(Latex(texA))
return
def printEquMatrices(*args):
"""Method which prints the list of input matrices.
.. note:: if two inputs are passed, they represent the list of coefficient matrices A and the list of rhs b
respectively; as a result the augmented matrices A|B are plotted. Otherwise, if the input is unique, just the
matrices A are plotted
:param args: input arguments; they could be either a list of matrices and a list of vectors or
a single list of matrices
:type args: list
"""
# list of matrices is M=[M1, M2, ..., Mn] where Mi=(Mi|b)
if len(args) == 2:
listOfMatrices = args[0]
listOfRhS = args[1]
texEqu = '$' + texMatrix(listOfMatrices[0], listOfRhS[0])
for i in range(1, len(listOfMatrices)):
texEqu = texEqu + '\\quad \\sim \\quad' + texMatrix(listOfMatrices[i], listOfRhS[i])
texEqu = texEqu + '$'
display(Latex(texEqu))
else:
listOfMatrices = args[0]
texEqu = '$' + texMatrix(listOfMatrices[0])
for i in range(1, len(listOfMatrices)):
texEqu = texEqu + '\\quad \\sim \\quad' + texMatrix(listOfMatrices[i])
texEqu = texEqu + '$'
display(Latex(texEqu))
return
# %% Functions to enter smth
def EnterInt(n): # function enter integer.
while type(n) is not int:
try:
n = int(n)
if n <= 0:
print("Le nombre ne peut pas être négatif!")
print("Entrez à nouveau : ")
n = input()
except:
print("Ce n'est pas un entier!")
print("Entrez à nouveau :")
n = input()
n = int(n)
return n
def EnterListReal(n): # function enter list of real numbers.
coeff = input()
while type(coeff) is not list:
try:
coeff = [float(eval(x)) for x in coeff.split(',')]
if len(coeff) != n + 1:
print("Vous n'avez pas entré le bon nombre de réels!")
print("Entrez à nouveau : ")
coeff = input()
except:
print("Ce n'est pas le bon format!")
print("Entrez à nouveau")
coeff = input()
# coeff[abs(coeff)<1e-15]=0 #ensures that 0 is 0.
return coeff
# %%Verify if sol is the solution of the equation with coefficients coeff
def SolOfEq(sol, coeff, i):
"""Method that verifies if `sol` is a solution to the linear equation `i`with coefficients `coeff`
:param sol: candidate solution vector
:type sol: list
:param coeff: coefficients of the linear equation
:type coeff: list
:param i: index of the equation
:type i: int
:return: True if `sol` is a solution, False otherwise
:rtype: bool
"""
try:
assert len(sol) == len(coeff)-1
except AssertionError:
print(f"La suite entrée n'est pas une solution de l'équation {i}; Les dimensions ne correspondent pas")
return False
A = np.array(coeff[:-1])
- isSol = abs(np.dot(A, sol) - coeff[-1]) < 1e-10
+ isSol = abs(np.dot(A, sol) - coeff[-1]) < 1e-8
if isSol:
print(f"La suite entrée est une solution de l'équation {i}")
else:
print(f"La suite entrée n'est pas une solution de l'équation {i}")
return isSol
def SolOfSyst(solution, A, b):
"""Method that verifies if `solution` is a solution to the linear system with left-hand side matrix `A` and
right-hand side vector `b`
:param solution: candidate solution vector
:type solution: list
:param A: left-hand side matrix of the linear system
:type A: list[list[float]] or numpy.ndarray
:param b: right-hand side vector of the linear system
:type b: list[float] or numpy.ndarray
:return: True if `sol` is a solution, False otherwise
:rtype: bool
"""
try:
assert len(solution) == (len(A[0]) if type(A) is list else A.shape[1])
except AssertionError:
print(f"La suite entrée n'est pas une solution du système; Les dimensions ne correspondent pas")
return False
A = [A[i] + [b[i]] for i in range(0, len(A))]
A = np.array(A)
isSol = [SolOfEq(solution, A[i, :], i+1) for i in range(len(A))]
if all(isSol):
print("C'est une solution du système")
return True
else:
print("Ce n'est pas une solution du système")
return False
# %%Plots using plotly
def drawLine(p, d): # p,d=vectors p-->"point" and d-->"direction",
blue = 'rgb(51, 214, 255)'
colors = [blue]
colorscale = [[0.0, colors[0]],
[0.1, colors[0]],
[0.2, colors[0]],
[0.3, colors[0]],
[0.4, colors[0]],
[0.5, colors[0]],
[0.6, colors[0]],
[0.7, colors[0]],
[0.8, colors[0]],
[0.9, colors[0]],
[1.0, colors[0]]]
vec = 0.9 * np.array(d)
if len(p) == 2:
data = []
t = np.linspace(-5, 5, 51)
s = np.linspace(0, 1, 10)
trace = go.Scatter(x=p[0] + t * d[0], y=p[1] + t * d[1], name='Droite')
peak = go.Scatter(x=d[0], y=d[1], marker=dict(symbol=6, size=12, color=colors[0]), showlegend=False)
vector = go.Scatter(x=p[0] + s * d[0], y=p[1] + s * d[1], mode='lines',
line=dict(width=5,
color=colors[0]), name='Vecteur directeur')
data.append(trace)
data.append(vector)
data.append(peak)
fig = go.FigureWidget(data=data)
plotly.offline.iplot(fig)
elif len(p) == 3:
data = [
{
'type': 'cone',
'x': [1], 'y': vec[1], 'z': vec[2],
'u': d[0], 'v': d[1], 'w': d[2],
"sizemode": "absolute",
'colorscale': colorscale,
'sizeref': 1,
"showscale": False,
'hoverinfo': 'none'
}
]
t = np.linspace(-5, 5, 51)
s = np.linspace(0, 1, 10)
trace = go.Scatter3d(x=p[0] + t * d[0], y=p[1] + t * d[1], z=p[2] + t * d[2], mode='lines', name='Droite')
zero = go.Scatter3d(x=t * 0, y=t * 0, z=t * 0, name='Origine', marker=dict(size=5), showlegend=False)
if all(dd == [0] for dd in d):
vector = go.Scatter3d(x=p[0] + s * d[0], y=p[1] + s * d[1], z=p[2] + s * d[2], marker=dict(size=5),
name='Point')
else:
vector = go.Scatter3d(x=p[0] + s * d[0], y=p[1] + s * d[1], z=p[2] + s * d[2], mode='lines',
line=dict(width=5,
color=colors[0], dash='solid'), name='Vecteur directeur', hoverinfo='none')
data.append(zero)
data.append(vector)
data.append(trace)
layout = {
'scene': {
'camera': {
'eye': {'x': -0.76, 'y': 1.8, 'z': 0.92}
}
}
}
fig = go.FigureWidget(data=data, layout=layout)
plotly.offline.iplot(fig)
return fig
-def Plot2DSys(xL, xR, p, A, b): # small values for p allows for dots to be seen
+def Plot2DSys(xL, xR, p, A, b):
+ """Function for the graphical visualization of a 2D system of equations, plotting the straight lines characterizing
+ the different equations appearing in the system
+
+ :param xL: left limit of the plot in both coordinates
+ :type xL: int or float
+ :param xR: right limit of the plot in both coordinates
+ :type xR: int or float
+ :param p: number of points used to draw the straight lines
+ :type p: int
+ :param A: matrix of the linear system
+ :type A: list[list[float]] or numpy.ndarray
+ :param b: right-hand side vector of the linear system
+ :type b: list[float] or numpy.ndarray
+ """
+
A = [A[i] + [b[i]] for i in range(0, len(A))]
A = np.array(A)
t = np.linspace(xL, xR, p)
data = []
for i in range(1, len(A) + 1):
if (abs(A[i - 1, 1])) > abs(A[i - 1, 0]):
# p0=[0,A[i-1,2]/A[i-1,1]]
# p1=[1,(A[i-1,2]-A[i-1,0])/A[i-1,1]]
- trace = go.Scatter(x=t, y=(A[i - 1, 2] - A[i - 1, 0] * t) / A[i - 1, 1], name='Droite %d' % i)
+ trace = go.Scatter(x=t, y=(A[i-1, 2] - A[i-1, 0] * t) / A[i-1, 1], name='Droite %d' % i)
else:
- trace = go.Scatter(x=(A[i - 1, 2] - A[i - 1, 1] * t) / A[i - 1, 0], y=t, name='Droite %d' % i)
+ trace = go.Scatter(x=(A[i-1, 2] - A[i-1, 1] * t) / A[i-1, 0], y=t, name='Droite %d' % i)
data.append(trace)
+
fig = go.Figure(data=data)
plotly.offline.iplot(fig)
+ return
-def Plot3DSys(xL, xR, p, A, b): # small values for p allows for dots to be seen
+def Plot3DSys(xL, xR, p, A, b):
+ """Function for the graphical visualization of a 3D system of equations, plotting the straight lines characterizing
+ the different equations appearing in the system
+
+ :param xL: left limit of the plot in all coordinates
+ :type xL: int or float
+ :param xR: right limit of the plot in all coordinates
+ :type xR: int or float
+ :param p: number of points used to draw the straight lines
+ :type p: int
+ :param A: matrix of the linear system
+ :type A: list[list[float]] or numpy.ndarray
+ :param b: right-hand side vector of the linear system
+ :type b: list[float] or numpy.ndarray
+ """
A = [A[i] + [b[i]] for i in range(0, len(A))]
A = np.array(A)
gr = 'rgb(102,255,102)'
org = 'rgb(255,117,26)'
- red = 'rgb(255,0,0)'
+ # red = 'rgb(255,0,0)'
blue = 'rgb(51, 214, 255)'
colors = [blue, gr, org]
s = np.linspace(xL, xR, p)
t = np.linspace(xL, xR, p)
tGrid, sGrid = np.meshgrid(s, t)
data = []
- for i in range(0, len(A)):
+ for i in range(len(A)):
colorscale = [[0.0, colors[i]],
[0.1, colors[i]],
[0.2, colors[i]],
[0.3, colors[i]],
[0.4, colors[i]],
[0.5, colors[i]],
[0.6, colors[i]],
[0.7, colors[i]],
[0.8, colors[i]],
[0.9, colors[i]],
[1.0, colors[i]]]
j = i + 1
if (abs(A[i, 2])) > abs(A[i, 1]): # z en fonction de x,y
x = sGrid
y = tGrid
surface = go.Surface(x=x, y=y, z=(A[i, 3] - A[i, 0] * x - A[i, 1] * y) / A[i, 2],
- showscale=False, colorscale=colorscale, opacity=1, name='Plan %d' % j)
+ showscale=False, showlegend=True, colorscale=colorscale, opacity=1, name='Plan %d' % j)
elif A[i, 2] == 0 and A[i, 1] == 0: # x =b
y = sGrid
z = tGrid
surface = go.Surface(x=A[i, 3] - A[i, 1] * y, y=y, z=z,
- showscale=False, colorscale=colorscale, opacity=1, name='Plan %d' % j)
+ showscale=False, showlegend=True, colorscale=colorscale, opacity=1, name='Plan %d' % j)
else: # y en fonction de x,z
x = sGrid
z = tGrid
surface = go.Surface(x=x, y=(A[i, 3] - A[i, 0] * x - A[i, 2] * z) / A[i, 1], z=z,
- showscale=False, colorscale=colorscale, opacity=1, name='Plan %d' % j)
+ showscale=False, showlegend=True, colorscale=colorscale, opacity=1, name='Plan %d' % j)
data.append(surface)
layout = go.Layout(
- showlegend=True, # not there WHY????
+ showlegend=True, # not there WHY???? --> LEGEND NOT YET IMPLEMENTED FOR SURFACE OBJECTS!!
legend=dict(orientation="h"),
autosize=True,
width=800,
height=800,
scene=go.layout.Scene(
xaxis=dict(
gridcolor='rgb(255, 255, 255)',
zerolinecolor='rgb(255, 255, 255)',
showbackground=True,
backgroundcolor='rgb(230, 230,230)'
),
yaxis=dict(
gridcolor='rgb(255, 255, 255)',
zerolinecolor='rgb(255, 255, 255)',
showbackground=True,
backgroundcolor='rgb(230, 230,230)'
),
zaxis=dict(
gridcolor='rgb(255, 255, 255)',
zerolinecolor='rgb(255, 255, 255)',
showbackground=True,
backgroundcolor='rgb(230, 230,230)'
)
)
)
fig = go.Figure(data=data, layout=layout)
plotly.offline.iplot(fig)
return
def Ex3Chapitre1_7():
systa = widgets.Select(
options=['Point', 'Droite', 'Plan', 'Incompatible'],
description='Système a):',
# layout=Layout(width='auto'),
disabled=False,
)
systb = widgets.Select(
options=['Point', 'Droite', 'Plan', 'Incompatible'],
description='Système b):',
disabled=False
)
systc = widgets.Select(
options=['Point', 'Droite', 'Plan', 'Espace', 'Incompatible'],
description='Système c):',
disabled=False
)
systd = widgets.Select(
options=['Point', 'Droite', 'Plan', 'Espace', 'Incompatible'],
description='Système d):',
disabled=False
)
choice = widgets.Dropdown(
options=['a)', 'b)', 'c)', 'd)'],
value='a)',
description='Système:',
disabled=False,
)
def plot(c):
if c == 'a)':
drawLine([[0], [0]], [[4], [1]])
if c == 'b)':
print("Le système est incompatible, donc il n'y a pas de solutions")
if c == 'c)':
drawLine([[-17], [5], [-10]], [[0], [0], [0]])
if c == 'd)':
drawLine([[1], [0], [0]], [[0], [-1], [1]])
def correction(a, b, c, d):
if 'Droite' in a and 'Incompatible' in b and 'Point' in c and 'Droite' in d:
print("C'est correct!")
out = interact_manual(plot, c=choice)
else:
print("C'est faux. Veuillez rentrer d'autres valeurs")
out = interact_manual(correction, a=systa, b=systb, c=systc, d=systd)
# %%Echelonnage
def echZero(indice,
M): # echelonne la matrice pour mettre les zeros dans les lignes du bas. M (matrice ou array) et Mat (list) pas le même format.
Mat = M[indice == False, :].ravel()
Mat = np.concatenate([Mat, M[indice == True, :].ravel()])
Mat = Mat.reshape(len(M), len(M[0, :]))
return Mat
def Eij(M, i, j): # matrice elementaire, echange la ligne i avec la ligne j
M = np.array(M)
M[[i, j], :] = M[[j, i], :]
return M
def Ealpha(M, i, alpha): # matrice elementaire, multiple la ligne i par le scalaire alpha
M = np.array(M)
M[i, :] = alpha * M[i, :]
return M
def Eijalpha(M, i, j, alpha): # matrice elementaire, AJOUTE à la ligne i alpha *ligne j. Attention alpha + ou -
M = np.array(M)
M[i, :] = M[i, :] + alpha * M[j, :]
return M
def echelonMat(ech,
*args): # Nous donne la matrice echelonnée 'E' ou reduite 'ER' d'une matrice des coeffs. ou augmentée.
if len(args) == 2: # matrice augmentée
A = np.array(args[0]).astype(float)
m = A.shape[0]
n = A.shape[1]
b = args[1]
if type(b[0]) == list:
b = np.array(b).astype(float)
A = np.concatenate((A, b), axis=1)
else:
b = [b[i] for i in range(m)]
A = [A[i] + [b[i]] for i in range(0, m)]
else: # matrice coeff
A = np.array(args[0]).astype(float)
m = A.shape[0]
n = A.shape[1]
b = np.zeros((m, 1))
A = np.concatenate((A, b), axis=1)
if ech == 'E': # Echelonnée
Mat = np.array(A)
Mat = Mat.astype(float) # in case the array in int instead of float.
numPivot = 0
for i in range(len(Mat)):
j = i
while all(abs(Mat[j:, i]) < 1e-15) and j != len(Mat[0, :]) - 1: # if column (or rest of) is 0, take next
j += 1
if j == len(Mat[0, :]) - 1:
if len(Mat[0, :]) > j:
Mat[i + 1:len(Mat), :] = 0
print("La matrice est sous la forme échelonnée")
if len(args) == 2:
printEquMatrices([A[:, 0:n], Mat[:, 0:n]], [A[:, n:], Mat[:, n:]])
else:
printEquMatrices([A, Mat])
break
if abs(Mat[i, j]) < 1e-15:
Mat[i, j] = 0
zero = abs(Mat[i:, j]) < 1e-15
M = echZero(zero, Mat[i:, :])
Mat[i:, :] = M
Mat = Ealpha(Mat, i, 1 / Mat[i, j]) # normalement Mat[i,j]!=0
for k in range(i + 1, len(A)):
Mat = Eijalpha(Mat, k, i, -Mat[k, j])
# Mat[k,:]=[0 if abs(Mat[k,l])<1e-15 else Mat[k,l] for l in range(len(MatCoeff[0,:]))]
numPivot += 1
Mat[abs(Mat) < 1e-15] = 0
# printA(np.array(Mat))
elif ech == 'ER': # Echelonnée réduite
Mat = np.array(A)
Mat = Mat.astype(float) # in case the array in int instead of float.
numPivot = 0
for i in range(len(Mat)):
j = i
while all(abs(Mat[j:, i]) < 1e-15) and j != len(
Mat[0, :]) - 1: # if column (or rest of) is zero, take next column
j += 1
if j == len(Mat[0, :]) - 1:
# ADD ZERO LINES BELOW!!!!!!
if len(Mat[0, :]) > j:
Mat[i + 1:len(Mat), :] = 0
print("La matrice est sous la forme échelonnée")
if len(args) == 2:
printEquMatrices([A[:, 0:n], Mat[:, 0:n]], [A[:, n:], Mat[:, n:]])
else:
printEquMatrices([np.asmatrix(A), np.asmatrix(Mat)])
break
if abs(Mat[i, j]) < 1e-15:
Mat[i, j] = 0
zero = abs(Mat[i:, j]) < 1e-15
M = echZero(zero, Mat[i:, :])
Mat[i:, :] = M
Mat = Ealpha(Mat, i, 1 / Mat[i, j]) # normalement Mat[i,j]!=0
for k in range(i + 1, len(A)):
Mat = Eijalpha(Mat, k, i, -Mat[k, j])
# Mat[k,:]=[0 if abs(Mat[k,l])<1e-15 else Mat[k,l] for l in range(len(MatCoeff[0,:]))]
numPivot += 1
Mat[abs(Mat) < 1e-15] = 0
Mat = np.array(Mat)
MatAugm = np.concatenate((A, b), axis=1)
# MatAugm = [A[i]+[b[i]] for i in range(0,len(A))]
i = (len(Mat) - 1)
while i >= 1:
while all(
abs(Mat[i, :len(Mat[0]) - 1]) < 1e-15) and i != 0: # if ligne (or rest of) is zero, take next ligne
i -= 1
# we have a lign with one non-nul element
j = i # we can start at pos ij at least the pivot is there
if abs(Mat[i, j]) < 1e-15: # if element Aij=0 take next one --> find pivot
j += 1
# Aij!=0 and Aij==1 if echelonMat worked
for k in range(i): # put zeros above pivot (which is 1 now)
Mat = Eijalpha(Mat, k, i, -Mat[k, j])
i -= 1
print("La matrice est sous la forme échelonnée réduite")
if len(args) == 2:
printEquMatrices([A[:, 0:n], Mat[:, 0:n]], [A[:, n:], Mat[:, n:]])
else:
printEquMatrices([A, Mat])
return np.asmatrix(Mat)
-# Generate random matrix
+
def randomA():
+ """Method which generates a random matrix with rows and columns within 1 and 10 and integer entries between -100
+ and 100
+
+ :return: generated random matrix
+ :rtype: numpy.ndarray
+ """
n = random.randint(1, 10)
m = random.randint(1, 10)
A = [[random.randint(-100, 100) for i in range(n)] for j in range(m)]
printA(A)
return np.array(A)
def dimensionA(A):
+ """Method which allows the user to enter the matrix dimensions and verifies whether they are correct or not
+
+ :param A: reference matrix
+ :type A: numpy.ndarray
+ """
m = widgets.IntText(
value=1,
step=1,
description='m:',
disabled=False
)
n = widgets.IntText(
value=1,
step=1,
description='n:',
disabled=False
)
display(m)
display(n)
def f():
if m.value == A.shape[0] and n.value == A.shape[1]:
print('Correcte!')
else:
print('Incorrecte, entrez de nouvelles valeurs')
- out = interact_manual(f)
+ interact_manual(f)
+ return
def manualEch(*args):
if len(args) == 2: # matrice augmentée
A = np.array(args[0]).astype(float)
m = A.shape[0]
b = args[1]
- # b=[b[i] for i in range(m)]
if type(b[0])is list:
b = np.array(b).astype(float)
A = np.concatenate((A, b), axis=1)
else:
b = [b[i] for i in range(m)]
A = [A[i] + [b[i]] for i in range(0, m)]
else:
A = np.array(args[0]).astype(float)
m = A.shape[0]
A = np.array(A) # just in case it's not
j = widgets.BoundedIntText(
value=1,
min=1,
max=m,
step=1,
description='Ligne j:',
disabled=False
)
i = widgets.BoundedIntText(
value=1,
min=1,
max=m,
step=1,
description='Ligne i:',
disabled=False
)
r = widgets.RadioButtons(
options=['Eij', 'Ei(alpha)', 'Eij(alpha)'],
description='Opération:',
disabled=False
)
alpha = widgets.Text(
value='1',
description='Coeff. alpha:',
disabled=False
)
print("Régler les paramètres et évaluer la cellule suivante")
print("Répéter cela jusqu'à obtenir une forme échelonnée réduite")
display(r)
display(i)
display(j)
display(alpha)
return i, j, r, alpha
def echelonnage(i, j, r, alpha, A, m, *args): # 1.5-1.6 Matrice echelonnées
m = np.array(m).astype(float)
if alpha.value == 0:
print('Le coefficient alpha doit être non-nul!')
if r.value == 'Eij':
m = Eij(m, i.value - 1, j.value - 1)
if r.value == 'Ei(alpha)':
m = Ealpha(m, i.value - 1, eval(alpha.value))
if r.value == 'Eij(alpha)':
m = Eijalpha(m, i.value - 1, j.value - 1, eval(alpha.value))
if len(args) == 2:
A = np.asmatrix(A)
MatriceList = args[0]
RhSList = args[1]
MatriceList.append(m[:, 0:A.shape[1]]) # ??????????
RhSList.append(m[:, A.shape[1]:]) # ??????????
printEquMatrices(MatriceList, RhSList)
else:
MatriceList = args[0]
A = np.asmatrix(A)
MatriceList.append(m[:, 0:A.shape[1]]) # ??????????
printEquMatrices(MatriceList)
return m
def manualOp(*args):
if len(args) == 2: # matrice augmentée
A = np.array(args[0]).astype(float)
- m = A.shape[0]
+ M = A.shape[0]
b = args[1]
- # b=[b[i] for i in range(m)]
if type(b[0]) is list:
b = np.array(b).astype(float)
A = np.concatenate((A, b), axis=1)
else:
- b = [b[i] for i in range(m)]
- A = [A[i] + [b[i]] for i in range(0, m)]
+ b = [b[i] for i in range(M)]
+ A = [A[i] + [b[i]] for i in range(0, M)]
else:
A = np.array(args[0]).astype(float)
- m = A.shape[0]
+ M = A.shape[0]
A = np.array(A) # just in case it's not
- j = widgets.BoundedIntText(
+
+ i = widgets.BoundedIntText(
value=1,
min=1,
- max=m,
+ max=M,
step=1,
- description='Ligne j:',
+ description='Ligne i:',
disabled=False
)
- i = widgets.BoundedIntText(
+
+ j = widgets.BoundedIntText(
value=1,
min=1,
- max=m,
+ max=M,
step=1,
- description='Ligne i:',
+ description='Ligne j:',
disabled=False
)
r = widgets.RadioButtons(
options=['Eij', 'Ei(alpha)', 'Eij(alpha)'],
description='Opération:',
disabled=False
)
alpha = widgets.Text(
value='1',
description='Coeff. alpha:',
disabled=False
)
+
print("Régler les paramètres et cliquer sur RUN INTERACT pour effectuer votre opération")
def f(r, i, j, alpha):
- m = np.concatenate((A, b), axis=1)
- MatriceList = [A]
- RhSList = [b]
- if alpha == 0:
+ m = A
+ MatriceList = [A[:, :len(A[0])-1]]
+ RhSList = [A[:, len(A[0])-1:]]
+ if alpha == 0 and r != 'Eij':
print('Le coefficient alpha doit être non-nul!')
if r == 'Eij':
- m = Eij(m, i - 1, j - 1)
+ m = Eij(m, i-1, j-1)
if r == 'Ei(alpha)':
- m = Ealpha(m, i.value - 1, eval(alpha))
+ m = Ealpha(m, i-1, eval(alpha))
if r == 'Eij(alpha)':
- m = Eijalpha(m, i - 1, j - 1, eval(alpha))
- MatriceList.append(m[:, 0:len(A[0])])
- RhSList.append(m[:, len(A[0]):])
+ m = Eijalpha(m, i-1, j-1, eval(alpha))
+ MatriceList.append(m[:, :len(A[0])-1])
+ RhSList.append(m[:, len(A[0])-1:])
printEquMatricesAug(MatriceList, RhSList)
+ return
interact_manual(f, r=r, i=i, j=j, alpha=alpha)
+ return
-########################################OBSOLETE
+######################################## OBSOLETE ####################################################################
def printEquMatricesAug(listOfMatrices, listOfRhS): # list of matrices is M=[M1, M2, ..., Mn] where Mi=(Mi|b)
- texEqu = '$' + texMatrixAug(listOfMatrices[0], listOfRhS[0])
+ texEqu = '$' + texMatrix(listOfMatrices[0], listOfRhS[0])
for i in range(1, len(listOfMatrices)):
- texEqu = texEqu + '\\quad \\sim \\quad' + texMatrixAug(listOfMatrices[i], listOfRhS[i])
+ texEqu = texEqu + '\\quad \\sim \\quad' + texMatrix(listOfMatrices[i], listOfRhS[i])
texEqu = texEqu + '$'
display(Latex(texEqu))
def echelonMatCoeff(A): # take echelonMAt but without b.
- b = [0 for i in range(len(A))]
- Mat = [A[i] + [b[i]] for i in range(0, len(A))]
+ b = [0 for _ in range(len(A))]
+ Mat = [A[i] + [b[i]] for i in range(len(A))]
Mat = np.array(Mat)
Mat = Mat.astype(float) # in case the array in int instead of float.
numPivot = 0
for i in range(len(Mat)):
j = i
while all(abs(Mat[j:, i]) < 1e-15) and j != len(
Mat[0, :]) - 1: # if column (or rest of) is zero, take next column
j += 1
if j == len(Mat[0, :]) - 1:
# ADD ZERO LINES BELOW!!!!!!
if len(Mat[0, :]) > j:
Mat[i + 1:len(Mat), :] = 0
print("La matrice est sous la forme échelonnée")
printEquMatrices(np.asmatrix(A), np.asmatrix(Mat[:, :len(A[0])]))
break
if abs(Mat[i, j]) < 1e-15:
Mat[i, j] = 0
zero = abs(Mat[i:, j]) < 1e-15
M = echZero(zero, Mat[i:, :])
Mat[i:, :] = M
Mat = Ealpha(Mat, i, 1 / Mat[i, j]) # normalement Mat[i,j]!=0
for k in range(i + 1, len(A)):
Mat = Eijalpha(Mat, k, i, -Mat[k, j])
# Mat[k,:]=[0 if abs(Mat[k,l])<1e-15 else Mat[k,l] for l in range(len(MatCoeff[0,:]))]
numPivot += 1
Mat[abs(Mat) < 1e-15] = 0
# printA(np.asmatrix(Mat[:, :len(A[0])]))
return np.asmatrix(Mat)
def echelonRedMat(A, b):
Mat = echelonMat('ER', A, b)
Mat = np.array(Mat)
MatAugm = np.concatenate((A, b), axis=1)
# MatAugm = [A[i]+[b[i]] for i in range(0,len(A))]
i = (len(Mat) - 1)
while i >= 1:
while all(abs(Mat[i, :len(Mat[0]) - 1]) < 1e-15) and i != 0: # if ligne (or rest of) is zero, take next ligne
i -= 1
# we have a lign with one non-nul element
j = i # we can start at pos ij at least the pivot is there
if abs(Mat[i, j]) < 1e-15: # if element Aij=0 take next one --> find pivot
j += 1
# Aij!=0 and Aij==1 if echelonMat worked
for k in range(i): # put zeros above pivot (which is 1 now)
Mat = Eijalpha(Mat, k, i, -Mat[k, j])
i -= 1
printA(Mat)
print("La matrice est sous la forme échelonnée réduite")
printEquMatrices(MatAugm, Mat)
return np.asmatrix(Mat)
def printEquMatricesOLD(listOfMatrices): # list of matrices is M=[M1, M2, ..., Mn]
texEqu = '$' + texMatrix(listOfMatrices[0])
for i in range(1, len(listOfMatrices)):
texEqu = texEqu + '\\quad \\sim \\quad' + texMatrix(listOfMatrices[i])
texEqu = texEqu + '$'
display(Latex(texEqu))
return
def texMatrixAug(A, b): # return tex expression of one matrix (A|b) where b can also be a matrix
m = len(A[0])
A = np.concatenate((A, b), axis=1)
texApre = '\\left(\\begin{array}{'
texA = ''
for i in np.asarray(A):
texALigne = ''
texALigne = texALigne + str(round(i[0], 3) if i[0] % 1 else int(i[0]))
if texA == '':
texApre = texApre + 'c'
for j in i[1:m]:
if texA == '':
texApre = texApre + 'c'
texALigne = texALigne + ' & ' + str(round(j, 3) if j % 1 else int(j))
if texA == '':
texApre = texApre + '| c'
for j in i[m:]:
if texA == '':
texApre = texApre + 'c'
texALigne = texALigne + ' & ' + str(round(j, 3) if j % 1 else int(j))
texALigne = texALigne + ' \\\\'
texA = texA + texALigne
texA = texApre + '} ' + texA[:-2] + ' \\end{array}\\right)'
return texA
def printAAug(A, b): # Print matrix (A|b)
texA = '$' + texMatrixAug(A, b) + '$'
display(Latex(texA))
return
def printEquMatricesOLD(*args): # M=[M1, M2, ..., Mn] n>1 VERIFIED OK
texEqu = '$' + texMatrix(args[0])
for i in range(1, len(args)):
texEqu = texEqu + '\\quad \\sim \\quad' + texMatrix(args[i])
texEqu = texEqu + '$'
display(Latex(texEqu))
return