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#!/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.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, cell=None):
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 strEq(n,coeff):# Latex str of one equation in format a1x1+ ...+anxn or with coefficients.
Eq=''
if str(coeff)=='' or coeff==[]:
if n==1:
Eq=Eq + 'a_1x_1 = b'
elif n==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 :
if n==1:
Eq=Eq + str(round(coeff[0],3) if coeff[0] % 1 else int(coeff[0]))+ 'x_'+str(1) +\
'='+ str(round(coeff[len(coeff)-1],3) if coeff[len(coeff)-1] % 1 else int(coeff[len(coeff)-1]))
else:
Eq=Eq + str(round(coeff[0],3) if coeff[0] % 1 else int(coeff[0]))+ 'x_'+str(1)
for i in range(1,n-1):
Eq=Eq + ' + ' + str(round(coeff[i],3) if coeff[i] %1 else int(coeff[i])) + 'x_' + str(i+1)
Eq=Eq + ' + ' +str(round(coeff[len(coeff)-2],3) if coeff[len(coeff)-2] % 1 else int(coeff[len(coeff)-2]))\
+ 'x_' + str(n) + '=' + str(round(coeff[len(coeff)-1],3) if coeff[len(coeff)-1] % 1 else int(coeff[len(coeff)-1]))
return Eq
def printEq(coeff,b, *args):# Latex Print of one equation in format a1x1+ ...+anxn or with coefficients.
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))
def printSyst(A,b, *args):# Latex Print of one system of m equation in format ai1x1+ ...+ainxn=bi or with coeff in MatCoeff.
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='$\\begin{cases}'
Eq_list=[]
if type(b[0])==list:
b=np.matrix(b).astype(float)
A=np.concatenate((A,b), axis=1)
else:
A = [A[i]+[b[i]]for i in range(0,m)] #becomes augmented matrix
A=np.array(A) #just in case it's not
for i in range(m):
if str(A)=='':
Eq_i=''
if n==1:
Eq_i=Eq_i + 'a_{' + str(i+1) + '1}' + 'x_1 = b_' + str(i+1)
elif n==2:
Eq_i=Eq_i + 'a_{' + str(i+1) + '1}' + 'x_1 + ' + 'a_{' + str(i+1) + '2}' + 'x_2 = b_' + str(i+1)
else:
Eq_i=Eq_i + 'a_{' + str(i+1) + '1}' + 'x_1 + \ldots +' + 'a_{' + str(i+1) +str(n) + '}' + 'x_'+ str(n) + '=b_' + str(i+1)
else:
Eq_i=strEq(n,A[i,:]) #attention A is (A|b)
Eq_list.append(Eq_i)
texSyst=texSyst+ Eq_list[i] + '\\\\'
texSyst =texSyst+ '\\end{cases}$'
display(Latex(texSyst))
else:
print("La matrice des coefficients n'a pas les bonnes dimensions")
def texMatrix(*args): #return tex expression of one matrix of A or (A|b) -- or (A|B) VERIFIED OK
if len(args)==2: # matrice augmentée
A=np.matrix(args[0]).astype(float)
m=A.shape[1] #??????
b=args[1]
#b=[b[i] for i in range(m)]
b=np.matrix(b).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.matrix(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")
return texA
def printA(*args) : #Print matrix VERIFIED OK
texA='$'+ texMatrix(*args) + '$'
# print(texA)
display(Latex(texA))
def printEquMatrices(*args): #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]
# print(listOfMatrices)
texEqu='$' + texMatrix(listOfMatrices[0])
for i in range(1,len(listOfMatrices)):
texEqu=texEqu+ '\\quad \\sim \\quad' + texMatrix(listOfMatrices[i])
texEqu=texEqu+ '$'
display(Latex(texEqu))
#%% Functions to enter smth
def EnterInt(n): #function enter integer.
while type(n)!=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)!=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):
A = np.asarray(coeff[0:len(coeff)-1])
if abs(np.dot(A,sol)-coeff[len(coeff)-1])<1e-10:
print("La suite entrée est une solution de l'équation", i)
else:
print("La suite entrée n'est pas une solution de l'équation",i)
return abs(np.dot(A,sol)-coeff[len(coeff)-1])<1e-10
def SolOfSyst(solution,A,b):
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(0,len(A))]
if all(isSol[i]==True for i in range(len(isSol))):
print("C'est une solution du système")
else:
print("Ce n'est pas une solution du système")
#%%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=Scatter(x=p[0]+t*d[0], y= p[1] + t*d[1],name='Droite')
peak=Scatter(x= d[0], y= d[1],marker=dict(symbol=6,size=12,color = colors[0]),showlegend=False)
vector=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 = 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=Scatter3d(x=p[0]+t*d[0], y= p[1] + t*d[1], z=p[2]+ t*d[2], mode='lines', name='Droite')
zero=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=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=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 = 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
A = [A[i]+[b[i]] for i in range(0,len(A))]
A=np.array(A)
t=np.linspace(xL,xR,p)
legend=[]
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)
else :
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)
def Plot3DSys(xL,xR,p,A,b): # small values for p allows for dots to be seen
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)'
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)):
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)
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)
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)
data.append(surface)
layout = go.Layout(
showlegend=True, #not there WHY????
legend=dict(orientation="h"),
autosize=True,
width=800,
height=800,
scene=go.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.matrix(args[0]).astype(float)
m=A.shape[0]
n=A.shape[1]
b=args[1]
if type(b[0])==list:
b=np.matrix(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.matrix(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():
n=random.randint(1,10)
m=random.randint(1,10)
A=np.empty([m,n],dtype=int)
A=[[random.randint(-100,100) for i in range(n)] for j in range(m)]
printA(A)
return np.matrix(A)
def dimensionA(A):
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)
def manualEch(*args):
if len(args)==2: # matrice augmentée
A=np.matrix(args[0]).astype(float)
m=A.shape[0]
b=args[1]
#b=[b[i] for i in range(m)]
if type(b[0])==list:
b=np.matrix(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.matrix(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.matrix(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.matrix(args[0]).astype(float)
m=A.shape[0]
b=args[1]
#b=[b[i] for i in range(m)]
if type(b[0])==list:
b=np.matrix(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.matrix(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 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:
print('Le coefficient alpha doit être non-nul!')
if r=='Eij':
m=Eij(m,i-1,j-1)
if r=='Ei(alpha)':
m=Ealpha(m,i.value-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]):])
printEquMatricesAug(MatriceList,RhSList)
interact_manual(f,r=r,i=i,j=j,alpha=alpha)
########################################OBSOLETE
def printEquMatricesAug(listOfMatrices, listOfRhS): #list of matrices is M=[M1, M2, ..., Mn] where Mi=(Mi|b)
texEqu='$' + texMatrixAug(listOfMatrices[0],listOfRhS[0])
for i in range(1,len(listOfMatrices)):
texEqu=texEqu+ '\\quad \\sim \\quad' + texMatrixAug(listOfMatrices[i], listOfRhS[i])
texEqu=texEqu+ '$'
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))]
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))
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='$'+ texMatrixAug(A,b) + '$'
# print(texA)
display(Latex(texA))
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))
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