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AL_Fct.py
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Mon, May 13, 08:07

AL_Fct.py
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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 plotly.plotly as py
import plotly.graph_objs as go
plotly.offline.init_notebook_mode(connected=True)
from IPython.core.magic import register_cell_magic
from IPython.display import HTML, display
@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(n,coeff,b):# Latex Print of one equation in format a1x1+ ...+anxn or with coefficients.
coeff=coeff+b
texEq='$'
texEq=texEq+ strEq(n,coeff)
texEq=texEq + '$'
display(Latex(texEq))
def printSyst(m,n,MatCoeff,b):# Latex Print of one system of m equation in format ai1x1+ ...+ainxn=bi or with coeff in MatCoeff.
if (str(MatCoeff)=='') or (len(MatCoeff[1])==n and len(MatCoeff)==m): #ensures that MatCoeff has proper dimensions
texSyst='$\\begin{cases}'
Eq_list=[]
MatCoeff = [MatCoeff[i]+[b[i]]for i in range(0,m)]
MatCoeff=np.array(MatCoeff) #just in case it's not
for i in range(m):
if str(MatCoeff)=='':
Eq_i=''
if n==1:
Eq_i=Eq_i + 'a_{' + str(i+1) + '1}' + 'x_1 = b_' + str(i+1)
elif n==2:
Eq_i=Eq_i + 'a_{' + str(i+1) + '1}' + 'x_1 + ' + 'a_{' + str(i+1) + '2}' + 'x_2 = b_' + str(i+1)
else:
Eq_i=Eq_i + 'a_{' + str(i+1) + '1}' + 'x_1 + \ldots +' + 'a_{' + str(i+1) +str(n) + '}' + 'x_'+ str(n) + '=b_' + str(i+1)
else:
Eq_i=strEq(n,MatCoeff[i,:])
Eq_list.append(Eq_i)
texSyst=texSyst+ Eq_list[i] + '\\\\'
texSyst =texSyst+ '\\end{cases}$'
display(Latex(texSyst))
else:
print("La matrice des coefficients n'a pas les bonnes dimensions")
def texMatrix(A): #return tex expression of one matrix
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)'
return texA
def printA(A) : #Print matrix
texA='$'+ texMatrix(A) + '$'
# print(texA)
display(Latex(texA))
def printEquMatrices(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))
#%% 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,MatriceCoeff,b):
MatriceCoeff = [MatriceCoeff[i]+[b[i]] for i in range(0,len(MatriceCoeff))]
MatriceCoeff=np.array(MatriceCoeff)
isSol=[SolOfEq(solution, MatriceCoeff[i,:],i+1) for i in range(0,len(MatriceCoeff))]
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 Plot2DSys(xL,xR,p,MatCoeff,b): # small values for p allows for dots to be seen
MatCoeff = [MatCoeff[i]+[b[i]] for i in range(0,len(MatCoeff))]
MatCoeff=np.array(MatCoeff)
t=np.linspace(xL,xR,p)
legend=[]
data=[]
for i in range(1,len(MatCoeff)+1):
if (abs(MatCoeff[i-1,1])) > abs (MatCoeff[i-1,0]) :
trace=go.Scatter(x=t, y= (MatCoeff[i-1,2]-MatCoeff[i-1,0]*t)/MatCoeff[i-1,1], name='Droite %d'%i)
else :
trace=go.Scatter(x=(MatCoeff[i-1,2]-MatCoeff[i-1,1]*t)/MatCoeff[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,MatCoeff,b): # small values for p allows for dots to be seen
MatCoeff = [MatCoeff[i]+[b[i]] for i in range(0,len(MatCoeff))]
MatCoeff=np.array(MatCoeff)
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(MatCoeff)):
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(MatCoeff[i,2])) > abs (MatCoeff[i,1]) : # z en fonction de x,y
x = sGrid
y = tGrid
surface = go.Surface(x=x, y=y, z=(MatCoeff[i,3]-MatCoeff[i,0]*x -MatCoeff[i,1]*y)/MatCoeff[i,2],
showscale=False, colorscale=colorscale, opacity=1, name='Plan %d' %j)
elif (MatCoeff[i,2]==0 and MatCoeff[i,1]==0): # x =b
y = sGrid
z = tGrid
surface = go.Surface(x=MatCoeff[i,3]-MatCoeff[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=(MatCoeff[i,3]-MatCoeff[i,0]*x -MatCoeff[i,2]*z)/MatCoeff[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)
#%%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[[i,j],:]=M[[j,i],:]
return M
def Ealpha(M, i, alpha): # matrice elementaire, multiple la ligne i par le scalaire alpha
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[i,:]=M[i,:] + alpha *M[j,:]
return M
def echelonMat(MatCoeff,b): #Nous donne la matrice echelonne d un system [A|b]
MatCoeff = [MatCoeff[i]+[b[i]] for i in range(0,len(MatCoeff))]
Mat=np.array(MatCoeff)
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([MatCoeff, 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(MatCoeff)):
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))
return np.asmatrix(Mat)
def echelonMatCoeff(MatCoeff): #take echelonMAt but without b.
b= [0 for i in range(len(MatCoeff))]
Mat = [MatCoeff[i]+[b[i]] for i in range(0,len(MatCoeff))]
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(MatCoeff), np.asmatrix(Mat[:, :len(MatCoeff[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(MatCoeff)):
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(MatCoeff[0])]))
return np.asmatrix(Mat)
def echelonRedMat(MatCoeff, b):
Mat=echelonMat(MatCoeff,b)
Mat=np.array(Mat)
MatAugm = [MatCoeff[i]+[b[i]] for i in range(0,len(MatCoeff))]
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 Mat
def f(a,x):
return x+ a

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