816.sagews

a = ComplexField()#e.g.: a(4,2) = 4+2i
b = .5 
c = -.5
d = a(.5,1) #.5+i
e = a(.5,-1)#-.5+i
TOL = 10^(-10)

def newton_con(p_0, constant):
    f_1 = derivative(f,x)
    return n(p_0 - (f(p_0)/f_1(p_0)))

f(x) = x^4 - 1
#the purpose method is to make the plot of the newtons method
def plot_evaluation(p_0, tol, n_0):
   var('i')
   var('p')
   data = []
   temp = []
   con = p_0# to pass constant
   i=1
   while i<=n_0:
       p = newton_con(p_0,con)
       data.append((p.real(),p.imag()))
       if abs(p - p_0) < tol:
           break
       p_0 = p
       i = i + 1
   return data

#getting the data of roots
pts_b = plot_evaluation(b,TOL,500)
pts_c = plot_evaluation(c,TOL,500)
pts_d = plot_evaluation(d,TOL,500)
pts_e = plot_evaluation(e,TOL,500)

#plot the points for the roots in red,green,purple, blue
plot_b = point(pts_b,rgbcolor=(1,0,0))
plot_c = point(pts_c,rgbcolor=(0,1,0))
plot_d = point(pts_d,rgbcolor=(.5,0,.5))
plot_e = point(pts_e,rgbcolor=(0,0,1))
graph = plot_b+plot_c+plot_d+plot_e
graph.show(xmin = -2, xmax=2,ymin=-2,ymax=2)

def evaluation(p_0, tol, n_0,root):
    #the extra parameter in the definition, root, makes things awesome later on...you'll see why
   var('i')
   var('p')
   data = []
   temp = []
   con = p_0
   i=1
   #I need to to an if-else statement in order to process the original p_0's iteration inside the while loop
   while i<=n_0:
       if i == 1:
           temp = (Integer((i-1)),p_0, f(p_0), abs(p_0-root), abs(p_0-root)/abs(root))
           data.append(temp)
           i = i +1
       else:
           p = newton_con(p_0,con)
           #store the other data
           temp = ((i - 1),p,f(p),abs(p-root), abs(p-root)/abs(root))
           data.append(temp)
           if abs(p - p_0) < tol:
               break
           p_0 = p
           i = i + 1
   return data

#vvvvvv  awesome-ness below  vvvvvvv 
data_b = evaluation(b,TOL,500,1)
data_c = evaluation(c,TOL,500,-1)
data_d = evaluation(d,TOL,500,a(0,1))
data_e = evaluation(e,TOL,500,a(0,-1))
#^^^^^^  awesome-ness above  ^^^^^^^

# put data into a matrix
mat_b = matrix(data_b)
mat_c = matrix(data_c)
mat_d = matrix(data_d)
mat_e = matrix(data_e)

#now to make a Latex array, If you want a table,
#i just modified \left(\begin{array}{rrrrr} 
#to
#\begin{tabular}{|r|r|r|r|r|} 
#and
#\end{array}\right) 
#to
#\end{tabular}
#in my latex editor
latex(mat_b)

\left(\begin{array}{rrrrr} 0.000000000000000 & 0.500000000000000 & -0.937500000000000 & 0.500000000000000 & 0.500000000000000 \\ 1 & 2.37500000000000 & 30.8166503906250 & 1.37500000000000 & 1.37500000000000 \\ 2 & 1.79991161247995 & 9.49553824779953 & 0.799911612479953 & 0.799911612479953 \\ 3 & 1.39280696583296 & 2.76325563488009 & 0.392806965832955 & 0.392806965832955 \\ 4 & 1.13713195528050 & 0.672027655272641 & 0.137131955280504 & 0.137131955280504 \\ 5 & 1.02287186525853 & 0.0946743271371449 & 0.0228718652585267 & 0.0228718652585267 \\ 6 & 1.00075576570494 & 0.00302649163762347 & 0.000755765704943512 & 0.000755765704943512 \\ 7 & 1.00000085569472 & 3.42278329013013 \times 10^{-6} & 8.55694724188893 \times 10^{-7} & 8.55694724188893 \times 10^{-7} \\ 8 & 1.00000000000110 & 4.39293046383682 \times 10^{-12} & 1.09823261595920 \times 10^{-12} & 1.09823261595920 \times 10^{-12} \\ 9 & 1.00000000000000 & 0 & 0.000000000000000 & 0.000000000000000 \end{array}\right)

latex(mat_c)

\left(\begin{array}{rrrrr} 0.000000000000000 & -0.500000000000000 & -0.937500000000000 & 0.500000000000000 & 0.500000000000000 \\ 1 & -2.37500000000000 & 30.8166503906250 & 1.37500000000000 & 1.37500000000000 \\ 2 & -1.79991161247995 & 9.49553824779953 & 0.799911612479953 & 0.799911612479953 \\ 3 & -1.39280696583296 & 2.76325563488009 & 0.392806965832955 & 0.392806965832955 \\ 4 & -1.13713195528050 & 0.672027655272641 & 0.137131955280504 & 0.137131955280504 \\ 5 & -1.02287186525853 & 0.0946743271371449 & 0.0228718652585267 & 0.0228718652585267 \\ 6 & -1.00075576570494 & 0.00302649163762347 & 0.000755765704943512 & 0.000755765704943512 \\ 7 & -1.00000085569472 & 3.42278329013013 \times 10^{-6} & 8.55694724188893 \times 10^{-7} & 8.55694724188893 \times 10^{-7} \\ 8 & -1.00000000000110 & 4.39293046383682 \times 10^{-12} & 1.09823261595920 \times 10^{-12} & 1.09823261595920 \times 10^{-12} \\ 9 & -1.00000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 \end{array}\right)

latex(mat_d)

\left(\begin{array}{rrrrr} 0 & 0.500000000000000 + 1.00000000000000i & -1.43750000000000 - 1.50000000000000i & 0.500000000000000 & 0.500000000000000 \\ 1 & 0.199000000000000 + 0.782000000000000i & -0.769771929767000 - 0.356006053656000i & 0.295169442862909 & 0.295169442862909 \\ 2 & -0.174237430404181 + 0.935443186698232i & -0.392751834733713 + 0.550704885322329i & 0.185812443871586 & 0.185812443871586 \\ 3 & 0.0216031068276627 + 0.948596404538570i & -0.192816192524941 - 0.0737216971357816i & 0.0557586213151812 & 0.0557586213151812 \\ 4 & -0.00377332053071798 + 1.00342102746120i & 0.0136684777211655 + 0.0152485006219681i & 0.00509326778383489 & 0.00509326778383489 \\ 5 & -0.0000385285289577100 + 0.999996460974189i & -0.0000141649347179529 + 0.000154112479366356i & 0.0000386907255984654 & 0.0000386907255984654 \\ 6 & 4.08921007549279 \times 10^{-10} + 0.999999997792076i & -8.83169404275463 \times 10^{-9} - 1.63568401936272 \times 10^{-9}i & 2.24547158064110 \times 10^{-9} & 2.24547158064110 \times 10^{-9} \\ 7 & -2.70859898669698 \times 10^{-18} + 1.00000000000000i & 1.08343959467879 \times 10^{-17}i & 2.70859898669698 \times 10^{-18} & 2.70859898669698 \times 10^{-18} \\ 8 & 1.00000000000000i & 0 & 0.000000000000000 & 0.000000000000000 \end{array}\right)

latex(mat_e)

\left(\begin{array}{rrrrr} 0 & 0.500000000000000 - 1.00000000000000i & -1.43750000000000 + 1.50000000000000i & 0.500000000000000 & 0.500000000000000 \\ 1 & 0.199000000000000 - 0.782000000000000i & -0.769771929767000 + 0.356006053656000i & 0.295169442862909 & 0.295169442862909 \\ 2 & -0.174237430404181 - 0.935443186698232i & -0.392751834733713 - 0.550704885322329i & 0.185812443871586 & 0.185812443871586 \\ 3 & 0.0216031068276627 - 0.948596404538570i & -0.192816192524941 + 0.0737216971357816i & 0.0557586213151812 & 0.0557586213151812 \\ 4 & -0.00377332053071798 - 1.00342102746120i & 0.0136684777211655 - 0.0152485006219681i & 0.00509326778383489 & 0.00509326778383489 \\ 5 & -0.0000385285289577100 - 0.999996460974189i & -0.0000141649347179529 - 0.000154112479366356i & 0.0000386907255984654 & 0.0000386907255984654 \\ 6 & 4.08921007549279 \times 10^{-10} - 0.999999997792076i & -8.83169404275463 \times 10^{-9} + 1.63568401936272 \times 10^{-9}i & 2.24547158064110 \times 10^{-9} & 2.24547158064110 \times 10^{-9} \\ 7 & -2.70859898669698 \times 10^{-18} - 1.00000000000000i & -1.08343959467879 \times 10^{-17}i & 2.70859898669698 \times 10^{-18} & 2.70859898669698 \times 10^{-18} \\ 8 & -1.00000000000000i & 0 & 0.000000000000000 & 0.000000000000000 \end{array}\right)