Figure 8.5:

Newton-type iterations for solution of R(z)=0 from Example~\ref {ex:num-newton-fifth-root}. Left: exact Newton method. Right: constant Jacobian approximation.

Code for Figure 8.5

Text of the GNU GPL.

main.py

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# [makes] Newton_Type_Solver.mat
# Generate figure for Example 8.5.
import numpy as np
from scipy.io import savemat

z0    = 2.0       # Initial guess.
Mk    = 5*z0**4   # Inexact Jacobian.
iter1 = 10        # Number of iterations for exact Jacobian solver.
iter2 = 100       # Number of iterations for inexact Jacobian solver.

# Exact Jacobian Newton's method on R(z) = z^5 - 2.
zk1 = np.zeros(iter1 + 1)
zk1[0] = z0
dashline1 = np.zeros((2*iter1 + 1, 2))
dashline1[0] = [z0, 0]

for i in range(iter1):
    zk1[i+1] = zk1[i] - (zk1[i]**5 - 2) / (5*zk1[i]**4)
    dashline1[2*i+1] = [zk1[i],     zk1[i]**5 - 2]
    dashline1[2*i+2] = [zk1[i+1],   0]

# Inexact Jacobian Newton's method.
zk2 = np.zeros(iter2 + 1)
zk2[0] = z0
dashline2 = np.zeros((2*iter2 + 1, 2))
dashline2[0] = [z0, 0]

for i in range(iter2):
    zk2[i+1] = zk2[i] - (zk2[i]**5 - 2) / Mk
    dashline2[2*i+1] = [zk2[i],     zk2[i]**5 - 2]
    dashline2[2*i+2] = [zk2[i+1],   0]

ztrue = 2**(1/5)
z = np.arange(1.0, 2.16, 0.01)
R = z**5 - 2

savemat('Newton_Type_Solver.mat', {
    'z': z, 'R': R, 'zk1': zk1, 'zk2': zk2,
    'dashline1': dashline1, 'dashline2': dashline2, 'ztrue': ztrue,
})