CHOOSE ONLY 1! 4A or 4B, up to you. We are programming in python. We are able to

Question

CHOOSE ONLY 1! 4A or 4B, up to you.

We are programming in python. We are able to use numpy and scipy

CHOOSE ONLY ONE 4a) Consider f(x)-(1-x)/x, which has a root at x-1. Write a code to find the root using the secant method with xo=01 and X1-10. Comment on the convergence. Try some other starting intervals. Consider the function glx)-(x-a)/(bx-c), where a,b, and c are free parameters. Use your 3 data points (x i,f(x_i) to construct an approximation for f and use it to numerically determine the root 4b) Let a root finding method be given by: Xn+1-Xn +d , where d is a positive integer and fíx) is a smooth function, and (g)d) means d derivatives of the function g. Give a proof of the convergence rate of the iteration near a simple root r (f(r) 0). Code the method up for d-2 efficiently and demonstrate the convergence rate for a given test function(you pick, no trivial cases).

Explanation / Answer

from math import *

def scrub(f):

nf=''

nums='0123456789'

  for i in range(len(f)):

nf += f[i]

  if f[i] in nums:

  if i+1 < len(f):

  if f[i-1] != '.' and f[i+1] != '.' and f[i-1] not in nums and f[i+1] not in nums:

nf += '.0'

  elif i+1 == len(f):

nf +='.0'

  return nf

def funct(f):

f=scrub(f)

  def aval(x):

  try:

x=float(x)

  return eval(f)

  except:

  return f

  return aval

def newton(f,d,n,guess,exact=None):

F=funct(f)

D=funct(d)

p=[guess]

  for i in range(1,n+1):

prv = p[i-1]

p.append(prv-F(prv)/D(prv))

  if not exact==None: exact=(len(p)-1)*[exact]

  else: exact=(len(p)-1)*[p[len(p)-1]]

error = [abs(p_i-exact_i) for p_i,exact_i in zip(p,exact)]

  return [p,exact,error]

#==============================================================================

f='x**2-329'

d='2*x'

n=16

guess=2

exact=sqrt(329)

[p,exact,error]=newton(f,d,n,guess,exact)

import matplotlib

matplotlib.use("TKAgg")

import matplotlib.pyplot as plt

import numpy as np

plt.ion()

plt.plot(p,'r')  

plt.plot(exact,'y')

#plt.plot(np.log(error),'b--')

#

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