Numworks: Quadratic Equation and Newton's Method - Do Guesses Matter?

Numworks: Quadratic Equation and Newton's Method - Do Guesses Matter?  

Introduction

Let A, B, C, be positive, non-zero coefficients of the quadratic equation:

Ax^2 + Bx - C = 0 with A > 0, B > 0, C > 0

First, start with an initial guess x0 = 0.  Count how many iterations it takes for the method to reach a suitable answer.   For this experiment, we will use a tolerance of 8 decimal places (tol = 1E-8), setting a maximum of 200 iterations.   

Consider a "reduced" equation to obtain guesses for the original equation:  

Ax^2 - C = 0

Hence, our initial guesses are x0 = √(C/A) and x0 = -√(C/A)

Using Newton's Method to determine the roots with a guess of x0:

x_n+1 = x_n - [Ax^2 + Bx - C]/[2Ax + B]

So which of the following guesses gets us to a solution faster?

x0 = 0

x0 = √(C/A) 

x0 = -√(C/A)

The Python script below tests the question.  

Python Test Script:  guesttest.py

from math import *

def testf(a,b,c,w0):

 n=1

 w1=1

 while abs(w1)>1e-8

  w1=(a*w0**2+b*w0-c)/(2*a*w0+b)

  w0=w0-w1

  n=n+1

  if n==201:

   print('Exceeded iterations')

   break

 print('Root: '+str(w0)

 print('Iterations: '+str(n))

 return

# EWS 2020-08-10

print("y=ax**2+bx-c=0")

print("a>0,b>0,c>0")

a=float(input('a? '))

b=float(input('b? '))

c=float(input('c? '))

# initial conditions

x1=0

x2=sqrt(c/a)

x3=-sqrt(c/a)

# results

print('x1: '+str(x1))

testf(a,b,c,x1)

print('----')

print('x2: '+str(x2))

testf(a,b,c,x2)

print('----')

print('x3: '+str(x3))

testf(a,b,c,x3)

print('----')

Results

Test Results - Iterations Used to get to a Solution

Ax^2 + Bx - C = 0 with A > 0, B > 0, C > 0

A	B	C	x0 = 0	x0 = √(C/A)	x0 =-√(C/A)
1	1	4	6	6	8
1	4	4	6	6	error reached
1	5	4	6	6	9
1	5	6	6	6	12
2	5	6	6	6	8
3	5	6	7	6	7
8	13	81	8	6	6
16	13	81	8	5	6
32	13	81	9	5	5
64	13	81	9	5	5
1	40	10	5	6	6
1	40	25	5	6	6
1	40	40	5	6	7
1	40	55	6	6	8
10	18	42	7	6	7
10	18	52	7	6	6
10	18	62	7	6	6
10	18	72	7	6	6

From the data presented, I think it is inconclusive of whether which guess generally makes for getting to the solution faster.   What do you find?  

Eddie 

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