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 |

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