Question

PLEASE USE PYTHON CODE

7. Use Newton's method to find the polynomial that fits the following points:

x = -3, 2, -1, 3, 1

y = 0, 5, -4, 12, 0

Answer 1

NEWTON'S METHOD

GRAPHICAL INTERPRETATION :Let the given equation be f(x) = 0 and the initial approximation for the root is x₀. Draw a tangent to the curve y = f(x) at x₀ and extend the tangent until x-axis. Then the point of intersection of the tangent and the x-axis is the next approximation for the root of f(x) = 0. Repeat the procedure with x₀ = x₁until it converges. If m is the slope of the Tangent at the point x ₀ and b is the angle between the tangent and x-axis then

m = tan b = f '(x0 ) = f(x0) x 0-x1 (x0-x1) * f '(x0) = f(x0 ) x1 = x0   - f(x0) f '(x0)

This can be generalized to the iterative process as

xi+1= xi  -	f(xi)	i = 0, 1, 2, . . .
f '(xi)

Use Newton's method to find the polynomial that fits the following points:

x = -3, 2, -1, 3, 1

y = 0, 5, -4, 12, 0

y=ax⁴+bx³+cx²+dx+e

y=a(-3)⁴+b(-3)³+c(-3)²+d(-3)+e

y=-81a-27b+9c-3d+e=0 (first equation)

y= y=ax⁴+bx³+cx²+dx+e

y=a(2)⁴+b(2)³+c(2)²+d(2)+e

y=16a+8b+4c+2d+e=5   (second equation)

y=ax⁴+bx³+cx²+dx+e

y=a(-1)⁴+b(-1)³+c(-1)²+d(-1)+e

y=a+b+c+d+e =-4 (third equation)

y=ax⁴+bx³+cx²+dx+e

y=a(3)⁴+b(3)³+c(3)²+d(3)+e

y=81a+27b+9c+3d+e=12 (fourth equation)

y=ax⁴+bx³+cx²+dx+e

y=a(1)⁴+b(1)³+c(1)²+d(1)+e

y=a+b+c+d+e=0 (fifth equation)

Python program to implement Newton’s method:

from scipy import misc

def NewtonsMethod(f, x, tolerance=0.000001):

    while True:

        x1 = x - f(x) / misc.derivative(f, x) 

        t = abs(x1 - x)

        if t < tolerance:

            break

        x = x1

    return x

def f(x):

    return (1.0/4.0)*x**3+(3.0/4.0)*x**2-(3.0/2.0)*x-2

x = 4

x0 = NewtonsMethod(f, x)

print('x: ', x)

print('x0: ', x0)

print("f(x0) = ", ((1.0/4.0)*x0**3+(3.0/4.0)*x0**2-(3.0/2.0)*x0-2 ))