Lab 9: Newton–Raphson Lab · Numerical Methods

Each step draws the tangent at xₙ. Try the failure presets and a good start for comparison. The derivative f′(x) is computed for you and shown below the table.

Type your equation and starting value. Switch methods to compare bisection, false position, Newton and secant on the same problem.

Newton–Raphson

Try this

Run it from 1.5, then from 0.6. How many iterations each?
Change f and df to 2x³ − 2x − 5 (Problem 1) and check the table.
Replace df by a central difference (f(x+h) − f(x−h))/(2h) with h = 1e-6. Does it still converge as fast?

Square root and reciprocal

Try this

Run it. How many correct digits after each step?
Change y = 0.1 to y = 0.3. What happens? (Hint: reciprocal iteration needs 0 < y₀ < 2/N.)

Failure gallery

Try this

Run it and match each line to a failure type.
Find a starting value for the cycle function that converges (try −1, −1.5).

Q1 Derive the Newton–Raphson formula and explain its geometrical meaning.

The tangent at

(x_n, f(x_n))

is

y = f(x_n) + f'(x_n)(x - x_n)

. Setting

y = 0

:

x_{n+1} = x_n - f(x_n)/f'(x_n)

. Geometrically,

x_{n+1}

is where the tangent line meets the

x

-axis.

Q2 Find a root of

2x^3 - 2x - 5 = 0

by Newton–Raphson.

Root in (1, 2).

x_0 = 1.5

:

x_1 = 1.608696

,

x_2 = 1.600645

,

x_3 = 1.600599

,

x_4 = 1.600599

. Root ≈ 1.600599.

Q3 Find

\sqrt{12}

by Newton's method from

x_0 = 3.5

, 6 decimal places.

x_{n+1} = \tfrac12(x_n + 12/x_n)

: 3.464286, 3.464102, 3.464102.

Q4 Show that Newton's method for

f(x) = x^3 - 2x + 2

from

x_0 = 0

never converges.

f'(x) = 3x^2 - 2

.

x_1 = 0 - 2/(-2) = 1

;

x_2 = 1 - 1/1 = 0

; so the iterates alternate 0, 1, 0, 1, … forever.

Q5 Why does Newton converge only linearly for

f(x) = (x-1)^2

?

f'(1) = 0

(double root). The iteration becomes

x_{n+1} = x_n - (x_n-1)/2

, so

e_{n+1} = e_n/2

: linear with rate ½. The quadratic error formula needs

f'(r) \ne 0

.

Q6 Give two advantages and two disadvantages of Newton–Raphson.

Advantages: quadratic convergence; one starting point and extends to systems. Disadvantages: needs

f'(x)

; may diverge or cycle, and fails when

f'(x) \approx 0

.