unit 4assignment

1. Unit-IV
ASSIGNMENT – 1
1. Find a real root of each of the following equations correct to three decimal places, by
using BISECTION METHOD:
(i) 𝑥2
− 2𝑥 − 3 cos 𝑥 = 0 (ii) cos 𝑥 − 𝑥𝑒 𝑥
= 0
2. Find a real root of each of the following equations correct to four decimal places, by
using NEWTON - RAPHSON METHOD:
(I)𝑥 = 𝑒−𝑥
(ii)3𝑥 = cos 𝑥 + 1
3. Find a real root of each of the following equations correct to three decimal places, by
using REGULA FALSI METHOD
(i) 𝑥3
+ 𝑥2
− 1 = (ii) 𝑥𝑒 𝑥
− 2 = 0.
4. Evaluate√12 to five decimal places by Newton iterative method.
5. Find all the roots of cos 𝑥 − 𝑥2
− 𝑥 = 0 to five decimal places by Newton Raphson
method.
6. Find the value of(
1
17
)
1
3
correct up to four decimal places using Newton – Raphson
method.
7. Show that the following two sequences, both have the convergence of the second
order with the same limit√ 𝑎
xn+1 =
1
2
xn (1 +
𝑎
𝑥 𝑛
2 ) , xn+1 =
1
2
xn (3 −
𝑥 𝑛
2
𝑎
) .
8. Explain the order of convergence and prove that Newton – Raphson method is
second order convergent.
9. Prove that Regula Falsi method is linearly convergent.
10. Prove that the bisection method is linearly convergent.
ANSWERS
1. (i)1.728 (ii)0.517
2. (i)0.56714 (ii)0.6071
3. (i)0.75488 (ii)0.852605
4. 3.46410
5. 0.55000,-1.25115
6. 0.3889