Maximum Marks: 20 Due Date: 24 -11-2019

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Question #1: Find the root of the equation, Perform three iteration of the equation,

ln (x −1) + sinx = 0 by using Newton Raphson method.

Question #2: Solve the system of linear equations with the help of Gaussian elimination method.

x + y + z = 6; 2x − y + z = 3; x + z = 4

Question #1: Find the root of the equation, Perform three iteration of the equation,

ln (x −1) + sinx =0 by using Newton Raphson method.

Ans: Let f(x) = ln(x+1) + sinx = 0 and f(x) = 1/(x-1) + cosx

F (1.5) = ln(0.5) + (1.5) = - 0.0667

F(2) = ln(1) + sin(2) = 0.035

Since f (1.5) f (2) < 0 so roots lies in interval [1.5, 2]

Let x0 = 1.75 . x0 can be taken in the interval any real number [ 1.5 , 2 ], we let mid point

of this interval .

As we know Newton Raphson method is

Xn+1 = xn – f ( xn ) / f(xn)

First iteration

X1 = x0 –f(x0) / f(x0) = 1.75 - f(1.75) / f(1.75)

= 1.75 – (-0.2571 / 2.3329) = 1.8602

Second iteration:

X2 = x1 - f(x) / f(x) = 1.8602 –[ f(1.8602) / f(1.8602)]

= 1.8602 - ( -0.1181 / 2.1620 ) = 1.9148

Third iteration:

X3 = x2- f(x2) / f(x2) = 1.9148 –f(1.9148) / f(1.9148)

= 1.9148 – [-0.0556/2.0926]

= 1.9414

Question #2: Solve the system of linear equations with the help of Gaussian elimination method.

x + y + z = 6;2x − y + z = 3;x + z = 4

ANS: In Gaussian elimination method we convert the augmented matrix into reduce

Echelon form therefore,

Augmented matrix is

R2- 2R1 , R3 – R1

-1R2 , -1R3

R23

R3-3R2

X + Y+ Z = 6 ;………………….(1)

Y = 2,

Z = 3

Put into eq (1),

we get X = 1 ,