Assignment No: 01
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 ,