Numerical Methods — Winter 2022

Construct the difference table for f(x)= (x+2)2,x = 1,2,3 and find ∇2f(3).

Write the Iterative formula of NR method. Using it find the first iteration of the solution of x3−x−1 = 0.

Find the approximate root of the equation x−cosx = 0 up to three decimal places using bisection method.

Prove that (1+∆)(1−∇)= 1.

Using trapezoidal rule evaluate ∫ 1 1 dx for ℎ = 1 . 0 1+x

Find the polynomial using Newton’s forward interpolation formula from the following data: x 1 2 3 f(x) -1 -1 1

Find Γ(1.01), using Newton’s backward interpolation formula from the following: x 1 1.02 1.04 Γ(x) 1.0000 0.9888 0.9784

Using inverse Lagrange’s interpolation formula find x for y = 5 from the following table. x 1 3 y 3 12 19

If f(x)= 1⁄ x , find the divided differences [a,b] and [a,b,c].

Construct a second degree polynomial using Lagrange’s interpolation formula from the following x 0.3 0.5 0.7 f(x) 0.61 0.69 0.72 Hence Compute f(0.4)

Using Simpson’s 1 rd rule evaluate ∫ 1.8 e−xdx for ℎ = 0.2.

Using Simpson’s 3 th rule evaluate ∫ 6 exdx for ℎ = 1.80

Using Newton’s divided difference formula, compute f(10.5) from the following: x 10 11 13 17 f(x) 2.3026 2.3979 2.5649 2.8332

Applying Budan’s theorem, calculate the number of real roots of x5−x3+ 1 = 0.

dy Apply Runge-Kutta method of order two, calculate y(1.1), given that = x− dx y with y(1) = 0 taking ℎ = 0.1. Page 1 of

dy Apply Runge-Kutta method of forth order to calculate y(0.2), given that = dx x+y with y(0) = 1 taking ℎ = 0.1.

Using Euler’s method, find y(0.02) from the initial value problem y′ = y,y(0)= 1 by taking step size ℎ = 0.01.

Find y(0.2) from dy = x+y with y(0) = −1 using Taylor’s series method. dx

Using Gauss Jacobi method, solve the following system up to three decimal places. 10x+y+z=6, x+10y+z=6 , x+y+10z=6

Fit a straight line using least square method from the following data: x 0 1 y 1 3

dy Using modified Euler’s method, find y(0.1) from the initial value problem = dx xy,y(0)= 1 by taking step size ℎ = 0.1.

Using Gauss Seidel method, solve the following system up to three decimal places. 5x+y+2z=19, x+4y-2z=-2 , 2x+3y+8z=39

Write down the condition for convergence of the system of three linear equations.

Write down the normal equations for the fitting of linear curve using least square method.

Fit a second degree polynomial using least square method from the following data: x 0 1 -1 y 12 9 Page 2 of