Numerical Methods — Winter 2023

State Budan’s Theorem.

Define Error. Explain different types of numerical errors with suitable example.

Derive an expression for Newton’s forward difference interpolation formula.

Define the following terms: Absolute Error, Relative Error, and Blunders.

Use Bisection method to find the root of the equation x3 – 5x + 1 = 0, in the interval [2, 3], correct upto three decimal places.

Give diagrammatic representation of the Successive Approximation method to find the root of the equation f(x) = 0, for the cases of convergence as well as divergence.

Solve the equation x4 − x – 10 = 10 by Newton Raphson method, taking initial guess as 2.0.

Find y(10) from the data given below using Lagrange’s interpolation. x 5 6 9 11 y 12 13 14 16

From the following data, find the value of y at x = 0.5, using Lagrange’s interpolation formula. x −2 −1 2 y −12 −8 3

Find y(46) and y(63) from the below given data using Newton’s interpolation: Age (x) 45 50 55 60 65 Premium (y) 114.84 96.16 83.22 74.48 68.48

Obtain Cubic Spline equation for subinterval [0, 1] for the data given in the table: x 0 1 2 f(x) 1 2 33 244

Fit a second degree parabola of the form y = ax2 + bx + c to the following data by using method of least squares: x 1 2 3 4 y 5 12 26 60 97

Determine the curve of the form y = a.xb, which is the best fit to the following data according to least square equation. x 1.0 1.5 2.0 2.5 3.0 3.5 y 0.01 0.405 0.693 0.916 1.098 1.252

From the data, find numerically the first and second order derivatives at x = 1.3. x 0.5 0.7 0.9 1.1 1.3 1.51 y 1.48 1.64 1.78 1.89 1.96 2.00

The table below gives the results of an observation, ‘θ’ is the observed temperature in degrees centigrade of a vessel of cooling water, ‘t’ is the time in minutes from the beginning of observation. Find the appropriate rate of cooling at t = 3 and t = 3.5. t 1 3 5 7 9 θ 85.3 74.5 67.0 60.5 54.3

Find the first two derivatives of ‘x1/3’ at x = 50 and x = 56 from the table below: x 50 51 52 53 54 55 56 y = x1/3 3.6840 3.7084 3.7325 3.7563 3.7798 3.8030 3.8259

From the data, find numerically the first and second order derivatives at x = 1.1. x 1.0 1.2 1.4 1.6 1.8 2.0 y 0 0.128 0.544 1.296 2.432 4.000

A Curve passes through the points (1, 2), (1.5, 2.4), (2.0, 2.7), (2.5, 2.8), (3, 3), (3.5, 2.6) and (4.0, 2.1). Obtain the area bounded by the curve, the X-axis and x = 1 and x = 4.

Ariver is 80 metres wide. The depth ‘d’ in metres at a distance ‘x’ metres from one bank is given by the following table. Calculate the area of cross-section of the river using Simpson’s 1/3 rule. x (distance in metres) 0 10 20 30 40 50 60 70 80 d (depth in metres) 0 4 7 9 12 15 14 8

Use Milne-Simpson’s Predictor corrector formula to solve y' = 2y − y2, for x = 0.2 and x = 0.25 if y(0) = 1 y(0.05) = 1.0499584 y(0.10) = 1.0996680 y(0.15) = 1.148850

Given the following differential equation y' = x2 + x4y, with y(0) = 3 and starting values y(0.1) = 3.0050, y(0.2) = 3.0202 and y(0.3) = 3.0465. Find y(0.4) using Adam–Bashforth–Moulton’s Predictor–Corrector method.

Solve the following system of linear equations using Gauss-Seidel method: 2x – 2x + 5x =123 13 2x + 3x + 4x =123 3x – x + 3x =123

Solve the following ordinary differential equation using Taylor series method: y' = y2 + x; given that y(0) = 0, find y(0.2).

Solve the following differential equation y' = x – 2y, y(0) = 1, using Runge– Kutta 4th order method to find y(0.1) and y(0.2).

Use Runge Kutta 4th order method to solve y(0.2) and y(0.4) when y' = (2xy + ex) / (x2 + x.ex) given that y(0) = 0 and h = 0.2.