Quantitative Methods Answers

Compute the largest eigenvalue in magnitude, and the corresponding eigenvector of

the matrix

A=1 −1 1

2 0 3

1 4 −1

by performing four iterations of the power method

Compute integral 0 to 4 f(x)dx using the Romberg integral technique on the trapezoidal integrals evaluated by the trapezoidal rule taking h = 1 and h = 0.5. The tabulated

values are given below.

x 0 0.5 1 1.5 2.0 2.5 3.0 3.5 4.0

f(x) 1 4 3 2 2.5 2.9 3.6 4 1.8

Use Euler’s method to solve the I.V.P.

y'=(x-y)/2, on [0,3], with y(0) = 1, and h = 0.5

Taking the endpoints of the last interval obtained in part a) above as the initial

approximations, perform two iterations of the secant method to approximate the

root.

Find the interval of unit length that contains the smallest positive root of the

equation f(x) = x^3 −5x^2+1 = 0. Starting with this interval, find an interval of

length 0.05 or less that contains the root, by Bisection method.

Find the inverse of the matrix

1 −1 4

2 9 8

6 5 2

using LU decomposition method.

Compute the largest eigenvalue in magnitude, and the corresponding eigenvector of the matrix

A =1 −1 1

2 0 3

1 4 −1

by performing four iterations of the power method.

Find the inverse of the following matrix, using Gauss Jordan method.

1 2 3 4

2 3 4 1

3 4 1 2

4 1 2 3

Given the following data, estimate the value of f(5) using

i) Lagrange’s interpolation

ii) Newton’s divided difference interpolation

x 1 4 6 7 10

f(x) -3 9 10 9 8

Using simulated business cases illustrate the following terms as used in linear cost volume profit analysis. 1. Market Equilibrium. 2. The margin of safety