Quantitative Methods Answers

Using the Horner’s method find the values of f(4) and f'(4) for the polynomial

f(x) = x^4 +2x^3 −x^2 +1.

From the data

x 1 1.5 2 2.5 3

f(x) 1 .5 1.3 .8 1.5

interpolate the value of f(2.8) using the Newton’s backward difference formula.

Find a root of the equation 3x^3+10x^2+10x+7 = 0 which is close to −2.0, using

the Berge -Vieta method. Perform two iterations of the method.

Consider the following data

x 1.0 1.3 1.6 1.9 2.2

f(x) 0.76519 0.62008 0.45540 0.28181 0.11036

Use stirling’s formula to approximate f(1.5) with x0 = 1.6

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

From the values of f(x) = xex given in the table

x 1.8 1.9 2.0 2.1 2.2

f(x) 10.8894 12.7032 14.7787 17.1489 19.8550

find f

00(2.0) using the central difference formula of O(h

2

) for h = 0.1 and h = 0.2.

Calculate T.E. and actual error.

Taking the endpoints of the last interval obtained in f(x) = x^3−5x^2 +1 = 0 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.

Perform four iterations of the inverse power method to compute the smallest

eigenvalue in magnitude, and the corresponding eigenvector of the matrix A given below above.

A=1 −1 1

2 0 3

1 4 −1