Quantitative Methods Answers

a)Use Newton's method to approximate the root of the equation xsinx =1 in [0,pi/2].Taking x=1 as the initial guess calculate x1,x2,x3(Work to at least 6 decimal places)
b)Show that the equation x^3-3x^2+2x=1 has no root in the interval [-1,1]

Q: The equation x^3+2x^2-5=0 has a positive real root in the interval (1, 2). Write a fixed point iteration method and show that it converges. Starting with initial approximation x=1.5 find the root of the equation. Perform two iterations.

A book binder has one printing press, one binding machine and manuscripts of seven
different books. The time required for performing printing and binding operations for
different printing and binding operations for different books are shown below:

Book 1 2 3 4 5 6 7

Printing Time 20 90 80 20 120 15 65
(Days)

Binding Time 25 60 75 30 90 35 50
(Days)

Find the optimum sequence of processing of the jobs that minimises the total time required.
Also compute the optimal time required.

1.a-Solve 9x^4-18x^3-31x^2 8x 12=0 by ferrari's method. 2.b-solve the equeation in(a) given that the sum of two of its roots is zero

The position ) f (x of a particle moving in a line at various times k
x is given in the
following table. Estimate the velocity and acceleration of the particle at x = 2.1 .

x 1.0 1.2 1.4 1.6 1.8 2.0 2.2
f (x) 2.72 3.32 4.06 4.96 6.05 7.39 9.02

Using finite differences, show that the data

x − 3 − 2 −1 0 1 2 3
f (x) 13 7 3 1 1 3 7
represents a second degree polynomial. Obtain this polynomial using interpolation
and find f(5.2) .

The function ) f (x) = ln(1+ x is to be tabulated at equispaced points in the interval
,2[ using linear interpolation. Find the largest s ]3 tep size h that can be used so that
the error 4
5 10−
≤ × in magnitude

Find all the roots of the equation 0 6 11 6
3 2
x + x + x + = by the Graeffe’s root
squaring method using three squaring.

Using synthetic division and perform two iterations of the Birge-Vieta method to find
the smallest positive root of the equation 0 3 3 3 2
4 3 2
x − x + x − x + = . Use the initial
approximation 5.0 x0 = .

Solve the system equations
4 5x − 2y + z =
6 7x + y − 5z =
10 3x + 7y + 4z =
using LU decomposition method.