Quantitative Methods Answers

b) Set up the Gaussi-Jacobi iteration scheme in matrix form for the linear system of equations

4 3
4 2
4 3
2 3
1 2 3
1 2
− + =
− + − =
− =
x x
x x x
x x

Show that the iteration scheme is convergent. Hence find the rate of convergence of this
method.

2. a) Find by Newton’s method the roots of the following equations correct to three places of
decimals.
i) log .4 772393 x 10 x = near x = 6 .
ii) f (x) = x − 2sin x near x = 2 .

b) Solve the IVP, ; )4(y 4
x 4y
1
y
2
=
−
′ = using Euler’s method. Find )2.4(y with h = 2.0
and 1.0 and extrapolate the value )

c) Obtain the cube root of 12 using Newton-Raphson formula.

Draw a flowchart that reads the names , sex and score of each student and outputs the names of the male students that scored above 75% and how many they are?

use newton's method to approximate SQRT of 11 to 5 decimal places

The technique of determining an approximate value of f(x) for a non-tabular value of x which lies outside the internal [a, b] is known as............

For the given functions f (x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct interpolation polynomials
of degree at most one and at most two to approximate f (0.45), and find the absolute error.
a. f (x) = cos x

In recording measured values, a digit is significant if and only if it affects the -------------- of the measurement

A. absolute error
B. relative error
C. inherent error
D. truncation error

8 The zeroeth divided difference of the function f, with respect to
x i
, denoted by
f[x i ]
is the same as



f[x i ]=f(x i )



f[x 0 ]=f(x i )



f[x]=f(x i )



f[x]=f(xi)





9 The quantity
L 0 (x)
of the Lagrange’s interpolating polynomial P(x) is equal to



(x−x 1 )(x−x 2 )(x−x 3 )(x 0 −x 1 )(x 0 −x 2 )(x 0 −x 3 )



(x−x 1 )(x1−x 2 )(x 0 −x 3 )(x 0 −x 1 )(x 0 −x 2 )(x 0 −x 3 )



(x−x 1 )(x−x 2 )(x−x 3 )(x−x 1 )(x−x 2 )(x−x 3 )



(x−x 1 )(x−x 0 )(x−x 3 )(x 0 −x 4 )(x 0 −x 2 )(x 0 −x 3 )





10 A square matrix is called ………….. if all the elements above the main diagonal vanish.

upper triangular
triangular

lower triangular

rectangular