Code: AC09 / AT09                                                         Subject: NUMERICAL COMPUTING

Time: 3 Hours                                                                                                     Max. Marks: 100

NOTE: There are 9 Questions in all.

· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1       Choose the correct or best alternative in the following:                                         (2x10)

a. The operator defined by

 is called

  (A)  Central difference operator            (B)  Backward difference operator

                    (C)  Simple Equation                            (D)  Averaging operator

b. Computers operate with a fixed number of digits. Any excess digits produced by multiplication or division are lost. This type of error is an example of

(A) Truncation Error                             (B)  Round-off Error

(C) Inherent Error (D) Percentage Error

c. For Simpson’s 3/8^th rule, the interpolating polynomial is a

(A) Straight-line                                  (B)  Parabola

(C)  Cubic Curve                                 (D)  None of the above

d. The value of  where is forwarding difference operator is

(A) (B)

(C) .                          (D)  .        

             e.   Which of the method is not used for finding solutions of differential equation             

(A) Runga-Kutta Method. (B)  Euler’ Method

(C) Trapezoidal Method                      (D)  Taylor’s Series Method.

f.    If λ is an eigen value of a Matix A, then is the eigen value of

(A) A² (B)  Transpose of A.

(C)  A^–1 (D)  None of the above.

g. Which interpolation method is used for unequal intervals

(A) Langrage’s Interpolation Formulae

(B) Bessel’s formula

(C) Gauss’s forward Interpolation formula

(D) None of the above

h.   is equal to

(A) (B)

(C)  (D) None of the above

i.     The rate of convergence of Regula-Falsi method is

(A)   Linear. (B)  Quadratic

(C)  Cubic. (D)  None of the above.

             j.    The value of   is

(A)  (B) .

(C) . (D) None of the above

                   where h is called interval of difference.

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   What do you mean by the following terms in numerical analysis?

(i) truncation error                                (ii) round-off error                                          (3)

             b.   If π is approximated as 3.14 instead of 3.14156, find the absolute, relative and percentage errors.   (6)

             c.   Find the convergence of Newton-Raphson method.                                                    (7)

Q.3       a.   Find a root of the equation using the bisection method correct to three decimal places.                                                                                                                             (6)

             b.   Solve the following system of non-linear equations                                                                                                                     

 and

                   Starting with using iteration method correct to three decimal places.                (6)

             c.   The system of equations Ax=b is to be solved iteratively by x_n+1=Mx_n+b.

Find a necessary and sufficient condition on k for convergence of the Jacobi Method.                                                                                               (4)

Q.4      a.     Solve the following equations by using Gauss-Jordan Method                                   (10)

            b.    If A is a strictly diagonally dominant matrix, then prove that the Jacobi iteration scheme converges for any initial starting vector.                                                                                                  (6)

Q.5      a.   For the Matrix A =  find all the eigenvalues and the corresponding eigen vectors.                                                                                       (8)

b.  Transform the matrix

    to tridiagonal form by Given’s method.                                  (6)

c.                                                         What is the advantage of special cyclic Jacobi Method over Jacobi Method for diagonal form.                                                                                                        (2)

Q.6    a. From the following table, estimate the number of students who obtained marks between 40 and 45.                                                                       (8)

Marks	30-40	40-50	50-60	60-70	70-80
Number of students	31	42	51	35	31

b.   Find the value of ƒ(4) from the following table                                                              (4)

x	0	1	2	3
ƒ(x)	1	2	1	10

           c.     Find the least squares approximation of second degree for the discrete data                (4)

x	–2	–1	0	1	2
ƒ(x)	15	1	1	3	19

Q.7   a.      A particle is moving along a straight line. The displacement x at some instances t are given below

t	0	1	2	3	4
x	5	8	12	17	26

                  Find the velocity and acceleration of the particle at t=4.                                                (6)

b.      For the method

                  determine the optimal value of h, using the criteria

                  (i)  

                  (ii) 

                  where RE & TE denote Round off error and Truncation error respectively.                 (10)

Q.8   a.    A solid of revolution, is formed by rotating about X-axis, the area between X axis, x=0 & x=1 and a curve through the points with following coordinates.                                                 (6)

X	0	0.25	0.5	0.75	1
Y	1	0.98	0.95	0.91	0.84

                  Find the volume generated by solid of revolution.

            b.   Determine a, b, c such that the formula  is exact for polynomials of as high order as possible.                                                                                   (6)

            c.   Evaluate using Trapezoidal rule.                                                                     (4)

Q.9      a.   Use the Romberg’s method to compute correct to 4 decimal places. (6)

b.   Apply Runge-Kutta method of 4^th order, to solve at x=0.2, 0.4                           (10)