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. Given’s method is used to reduce the matrix  to tri-diagonal form.  The angle of orthogonal transformation is

  (A) .                                 (B) .

                    (C) .                                                (D)  .

b. If then is

(A) (B)

(C)  (D)

c. The order of the numerical differentiation method

(A) 1.                                                 (B)  2.

(C)  3.                                                 (D)  4.

d. The n-point Gauss Quadrature formula is exact for all polynomials of degree upto

(A)  n.                                                (B)  2n.

(C)  2n – 1.                                        (D)  2n + 1.          

             e.   If  is an eigenvalue of A, then the eigenvalue of A^–1 is        

(A) .                                              (B)  .

(C) .                                              (D) .

f.    The least squares approximation to the data

x	1	2	3	4
f(x)	6	9	14	21

                    is given as f(x)=5x. Then, the least squares error is given as

(A)  0.04.                                            (B)  4.

(C)  6.                                                 (D)  0.004.

             g.   The value of the integral

 using Simpson's  rule is

(A) (B) 

(C)  (D)

             h.   The integration rule is given. The error is of the form . Then, the value of c is given by

(A) .                                               (B)  .

(C)  .                                               (D) .

i.      An integral I is being evaluated by the composite trapezoidal rule. The values of I for two different step lengths and  are obtained as I()=0.6123,  I()=0.6011. A better approximation using Romberg integration is                                                         

(A)  0.5974.                                        (B)  0.6004.

(C)  0.1893.                                        (D)  0.5867.

             j.    Gauss-Seidel method is applied to solve the system of equations  real constant. The method converges for

(A)  .                                         (B)  all .

(C) .                                         (D) .

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Let  be a simple root of the equation f(x)=0. We try to find the root by means of the iteration formula

                   Find the order of convergence.                                                                               (10)

             b.   Find the interval in which the smallest positive root of the following equation

 lies. Determine the root correct to two decimals using the Secant method.                                                                       (6)

Q.3     a.   Find the error term of the method

                   as a power series in h. Derive the corresponding Richardson's Extrapolation scheme. Using this method and the Richardson's extrapolation, find the best value of  when  is given in tabular form as     (9)

x:	1	2	3	4	5
:	1	32	243	1024	3125

             b.   Using LU decomposition method, solve the following system

(7)

Q.4      a.     Find the iteration matrix for Gauss-Jacobi method and hence find the rate of convergence for the following system of equations:                                                                                                   (8)

            b.    Starting with the set , generate a set of orthogonal polynomials on  with the weight function . Using these polynomials, find a Least Square Approximation of the form  for the function  .                                                                         (8)

Q.5      a.   Find the eigenvalue of the matix A =  which is nearest to 6 using three iterations of the inverse power method. Take the initial vector as .                               (8)

b.  Find the Lagrange Interpolating Polynomial for the following:

x	0	1	2	5
ƒ(x)	2	1	12	147

       Hence find f(4).                                                                                                          (8)

Q.6      a. Evaluate the integral by two point Gauss-Hermite formula.            (8)

              b.  Using Newton's backward differences, find the Interpolating Polynomial for the following:                (8)

x	–1	0	1	2
f(x)	1	3	2	5

  Q.7     a. Determine an appropriate step size to use in the construction of a table of  such that the error for linear interpolation is to be bounded by .                                             (8)

             b.  Using Jacobi's method, find all the eigenvalues and the corresponding eigenvectors of the following matrix.                                                                (8)

Q.8   a.    Using Newton’s iterative Method, solve the system of equation                                           

                  Iterate two times starting with x₀ = 0.5 and y₀ = 0.5.                                                       (8)

            b.   Using Simpson’s  ‘3/8’ rule compute, (8)

Q.9      a.   Solve the initial value problem  with h = 0.2 on  using the Euler method.                   (8)

b.   The initial value problem is given. Find an approximation to , when h = 0.2, using the Runge-Kutta method

                  for the solution of differential equation .                                                     (8)