AMIETE – CS/IT (OLD SCHEME)

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 the best alternative in the following:                                 (2 10)

a. If  , then the valued of f[a, b, c] will be 

  (A)  a+b+c.                                          (B)  a-b+c.

                    (C)  a+b-c.                                           (D)  a-b-c.

b. The order of convergence for Newton-Raphson Method and Secant Method respectively are

(A)  2, (B)  2,  

(C)  2,  (D)  2,

c. The order of convergence for the iteration given by  to  is

(A) 1.                                                 (B)  2.

(C)  3.                                                 (D)  4.

d. If  is an eigen value of A, then the eigen value of A^-1 is

(A) (B)

(C) - (D) 

             e.   The Langranges interpolation polynomial which passes through the point (0,1), (-1,2) and (1,3) is     

(A) (B)  

(C) (D)

f.    For the Simpson’s th rule, the interpolating polynomial is a

(A)  Straight-line                                  (B)  Parabola

(C)  Cubic Curve                                (D)  None

             g.   The value of the integral  using Gaussian integration formula for n=2 is

(A) (B) 

 (C)  (D)  None

             h.   The value of  is

(A) (B) 

(C)  (D)  None

i.      The value of y corresponding to x =0.1 for the differential equation  using Euler’s Method is                                                                         

(A)  1.10                                             (B)  1.36

(C)  1.94                                             (D)  2.19

             j.    Given (n+1) distinct points  and (n+1) ordinates , there is a polynomial p(x) that interpolates y_i at x_i,
i = 0, 1, 2,...,n. This polynomial p(x) is unique among the set of all polynomials of degree

(A)  n                                                  (B)  at least n

(C)  at most n                                      (D)  None

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Show that the given sequence has Convergence of the second order.

    (8)

             b.   Using Newton-Raphson Method, solve x = ,  where N is a positive real number. Apply the methods to N =18 to obtain the results Correct to two decimal places.                                                     (8)

  Q.3     a.   Solve the system of equations

                   using Decomposition (LU) Method                                                                     (8)

             b.   Find the inverse of the matrix by the Cholesky method.       (8)

Q.4      a.     Find the largest eigen value in modulus and Corresponding eigen vector of the Matrix.

using power method.                                                     (8)

            b.    Find all the eigen values and eigen vector of the matrix.

by Jacobi Method.                                                                       (8)

Q.5      a.   Given that f (0) = 1, f (1) = 3, f (3) = 55, find the unique polynomial of degree 2 or less, which fits the given data. Find the bound on the error.                                                                       (8)

b.  Using the data sin(0.1) = 0.09983 and sin(0.2) = 0.19867, find an approximate value of sin(0.15) by Lagrange interpolation. Obtain a bound on the truncation error.                             (8)

  Q.6     a.  A differentiation rule of the form where  is given. Find the values of and  so the rule is exact for. Find the error term.                         (8)

              b.  Evaluate the integral  using Composite trapezoidal rule with

                  N = 2,4 equal subinterval.                                                                                   (8)

  Q.7     a. Evaluate the integral   using Gauss-Legendre three point formula.

(8)

             b.  Apply Runge-Kutta method to find an approximate value of y at x = 0.2 in the step of 0.1, if ; given that y =1 when x = 0.                   (8)

Q.8   a.    Evaluate using the Gauss-Laguerre two- point formula compare with the exact solution.                                                                                                         (8)

b.   Perform five iterations of the bisection method to obtain the smallest positive root of the equation      f (x) = x³ – 5x + 1 = 0.                                                          (8)

Q.9      a.   For the following data, calculate the differences and obtain the forward and backward difference polynomials. Interpolate at x = 0.25 and x = 0.35.                                                                   (8)

x	0.1	0.2	0.3	0.4	0.5
f (x)	1.40	1.56	1.76	2.00	2.28

b.   Obtain a linear polynomial approximation to the function f(x) = x³ on the interval [0,1] using least squares approximation with W(x) =1.                                                                         (8)