2008 A.M.I.E.T.E CSE - NUMERICAL COMPUTING (DEC 08) Question paper

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2008 A.M.I.E.T.E CSE - NUMERICAL COMPUTING (DEC 08) Question paper

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 exact solution of the Initial Value Problem is
(A) (B)
(C) (D)

b. The expression is equal to

(A) (B)
(C) (D)

c. The equation has a multiple root x = 1 of multiplicity 2. The other 2 roots will be

(A) both real but different (B) a pair of real roots
(C) a complex pair (D) one real and one complex

d. Which of the following is a Composite Simpson's Rule?

(A)
(B)
(C)
(D)

e. The nth divided difference of is given by

(A) (B)
(C) (D)

f. Attempt is made to solve the system of equations where and by the Gauss–Jacobi iteration method. Then, the iteration

(A) has rate of convergence 0.5634. (B) has rate of convergence 0.235.
(C) has rate of convergence 1.234. (D) diverges.

g. The least squares straight line approximation to the data

is given by

(A) (B)
(C) (D)

h. The integration formula is to be used. The value of for which the method is of highest order, is given by

(A) 1 (B) 2/3
(C) 1/3 (D) 1/2

i. The order of Newton-Raphson method for finding out a multiple root of multiplicity 3 of the equation , is

(A) 1 (B) 2
(C) 3 (D) 4

j. What should be the condition on such that the method where , converges.
(A) (B)
(C) (D) None of above

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Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.

Q.2 a. Obtain a second degree polynomial approximation to using the Taylor series expansion about . Use the expansion to approximate and find a bound of the truncation error. (6)

b. Use the classical Runge-Kutta fourth order method to obtain an approximate value for the initial value problem with . (10)

Q.3 a. How should the constant be chosen to ensure the fastest possible convergence with the iteration formula
. (

b. Obtain a bound on the truncation error in linear interpolation based on the nodal points . Determine the maximum step-size that can be used in the tabulation of in so that the error in linear interpolation will be less than . (

Q.4 a. Solve the following system of equations using the Gauss-Seidel iteration method

with the initial approximation as and . Perform two iterations. Find the iteration matrix and hence find the rate of convergence of the method. (

a. Use the method of least squares to fit a function of the form to the following data
(
Q.5 a. Determine all the eigenvalues of the matrix , using Jacobi's method. (Use exact arithmetic). (

b. The equation has a simple root in the interval . The function is such that and for all in . Assuming that the Newton-Raphson's method converges for all initial approximations in , find the number of iterations required to obtain the root correct to . (

Q.6 a. Find the Lagranges interpolation polynomial which fits the following data.

and use the same to estimate the value of . (

b. Use the two point Gauss-Legendre quadrature formula to evaluate . (


Q.7 a. The following table of values is given


Find all the possible approximations for using the differentiation formula .
Perform Richardson's extrapolation to obtain a better estimate. (

b. The polynomial interpolates the first four points in the table:


By adding one additional term to , find a polynomial that interpolates the whole of table. (

Q.8 a. Calculate using Trapezoidal rule with number of points as 3, 5 and 9. Improve the results using Romberg Integration. (

b. Find the inverse of the matrix using the Choleski factorization method, taking , where U is an upper triangular matrix. (

Q.9 a. Use Taylor's series method of order four to obtain the approximate value of for the initial value problem .
Take the step size . (

b. Perform four iterations of power method to find the largest eigen value in magnitude and the corresponding eigen vector of the matrix

Take the initial approximation to the eigenvector as . (