NUMERICAL COMPUTING University Question paper

[NAREN]

NUMERICAL COMPUTING QUESTION PAPER

Time: 3 Hours June 2006 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 approximation to is correct to n significant digits. The value of n is

(A) 7. (B) 6.

(C) 5. (D) 4.


b. The equation f (x) = 0 has a simple root in the interval (1, 2). This root is to be determined correct to two decimal places using bisection method. The required number of iterations is

(A) 5. (B) 6.

(C) 7. (D) 8.



c. The system of equations has


(A) no solution.

(B) two parameter family of solutions.

(C) one parameter family of solutions.

(D) unique solution.



d. Gauss-Seidel iteration method is used to solve the system of equations

The rate of convergence of this method is


(A) 0.7659. (B) –0.7659.

(C) 1.7635. (D) –1.7635.


e. The value of where is the forward difference operator and E is the shift operator, with differencing h =1 is

(A) . (B) .

(C) 6 x. (D) 6




f. The data represents a third degree polynomial. It is known that f (3) is in error. The correct value of f (3) is

(A) 37. (B) 39.

(C) 41. (D) 43.



g. The least squares polynomial approximation of degree 1 to on [0, 1] is . The least squares error is

(A) . (B) .

(C) . (D) .

h. The truncation error in the method



is of . The value of p is

(A) 1. (B) 2.

(C) 3. (D) 4.


i. The minimum number of sub-intervals that will be required, so that the error in evaluating by trapezoidal rule is , is

(A) 8. (B) 9.

(C) 12. (D) 13.



j. The value of y(1.4) for the initial value problem using Euler method with h=0.2 is

(A) 4.6. (B) 22.5552.

(C) 25.1552. (D) 26.2432.


Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. The method is used to find a multiple root of multiplicity two of the equation f (x)=0. Determine the rate of convergence of the method. Also obtain the asymptotic error constant. (10)

b. Perform two iterations of the method to determine a root of with . (6)


Q.3 a. Set up the Newton-Raphson method in matrix form to solve the system of equations

Perform two iterations of the method with the initial approximation . (8 )

b. Solve the system of equations  using Gauss-Jordan method with partial pivoting, if necessary. (8 )


Q.4 a. Using decomposition method, find the inverse of the matrix Take . (8 )


b. Perform four iterations of the power method to obtain the largest eigenvalue in magnitude of the matrix . Take the initial approximation vector as . (8 )


Q.5 a. Using Jacobi method, find all the eigenvalues and the corresponding eigenvectors of the matrix . (Use exact arithmetic) (10)


b. Obtain the least squares approximation of the form to the data (6)



Q.6 a. Jacobi iteration method is used to solve the system of equations determine the rate of convergence of the method. Starting with the initial vector perform two iterations. (8 )


b. Prove the following operator relations

(i) .

(ii) .

(iii) .

The operators have their standard meaning. (2+3+3)


Q.7 a. Determine the largest step size h that can be used in the tabulation of a function f(x), , at equally spaced nodal points so that the truncation error
of the quadratic interpolation is less than in magnitude. Find h, when and . (8 )

b. For the method determine the optimal value of h, so that =minimum where is the maximum round off error in function values and is the maximum value of in the given interval. (8 )

Q.8 a. Evaluate using Simpson's rule with 3 and 5 nodalp oints. Find the improved value of I using Romberg integration. (8 )

b. Determine the constants A, B and C in the method so that the method is of highest possible order. Obtain the truncation error and the

order of the method. Use this method to obtain the value of . (8 )

Q.9 a. Find the truncation error and the order of the Heun's method for solving the initial value problem . (8 )



b. Using classical Runge-Kutta method of order four with h=0.1, obtain the

approximate value of y (1.2) for the initial value problem. (8 )