LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2011

MT 6605 – NUMERICAL METHODS

 

 

 

Date : 09-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL questions.                                                                                                 (10 ´ 2 = 20)

1. What is the condition of convergence for solving a system of linear equations by iteration procedure?
2. What do you mean by partial pivoting?
3. Explain the method of successive approximation.
4. What is the order of convergence in regula falsi method?
5. Write a short note on interpolation.
6. Write the Gregory-Newton backward interpolation formula.
7. State the relation between Bessel’s and Laplace-Everett’s formulae.
8. Write Bessel’s central difference interpolation formula.
9. What is the order of error in Simpson’s ¹/₃
10. Using Euler’s method, Solve y¢ = x + y, given y(0) = 1 for x = 0.2

 

PART – B

Answer any FIVE questions.                                                                                      (5 ´ 8 = 40)

11. Using Gauss elimination method, solve the system

                  10x + y + z = 12,        2x + 10y + z = 13,      2x + 2y + 10z = 14

12. Find an approximate root of x log₁₀ x – 1.2 = 0 by regula falsi method.
13. Find a real root of the equation cos x = 3x – 1 correct to 3 decimal places.
14. Find a polynomial which takes the following values and hence compute y_x at x = 2, 12

x:         1          3          5          7          9          11

y_x:        3          14        19        21        23        28

15. Obtain Newton’s divided difference formula for unequal intervals.
16. The population of a certain town (as obtained from census data) is shown in the following table. Find the rate of growth of the population in the year 1981.

Year:                           1951                1961                1971                1981                1991

Population:                 19.96               36.65               58.81               77.21               94.61

(in thousands)

17. Evaluate using (i) Simpson’s ¹/₃ rule and (ii) Simpson’s ³/₈
18. Using Modified Euler method, find y(0.1), y(0.2) given

PART – C

Answer any TWO questions.                                                                                   (2 ´ 20 = 40)

19. (a) Solve by Gauss-Seidel method, the following system of equations

            10x – 5y – 2z = 3,                   4x – 10y + 3z = –3,                 x + 6y + 10z = –3

(b)  Find the positive root of f(x) = 2x² – 3x – 6 = 0 by Newton-Raphson method correct to 3 decimal places.                                                                                                                                  ( 12 + 8)

20. (a) Using Lagrange’s formula of interpolation find y(9.5) given

x:         7          8          9          10

y:         3          1          1          9

(b)  The population of as town is a follows

Year     x:                                 1941       1951       1961          1971       1981     1991

Population in lakhs y:               20           24           29              36           46         51

(10 + 10)

21. The following table gives the values of the probability integral for certain values of x.  Find the values of this integral when x = 0.5437 using (i) Stirling’s formula (ii) Bessel’s formula and (iii) Laplace-Everett’s formula.

x:                           0.51                 0.52                 0.53                0.54                  0.55

y = f(x):           0.5292437       0.5378987       0.5464641       0.5549392       0.5633233

x:                           0.56                 0.57

y = f(x):           0.5716157       0.5798158

22. (a) Develop a C-program to implement Trapezoidal rule.

(b)  Using Runge-Kutta method of fourth order, solve given y(0) = 1 at
x = 0.2, 0.4                                                                                                               (8 + 12)

 

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