LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 38

SIXTH SEMESTER – April 2009

MT 6605 – NUMERICAL METHODS

 

 

 

Date & Time: 23/04/2009 / 9:00 – 12:00 Dept. No.                                                     Max. : 100 Marks

 

 

 

PART – A (10 ´ 2 = 20)

Answer ALL questions.

 

1. What do you mean by partial pivoting?
2. State Cramer’s rule.
3. What is the order of convergence in Newton-Raphson method?
4. Explain bisection method.
5. What is meant by interpolation?
6. Write the error polynomial in the Newton’s forward interpolation formula?
7. Write the Stirling’s central difference interpolation formula.
8. Write any two advantages of central difference interpolation formula.
9. What is the order of error in the Trapezoidal rule?
10. Write the formula for third order Range-Kutta method.

 

PART – B (5 ´ 8 = 40)

Answer any FIVE questions.

11. Solve the following system of equations by Gauss elimination method

28x + 4y – z = 32,       x + 3y + 10z = 24    and     2x + 17y +4z = 35.

12. Solve for a positive root of x³ – 4x + 1 = 0 by Regula Falsi method.
13. Write a C program to for Lagrange’s interpolation formula.
14. Obtain Newton’s forward interpolation formula for equal intervals.
15. Find the first two derivatives of at x = 50 and x = 56 given the table below:

x                        :              50            51              52          53           54           55             56

:           3.6840     3.7084      3.7325    3.7563   3.7798     3.8030      3.8259

 

16. Use Laplace-Everett’s formula to obtain f(1.15) given that f(1) = 1, f(1.1) =1.049,  f(1.2) = 1.096, f(1.3) = 1.14.
17. Evaluate by        (i) Trapezoidal rule      (ii) Simpson’s 1/3  rule   and

(iii) Simpson’s 3/8 rule.

18. Solve in the range 0 £ x £2 using (i) Euler’s method  (ii) improved Euler’s method

 

 

 

 

 

PART – C (2 ´ 20 = 40)

Answer any TWO questions.

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

                                    28x + 4y – z = 32,      

                                    x + 3y + 10z = 24,     

                        and      2x + 17y + 4z = 35

(b) Find the real positive root of 3x – cos x – 1 = 0 by Newton-Raphson method correct to 6 decimal places.

20. (a) From the following table find f(x) and hence f(6) using Newton’s divided difference formula.

x          :           1             2            7             8

                        f(x)       :           1             5            5             4

• The following table gives the value of density of saturated water for various temperatures of saturated stream.

Temp^o C ( = T)            :           100      150      200      250      300

Density hg/m³ (= d)     :           958      917      865      799      712

      Find by interpolation, the densities when the temperatures are 130^oC and 275^oC respectively.

 

21. (a) Using Gauss’s forward interpolation formula, find the value of log 337.5 from the following table.

x                 :              310           320            330            340          350          360

       y_x = log x        :           2.4914       2.5052       2.5185      2.5315      2.5441   2.5563

• Using Bessel’s formula, find the derivative of f(x) at x = 3.5 from the following table.

x          :           3.47     3.48     3.49     3.50     3.51     3.52     3.53

f(x)       :           0.193   0.195   0.198   0.201   0.203   0.206   0.208

22. (a) Using Range-Kutta method of fourth order, solve for y(0.1) and y(0.2) given that                 y¢ = xy + y², y(0) =1.

(b)  Develop a C program to implement Simpson’s 3/8 rule.

 

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