COMPUTER ORIENTED NUMERICAL ANALYSIS

ASSIGNMENT-A

Solve the following set of equation by Gauss elimination method.
               2x +3y+4z=12
               3x + y +z=4
               x+4y+z=5

                       
                                 

Find the rate of convergence of Newton Raphson method. Solve by www.solvezone.in 

4.           Define error, relative error and absolute error, give example of each.

ASSIGNMENT-B
1. a) Round off and truncate the following numbers to the four decimal places.
                  a) 132.59839
                             Solution     round off - 132.5984
                                                 truncate   - 132.5983
                  b) 0.073729
                             Solution     round off - 0.0737
                                                 truncate   - 0.0737

                   c) 9528.26058
                             Solution      round off - 9528.2606
                                                 truncate   - 9528.2605

    
b) Find the route of equation 2x² -5x-1=0 by using bisection method. Perform 5 steps.

2. Solve the following equation by Jacobi method. Perform three steps.
 x₁-2x₂-x₃-x₄=3
-2x₁ + x₂-x₃ -x₄=15
x₁ – x₂ + x₃ -2x₄ =27
-x₁ -x₂-2x₃+x₄ =-9

For a given data
 
X    F(x)       
1    33       
2    50       
3    69       
4    90       
5    129     

Find the value of f(x) at 1.4

Choose 4 card at random from standard 52 card deck. What is the probability that two kings and two ace will be chosen.

 
             
ASSIGNMENT C

1. The order of convergence of Newton-Raphson method is
 (A): 2
 (B): 3
 (C): 0
 (D): 1

2. One of the roots of the equation x3-3x2+x-3=0  is
 (A): -1
 (B): 1
 (C): √3
 (D): 3

3. The solution to the set of equations:
25x+y+z=25; 64x+8y+z=71; 114x+12y+z=155
The most nearly value of ??,??,??=
 (A): (1,1,1)
 (B):  (1,-1,1)
 (C):  (1,1,-1)
 (D):  Does not have a unique solution

4. The order of convergence of Regula-falsi method is
 (A): 1.235
 (B): 3.141
 (C): 1.618
 (D): 2.792

5. Newton’s iterative formula to find the value of √N is
  
6. The Newton-Raphson method fails when
 
7. The one of the root of the equation x^3-4x-9=0 using bisection method is
 (A): 2.5065
 (B): 2.6066
 (C): 2.7066
 (D): 3.5066

8. Which of the following is increasing order of convergence method for finding roots
 (A):  Bisection, regula-falsi, newton Raphson
 (B): Regula-falsi, Bisection, Newton Raphson
 (C): Newton Raphson, Regula falsi, bisection
 (D): Bisection, Newton Raphson, regula-falsi

9. If X is a true value and  X is its approximate value then absolute error ea  is
  
10. If X is a true value and  X is its approximate value then Relative error er is
  
11. Any quadratic equation f(x)=0 has
 (A): One root
 (B): Two roots
 (C): Three roots
 (D): Four roots

12. An equation such as tanx=x has
 (A): Zero roots
 (B): One root
 (C): Two roots
 (D): Infinite roots

13. The roots of an equation f(x)=x^3-x-1=0 lies between
 (A): f(2)  and f(3)
 (B): f(3)  and f(4)
 (C): f(1)  and f(2)
 (D): All of these

14. The real roots of an equation f(x)=xex-2=0 using newton Raphson method is
 (A): 0.5682
 (B): 0.8526
 (C): 0.3525
 (D): 1.5000

15. Given n+1 data pairs (x_0,y_0),(x_1,y_1),… (x_n,y_n). The process of finding the value of y corresponding to any value of x (between x_0 and x_n) is called
 (A): Extrapolation
 (B): Interpolation
 (C): Differentiation
 (D): Integration

16. Given n+1 data pairs (x_0,y_0),(x_1,y_1),… (x_n,y_n). The process of finding the value of y corresponding to any value of x (outside of the range  x_0 and x_n) is called
 (A): Extrapolation
 (B): Interpolation
 (C): Differentiation
 (D): Integration

17. Given n+1 data pairs (x_0,y_0),(x_1,y_1),… (x_n,y_n). The n^(th )  forward difference is
 (A): ?n yr=?(n-1) y(r+1)-?(n-1) yr
 (B): ?n yr=?(n-1) y(r+1)-?(n-1) yr
 (C): ?^n y_r=?^n y_(r+1)-?^n y_r
 (D): ?^n y_r=?^n y_r-?^(n-1) y_(r-1)

18. If ? is forward difference operator and E is a shift operator then
 (A): ?=E-1
 (B): ?=E+1
 (C): E=?-1
 (D): None of these

19. If δ is central difference operator and E is a shift operator then
  
20. The rate of convergence of Bisection method for finding roots is
 (A): 2
 (B): 3
 (C): 1.5
 (D): 1.0

21. A second degree polynomial passes through (0,1),(1,3), (2,7), (3,13), then the polynomial f(x) is
  
22. Given data pairs (1,7),(2,x),(3,13),(4,21),(5,37).  The value of x is
 (A): 10.5
 (B): 9.5
 (C): 12.5
 (D): 12.0

23. Which of the following Interpolation formula is used for unequally spaced points
 (A): Newton forward
 (B): Newton backward
 (C): Lagrange formula
 (D): Euler formula

24. 1)=1, P(3)=27, P(4)=64, Using Lagrange interpolation formula, the polynomial P(x) of degree 2 is:
  
25. Putting n=1 in the newton-Cote’s quadrature formula, we have
 (A): Trapezoidal rule
 (B): Simpson’s 1/3 formula
 (C): Simpson’s 3/8 formula
 (D): Euler’s formula

26. Putting n=3 in the newton-Cote’s quadrature formula, we have
 (A): Trapezoidal rule
 (B): Simpson’s 1/3 formula
 (C): Simpson’s 3/8 formula
 (D): Euler’s formula

27. Which of the following method required odd number of points for integration
 (A): Trapezoidal rule
 (B): Simpson’s  formula
 (C): Both Trapezoidal & Simpson
 (D): None of these

28. Using Trapezoidal rule, the value of     
 (A): 1.3662
 (B): 1.4107
 (C): 1.3570
 (D): 1.5706

29. Which of the following symbol is called backward difference operator
 (A): ?
 (B): ∇
 (C): δ
 (D): E

30. Newton divided difference method for interpolation can be used for
 (A):  Equal spaced points
 (B): Unequal spaced points
 (C): Not well defined
 (D): All of these

31. Which of the following is NOT a method to solve ordinary differential equation
 (A): Euler’s method
 (B): Picard’s method
 (C): Taylor series method
 (D): Romberg’s method

32. Euler’s formula for solving ordinary differential equation is
 
33. Which of the following is a Runga kutta 2nd order formula
 
34. Taylor series method is used for
 (A): Integration
 (B): Differentiation
 (C): Ordinary differential equation
 (D): Roots finding

35. Polynomials are most commonly used functions for interpolation because they are easy to
 (A): Evaluate
 (B): Differentiate
 (C): Integrate
 (D): Evaluate, differentiate, and integrate

36. To solve the ordinary differential equation using Runga kutta 2nd order, we need to write the equation  
  
37. Picard’s method is used to solve
 (A): Integration
 (B): Differentiation
 (C): Ordinary Differential equation
 (D): Roots finding

38. The inverse of a matrix A is written as   
      
 (A): Identity matrix
 (B): Null matrix
 (C): Singular matrix
 (D): Inverse matrix

39. he second forward difference 
40.