DEPARTMENT OF MATHEMATICS

[Pages:9]DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis

Question Bank

Unit-1 [A] 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27. 28. 29. 30.

Error analysis and solutions of algebraic and transcendental equations

5 ? Marks Questions

Find the real root of the equation, x3- 3x-5 = 0 by the bisection method.

Find the real root of the equation, x3- 4x-9 = 0 by the bisection method. Find the real root of the equation, x3- x2-1 = 0 by the bisection method. Find the real root of the equation, x3- x-1 = 0 by the bisection method. Find the real root of the equation, x3+x2+x+7 = 0 by the bisection method. Find the real root of the equation, x3- 5x+3 = 0 by the bisection method. Use Regula-Falsi Method to Compute the Root of the equation x3-3x-3 = 0 Use Regula-Falsi Method to Compute the Root of the equation x3+x-3 = 0

Find the real root of the equation, x3- x- 4 = 0 by the Regula-Falsi method. Use Regula-Falsi Method to Compute the Root of the equation x3-2x-6 = 0 Use Regula-Falsi Method to Compute the Root of the equation x3-x+7 = 0 Find the real root of the equation, x3- 18 = 0 by the Regula-Falsi method. Find the real root of the equation, x3- 2x2-1 = 0 by the Regula-Falsi method. Find the real root of the equation, x3+x-1 = 0 by the Regula-Falsi method. Find the root of the equation sin2x = x2 - 1, correct to three decimal places using iteration method. Find the root of the equation e-x = 10x, correct to three decimal places using iteration method. Find the root of the equation 1+x2 = x3, correct to three decimal places using iteration method. Find the root of the equation 5x3-20x +3 = 0, correct to three decimal places using iteration method. Find the root of the equation sinx = 10(x-1), correct to three decimal places using iteration method. Find the root of the equation 2x = cosx + 3, correct to three decimal places using iteration method. Find the root of the equation cosx = 3x - 1, correct to three decimal places using iteration method. Compute the root of the equation xsin2 - 4 =0, correct to three decimal places using Newton-Raphson method. Compute the root of the equation sinx = 1- x, correct to three decimal places using Newton-Raphson method. Compute the root of the equation x3 -5x+3 =0, correct to three decimal places using Newton-Raphson method. Compute the root of the equation x4 +x2-80 =0, correct to three decimal places using Newton-Raphson method. Compute the root of the equation x-cosx =0, correct to three decimal places using Newton-Raphson method. Using Secant method determine the root of the equation xex = 1.

Using Secant method determine the root of the equation x3-2x-5=0

Solve the following system of nonlinear equations; x2 - y2 = 4, x2 + y2 = 16.

Solve the following system of nonlinear equations; x2 +y = 11, y2 + x = 7.

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis

Unit-2 [A] 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Numerical solution of linear system equations and matrix Inversion

5 ? Marks Questions

Solve the system of equation by Gauss elimination method: 2x+3y ?z = 5 4x+4y-3z=3 -2x+3y-z=1

Solve the system of equation by Gauss elimination method: x + y + z = 7 3x +3y + 4z =24 2x + y + 3z = 16

Solve the system of equation by Gauss elimination method: 4x + y + z = 4 x + 4y -2z = 4 3x + 2y ? 4z = 6

Solve the system of equation by Gauss elimination method: x + y/2 + z/3 = 1 x/2 + y/3 + z/4 = 0 x/3 + y/4 + z/5 =0

Solve the system of equation by Gauss elimination method: 2.5x ? 3y + 4.6z = -1.05 -3.5x + 2.6y + 1.5z = -14.46 -6.5x ? 3.5y + 7.3z = -17.735

Solve the system of equation by Gauss elimination method: 4x + y + 2z = 12 2x ? 3y + 8z = 20 -x + 11y + 4z = 33 Solve the system of equation by Gauss elimination method: x + 5y + z = 14 2x + y + 3z = 13 3x + y + 4z = 17

Solve the system of equation by Gauss elimination method: 2x ? 3y + 10z = 3 -x + 4y + 2z = 20 5x + 4y -3z = 2

Solve the system of equation by Gauss elimination method: 10x + y + z = 12 2x + 10y + z = 13 x + y + 5z = 7 Solve the system of equation by Gauss elimination method: 2x + y + 4z = 12 8x + 3y + 2z = 20 4x + 11y + z = 33

Solve following system of equation by Gauss-Jordan Method. 10x + y + z = 12 x + 10y + z = 12 x + y +10z = 12 Solve following system of equation by Gauss-Jordan Method.

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis

x + y + z = 7 3x + 3y + 4z = 24 2x + y + 3z = 16 13. Solve following system of equation by Gauss-Jordan Method. 4x + y ? z = 10 x ? 3y +2z = -5 2x + 5y ? z = 18 14. Solve following system of equation by Gauss-Jordan Method. x - y + z = 1 -3x + 2y -3z = -6 2x -5y +4z = 5 15. Solve following system of equation by Gauss-Jordan Method. x-y+z=1 -3x+2y-3z=-6 2x-5y+4z=5 16. Solve following system of equation by Gauss-Jordan Method. x+3y+10z =24 2x+17y+4z=35 28x+4y-z=32 17. Solve following system of equation by Gauss-Jordan Method. 3x-y+2z=12 x+2y+3z=11 2x-2y-z=2 18. Solve following system of equation by Gauss-Jordan Method. x+y+2z=4 3x+y-3z=-4 2x-3y-5z=-5 19. Using LU decomposition method solve the system of equation. 5x ? 2y + z = 4 7x + y -5z = 8 3x + 7y + 4z = 10 20. Using LU decomposition method solve the system of equation. 2x + 3y + z =9 x + 2y + 3z = 6 3x + y + 2z = 8 21. Using LU decomposition method solve the system of equation. 4x -3y + 2x = 11 2x + y + 7z = 2 3x ? y + 5z = 8 22. Using LU decomposition method solve the system of equation. x-y+z=6 2x+4y+z=3 3x+2y-2z=-2 23. Using LU decomposition method solve the system of equation. 4x -3y + 2x = 11 2x + y + 7z = 2 3x ? y + 5z = 8 24. Using LU decomposition method solve the system of equation. 2x+y+4z=12

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis

8x-3y+2z=20 4x+11y-z=33 25. Solve following system of equation by Gauss-Seidel Iteration Method and perform first three iteration. 20x + y ? 2z = 17 3x + 20y ?z = -18 2x -3y + 20z = 25 26. Solve following system of equation by Gauss-Seidel Iteration Method and perform first-five iterations. 2x ? y = 7 -x + 2y ? z = 1 -y + 2x = 1 27. Solve following system of equation by Gauss-Seidel Iteration Method and perform first three iteration. 8x-y+z=18 2x+5y-2z=3 X+y-3z=-16 28. Solve the system of equation by using Relaxation method. 8x + y ? z = 8 2x + y + 9z = 12 X ? 7y + 2z = -4 29. Solve the system of equation by using Relaxation method. 4x+3y ?z =18 x ? y + z = -5 2x + y + 3z = 0 30. Solve the system of equation by using Relaxation method. 6x-y+z=13 x+y+z=9 10x+Y-z=19 31. Solve the system of equation by using Relaxation method. 10x-2y+z=12 x+9y-z=10 2z-y+11z=20 32. Solve the system of equation by using Relaxation method. 2x-y+z=3 2x+y-z=1 x+y+z=0 33. Solve following system of equation by Jacobi Method. 4x + 8y ? 3z = 20 12x ? 6y + 3z = 18 9x + y ? 6z = -3 34. Solve following system of equation by Jacobi Method. 5x-2y+z=-4 x+6y-2z=-1 3x+y+5z=13 35. Solve following system of equation by Jacobi Method. 2x+y+z=4 x+2y+z=4 x+y+2z=4

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis 36. Find the inverse of a matrix by Gauss-Jordan Elimination method.

[

]

37. Solve following system of equation by Gauss-Seidel Iteration Method. 2x + y ?z = 1 3x ? 2y + z = 2 x + y + z =6

38. Solve following system of equation by Gauss-Seidel Iteration Method. 4x+2y+z=14 x+5y-z=10 x+y+8z=20

39. Find the inverse of a matrix by Gauss Elimination method.

[

]

40. Find the inverse of a matrix by Gauss-Jordan Elimination method.

A = [

]

41. Find the inverse of a matrix by Gauss-Jordan Elimination method.

[

]

Unit-3 [A] 1.

Eigen value problems 5 ? Marks Questions Find the eigenvalue of largest modulus and the associated eigenvector of the matrix by power method;

A = [

]

2. Find the eigenvalue of largest modulus and the associated eigenvector of the matrix by power method after sixth iteration;

A = [

]

3. Find the eigenvalue of largest modulus and the associated eigenvector of the matrix by power method after fourth iteration;

A = [

]

4. Using Jacobi method find all eigen values and corresponding eigenvector of the matrix:

A = [

]

5. Using Jacobi method find all eigen values and corresponding eigenvector of the matrix after third rotation:

A = [

]

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis

6. Find the eigenvalue of largest modulus and the associated eigenvector of the matrix by power method after fourth iteration;

A = [

]

7. Find the eigenvalue of largest modulus and the associated eigenvector of the matrix by power method;

A = [

]

8. Find the eigenvalue of largest modulus and the associated eigenvector of the matrix by power method after sixth iteration;

A = [

]

9. Using Jacobi method find all eigen values and corresponding eigenvector of the matrix:

A = [

]

10. Using Jacobi method find all eigen values and corresponding eigenvector of the matrix after third rotation:

A = [

]

Unit-4 [A] 1. 2.

Interpolation 5 ? Marks Questions

Show that the operator With the equal spacing h, show that:

(i)

(

)

commute with one another.

(ii)

3. With the equal spacing h, show that: (i)

(ii)

( )

4. With the equal spacing h, show that:

(i)

( ) and

(ii)

( )

5. The sales in a particular department store for the last five years is given in the

following table:

Year

1974 1976 1978 1980 1982

Sales (in 40

43

48

52

57

lakhs)

Estimate the sales in the year 1979.

6. Construct the forward difference table for the following values of x and y.

x 0.1

0.2

0.3

0.4

0.5

0.6

0.7

y 0.0032 0.0167 0.1348 0.5248 0.2370 0.1518 0.1697

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis

7. The sales in a particular department store for the last five years is given in the

following table:

Year

1974 1976 1978 1980 1982

Sales (in 12

34

25

10

24

lakhs)

Estimate the sales in the year 1979.

8. Estimate the missing figure in the following table:

x

1

2

3

4

5

y = f(x) 2

5

7

--

32

9. Evaluate f(17), given the following table of values.

x

15

20

25

30

35

y = f(x) 37

56

90

107

120

10. Evaluate f(15), given the following table of values.

x

10

20

30

40

50

y = f(x) 46

66

81

93

101

11. Construct the Backward difference table for the following values of x and y.

x

1

2

3

4

5

6

7

y

12

20

35

55

76

80

102

12. Estimate the missing figure in the following table:

x

2

4

6

8

10

y = f(x) 12

18

26

--

42

13. Construct the forward difference table for the following values of x and y.

x

2

4

6

8

10

12

14

Y

15

26

32

38

40

45

54

14. Construct the Backward difference table for the following values of x and y.

x

1.2 1.4 1.6 1.8 2.0 2.2 2.4

y

2.44 3.45 3.78 4.12 4.38 5.00 5.46

15. For the following table of values estimate f(7.6).

x

12 3 4 5 6 7 8

y = 23 34 42 55 64 78 89 96

f(x)

16. From the following table, find y when x = 1.45

x

1.0

1.2

1.4

1.6

1.8

2.0

y

0.0

- 0.112 - 0.016 0.336 0.992 2.0

17. From the following table, find y when x = 1.5

X

1.0

1.2

1.4

1.6

1.8

2.0

Y

0.123

0.453 -0.123 0.432 0.187 0.992

18. Find the interpolating polynomial for function f(x) given by.

X

0

1

2

5

y=f(x) 1

34

80

98

19. Find the interpolating polynomial for function f(x) given by.

x

0

1

2

5

y=f(x) 2

3

12

147

20. The following values of x and y are given. Find y(0.443):

x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y(x) 2.631 3.328 4.097 4.944 5.875 6.896 8.013

21. Use Stirling's formula to find u32 from the following table:

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

DEPARTMENT OF MATHEMATICS

Semester ? III : 060090303 - CC7 Fundamentals of Numerical Analysis

u20 = 14.035 U35 = 12.374

u25 = 13.674 U40 = 12.089

U30 = 13.257 U45 = 11.309

22. Use Stirling's formula to find u32 from the following table:

u20 = 13.24

u25 = 16.134

U30 = 15.123

U35 = 16.146

U40 = 12.156

U45 = 11.239

23. Find the interpolation polynomial for the following data using Lagrange's formula.

x

1

2

-4

y=f(x) 3

-5

4

24. Find the interpolation polynomial for the following data using Lagrange's formula.

x

1

2

3

5

y=f(x) 0

7

26

124

25. Determine the interpolating polynomial of degree three using Lagrange's formula.

x

-1

0

1

3

y=f(x) 2

1

0

-1

26. Determine the interpolating polynomial of degree three using Lagrange's formula.

x

-1

0

1

3

y=f(x) 2

-3

4

-1

27. Find the interpolation polynomial for the following data using Lagrange's formula.

x

3

4

7

12

y=f(x) 12

34

55

76

28. Find the interpolation polynomial for the following data using Newton's divided

difference formula.

x

1

2

3

5

y=f(x) 0

7

26

124

29. Find the hermit polynomial of third degree approximation the function y(x) such that

y(x0) = 1, y(x1) = 0, and y'(x0) = y'(x1) = 0. 30. The natural logarithm and its derivative is given in the following table. Estimate the

value of ln(0.6) using Hermit interpolation formula.

31. Using Hermit interpolation, estimate the value of y(1.3) from the following data.

x

0.5

1.0

1.5

2.0

y

0.4794

0.8415

0.9975

0.9093

y'

0.8776

0.5403

0.7074

-0.4162

32. Using Hermite's interpolation formula, estimate the value of ln (3.2) from the

following table.

x

y =ln x

y'= 1/x

3.0

1.09861

0.3333

3.5

1.25276

0.28571

4.0

1.38629

2.25000

33. Find the hermit polynomial of third degree approximation the function y(x) such that

y(x0) = 2, y(x1) = 1, and y'(x0) = y'(x1) = 0. 34. Using Gauss's forward formula, find the value of f(32) given that

f(25) = 0.2707

f(30) = 0.3027

f(35) = 0.3386

f(40) = 0.3794

35. Using Newton's divided difference formula, evaluate f(2) and f(15) from the following

table of values:

x

4

5

7

10

11

13

f(x)

48

100

294

900

1210 2028

Uka Tarsadia University Maliba Campus, Gopal Vidyanagar, Bardoli-Mahuva Road-394350

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