UNIT I MA2264 NUMERICAL METHODS

SOLUTION OF EQUATIONS AND EIGEN VALUE PROBLEMS

PART-A

1.  Derive the formula for Newton Raphson method.

2.  What is the condition for convergence in a) fixed point iteration b)Newton Raphson method

3.  Show the Newton Raphson formula for is

4.  Show the Newton Raphson formula for is

5.  What is the order of convergence for a) fixed point iteration b)Newton Raphson method

6.  Distinguish between direct and iterative methods for solving a system of linear algebraic equations.

7.  Explain a) partial pivoting b)diagonal dominance

8.  State the condition for convergence of iterative methods for solving a system of linear algebraic equations.

9.  How will you find the smallest eigenvector of a square matrix numerically using the power method

10.  Find the largest eigenvalue of [5.3722]

11.  Solve 5x+4y=15, 3x+7y=12 using Gauss Jordan method [x=2.4783,y=0.6522]

12.  Solve 3x+y=2, x+3y=-2 using Gauss Seidal method [x=1,y=-1]

13.  State the principle used in Gauss Jordan method and Gauss Elimination method

14.  Compare Gauss Elimination with Gauss Seidal method

PART-B

15.  Use fixed point iteration to find an approximate root of the following equations correct to 4 decimal places: a) b) c)

16.  Use Newton Raphson method to find an approximate root of the following equations correct to 4 decimal places: a) b)

17.  Use Newton Raphson method to find an approximate value of a) and b) correct to 4 decimal places

18.  Solve using Gaussian elimination and Gauss- Jordan method

(a) (b) (c)

19.  Solve using Gauss- Seidel method correct to three places of decimals

(a)  (b) (c)

20.  Using Gauss Jordan method find the inverse of the following

a ) b) c)

21.  Find the numerically largest eigenvalue and the corresponding eigenvector of the following matrices using the power method

(a) (b) (c) (starting vector (1,1,1) for c)

22.  Find all the eigenvalues and eigenvectors of the following matrices using Jacobi’s method

(a) (b) (c) (d) (e)

Answers

15 a) 0.6823 b) 2.1080 c) 0.3604

16 a) 1.8558 b) 0.5178] 13 a) 3.4641 b) 0.0562

18 a) b) c)

19 a) b)

c)

20 a) b)

c)

21a) 11.6619, b) 25.1821, c) 24.9975,

22a) 5,1,-1 b)0.5857,2,3.4142

c) 1,2,4 d) 3,-2 (2,1) , (-1,2)e) 5, -1 (1,1), (-1,1)