__Anna University__

__NUMERICAL METHODS __

__ __

__UNIT-I__

__SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS__

__Two Marks with Answes __

__ __

1. If a function f(x) = 0 is continuous in the interval (a, b) and if f (a) and f (b) are of opposite signs. Then one of the root lies between a and b.

2. Example of

**Algebraic equation**:

(i) x^{3} – 2x + 5 = 0;

(ii) 2x^{3} – 3x – 6 = 0.

3. Example of

**Transcendental equation**:

(i) x – cosx = 0;

(ii) xe^{x} -2 = 0;

4.

**:**

__Regula Falsi Method__(First iteration of Regula Falsi Method).

5.

**:**

__Iterative Method__

6. Convergence condition of iterative method is

7. Order of convergence of iterative method is linear (i.e.) 1.

8.

__Newton__**:**

__Raphson’s Method (Method of Tangents)__

9. Convergence condition of

**N-R**method is

10. Order of convergence of

**Newton**

**’s method**is quadratic (i.e.) 2.

11. ** Direct Method**:

(i) Gauss Elimination Method,

(ii) Gauss Jordan Method.

12.

**:**

__Indirect Method (or) Iteration Method__(i) Gauss-Jacobi Method,

(ii) Gauss-Seidel Method.

13.

**: To reduce the augmented matrix [A, B] to upper triangular matrix. Then, using Back Substitution method we’ve to find the unknowns.**

__Gauss Elimination Method__

14.

**: To reduce the augmented matrix [A, B] to diagonal matrix. Finally this system of equation each has only one unknown and then we’ve to find the unknowns directly.**

__Gauss Jordan Method__

15.

**: An nn matrix A is said to be diagonally dominant if the absolute value of each leading diagonal element is greater than or equal to the sum of the absolute values of the remaining elements in that row.**

__Diagonally Dominant__Given system of equations is

is a diagonal system is if

16.

**: If the given system of equation is diagonally dominant then**

__Gauss Jacobi Method__

17.

**: If the given system of equation is diagonally dominant then**

__Gauss Seidel Method__

18. Sufficient condition for iterative methods (Gauss Seidel Method & Gauss Jacobi Method) to convergence is the coefficient matrix should be diagonally dominant.

19. The iteration method is a self correcting method since the round off error is smaller.

20. Why Gauss Seidel iteration is a method of successive corrections?

Ans: Because we replace approximations by corresponding new ones as soon the latter have been computed.

21. Compare

**Gauss Elimination Method**and

**Gauss Jordan Method**

Gauss Elimination Method | Gauss Jordan Method |

1. Direct Method 2. Coefficient matrix is transformed into upper triangular matrix. 3. We obtain the solution by back substitution method. | 1. Direct Method 2. Coefficient matrix is transformed into diagonal matrix. 3. No need of back substitution method. Since finally this system of equation each has only one unknown. |

22.

**: Let A be an nn nonsingular matrix. If X is the inverse of the matrix A then AX = I (i.e.) X = I A**

__Inverse of a Matrix__^{-1}. We start with augmented matrix of A with identity matrix I of the same order and convert A into the required form (i.e.) identity then the inverse is formed. [A / I ] [I / A

^{-1}].

23. Compare

**Gauss Elimination Method**and

**Gauss Seidel Method**.

Gauss Elimination Method | Gauss Seidel Method |

1. Direct Method 2. It has the advantage that it is finite and works in theory for any non-singular set of equation. 3. We obtain exact value. | 1. Indirect Method 2. It converges only for diagonally dominant. 3. Approximate value which is self correct method. |

24. Compare

**Gauss Jacobi**and

**Gauss Seidel Methods**.

Gauss Jacobi Method | Gauss Seidel Method |

1. Indirect Method 2. Convergence rate is slow. 3. Condition for convergence is diagonally dominant. | 1. Indirect Method 2. The rate of convergence of this method is roughly twice that of Jacobi. 3. Condition for convergence is diagonally dominant. |

25. Why

**Gauss Seidel method**is better method than

**Jacobi’s method**?

Ans: Since the current value of the unknowns at each stage of iteration are used in proceeding to the next stage of iteration, the convergence in Gauss Seidel method will be more rapid than in Gauss Jacobi method.