__Anna University__

__Numerical Methods____ __

__ __

UNIT-II

__INTERPOLATION AND APPROXIMATIONT__

__Two Marks With Answers __

__ __

1. Explain briefly **Interpolation**.

Ans: Interpolation is the process of computing the values of a function for any value of the independent variable within an interval for which some values are given.

2. Definition of **Interpolation** and **extrapolation.**

Ans: ** Interpolation**: It is the process of finding the intermediate values of a function from a set of its values specific points given in a tabulated form. The process of computing y corresponding to x is interpolation.

** Extrapolation: **If then the process is called extrapolation.

3. State **Newton****’s Forward interpolation** formula.

4. State

**Newton**

**’s Backward interpolation**formula.

5. Error in

**Newton**

**’s forward**:

6. Error in

**Newton**

**’s Backward**:

7. State

**Newton**

**’s divided difference**formula.

8. Show that the divided differences are symmetrical in their arguments.

9. Show that divided difference operator is linear.

10. Divided difference table:

X | Y | |||

X
X X X
| Y
Y Y Y |

11. Write

**.**

__Lagrangian’s polynomial formula__

12. What is the assumption we make when Lagrange’s formula is used?

Ans: It can be used whether the values of x, the independent variable are equally spaced or not whether the difference of y become smaller or not.

13. Write ** Lagrangian inverse interpolation formula**.

14. Define

__Cubic Spline__Ans: Let , i = 0, 1, 2... n be the given (n +1) pairs of a data. The third order curves employed to connect each pair of data points are called cubic splines. (OR) A smooth polynomial curve is known as cubic spline.

A cubic spline function f(x) w.r.t. the points x_{0}, x_{1, .....}x_{n }is a polynomial of degree three in each interval (x_{i-1,} x_{i}) i = 1, 2, ...n such that , and are continuous.

15. Write down the formula of

**Cubic Spline**.

(OR)