Numerical Methods–Unit 2–2 Marks with Answers

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 clip_image002 is interpolation.

Extrapolation: If clip_image004 then the process is called extrapolation.


3. State Newton’s Forward interpolation formula.

clip_image006


4. State Newton’s Backward interpolation formula.

clip_image008


5. Error in Newton’s forward:

clip_image010


6. Error in Newton’s Backward:

clip_image012


7. State Newton’s divided difference formula.

clip_image014


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

clip_image016


9. Show that divided difference operator clip_image018is linear.

Ans: clip_image019[f(x)clip_image021g(x)] = clip_image023

= clip_image025 = clip_image019[1]f(x)clip_image021[1] clip_image019[2]g(x).


10. Divided difference table:

X

Y

clip_image019[3]Y

clip_image019[4]2 Y

clip_image019[5]3Y

X0

X1

X2

X3

Y0

Y1

Y2

Y3

clip_image028

clip_image030

clip_image032

clip_image034

clip_image036

clip_image038


11. Write Lagrangian’s polynomial formula.

clip_image040


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.

clip_image042


14. Define Cubic Spline

Ans: Letclip_image044 , 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 x0, x1, .....xn is a polynomial of degree three in each interval (xi-1, xi) i = 1, 2, ...n such that clip_image046, clip_image048 and clip_image050 are continuous.


15. Write down the formula of Cubic Spline.

clip_image052

and clip_image054

where M = clip_image056

(OR)

clip_image058

whereclip_image060 and clip_image062; i = 1,2,3,....