Numerical Methods–Unit 2–2 Marks with Answers

Anna University

Numerical Methods

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UNIT-II

INTERPOLATION AND APPROXIMATIONT

Two Marks With Answers

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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.

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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.

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3. State Newton’s Forward interpolation formula.

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4. State Newton’s Backward interpolation formula.

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5. Error in Newton’s forward:

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6. Error in Newton’s Backward:

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7. State Newton’s divided difference formula.

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

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9. Show that divided difference operator is linear.

Ans: [f(x)g(x)] =

= = f(x) g(x).

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10. Divided difference table:

X	Y	Y	2 Y	3Y
X0 X1 X2 X3	Y0 Y1 Y2 Y3

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11. Write Lagrangian’s polynomial formula.

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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.

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13. Write Lagrangian inverse interpolation formula.

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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 x₀, 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.

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15. Write down the formula of Cubic Spline.

and

where M =

(OR)

where and ; i = 1,2,3,....

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