Interpolation Formula Interpolation Formula In mathematics, Interpolation is defined as the process of finding the value of function that is declared on the specific values at specific points . It will also be defined as the method for finding the values of new data points that is define within the range of declared data points. It is not the experimentation or approximation because in the interpolation the value for the specific point is not assumed but it is calculated through the interpolation formula. Interpolation Formula will be defined for the given values of the coordinate as (x1 , y1) and (x2 , y2) so for calculate the value of y at any point x within the range of coordinates (x1 , x2) and (y1 , y2) and x1 is not equal to x2 . y – y1 / x – x1 = y2 - y1 / x2 – x1 . Then by using above formula we can find the value y at the given point x as: y = y1 + (x – x1) y2 – y1 / x2 – x1
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or y = y1 + ((x – x1) y2 – (x – x1) y1) / x2 – x1 . We can define the interpolation formula by an example as if there is table that has the values for the function that is unknown as a:0123456 f (a) : 0 0.8415 0.0993 0.1411 -0.7658 -0.9589 -0.2794 So based on these values find the answer for a = 2.5 by interpolation? Solution: the above problem can be solved through different types of interpolation methods that are explained as follows: (1) Constant interpolation : In this method find the nearest data value and assign it to the value for which we find the answer .It is the simplest way for finding the answer for the given problem but it is rarely used . (2) Linear interpolation : It is also simple method for finding the value for a given data point as it can be solved as that function f (a) = f (2.5) where we have to find the value for 2.5 that is in between 2 and 3 and for f (2 ) = 0.9093 and f (3) = 0.1411 so by using the formula y = y1 + (y2 – y1) (x – x1) / (x2 – x1) So by putting the value in the above formula we can obtain value for the function f (2.5) = 0.5252 . There are some other interpolation techniques that are also used for finding the value that are known as polynomial interpolation and spline interpolation etc .
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Polynomial interpolation can be defined as the generalization of the linear interpolation .So the polynomial interpolation of higher degree is defined as for the given problem f (2.5) f (2.5) = f (a) = -0.0001521 a^6 – 0.003130 a^5 +0.07321 a^4 - 0.3677 a^3 + 0.2255 a^2 + 0.9038 a . So answer for the function f (a) = f (2.5) is 0.5965 . So the answer for the function f (2.5) from different methods is nearly equal to each other and these provide the answer that is nit approximate value but it is a well calculated value for the defined function.
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