11 Lagrange Interpolation
V.D. Pathak
1. Why and what is Interpolation?
There are many real situations in which a process / phenomena is observed at certain intervals of time, as it may be difficult or impossible to observe and record such process continuously. However, for certain purposes, it is required to know the status of the process at some instance within the period, over which the observations are recorded.
For example, population senses of a country are taken after every 5 / 10 years, say, in 1995, 2000, 2005, 2010, and 2015 and suppose the government wants to know the effectiveness of certain populist measures in 2009, for which the estimate of population in 2009 is required.
As another example, in a black & white digital image various grey levels z are recorded at certain grid points depending on the resolution of the image. To get exact analogue image, grey level estimates at the intermediate positions or a functional representation of a grey level, say, z = g(x, y) at each positional point (x, y), is required.
In the above examples, there are two types of variables associated with the process, a dependent variable (population / grey level) and an independent variable (year/ co-ordinates of a point). The observations are recorded in the form of values of dependent variables corresponding to certain discrete values of independent variables.
The technique of determining the values of a dependent variable at some intermediate values of independent variables (i. e. Values in the domain over which the observations are made) from the observed discrete data points is called Interpolation. If the values of the dependent variables are required to be determined at the values of the independent variables outside the domain of observation, then the technique is called Extrapolation.
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