interpolation using roots values

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Hi, im not pretty sure this thread belong to this category but it mixes math and statistics.
My advisor told me to use one of his codes to estimate an interpolation of data. But i really dont understand too much about the explanation he had given to me in the script. He writes:

"This script estimates the roots, f(xroot) = 0, of a tabulated real number function (x(i),y(i)) from which a real number function y = f(x) can be calculated by interpolation;in the simplest case the data table contains roots, i.e. y(k) = 0;other roots exist if y(k)*y(k+1) < 0 and can be estimated by interpolation; in this case, for each root to be estimated, this function 1) interpolates according to 'method' at subgrid resolution over the longest possible abscissa range, producing a subgrid table (x'(i),y'(i)), and then 2) estimates the root by linearly intepolating within y'(k) and y'(k+1), given y'(k)*y'(k+1) < 0"

I would like to understand mathematically how can you interpolate using roots because i had not found it anywhere.

Thanks for your help,

J
 
Hi, im not pretty sure this thread belong to this category but it mixes math and statistics.
My advisor told me to use one of his codes to estimate an interpolation of data. But i really dont understand too much about the explanation he had given to me in the script. He writes:

"This script estimates the roots, f(xroot) = 0, of a tabulated real number function (x(i),y(i)) from which a real number function y = f(x) can be calculated by interpolation;in the simplest case the data table contains roots, i.e. y(k) = 0;other roots exist if y(k)*y(k+1) < 0 and can be estimated by interpolation; in this case, for each root to be estimated, this function 1) interpolates according to 'method' at subgrid resolution over the longest possible abscissa range, producing a subgrid table (x'(i),y'(i)), and then 2) estimates the root by linearly intepolating within y'(k) and y'(k+1), given y'(k)*y'(k+1) < 0"

I would like to understand mathematically how can you interpolate using roots because i had not found it anywhere.

Thanks for your help,

J
This is a Mean Value Theorem simple Linear Interpolation.

1) y(k) * y(k+1) < 0 means only that you have evaluated your function in two places, and have found yourself on both sides of the x-axis. This MUST mean, assuming continuity, that there is a root between k and k+1
2) Using the four values, k, k+1, y(k), an y(k+1), there is then a linear interpolation to find k+2, which, it is hoped, will come closer to producing y(k+?) = 0

3) It is important to note that the notation "y'(k)" is NOT a derivative. It's just the next function value after y(k).

4) After seeing that y(k+2) isn't quite zero, it is time to do it all again. You must decide whether to use k and k+2 or k+1 and k+2. The criterion y(k+?) * y(k+??) < 0 tells you which to use.

5) Lookup the "Bisection Method". It is characterized as reliable but slow.

6) I do not understand "longest possible abscissa range". You have the product less than zero criterion or you don't. Perhaps we're talking about multiple roots and deciding which to hunt down first.
 
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