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# Program to Numerically Differentiate Data

IP.com Disclosure Number: IPCOM000053073D
Original Publication Date: 1981-Aug-01
Included in the Prior Art Database: 2005-Feb-12
Document File: 1 page(s) / 12K

IBM

## Abstract

In fields such as numerical analysis, statistics, reliability, or engineering there is often the need to differentiate a set of data. For example given X, Y values, it may be necessary to determine the minimum rate of change of Y versus X. A standard procedure would be to plot a graph of Y versus X, assume a function to represent the data, and use the derivative of the function to yield the rate of change of Y versus X. An alternative method, very simple in concept, would be to place a ruler by eye tangent to the empirical curve at various points and obtain an approximation to the derivative by the standard (delta X)/(delta Y) estimator. However, this process can be very tedious and time-consuming to do by hand, especially if many points are desired for the derivative.

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Program to Numerically Differentiate Data

In fields such as numerical analysis, statistics, reliability, or engineering there is often the need to differentiate a set of data. For example given X, Y values, it may be necessary to determine the minimum rate of change of Y versus X. A standard procedure would be to plot a graph of Y versus X, assume a function to represent the data, and use the derivative of the function to yield the rate of change of Y versus X. An alternative method, very simple in concept, would be to place a ruler by eye tangent to the empirical curve at various points and obtain an approximation to the derivative by the standard (delta X)/(delta Y) estimator. However, this process can be very tedious and time-consuming to do by hand, especially if many points are desired for the derivative. A program is described to numerically differentiate a curve by such a procedure.

Suppose the data consists of n (X,Y) pairs of data points. We choose the first p points on which to perform a least squares fit of Y versus X. In the least squares fit the sum of the squares of the vertical deviations of Y from the linear regression line is minimized. Thus, a regression line is fit to the first p data points, and the slope of the regression line is used as the estimator of the derivative. The point at which the derivative is considered evaluated is the average of the first p values of X, that is, X(p). The program then steps up by one to the next p data points...