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# Modified Least Squares Convergence Program

IP.com Disclosure Number: IPCOM000077727D
Original Publication Date: 1972-Sep-01
Included in the Prior Art Database: 2005-Feb-25
Document File: 1 page(s) / 12K

IBM

## Related People

Frei, RV: AUTHOR [+2]

## Abstract

Many functions, especially exponential types, are desirable for expressing the relationship among variables, i.e., curve fitting certain data patterns; however, when applying the method of least squares to exponential functions in order to determine the coefficients of the desired function, the resulting normal equations are nonlinear and not solvable except through trial and error. Since trial and error affords no certainty of solution, it is not possible to find the coefficients for the best curve fit for these functions. Described is a method of circumventing the problem of normal equation nonlinearity and hence, produces a method for fitting data with functions of this type.

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Modified Least Squares Convergence Program

Many functions, especially exponential types, are desirable for expressing the relationship among variables, i.e., curve fitting certain data patterns; however, when applying the method of least squares to exponential functions in order to determine the coefficients of the desired function, the resulting normal equations are nonlinear and not solvable except through trial and error. Since trial and error affords no certainty of solution, it is not possible to find the coefficients for the best curve fit for these functions. Described is a method of circumventing the problem of normal equation nonlinearity and hence, produces a method for fitting data with functions of this type.

Consider the function y = Alpha+Beta e/Gamma x/ (1). which has three parameters, Alpha, Beta and Gamma which must be evaluated when fitting this function to observed data. If it is assumed that Gamma has some domain of values bounded both negatively and positively, for example, +10, a combination search technique can be used for evaluating Gamma, and then values for Alpha and Beta can be obtained using the method of least squares. By reducing Gamma to a "known" value in this manner, equation 1 above reduces to a simple linear form solvable by the method of least squares. Consequently, equation 1 reduces to: y = Alpha+Beta c (2) where

c = e/Gamma x/ (3).

The boundaries on Gamma are found by inputting the vector of observed values on x into equatio...