Using Least Squares To Solve Systems of Equations

Abstract

The method of least squares (LS) yields exact solutions for the adjustable parameters when the number of data values n equals the number of parameters p. This holds also when the fit model consists of m different equations and m = p, which means that LS algorithms can be used to obtain solutions to systems of equations. In particular, nonlinear LS solves systems of nonlinear equations. An important example in chemistry is the case of reagents whose concentrations are coupled through multiple equilibrium relations. The capability of nonlinear LS in this application is examined for three programming environments, Excel Solver, FORTRAN, and KaleidaGraph, on a number of equilibrium problems having up to 10 unknown concentrations. FORTRAN and KaleidaGraph perform well in all the examples, but Solver presents difficulties that render it inadequate in several cases unless the problem is reformulated in terms of a smaller number of adjustable concentrations. When the input quantities (equilibrium constants, prepared concentrations) have uncertainty, the calculations can also be used to propagate these uncertainties into the derived quantities.

Least-Squares Analysis of Data with Uncertainty in y and x: Algorithms in Excel and KaleidaGraph

“…Many readers will be less familiar with KaleidaGraph (KG), so I will provide more procedural explanation. I also refer readers to my earlier papers in this Journal on the use of KG. − As regards the SEs, one point requires special attention, namely, the difference between a priori and a posteriori values . The former are appropriate when the data errors are considered known in an absolute sense, the latter when they are known in only a relative sense.…”

“…However, the EV 2 method normally requires access to at least 4 columns of data: x , y , and their weights or σ values. To accomplish this in the manner indicated by eqs and , I use KG’s cell­(row, column) function, where both indices are absolute, starting with 0. , The “ x ” value becomes the row entry, enumerating the points in the data set and normally starting with 1 in the second row. A column of 0s becomes the formal “ y ” for plotting, and designated columns contain the actual x , y , and weighting data.…”

“…Figure shows the Excel spreadsheet for the solution, which can be seen to be identical to the values in the fourth line of Table 6 in ref (except the SEs, which are post here, hence smaller by the factor in E6). However, achieving these results did require dealing with an idiosyncrasy in Solver that I also encountered when using it to solve systems of equations in ref , namely, its difficulty in handling adjustable parameters that differ greatly in magnitude, particularly when one or more are very small. Thus, with k in its native s –1 units, Solver seemed to satisfy its convergence criteria long before it had actually converged on the solution.…”

“…Implicit fit models can be useful in situations where the nominal dependent variable seems to require solving a quadratic or higher-order equation, for example, in binding and equilibrium studies. The LS algorithm can solve these equations in the course of obtaining the minimum-variance estimates of the parameters, just as when it is used to solve systems of equations for exactly fitting data . In some such studies, casting the equations in the commonly used linear forms makes the “dependent” variable a function of the measured and hence uncertain “independent” variable, making the two correlated.…”

“…The LS algorithm can solve these equations in the course of obtaining the minimum-variance estimates of the parameters, just as when it is used to solve systems of equations for exactly fitting data. 20 In some such studies, casting the equations in the commonly used linear forms makes the "dependent" variable a function of the measured and hence uncertain "independent" variable, making the two correlated. Both correlation and weighting are usually ignored in such work.…”