The Solution of Large Sparse Unsymmetric Systems of Linear Equations

We describe in moderate detail an algorithm for the solutionof a general sparse set of linear equations. It aims to maintainboth stability and sparseness, and allows further systems withthe same matrix, or another matrix having the same sparsitypattern, to be solved economically. It is compared, in testson several non-trivial problems, with two earlier algorithmsfrom which it was derived, and appears to combine the best featuresof both.     

On the Automatic Scaling of Matrices for Gaussian Elimination

The usual pivotal strategies for Gaussian elimination are basedon the assumption that all the matrix elements are of comparablesize. We propose an algorithm for scaling based on the assumptionthat the given matrix can be scaled to this required form. Somenumerical experiments are presented to show that it producesmuch better results than straightforward row and column normequilibration, particularly in the sparse case, and that thecomputing cost is moderate.     

On the Estimation of Sparse Jacobian Matrices

We show how to use known constant elements in a Jacobian matrixto reduce the work required to estimate the remaining elementsby finite differences.     

Jacobian Matrix Properties and Their Impact on Choice of Software for Stiff ODE Systems

Information is presented about the spectral and other propertiesof Jacobian matrices occurring in the numerical solution ofa number of large, very stiff ODE problems, arising from massaction kinetics. These properties demonstrate that the conceptof a few "stiff" eigenvalues, the rest being "non-stiff", isnot valid for such problems; consequently, it is argued thatpartitioning and exponential-fitting methods are inappropriatefor use in general-purpose software for stiff systems. Moreover,second-derivative methods and all but a very few formulationsof implicit Runge-Kutta methods would be at a grave disadvantagewhen applied to large, very stiff problems.     

Spectroscopic, Chromatographic and Visual Investigation of Organic Dyes

An introductory level laboratory experiment is presented in which students learn about color using spectroscopy and chromatography. The pedagogical approach is discovery-based; students are given only enough background information to enable them to take good data. Commercially available dyes are dissolved in water to make concentrated stock solutions, which students then dilute to prepare solutions of primary, secondary and tertiary colors. The class works as a team to study colored solutions representing a range of concentrations and combinations of the three primary colors: cyan, yellow, and magenta. Students record transmission and absorption spectra and compare the results with human perception. They show the number of components in each solution by paper chromatography. They explore the filter nature of dyes, the effects of concentration, and the cumulative effects of mixing dyes. From this information they deduce the principles of color printing.The experiment is best performed with a photodiode array spectrophotometer; alternative approaches include spectrometers and simple spectroscopes which the students make from cereal boxes. The experiment can be performed in either a single three-hour laboratory period, or alternatively, three one-hour sessions.     

The Choice of Step Lengths When Using Differences to Approximate Jacobian Matrices

We propose an algorithm for automatic choice of step lengthsfor use when approximating derivatives by differences. We envisagethat its principal application will be the approximation ofJacobian matrices and discuss this case explicitly.     

Analysis of Covariance after Nonlinear Least-Squares Fitting

After linear least-squares fitting (multiple regression) ofa number of parameters, tests of significance of fitted parametervalues may involve repeating the fit with various combinationsof parameters omitted (i.e. forced to have zero values). Thisis often inappropriate in the nonlinear case for two reasons:(i) zero may not be a reasonable value for an ill-determinedparameter, and (ii) the cost of doing additional fits may beprohibitive. Also, it can be difficult to automate the process of rejectingill-determined parameters in cases where, because of high correlation,some of them, all of which are apparently ill determined, havein fact one or more well-determined combinations. Such casesare by no means rare in some applications, for example fittingof rate coefficients to mass-action kinetic data. A method is described for automatic decision-making in thiscase, based on a multiparameter least-squares fitting process.     

High-order Explicit Runge-Kutta Formulae, Their Uses and Limitations

On page 53, the value given for the coefficient c16, approximatelyhalf-way down the right-hand column, is incorrect as printed.The correct figure is as follows: