Chebyshev Approximation to Data by Positive Sums of Exponentials

The problem is considered of calculating Chebyshev approximationsto given data by sums of exponentials with positive coefficients,where the number of terms in the sum has to be obtained as partof the process. An exchange procedure based on linear programmingis developed for the estimation of the exponents, and this ismade efficient by the use of postoptimality theory and the applicationof the dual simplex algorithm. Rapid convergence to a best approximationcan then be obtained by the application of Newton's method tothe characterization conditions interpreted as a nonlinear systemof equations. The Newton step can be determined through thesolution of a quadratic programming problem, and advantage istaken of the structure so that the calculation can be simplifiedwithout inhibiting a second-order convergence rate. Numericalresults are presented for the application of an algorithm basedon these ideas to a number of data sets which have appearedin the literature.