On the condition of a matrix arising in the numerical inversion of the Laplace transform

Bellman, Kalaba, and Lockett recently proposed a numerical method for inverting the Laplace transform. The method consists in first reducing the infinite interval of integration to a finite one by a preliminary substitution of variables, and then employing an n n -point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of n n linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters α α and β β ) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters α α , β β may have a beneficial influence on the condition of the linear system of equations. The object of this note is to investigate the condition number cond ( n , α , β ) (n,α ,β ) of this system as a function of n n , α α , and β β . It is found that cond ( n , α , β ) (n,α ,β ) is usually larger than cond ( n , β , α ) (n,β ,α ) if β > α β > α , at least asymptotically as