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

Abstract:

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 $n$-point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of $n$ linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters $\alpha$ and $\beta$) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters $\alpha$, $\beta$ 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 $(n,\alpha ,\beta )$ of this system as a function of $n$, $\alpha$, and $\beta$. It is found that cond $(n,\alpha ,\beta )$ is usually larger than cond $(n,\beta ,\alpha )$ if $\beta > \alpha$, at least asymptotically as $n \to \infty$. Lower bounds for cond $(n,\alpha ,\beta )$ are obtained together with their asymptotic behavior as $n \to \infty$. Sharper bounds are derived in the special cases $n$, $n$ odd, and $\alpha = \beta = \pm \frac {1} {2}$, $n$ arbitrary. There is also a short table of cond $(n,\alpha ,\beta )$ for $\alpha$, $\beta = - .8(.2)0,.5,1,2,4,8,16,\beta \leqq \alpha$, and $n = 5,10,20,40$. The general conclusion is that cond $(n,\alpha ,\beta )$ grows at a rate which is something like a constant times ${(3 + \surd 8)^n}$, where the constant depends on $\alpha$ and $\beta$, varies relatively slowly as a function of $\alpha$, $\beta$, and appears to be smallest near $\alpha = \beta = - 1$. For quadrature rules with equidistant points the condition grows like $(2\surd 2/3\pi ){8^n}$.