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 a and ß) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters a, ß 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 On, a, ß) of this system as a function of n, a, and ß. It is found that cond (n, a, ß) is usually larger than cond On, ß, a) if ß > a, at least asymptotically as n —> °°. Lower bounds for cond On, a, ß) are obtained together with their asymptotic behavior as n —> °°. Sharper bounds are derived in the special cases a = ß, n odd, and a — ß = ± §, n arbitrary. There is also a short table of cond On, a, ß) for a, ß = — .8(.2)0, .5, 1, 2, 4, 8, 16, ß 5Í a, and 71 — 5, 10, 20, 40. The general conclusion is that cond (n, a, ß) grows at a rate which is something like a constant times (3 + V 8)", where the constant depends on a and ß, varies relatively slowly as a function of a, ß, and appears to be smallest near a = ß = — 1. For quadrature rules with equidistant points the condition grows like (2V2/3tt)8". ■