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


Author: Walter Gautschi
Journal: Math. Comp. 23 (1969), 109-118
MSC: Primary 65.25
MathSciNet review: 0239729
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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}$.


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DOI: https://doi.org/10.1090/S0025-5718-1969-0239729-8
Article copyright: © Copyright 1969 American Mathematical Society