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

DOI:
https://doi.org/10.1090/S0025-5718-1969-0239729-8

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 -point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters and ) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters , 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 of this system as a function of , , and . It is found that cond is usually larger than cond if , at least asymptotically as . Lower bounds for cond are obtained together with their asymptotic behavior as . Sharper bounds are derived in the special cases , odd, and , arbitrary. There is also a short table of cond for , , and . The general conclusion is that cond grows at a rate which is something like a constant times , where the constant depends on and , varies relatively slowly as a function of , , and appears to be smallest near . For quadrature rules with equidistant points the condition grows like .

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DOI:
https://doi.org/10.1090/S0025-5718-1969-0239729-8

Article copyright:
© Copyright 1969
American Mathematical Society