On the condition of a matrix arising in the numerical inversion of the Laplace transform
Author:
Walter Gautschi
Journal:
Math. Comp. 23 (1969), 109118
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 point GaussLegendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of linear algebraic equations. Luke suggests the possibility of using GaussJacobi quadrature (with parameters and ) in place of GaussLegendre 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:
http://dx.doi.org/10.1090/S00255718196902397298
PII:
S 00255718(1969)02397298
Article copyright:
© Copyright 1969
American Mathematical Society
