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 .

**[1]**A. B. Bakušinskiĭ, ``On a numerical method for the solution of Fredholm integral equations of the first kind,''*Ž. Vyčisl. Mat. i Mat. Fiz.*, v. 5, 1965, pp. 744-749. (Russian)**[2]**A. B. Bakušinkiĭ, ``On a certain numerical method of solution of Fredholm integral equations of the first kind,''*Comput. Methods Programming*. Vol. V, Izdat. Moskov. Univ., Moscow, 1966, pp. 99-106. (Russian) MR**35**#6386.**[3]**R. Bellman, R. Kalaba, and J. Lockett,*Dynamic programming and ill-conditioned linear systems. II*, J. Math. Anal. Appl.**12**(1965), 393–400. MR**0191702**, https://doi.org/10.1016/0022-247X(65)90006-5**[4]**Richard Bellman, Robert E. Kalaba, and Jo Ann Lockett,*Numerical inversion of the Laplace transform: Applications to biology, economics, engineering and physics*, American Elsevier Publishing Co., Inc., New York, 1966. MR**0205454****[5]**Walter Gautschi,*On inverses of Vandermonde and confluent Vandermonde matrices*, Numer. Math.**4**(1962), 117–123. MR**0139627**, https://doi.org/10.1007/BF01386302**[6]**Walter Gautschi,*Construction of Gauss-Christoffel quadrature formulas*, Math. Comp.**22**(1968), 251–270. MR**0228171**, https://doi.org/10.1090/S0025-5718-1968-0228171-0**[7]**V. I. Krylov, V. V. Lugin, and L. A. Janovič,*\cyr Tablitsy dlya chislennogo integrirovaniya funktsiĭso stepennymi osobennostyami ∫₀¹𝑥^{𝛽}(1-𝑥)^{𝛼}𝑓(𝑥)𝑑𝑥*, Izdat. Akad. Nauk. Belorussk. SSR, Minsk, 1963 (Russian). MR**0157012****[8]**Y. L. Luke, Review 6,*Math. Comp.*, v. 22, 1968, pp. 215-218.**[9]**David L. Phillips,*A technique for the numerical solution of certain integral equations of the first kind*, J. Assoc. Comput. Mach.**9**(1962), 84–97. MR**0134481**, https://doi.org/10.1145/321105.321114**[10]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. PubL., Vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR**21**#5029.**[11]**A. N. Tihonov and V. B. Glasko,*An approximate solution of Fredholm integral equations of the first kind*, Ž. Vyčisl. Mat. i Mat. Fiz.**4**(1964), 564–571 (Russian). MR**0169404****[12]**John Todd,*Introduction to the constructive theory of functions*, Academic Press, Inc., New York, 1963. MR**0156129****[13]**S. Twomey,*On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature*, J. Assoc. Comput. Mach.**10**(1963), 97–101. MR**0148249**, https://doi.org/10.1145/321150.321157**[14]**P. N. Zaikin,*The numerical solution of the inverse problem of operational calculus in the real domain*, Ž. Vyčisl. Mat. i Mat. Fiz.**8**(1968), 411–415 (Russian). MR**0238471**

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

Article copyright:
© Copyright 1969
American Mathematical Society