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

Author:

Gautschi Walter

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 n -point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of n n linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters α \alpha and β \beta ) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters α \alpha , β \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 , α , β ) (n,\alpha ,\beta ) of this system as a function of n n , α \alpha , and β \beta . It is found that cond ( n , α , β ) (n,\alpha ,\beta ) is usually larger than cond ( n , β , α ) (n,\beta ,\alpha ) if β > α \beta > \alpha , at least asymptotically as n n \to \infty . Lower bounds for cond ( n , α , β ) (n,\alpha ,\beta ) are obtained together with their asymptotic behavior as n n \to \infty . Sharper bounds are derived in the special cases n n , n n odd, and α = β = ± 1 2 \alpha = \beta = \pm \frac {1} {2} , n n arbitrary. There is also a short table of cond ( n , α , β ) (n,\alpha ,\beta ) for α \alpha , β = .8 ( .2 ) 0 , .5 , 1 , 2 , 4 , 8 , 16 , β α \beta = - .8(.2)0,.5,1,2,4,8,16,\beta \leqq \alpha , and n = 5 , 10 , 20 , 40 n = 5,10,20,40 . The general conclusion is that cond ( n , α , β ) (n,\alpha ,\beta ) grows at a rate which is something like a constant times ( 3 + 8 ) n {(3 + \surd 8)^n} , where the constant depends on α \alpha and β \beta , varies relatively slowly as a function of α \alpha , β \beta , and appears to be smallest near α = β = 1 \alpha = \beta = - 1 . For quadrature rules with equidistant points the condition grows like ( 2 2 / 3 π ) 8 n (2\surd 2/3\pi ){8^n} .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

Reference14 articles.

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. Dynamic programming and ill-conditioned linear systems. II;Bellman, R.;J. Math. Anal. Appl.,1965

4. On inverses of Vandermonde and confluent Vandermonde matrices;Gautschi, Walter;Numer. Math.,1962

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