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1 Numer. Math. 44, 53-6 (1984) Numerische MathemalJk 9 Springer-Verlag 1984 Discrete Approximations to Spherically Symmetric Distributions* Dedicated to Fritz Bauer on the occasion of his 6th birthday Walter Gautschi Purdue University, Department of Computer Sciences, Mathematical Sciences Building, Room 442, West Lafayette, Indiana 4797, USA Summary. We consider the approximation of spherically symmetric distributions in ~ by linear combinations of Heaviside step functions or Dirac delta functions. The approximations are required to faithfully reproduce as many moments as possible. We discuss stable methods of computing such approximations, taking advantage of the close connection with Gauss- Christoffel quadrature. Numerical results are presented for the distributions of Maxwell, Bose-Einstein, and Fermi-Dirac. Subject Classifications: AMS(MOS): 65D15 CR: Introduction There is some interest among physicists in approximating the distribution functions of statistical mechanics by discrete functions - either linear combinations of Dirac delta functions or linear combinations of Heaviside step functions. For the Maxwell velocity distribution, Laframboise and Stauffer [9] and Calder, Laframboise and Stauffer [1] construct such approximations which are optimal in the sense of matching as many moments as possible. The resulting equations are solved in [9] by what amounts to Prony's method and in [1] by a reduction to an eigenvalue problem involving Hankel matrices. Both methods are classical; for the former, see, e.g. Hildebrand [8, w for the latter, Szeg6 [1, Eq. (2.2.9)]. They are subject to severe ill-conditioning, however, as is well-known. Here we point out that both approximation problems can be formulated in terms of Gauss-Christoffel quadrature, and can therefore be brought into the realm of stable modern methods of constructing orthogonal polynomials; see Gautschi [5]. In particular, algorithmic implementations of Christoffel's theorem (Galant [3], Gautschi [6]) find application here. We use these methods to generate numerical data for the distributions of Maxwell, Bose-Einstein, and Fermi-Dirac. * Work supported in part by the National Science Foundation under Grant MCS A1

2 54 W. Gautschi 2. Approximation by Step Functions We consider a function f which is spherically symmetric in F,. a, hence a function only of the radial distance, f=f(r), <r< ~. We impose the following conditions on f: (i) fe C 11-, (:x3] and f'(r)<o on [, ~]. (ii) The integrals ~f(r)r"dr, ~f'(r)r"dr, m=,1,2... all exist and are finite, o o In particular, f has to be nonnegative on [, ~]. This will not be required in this section. Noting that S f(r)rmdr= 1 f(r)rm+l]~ - 1 T f'(r)w+l dr, m=,1,2,..., o m+l m+l o it follows from (ii) that rm+lf(r) has a limit as r~, which of course must be as follows from (i) and (ii). Thus, equal to zero. Likewise, f(r)~o as r~, lim rmf(r) =, m =, 1, 2... (2.1) We wish to approximate f by a linear combination of Heaviside step functions, tl /(r)= ~ a~h(rv-r), (2.2) ~=1 where H(t)= if t<, H(t)=l if t>. Defining the moments of a function g(r), as in [1], [9], by Sg(r)r~dV, j=o, 1,2... where dv=[2nd/2/f(d/2)]ra-ldr is the volume element of the spherical shell in p,a, d>l, and dv=dr if d=l, we require that f and f have the same moments of orders up to 2n-l, i.e., co ~ avh(rv-r)rj+d-ldr=sf(r)r~+d-ldr, j=,1... 2n-l, v=l or, which is the same, rv av~ rj+a-ldr=~f(r)ri+a-'dr, v=l j=,1... 2n-1. Carrying out the integration on the left, and integrating by parts on the right, we get, upon using (2.1), d j co.. (a~rv)r~= ~ [-rnf'(r)]rjdr, j=, 1,.,2n-1. (2.3) v=l These are precisely the equations for n-point Gauss-Christoffel quadrature relative to the (nonnegative) integration measure d2(r)= -rdf'(r)dr on [, oo]. (2.4)

3 Discrete Approximations to Spherically Symmetric Distributions 55 Hence, r, in (2.2) are the Gaussian abscissas relative to the measure d2(r) in (2.4) and ;t~=avr ~ are the corresponding Christoffel numbers. Once the nodes rv and weights 2~ have been computed, the coefficients a v in (2.2) are simply obtained by a~ = 2,. r;- a, v = 1,2... n. If d= 1, the functions f(r) and f(r) are to be extended to negative values of r symmetrically with respect to the origin. To compute the n-point Gauss-Christoffel formula in question, it suffices to generate the coefficients ak, ilk' k =, 1... n - 1, in the recursion formula 7Ck+l(t):(t--O~k)7~k(t)--flkI~k_l(t), k=o, 1... n-l, (2.5) []i rc l(t)=o, rto(t)=l, for the respective (monic) orthogonal polynomials rck(')= nk(" ; d2). The desired nodes r~ are then the eigenvalues of the Jacobi matrix _ ] Jn = 1 ~1 r.. ' r O~n--1 3 oo and the weights 2v are expressible as 2~-flo -- v~,l, 2 flo = S d2(r), in terms of the o first components v~,l of the corresponding normalized eigenvectors. They all can be readily computed using standard software of linear algebra; see, e.g., Gautschi I-4, w 5.1]. 3. Examples Example3.1. The Maxwell distribution (cf. [9, 1])f(r)=n-a/2e -r2 on [, oc]. In this case, (2.4) yields o 2 dz(r)=~vffra+te-'~dr on [, oo]. (3.1) 7~' The recursion coefficients c~ k, fig in (2.5) that correspond to this measure can be computed in two different ways. We can start with the recursion coefficients co, flo for the measure d2~ on [, oe], which are available in Galant [2] for <k< 19 to 2 significant decimal digits, and then keep multiplying the measure by r (d + 1 times), each time generating the corresponding recursion coefficients by the algorithms described in Galant [3] or Gautschi [6]. The coefficient flo, at the end, is then adjusted to conform to the normalization adopted in (3.1). Alternatively, we may compute the ek, fig directly from (3.1), using the discretized Stieltjes procedure as described in Gautschi [5,

4 56 W. Gautschi Example 4.6]. We have used both these approaches, at the same time extending Galant's table up to k = 49 and recomputing it to 25 decimal places. Using double precision on the CDC 65 (ca. 29 significant decimal digits) we observed agreement to 25 decimal places. The results are listed in Table 1 of the Appendix. They can be used, as described at the end of Sect. 2, to produce Gauss-Christoffel formulae with as many as 5 terms. For reasons of space, we refrain from tabulating any of them. Example 3.2. The Bose-Einstein distribution f(r) = (fle r- 1)- 1 on [, oo], fi > 1, d > 2. The measure of interest now is d2(r)=flr a-2 e-'dr on [, oo]. (3.2) Since for large r the distribution behaves like d2(r)~fl-1 rde-rdr, we generated the recursion coefficients by the discretized Stieltjes procedure, using the Gauss-Laguerre quadrature rule to carry out the discretization (see [7] for a similar application). As a check, we also used a discretization based on the Fej6r quadrature rule applied to each subinterval of the decomposition [, oo] = [, 1] w [1, 1] w [1, 5] w [5, oo] (cf. [5, w In the case fl= 1, d=3 that we computed, the largest discrepancy observed was 1 unit in the 25th decimal place. The results are shown in Table 2 of the Appendix. Example 3.3. The Fermi-Dirac distribution f(r)=(fler2+ 1) 1 on [, oo], fl>. Here, rd+ 1 e--r 2 d2(r)=2fl (fl+e_r2)2 dr on [, ~]. For large r, this measure behaves similarly as the one in Example 3.1. Therefore, we used the same method as in Example 3.1 (the second one described there) to generate the recursion coefficients ~k, ilk' The results for fl= 1, d--3, are shown in Table 3 of the Appendix. 4. Approximation by Dirae Delta Functions We now assume that f~c[, oo], f(r)>o on [, oo], and that the integrals S f(r)r m dr, m =, 1, 2..., all exist and are finite, with f(r) dr >. Approximating f by a linear combination of Dirac delta functions, f(r),~f(r), f(r)= ~ a~(r-rv), (4.1) v=l and using the same moment-matching procedure as in Sect. 2, one is led immediately to the equations oo a-i j_ 1] (a,r,)r,- I [f(r)r"- #dr, j=o, n-1. (4.2) v=l

5 Discrete Approximations to Spherically Symmetric Distributions 57 Thus, r~ are the Gaussian abscissas relative to the (nonnegative) measure dp(r) =rd-lf(r)dr, and a~-14rv - 1-d, v=l,2... n, where Pv are the corresponding Christoffel numbers. For the Maxwell distribution of Example 3.1 this yields dp(r) =z-d/zra-1 e-'2dr, so that, when d=3, we can use the results of Example 3.1 relative to the case d=l, with an obvious modification of rio. For the Bose- Einstein distribution, one has d#(r)= r a- 1 (fie'-1)-1 dr, to which the numerical results of [7] apply, if d = 2 and fl--1. In other cases, one can easily adapt the methods used in Example 3.2. The same holds for the Fermi-Dirac distribution of Example 3.3. Appendix (Tables 1-3) see pages 58-6 References 1. Calder, A.C., Laframboise, J.G., Stauffer, A.D.: Optimum step-function approximation of the Maxwell distribution ~unpublished) oo 2. Galant, D.: Gauss quadrature rules for the evaluation of 2n ~S exp(-x:)f(x)dx. Math. Comput. 23 (1969), Review 42, Loose microfiche suppl. E o 3. Galant, D.: An implementation of Christoffel's theorem in the theory of orthogonal polynomials. Math. Comput (1971) 4. Gautschi, W.: A survey of Gauss-Christoffel quadrature formulae. In: E.B. Christoffel, The Influence of his Work on Mathematics and the Physical Sciences (P.L. Butzer, F. Feh6r, eds.), pp Basel: Birkh~iuser Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Statist. Comput. 3, (1982) 6. Gautschi, W.: An algorithmic implementation of the generalized Christoffel theorem. In: Numerische Integration (G. H~immerlin, ed.), pp Intern. Ser. Numer. Math. 57. Basel: Birkhauser Gautschi, W., Milovanovi6, G.V.: Gaussian quadrature involving Einstein and Fermi functions with an application to convergence acceleration of series, submitted for publication. 8. Hildebrand, F.B.: Introduction to Numerical Analysis, 2nd ed., New York: McGraw-Hill Laframboise, J.G., Stauffer, A.D.: Optimum discrete approximation of the Maxwell distribution. AIAA Journal 7, (1969) 1. Szeg6, G.: Orthogonal Polynomials, AMS Colloq. Publications, Vol. 23, 4th ed. 2nd printing. Providence, R.I.: Amer. Math. Soc Received July 5, 1983/September 7, 1983

6 58 W. Gautschi ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o,,,,, ~++++++g~+ c~ II ~d o o o o o o o o o o o ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o I I II o,...,,.., ~ ,,. O, ~C m ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o VII 2, ~ o o o o o o o o o o ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oo 2 m,..,,,,,,,,.....,,. ~ ,, ~ 1 7 6

7 Discrete Approximations to Spherically Symmetric Distributions 59 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o VII o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ~ ~ ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ~ I I I ~ o o o o o o o o o o o o o o o o o ~ o o o o o ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o m..,,, ~

8 6 W. Gautschi Table 3. Recursion coefficients ~k, ilk, O< k<49, for the orthogonal polynomials relative to d2(r)=2rn+l(1 +e-'~)-2e-'2dr on [, ~], d=3 k alpha(k) d d+O d d d d d d d+ 9 2.S d+O d+ II d d d d d d d d d d+O d d d S d+O d+OO d+O d d d+O d d d d d S d+O d+O d d d+O d d d Sd d d+O d d+ 48 S.89SO d+O d+ d=3 beta(k) d d d d d d SO d+O d+O 1, d IO id+O d d d d d d d+O d+ 3, d+ 3, d d+ 3, d d d d iI d d d d d d+O d d+O d+ 5, d d Sld+O iOlS d d d d d d+ 7, d+O d d d d d d+O