Gaussian Quadrature Involving Einstein and Fermi … of computation volume 44, number 169 Gaussian Quadrature Involving Einstein and Fermi Functions with an Application to Summation

MATHEMATICS OF COMPUTATIONVOLUME 44, NUMBER 169JANUARY 1985, PAGES 177-190

Gaussian Quadrature Involving Einstein and Fermi

Functions With an Application to

Summation of Series*

By Walter Gautschi and Gradimir V. Milovanovic

Abstract. Polynomials trk() = trk(; d\), k = 0,1,2,..., are constructed which are orthogo-

nal with respect to the weight distributions d\(t) = (t/(e' - l))r dt and d\(t) =

(I/O' + V)Y dt, r = 1,2, on (0, oo). Moment-related methods being inadequate, a discre-

tized Stieltjes procedure is used to generate the coefficients ak, ßk in the recursion formula

"¡t+i(0 = (' - «*K(0 - /V*-i(0. * - 0,1,2,..., ir0(f) = 1, «■_!(<) - 0. The discretiza-tion is effected by the Gauss-Laguerre and a composite Fejér quadrature rule, respectively.

Numerical values of ak, ßk, as well as associated error constants, are provided for 0 < k ^ 39.

These allow the construction of Gaussian quadrature formulae, including error terms, with up

to 40 points. Examples of n-point formulae, n = 5(5)40, are provided in the supplements

section at the end of this issue. Such quadrature formulae may prove useful in solid state

physics calculations and can also be applied to sum slowly convergent series.

1. Introduction. We are interested in Gaussian quadrature on [0, oo] relative to the

weight functions er(t) = (t/(e' - l))r and <pr(f) = (l/(e' + l))r. These functions

arise, for example, in solid state physics and are referred to, when r = 1, as Einstein

and Fermi functions, respectively. Integrals with respect to the measure dX(t) =

er(t) dt, r = 1 and 2, are widely used in phonon statistics and lattice specific heats

[7, §10], [1, §2.4], and occur also in the study of radiative recombination processes [9,

§9.2]. Specifically, the average energy of a quantum harmonic oscillator of frequency

« at temperature T (representing thermal vibrations of crystal lattice atoms) is given

by

— h<¿(1.1) u =

exp(hu/kT) - 1 '

where h = h/2m (h is the Planck constant) and k is the Boltzmann constant.

Therefore, U = kTex(hu/kT), where ex is Einstein's function. Letting g(u) denote

the phonon density of states function, and integrating (1.1) over the entire frequency

range, the total energy of thermal vibration of the crystal lattice becomes

Received April 11, 1983; revised November 14, 1983 and April 3, 1984.

1980 Mathematics Subject Classification. Primary 33A65, 65D32; Secondary 65A05, 65B10.* The work of the first author was sponsored in part by the National Science Foundation under grant

MCS-7927158A1.

©1985 American Mathematical Society

0025-5718/85 $1.00 + $.25 per page

177

178 WALTER GAUTSCHI AND GRADIMIR V. MILOVANOVIC

Furthermore, differentiating U with respect to temperature T, yields the crystal

lattice heat capacity at constant volume,

■ - —= f (hu/kT)2

0 (exp(r2w//cr) - 1);kg(o>) exp(hu/kT) du,

that is,

(1.3) C =k2T /■<»/•CO

/ e2(t)f(t) dt where f(t) = e'g((kT/h)t).

Similarly, integrals of the type /0°° q>x(t)f(t) dt (in fact, more generally

/o° <Pi(* ~ to)f(t) dt, for some parameter t0) are encountered in the dynamics of

electrons in metals, where they express the specific heat of the electron gas [8, §9.17].

For an application to heavy doped semiconductors, see also [10].

It will be shown in Section 4 that integration with respect to dX(t) = ex(t) dt and

dX(t) = <px(t) dt is also useful in summing slowly convergent series whose general

term is expressible in terms of a Laplace transform or its derivative.

Gaussian quadrature relative to the weight functions er and <pr will be particularly

attractive in the case r = 1, since both weight functions then have poles with

nonzero residues, the former at 2mri, n = ±1, ±2,..., the latter at (2n + l)wi,

n = 0, +1, ±2,_Classical integration, even based on Gauss-Laguerre quadrature,

will often be too slow, but will be used to generate the desired quadrature rules, at

least in the case of the weight function er; see (2.3) below.

The problem, basically, amounts to generating the coefficients ak, ßk in the

recursion formula

«*+i(0 - ('- «*K(0 - rV*-i(')> k = 0,1,2,...,n_x(t) = 0, *0(0 = 1

for the (monic) orthogonal polynomials trk(-) = irk(-; dX), k = 0,1,2,— (/?0 is

arbitrary, but will be defined as ß0 = /0°° dX(t).) Once the first n of these coeffi-

cients, ak, ßk, k = 0,1,... ,n - 1, and ßn, are known, the w-point Gaussian quadra-

ture formula

(1.4)

/•CO

/ f(t)dX(t)= £ V(t„) + *m(/)>•^0 ,1-1

Rjf) = yJ(2m)(r), o<t<^,

including the error constant ym, can easily be obtained for any m with 1 < m < n.

The nodes tm = r¿m), indeed, are the eigenvalues of the (symmetric tridiagonal)

Jacobi matrix

/_.=

Wi«1 {K

0

yPm-l am-\

GAUSSIAN QUADRATURE INVOLVING EINSTEIN AND FERMI FUNCTIONS 179

while the weights X^ = X^"1' are given by X^ = ß0v2x in terms of the first components

v^ of the corresponding normalized eigenvectors; see, e.g., [3, §5.1]. Moreover,

/Vi • • ■ ft,(2m)!

1,2,...,«.

A number of methods are available, and have been analyzed in [4], for computing

the recursion coefficients ak, ßk. They are either based on the moments (or modified

moments) of the weight distribution dX(t), or else use a suitable discretization of the

inner product

/•OO

(u,v)=f u(t)v(t) dX(t), u,vJn

P.

where u, v are polynomials. In the next section we discuss the relative merits of these

methods for constructing the particular orthogonal polynomials at hand.

2. Generation of the Recursion Coefficients. We consider first the weight function

er. The classical approach (for example, the Chebyshev algorithm; see [4, §2.3])

departs from the moments of the weight function, which, in the case at hand, can be

expressed in terms of the Riemann zeta function f(z),

/•OO

M* = / t\(t) dtJo

(2.1) (k + 1)!£(* + 2), r=l,

(* + 2)![?(* + 2)-£(* + 3)], r = 2,Jfc = 0,1,2,.

It is well-known, however, that the moments pk define the recursion coefficients <xk,

ßk and the associated Gaussian quadrature formulae poorly in a numerical sense.

The map Gn from the first 2« moments pk,k = 0,\,2,...,2n - 1, to the weights X'J1'

and nodes T„(n) of the «-point Gaussian formula, in fact becomes progressively more

ill-conditioned as n increases. This is illustrated in Table 2.1, where in the second

and third column the condition number of this map (in the sense of [4, Eq. (3.1 l)f])

is shown as a function of « for r = 1 and r = 2, respectively. (Numbers in

parentheses indicate decimal exponents.)

Table 2.1

The numerical condition of various maps relating to the construction

of orthogonal polynomials irk( ■ ; erdt), r = 1,2.

cond Gn

r= 1

cond Gn

r = 2

cond G„

r= 1

cond Hn

r= 1

cond Hn

r = 2

3

69

12

15

18

1.686(2)

1.288(5)

9.092(7)

6.334(10)4.409(13)

3.076(16)

7.620(1)

7.171(4)

5.820(7)

4.471(10)

3.344(13)

2.465(16)

1.415(1)

2.934(3)

9.795(5)

4.039(8)

1.878(11)

9.441(13)

5

10

15

20

30

40

7.665(0)

2.163(1)

3.299(1)

4.256(1)

6.761(1)

8.531(1)

7.980(0)

2.161(1)

3.356(1)

4.244(1)

6.783(1)

8.556(1)


Sometimes the condition can be improved by employing modified moments

vk = j™pk(t)er(t) dt, where {pk} is a system of (monic) polynomials suitably

chosen. If the support interval is infinite, however, as in the case at hand, the

improvement is often not sufficient to make this a viable alternative. The fourth

column of Table 2.1 indeed confirms this. Here, r = 1, and ex(t) ~ te'' as t -» oo, so

that the monic Laguerre polynomials with parameter a — 1,

pk(t) = (-1)**!L£>(/), k- 0,1,2,...,

appear to be the most natural choices for the polynomials pk. The corresponding

modified moments can be shown to be expressible in terms of the (forward)

differences, with unit step, of the Riemann zeta function at z = 2,

/•OO

(2.2) vk= pk(t)ex(t)dt=(k + \)\Akï(z)\^2, k = 0,1,2,....•"o

The map Gn in Table 2.1 is the map from the first 2« normalized modified moments

*k = vk\\oPk\t)ie'tdt]'x/2, k « 0,1,...,2« - 1, to the weights X<;> and nodes t,(b)

of the «-point Gaussian quadrature formula (1.4) (with dX(t) = ex(t) dt). The

entries in the fourth column exhibit the estimate derived in [4, Eq. (3.15)] for the

condition of G„. It can be seen that cond G„, while somewhat smaller than cond Gn,

is still growing unacceptably fast.

The modified Chebyshev algorithm (cf. [4, §2.4]), run in single precision on the

CDC 6500 (machine precision: ca. 15 significant decimal digits), with the modified

moments (2.2) computed in double precision, indeed loses accuracy very much in

accordance with the growth of cond Gn; the number of correct decimal digits

observed in ak, ßk, k = 3(3)18, is 14, 11, 10, 6,2,0, respectively, not a single digit

being correct for k = 18, not even the sign of ßxs.

An additional complication arises in connection with the high-order differences in

(2.2), which are subject to considerable cancellation errors. For k = 40, for example,

one must expect a loss of approx. 12 decimal digits.

In contrast, methods based on discretization appear to be incomparably more

effective, particularly if one uses the natural discretization based on the Gauss-

Laguerre quadrature rule,

" fc-1 \l-e-rt/rj

Here, rk are the zeros of the Laguerre polynomial of degree N, and XLk the associated

Christoffel numbers. The discretized Stieltjes procedure (cf. [4, §2.2]), based on (2.3),

then converges fairly rapidly as N -» oo. For example, to obtain the first 12

recursion coefficients (« = 12) to 12 correct decimals [25 correct decimals] requires

N = 49 [N = 127], when r = 1, and N = 41 [N = 85], when r = 2. The first 40

coefficients (« = 40) can be obtained accurately to 25 decimal digits with N = 281

for r = 1 and N = 201 for r = 2. Also the numerical stability of the procedure,

(2.3)


which is roughly determined (cf. [4, §3.4]) by the condition of the map

Hn: (Xx,...,Xn,Tx,...,Tn) -» (a0,...,an_x,ß0,...,ß„_x),

is rather good. This can be seen from the last two columns of Table 2.1, which

display the condition number of Hn in the sense of [4, Eq. (3.8)f].

One might expect that a discretization analogous to the one in (2.3) works well

also for the weight function <pr. This, however, is not the case, at least not if r = 1.

Convergence turns out to be slow on account of the poles at ± iir, which are twice as

close to the real axis as in the case of ex. For example, to obtain 20 correct decimal

digits (in the case r — 1 and for « = 40), requires N = 361. We used instead a

discretization procedure based on the composite Fejér quadrature rule, decomposing

the interval of integration into four subintervals, [0, oo] = [0,10] U [10,100] U

[100,500] U [500, oo]. Using A* = 201 for r = 1, and N = 281 for r = 2, on each

subinterval, then yields the first « = 40 recursion coefficients accurately to 25

decimal digits. The same procedure (with n = 50 and A* = 251) was also used to

check the computation based on (2.3) for er, r = 1,2. The maximum discrepancy

observed was 1 unit in the 25th digit.

3. Numerical Results. Using the procedures outlined in the last two paragraphs of

Section 2, we obtained the results shown in Tables 1-4 of the Appendix. The

coefficients ft0 = f(2) = ir2/6 in Table 1 and ft0 = 2[f(2) - f(3)] in Table 2 (cf. Eq.

(2.1)), agree to all 25 decimal digits with the values obtained from the 41S-table of

the Riemann zeta function in [6]. Similarly, one checks the values ft0 = In 2 in Table

3 and ft, = In 2 - \ in Table 4. Corresponding «-point Gaussian quadrature for-

mulae, « = 5(5)40, for r = 1 and r = 2, can be found to 25 significant decimal digits

in Tables 5-8 in the supplements section at the end of this issue.

We illustrate these formulae by computing the integrals

/i-/"«-'T7*-ï(2)-l,Jo e — 1

h = jT e'\^h\\dt = 2^2>2) - 2f(3'2)]>

r00 . dtI3= I e'-= 1 -ln2,3 h e'+l

f°° dt 3h= e-> -" 2 = |-21n2,

Jo (e'+l)2 2

where f(z, a) is the generalized zeta function. Table 3.1 shows the «-point approxi-

mations /,(«) to I¡, i = 1,2, together with the relative errors r¡(n), for n = 5(5)25.

Similarly, /,(«) and r,(«), / = 3,4, are shown in Table 3.2. In each entry the first

digit in error is underlined. The error term in (1.4) yields the bounds 7„//,,

/ = 1,2,3,4, for the relative error, which can be computed with the help of Tables

1-4 of the Appendix and the limit values in Tables 3.1, 3.2; they are summarized in

Table 3.3 for « = 5(5)25. The bounds for I2,14 are seen to be considerably sharper

than those for IX,I3.


Table 3.1

Gaussian approximations of the integrals Ix and I2 and relative errors.

/,(«) r,(n) I2(n) r2(n)

5 .644742 3.0(-4)

10 .6449340594 l.l(-8)15 .644934066848017 3.2(-13)

20 .64493406684822643131 8.0(-18)

25 .6449340668482264364772 1.8(-22)

.4816385 4.1(-6)

.48164052105737 1.5(-12)

.481640521058075731184 3.3(-19)

.4816405210580757313458776 l.l(-25)

Table 3.2

Gaussian approximations of the integrals I3 and I4 and relative errors.

I3(n) r3(«) 74(«) r4(n)

5 .3068171 l-2(-4)10 .30685281854 2.9(-9)

15 .3068528194400358 6.2(-14)20 .306852819440054690208 1.2(-18)

25 .3068528194400546905827607 2.3(-23)

.113705528 9.7(-7)

.113705638880091 1.5(-13)

.11370563888010938116323 2.0(-20)

.1137056388801093811655358 6.1(-26)

Table 3.3

Relative error bounds for I¡(n),i = 1,2,3,4.

7n/h(") Y„A(«) y„/h(n) y„/h(")5

1015

20

25

4.8(-2)l.l(-4)

1.9(-7)

2.7(-10)3.6(-13)

8.3(-5)3.0(-10)6.7(-16)

1.2(-21)1.8(-27)

1.2(-2)

1.6(-5)2.0(-8)2.3(-ll)2.5(-14)

1.3(-5)1.9(-11)2.3(-17)

2.5(-23)2.7(-29)

4. Summation of Series. As an application of Gaussian integration with weight

functions e, and <px, we consider the summation of series whose general term is

expressible in terms of the derivative of a Laplace transform, or in terms of the

Laplace transform itself. Suppose, for example, that ak = -F'(k), where

(4.1)

Then

that is,

(4.2a)

JfOOe-p'f(t)dt, Re/7>1.

n

00 ™ r°o r™ t

Lak=L te-k'f(t)dt= -^—f(t)dt,:=i k-iJ° J° e'-lk-

ÎF'(k)=rf(t)ex(t)dt., . ■'nfc-1 ■'o

Similarly, for "alternating" series, one obtains

(4.2b) -t(-l)k-1F'(k)=ff(t)t<px(t) dtk = l


and

(4.2c) l(-l)k-1F(k)=rf(t)<px(t)dt.k-i Jo

If the series on the left are slowly convergent and the respective function / on the

right is smooth, then low-order Gaussian quadrature applied to the integrals on the

right provides one (among many other) possible summation procedure. If / exhibits

singularities, then the weight function may have to be modified if the effectiveness of

the procedure is to be preserved (cf. Example 4.4). In the following examples, "(a)",

"(b)", "(c)" refers to a, b, c of Eq. (4.2).

Example 4.1. (a) !**_,(*: + l)"2 = 7r2/6 - 1. Here, -F'(p) = (p + l)-2, and

F(p) = (P + I)-1) the integration constant being zero on account of F(p) -» 0 as

p -» oo. Thus, f(t) = e'1, and the second column of Table 3.1 shows the rapid

convergence of Gaussian quadrature in this case. For example, 15-point quadrature

already yields 12 correct decimal digits. In contrast, 10 000 terms of the series would

give only 3-digit accuracy.

(b) L"_1(-l)*~1(/c + l)"2 = 1 - tt2/12. Applying Gaussian quadrature to the

integral in (4.2b) yields the results shown in Table 4.1. To guarantee, on the basis of

Leibniz' convergence criterion, the same accuracy as the one achieved for n = 15

would require the summation of approximately 690 000 terms.

(c) ¿Zf-x(-l)k~l(k + l)"1 = 1 - ln2. The convergence behavior of Gaussian

quadrature applied to the integral in (4.2c) is evident from the second column of

Table 3.2.Table 4.1

Gaussian approximations K(n) of the integral

i"00 \-iK=j e~'t(e' + 1) dt and relative errors.

n K(n) r(n)

5 .177753 L2(-3)

10 .1775329780 6.5(-8)

15 .17753296657625 2.1(-12)

20 .1775329665758867915 5.5(-17)

25 .177532966575886781764027 1.3(-21)

30 .177532966575886781763792 1.4(-25)

Example 4.2. F(p) = p-lcxp(-l/p), -F'(p) = (p - l)p~3 exp(-l/p). The origi-

nal function is here /(f) = J0(2<JJ). This being an entire function, we expect

Gaussian quadrature in (4.2) to converge rapidly. This is confirmed in Table 4.2,

which shows the relative errors for the «-point formula, « = 2(2)12. The exact sums

(to 24 significant digits), as determined by Gaussian quadrature, are, respectively,

OO

£ (k - l)k-3exp(-l/k) = .342918943844609780961838,k = l

oo

£ (-l)k~\k - l)fc-3exp(-lA) = -.0441559381340836052736928,fc-i

00

E (-l)/c_1/i"1exp(-l/A:)= .197107936397950656955672.*: = !


The first 10 000 terms of the series yield, respectively, 3,7 and 4 correct decimal

digits. The Bessel function J0 was evaluated by means of the rational approximations

in [5] indexed 5852, 6553 and 6953.

Table 4.2

Relative errors in Gaussian approximation of the integrals

in (4.2, a-c), whereF(p) = p~lexp(-\/p).

_n_(a)_(b)_(c)

2 4.48(-2) 8.95(-l) 1.77(-2)

4 3.80(-6) 2.39(-4) 9.65(-7)

6 3.35(-ll) 3.71(-9) 6.32(-12)

8 7.01(-17) 1.13(-14) 1.05(-17)

10 6.86(-23) 1.08(-20) 1.27(-23)

12 1.93(-23)

Example 4.3. F(p) = (1 + p2y1/2, -F'(p) = p(l + p2)~3/2. Here, f(t) =

J0(t)—again an entire function. Convergence of Gaussian quadrature in (4.2), while

still satisfactory, is not quite as fast as in Example 4.2, presumably since the

argument of JQ is now essentially squared. The relative errors are shown in Table 4.3.

For reference, we list the exact values of the respective sums:

00

£ A:(l + k2)~3/2 = .900524735348125924300853,*=i

£ (-1) k(l + k2) ' = .234771442466894018686113,k = l

oo

£ (-1) (1 + k2)\ ' = .440917473865185397183787.k = l

The first 10 000 terms yield 3,8 and 4 correct decimal digits, respectively.

Table 4.3

Relative errors in Gaussian approximation of the integrals

in (4.2, a-c), where F(p) = (l + p2)~l/2.

_n_(a)_(b)_(c)

5 1.68(-3) 8.15(-3) 1.99(-4)

10 8.03(-7) 4.75(-6) 2.11(-7)

15 4.05(-10) 2.99(-9) 9.01(-12)

20 1.64(-13) 7.61(-13) 2.26(-14)

25 5.79(-17) 5.64(-16) 2.52(-19)

30 2.69(-20) 8.25(-20) 2.33(-21)35 5.11(-24) 8.01(-23) 3.88(-25)


Example 4.4. F(p) = 2p'\p + 1)"1/2, -F'(p) = (3p + 2)p-2(p + l)"3/2. Here

we have an example in which the original function, f(t) = 2 erf \ft, is no longer

smooth, having a square root singularity at t = 0.

Table 4.4

Relative errors in Gaussian approximation of the integrals in (4.2, a-c), where

F(p) = 2p~x(p + 1)~1/2, using weight functions t1/2ex(t) andtl/2<px(t).

n (a)_(b)_(c)

5 2.10(-5) 4.69(-5) 9.20(-6)

10 5.47(-10) 1.54(-9) 1.58(-10)

15 1.24(-14) 3.88(-14) 2.70(-15)

20 2.62(-19) 8.78(-19) 4.61(-20)

25 5.17(-24) 1.85(-23) 4.91(-25)

This severely retards convergence of Gaussian quadrature in (4.2). When « = 40, for

example, the relative errors are still 4.29(-4), 7.19(-6) and 4.80(-4), respectively.

Better results can be obtained by integrating with respect to the measures t1/2ex(t) dt

and ti/2(px(t) dt. The recursion coefficients for the corresponding orthogonal poly-

nomials have been computed by a discretized Stieltjes procedure [4, §2.2] involving

two different discretizations, one on the interval [0,10], where a Gauss-Jacobi

quadrature with parameters a = 0, ft = \ (to account for the r1/2-singularity at

t = 0) was used, and one on the interval [10, oo], where we used Gauss-Laguerre

quadrature as before. With the special Gaussian quadrature rules thus obtained,

convergence of the summation process is very satisfactory; see Table 4.4. The exact

values of the sums are:oo

£ (3A: + 2)k~2(k + l)~3/2 = 2.571949632310480570278028,k = l

oo

£ (-l)*_1(3Jfe + 2)k'2(k + \)'3/2 = 1.485761529223412110869727,/c=l

oo

£ (-1)*"12*-1(A: + 1)"1/2 = 1.039526533711568982971620.

fc-i

The error function was evaluated by a double precison version of the incomplete

gamma function routine in [2], observing that erf ]ft - g*(\, t)\n the notation of [2].

Acknowledgment. The authors are indebted to Professors R. Colella and N. D.

Stqjadinovic for discussions of solid state physics applications and for the references

[1], [7]-[10].


Appendix AI: Recursion coefficients ak, ßk and error constants yk for orthogonal

polynomials irk(-\ erdt), r = 1,2.

Table 1

Recursion coefficients and error constants in the case r = 1

alpha(k) beta(k) gamma(k)

01234567

89

101112131415

161718192021222324252627282930313233343536373839

.461525938802876997452073d+0 0,704191444329339513502262d+0 0.770998656928091792159882d+0 0.806080700729485272611407d+00.828660553952942783922551d+00.184477941123603649109013d+01.385703960780471652429146d+01.586677509199924741103502d+01,787475083747655673320589d+01.988144159703517937644880d+01.188715988022024871783867d+01.389212101665985380091942d+01.589647889789580024594387d+01.790034684747346719114197d+01,990381042491900192595802d+01,190693560926138060674934d+01,390977421289520548874486d+01,591236757022591485989598d+01,7914749115710 57075454825d+01,991694622623751660309063d+01,191898156385961846196465d+01,392087407159060452589958d+01,592263972351102829585861d+01,792429209778631102387506d+01,992584281999713000914784d+01,192730191011571674407895d+01.392867805694775205910431d+01,592997883731142093202883d+01,793121089264575994179349d+01,9932380072490567523374lld+01,193 3 49155194 2 53981710560d+01,393454992848980916810097d+01,593555930237275729008440d+01.793652334368491309339130d+01,993 744 53 487 2 51859393 0477d+01,193832828757930287545616d+01,393917484449982621596683d+01593998745233871150386916d+01

,794076832204093652786452d+01994151946801537136949917d+01

644934066848226436472415d+0 0811783690642112489289654d+00676288706260277787680856d+00156744408657869148844530d+0194 73894 4 50164 51747566531d+01939056708923949547049877d+01131469329707496049141402d+01524456680862040935918879d+01117904800086908599819680d+01911733086545634278259615d+01090588203798980126014925d+02310030623930625789347323d+025494 9700904 517282 0025340d+0 2808984506906222174100004d+02088490790318381268700652d+02388013930944475277549535d+02707552309641486213037546d+02047104551169241967842131d+02406669475630502474295334d+027862460616812148410 53902d+02185833418200242159627432d+02605430762155397308587299d+02045037401085301054018693d+02504652719072102149716323d+02984276165390506043752540d+02483907245234129654656713d+02003545512072496003149494d+02543190561301266397056653d+021028 4202492 78511271 49421d+0 26 8249956 7093196335570932d+0 2282162880274321890289169d+029018316 8204 5221630 5082 58d+02054150571229892590247571d+03120118473085291539833540d+031880868 51 53751 33 52077144d+0 3258055685957963670922452d+03330024957165022879643043d+03403994647285782417304167d+03479964739634320690278091d+03557935218604162422993492d+03

1.6449d+001.4901d+007.0487d-012.7178d-019.4512d-023.0864d-029.6602d-032.9323d-038.6965d-04

5327d-042688d-050611d-057857d-066102d-064482d-072210d-073325d-080503d-094469d-09

6.5894d-101.7681d-10

7287d-ll2609d-ll3531d-128944d-133539d-132163d-14

1.6384d-144.3103d-15

1320d-159682d-167711d-170317d-173049d-183834d-186032d-193747d-204365d-203263d-216410d-21


Table 2

Recursion coefficients and error constants in the case r

alpha (k) beta(k) gamma (k)

0 8.110623843271969462056717d-01 81 2.082489563360933113678297d+00 52 3.180472732492342276049514d+00 13 4.230464687232567728899659d+00 34 5.261814180222519507249016d+00 55 6.283970335641423071715338d+00 86 7.300777826737402137770257d+00 17 8.314124893536196732880721d+00 18 9.325070412859877386826831d+00 19 1.033426456179618464586238d+01 2

10 1.134213320133091970071013d+01 211 1.234896912257135049629721d+01 312 1.335498141987434463665121d+01 413 1.436032407203413184648565d+01 414 1.536511338808887561282313d+01 515 1.636943912406203612259834d+01 616 1.737337182454617157330816d+01 717 1.837696782017803014429121d+01 718 1.938027271912508051564886d+01 819 2.038332390222895035094720d+01 920 2.138615234191184776508271d+01 121 2.238878395166622155008636d+01 122 2.339124060312416766916321d+01 123 2.439354090348347602881194d+01 124 2.539570079737826452062748d+01 125 2.639773403826193082384386d+01 126 2.739965256151148169598840d+01 127 2.840146678261384239170508d+01 128 2.940318583760677307621679d+01 229 3.040481777855517934134407d+01 230 3.140636973368393222174353d+01 231 3.240784803948646086438457d+01 232 3.340925835043190608386017d+01 233 3.441060573062978667260308d+01 234 3.541189473086012145546484d+01 335 3.641312945365461559438722d+01 336 3.741431360856105438919087d+01 337 3.841545055929548135777313d+01 338 3.941654336415387638822144d+01 339 4.041759481079403079082453d+01 3

8 57 54 3 27 37726430214 53540d-01721932339461376964598358d-01759962195010899092957389d+00452384919203117354550957d+00646769389936900623699487d+0 0344355062173523581471496d+00154523064520239591047982d+01524904924910513826966172d+01945541436638288259572407d+01416397877514391584300045d+01937445596792765549373146d+01508661182861105008895423d+01130025409989901518132003d+01801522341847326505433396d+01523138632084691795947509d+01294862990488389566003207d+01116685775941257898687472d+01988598684014213308377918d+01910 594504902261454319077d+01882666 93390 5573379469390d+01090481042149328176517281d+02197702005348084683811837d+02309929145435295611827254d+02427162070855731132915232d+02549400429588849232857666d+02676643903802234140414907d+02

,808892205395246706020251d+02,946145072259288312221530d+02,0884022651194356523440lld+02,235663564851183557323020d+02,387928770188127033667876d+02,545197695753403827498683d+02,707470170360892004885838d+02, 874746035542450893092687d+02,047025144265597523463770d+02,224307359812435482146968d+02406592554795780835373940d+02593880610292548417862187d+02786171415077789828663518d+02983464864945479141401659d+02

8.8575d-012.5341d-013.7166d-024.2771d-034.3128d-043.9986d-053.4973d-062.9303d-072.3754d-081.8758d-091.4500d-101.1012d-ll8.2392d-136.0862d-144.4465d-153.2172d-162.3081d-171.6433d-181.1621d-198.1686d-215.7101d-223.9715d-23

.7497d-24

.8958d-25

.3020d-26

.9102d-28

.0775d-29

.1327d-30

.8022d-31

.8950d-32

.2783d-338.6024d-355.7764d-363.8708d-372.5888d-381.7282d-391.1516d-407.6617d-425.0892d-433.3754d-44


Appendix A2: Recursion coefficients ak, ßk and error constants yk for orthogonal

polynomials Trk(-;<prdt),r = 1,2.

Table 3

Recursion coefficients and error constants in the case r = 1


0 1.186569110415625452821723d+0O1 3.096354215396777385868O97d+002 5.072227279535603748526280d+003 7.060122627907580237773362d+004 9.052561940107377063359596d+005 1.104727510638861953173980d+016 1.304331396591818524168138d+017 1.504020434367328423625343d+018 1.703767964046364214939868d+019 1.903557704350381572369641d+01

10 2.103379079950742683676929d+0111 2.303224885358531450836922d+0112 2.5O3O90O1997975271673O524d+0113 2.702970758679743587962983d+0114 2.902864309137106670368192d+0115 3.102768531646223714519018d+0116 3.302681755330956598650822d+0117 3.502602653879355465420772d+0118 3.702530159288482391627409d+0119 3.902463400604857779043088d+0120 4.102401659529770734984583d+0121 4.302344337664627204795512d+012 2 4.502290931954042200448238d+0123 4.702241016007743824559445d+0124 4.902194225707601499580970d+0125 5.102150247984585443741200d+0126 5.302108811972419260339366d+0127 5.502069681965214136421960d+0128 5.702032651759905847302053d+0129 5.901997540072814624028685d+0130 6.101964186797382815216901d+0131 6.301932449926549963911066d+013 2 6.501902203004635984103630d+0133 6.701873333004339765889443d+0134 6.901845738547507308111824d+0135 7.101819328405769019780267d+0136 7.301794020230469238292916d+0137 7.501769739471571379108481d+0138 7.701746418453185730908012d+0139 7.901723995579593585500405d+01

6.931471805599453094172321d-011.193356045789508659178946d+004.191806424549042451590083d+009.215367385434641976544932d+001.623913244828318399783906d+012.526147321685195747738024d+013.628235105123771646231038d+014.930193771925747304811208d+016.432041306075684261406964d+018.133793106899114661883749d+011.003546181232714737844363d+021.213705773374930439179113d+021.443858933064312247341199d+021.694006360695312561343422d+021.964148642010325651995636d+022.254286271736712212879007d+022.564419671643648677565733d+022.894549204416899668238851d+023.24467 5184414076026002832d+023.614797886084236427881509d+024.00491755062552531702251ld+024.415034391301419740031332d+024.845148597725808406593359d+025.295260339347629377708534d+025.765369768308248977038621d+026.255477021802805949656030d+026.765582224045888408376874d+027.295685487919013322017815d+027.845786916360230644686497d+028.415886603543216590530846d+029.005984635883343186160464d+029.616081092900618200644516d+021.024617604796350530317238d+031.089626956893303904655591d+031.156636171872303425240697d+031.225645255578932558332990d+031.296654213455868932332327d+03

1.369663050580626665464266d+031.444671771698882809763324d+031.521680381254001740382063d+03

6.9315d-014.1359d-011.4447d-014.4379d-021.2869d-023.6122d-039.9286d-042.6896d-047.2081d-051.9160d-055.0599d-061.3293d-063.4770d-079.0616d-082.3543d-086.1002d-091.5770d-094.0683d-101.0476d-102.6935d-ll6.9148d-121.7729d-124.5401d-131.1614d-132.9681d-147.5782d-151.9333d-154.9283d-161.2554d-163.1958d-178.1303d-182.0672d-185.2532d-191.3343d-193.3873d-208.5956d-212.1803d-215.5280d-221.4011d-223.5498d-23


Table 4

Recursion coefficients and error constants in the case r


0 6.695404638538438232387227d-011 1.664686133009829680011075d+002 2.619431841015758476604192d+003 3.593865366974072299756279d+004 4.579560611244422509231409d+005 5.570393244889081013183465d+006 6.563863856463274821282662d+007 7.558890435710582391017287d+008 8.554931506945101735350174d+009 9.551680424073512844025433d+00

10 1.054894749426849736758553d+0111 1.154660774499827949400336d+0112 1.254457487195045007921430d+0113 1.354278703591851122550789d+0114 1.454119858463881673190978d+0115 1.553977497041409480722936d+0116 1.653848949597723105885520d+0117 1.753732115701126023268357d+0118 1.853625316885666471251907d+0119 1.953527193452852551824561d+0120 2.053436630566769258684606d+0121 2.153352704286241009913773d+0122 2.253274641468520667332283d+0123 2.353201789515210436373844d+0124 2.453133593225018788013906d+012 5 2.553069576859866483809419d+0126 2.653009330090433519953643d+0127 2.7 52952496866371491931937d+0128 2.852898766517794626792057d+0129 2.952847866577761073760216d+0130 3.052799556945594577910743d+0131 3.152753625104640930157417d+0132 3.252709882176421757703103d+0133 3.352668159643583823594838d+0134 3.452628306611646621975720d+0135 3.552590187507868194832644d+0136 3.652553680137070829557449d+0137 3.752518674030767687262801d+0138 3.852485069038683713846996d+0139 3.952452774121695980565398d+01

1.931471805599453094172321d-013.705278710851684856806622d-011.154907853652601365365824d+002.416782871937569027723998d+004.178983363598481071166374d+006.442373341823139342338233d+009.205824543736968972236729d+001.246885990410737186232674d+011.623136041040844992228737d+012.049332827585098989839084d+012.525480038308609252223979d+013.051582056896307516793924d+013.627643102788570772578227d+014.253666988039415434188650d+014.929657079160519734344687d+015.655616326612624756002118d+016.431547313435234317081962d+017.257452305464815707363551d+018.133333297094291496385707d+019.059192050950994107419572d+011.003503013155485508754740d+021.106084893360278480033078d+021.21366497O566454120865567d+021.326243357004168602197873d+021.443820153944878713298334d+021.566395453107226933490024d+021.693969337846574069748673d+021.826541884165723339055932d+021.964113161577435259558123d+022.106683233843655691641098d+022.254252159611781753560632d+022.406819992964582304786437d+022.564386783897402174805713d+022.726952578733870106688784d+022.894517420489383379229000d+023.067081349190064005746629d+023.244644402153598017416908d+023.427206614237321739508610d+023.614768018058060537519819d+023.807328644187519319245180d+02

1.9315d-013.5783d-023.4439d-032.7744d-042.0704d-051.4820d-061.0336d-077.0810d-094.7889d-103.2072d-ll2.1315d-121.4079d-139.2524d-156.0549d-163.9482d-172.5666d-181.6640d-191.0764d-206.9479d-224.4767d-232.8797d-241.8497d-251.1865d-267.6022d-284.8653d-293.1106d-301.9869d-311.2681d-328.0864d-345.1529d-353.2813d-362.0882d-371.3281d-388.4422d-405.3635d-413.4059d-422.1617d-431.3715d-448.6975d-465.5135d-47


Department of Computer Sciences

Purdue University

West Lafayette, Indiana 47907

Faculty of Electronic Engineering

Department of Mathematics

University of Ni5

18000 NiS, Yugoslavia

1. J. S. Blakemore, Solid State Physics, 2nd ed., Saunders, Philadelphia, Pa., 1974.

2. W. Gautschi, "Algorithm 542—Incomplete gamma functions," ACM Trans. Math. Software, v. 5,

1979, pp. 482-489.

3. W. Gautschi, "A survey of Gauss-Christoffel quadrature formulae," in E. B. Christoffel—The

Influence of his Work in Mathematics and the Physical Sciences (P. L. Butzer and F. Fehér, eds.),

Birkhäuser Verlag, Basel, 1981, pp. 72-147.

4. W. Gautschi, "On generating orthogonal polynomials," SI AM J. Sei. Statist. Comput., v. 3, 1982,

pp. 289-317.5. J. F. Hart et al., Computer Approximations, Wiley, New York-London-Sydney, 1968.

6. A. McLellan IV, "Tables of the Riemann zeta function and related functions," Math. Comp., v.

22,1968, Review 69, pp. 687-688.

7. S. S. Mitra & N. E. Massa, "Lattice vibrations in semiconductors," Chapter 3 in: Band Theory and

Transport Properties (W. Paul, ed.), pp. 81-192. Handbook on Semiconductors (T. S. Moss, ed.), Vol. 1.

North-Holland, Amsterdam, 1982.

8. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York, 1965.

9. R. A. Smith, Semiconductors, 2nd ed., Cambridge Univ. Press, Cambridge, 1978.

10. R. J. Van Overstraeten, H. J. DeMan & R. P. Mertens, "Transport equations in heavy doped

silicon," IEEE Trans. Electron Dev., v. ED-20, 1973, pp. 290-298.

mathematics of computationvolume 44, number 169JANUARY 1985, PAGES Sl-Sll

Supplement to

Gaussian Quadrature Involving Einstein

and Fermi Functions with an

Application to Summation of Series

By Walter Gautschi and Gradimir V. Milovanovic

TABLE 5. Nodes and weights for n-point Gaussian quadrature with respect to the

weight function e,(t)=t/(e -1) on (0,»), n=5(S)40.

zero(n) weight(n)

1 3.4795351312035335358889O6d-012 1.789519270251396059078587d+û03 4.317073039784563286568702d+004 8.124316435674601205701546d+005 1.399259503591182245471045d+01

7.413801893864949064648418d-017.112793968534230324727631d-011.810544213015127812690667d-011.113156581019995660932093d-028.849349659575965642269187d-05

zerotn) weight(n)

1 1.712764587800172362970467d-012 8.916728564071628155969295d-013 2.154696241995276926728150d+004 3.940962194432075308546014d+005 6.27305497812020O583717756d+006 9.219833208404748987213936d+007 1.289612902426177067770092d+018 1.749262020229698453921089d+019 2.337506876689075787489480d+01

10 3.148092990870547794581304d+01

4.017581983871970550764847d-016.178151502068598877652466d-014.309238491671243158403736d-011.601831853477 29222338 2136d-013.111600156831707548706688d-023.0O2950279906314058385O99d-031.324400356318608169220646d-042.280734015322767264428365d-061.111475587288852659683946d-086.689509433931585817256954d-12

zerotn) weight(n)

1 1.122689742882840428560134d-012 5.872942570832169304369100d-013 1.427134594172779508922935d+O04 2.616230204390486070960179d+005 4.148505367683641117556727d+006 6.033516951668504802085092d+007 8.294299990936086527007386d+008 1.096384343614386146477958d+019 1.408587628929527609167703d+01

10 1.7719996911567692 33871477d+0111 2.195126297803758949286577d+0112 2.690847821599200481688143d+0113 3.280454835374218314720542d+0114 4.004648974077095940531751d+0115 4.967641464169027902025467d+01

2.717674922677317952221318d-014.852206727225164096245477d-014.581586361360587417881249d-012.806056660239435321238224d-011.135140668971388075471236d-012.998304918534276590192265d-025.098666065215486743990180d-035.482040128082603574763532d-043.618564942449709026477206d-051.398222595231963340037460d-062.936603420052541839075062d-082.982123035751269147999240d-101.203079310277072656860401d-121.324544206896025265830478d-151.546415245123910288693076d-19

SI

©1985 American Mathematical Society

0025-5718/85 $1.00 + $.25 per page

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Gaussian Quadrature Involving Einstein and Fermi … of computation volume 44, number 169 Gaussian Quadrature Involving Einstein and Fermi Functions with an Application to Summation