orthogonal polynomials arising in the numerical evaluation of … · 2018. 11. 16. · orthogonal...

164 orthogonal polynomials in inverse laplace transforms

four digits of the final result. The result has been tabulated to 3089D; the final

digit is unrounded.

Running time for the 3093D was approximately thirteen minutes. The pro-

gramming takes account of the number of zeros generated to the right of the

decimal point in each factor, so that the number of operations required for each

term in the series decreases. This leads to the following statement—if the time

to compute x to m digits is t units, then the time to produce km digits is roughly

kH units; this holds true as long as the calculation is contained in high-speed

storage.

The following table gives a count of each of the digits in x.

(1) (2) (3) (4) (5)1-3090 1-2036 2037-3090 (4)/(3)

0 269 184 85 .461 315 213 102 .472 314 210 104 .503 276 191 85 .454 322 198 124 .635 326 211 115 .546 311 204 107 .527 297 200 97 .498 318 207 111 .549 342 218 124 .57

£ 3090 2036 1054 .52

S. C. Nicholson

J. Jeenel

Watson Scientific Computing Laboratory

612 West 115th Street

New York 27, New York

L The IBM-Naval Ordnance Research Calculator, now located at Naval Proving Ground,Dahlgren, Virginia.

2. George W. Reitwiesner, "An ENIAC determination of jt and e to more than 2000 decimalplaces," MTAC, v. 4, 1950, p. 11-15.

3. Fora description of the NORC checking system, see W. J. Eckert & R. B. Jones, Faster,Faster, McGraw-Hill Book Company, New York, 1955, p. 98-104.

Orthogonal Polynomials Arising in the NumericalEvaluation of Inverse Laplace Transforms

Abstract. In finding f(t), the inverse Laplace transform of F(p), where (1)

f(t) = (l/2xj) I eptF(p)dp, the function F(p) may be either known onlyJ C—JOO

numerically or too complicated for evaluating f(t) by Cauchy's theorem. When

F(p) behaves like a polynomial without a constant term, in the variable 1/p,

along (c — joo, c + j°°)> one may find f(t) numerically using new quadrature

formulas (analogous to those employing the zeros of the Laguerre polynomials

in the direct Laplace transform). Suitable choice of pi yields an «-point quadrature

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orthogonal polynomials in inverse laplace transforms 165

formula that is exact when p2n is any arbitrary polynomial of the (2«)th degreePc+jx

in x = 1/p without a constant term, namely: (2) (l/2xj) I epp2n(l/p)dpJ C—jx

n

= Y. -4i(")P2n(l/£.). In (2), Xi = 1/p, are the zeros of the orthogonal polynomials»=i

Pn(x) m f[ (x - where (3) (l/2xj) f*'" ep{\/p)pn{\/p) (X/pYdp = 0, t = 0,

1, •••,« — 1 and -d^"' correspond to the Christoffel numbers. The nor-

malization Pn{\/p) = (4re — 2)(4w — 6) • • ■ bpn(\/p), n > 2, produces all in-

tegral coefficients. P„(l/p) is proven to be ( — l)"e-pp"d"(ep/p")/dpn. The

normalization factor is proved, in three different ways, to be given by (4)

(1/2x7) ep(l/p)tPn(l/p)Jdp = K-l)n- Proofs are given for the recur-J C—j 00

rence formula (5) (2n - 3)Pn(x) = [(4» - 2) (2 w - 3)x+2]Pn_i (x) + (2« -1 )P„_2 (x),

for n>3, and the differential equation (6) x2P„"(x) + (x-l)Pn'(x)-M2P„(x) =0.

The quantities pi{n), l/pi(n) and -4/"' were computed, mostly to 6S — 8S, for

i = 1(1)«, n = 1(1)8.I. Introduction: Occurrence of inverse Laplace transforms. For a given func-

tion of p, F(p), which is the direct Laplace transform of some unknown function

f(i), for I > 0, one usually finds the /(<) from the following explicit expression:

1 rc+jx

(1) /(/) = — ep'F(p)dp.Zirj Jc-jx

Formula (1) is known as the inverse Laplace transform of F{p). In (1) the quan-

tity c is a real constant ±0 that is greater than the real part of all the singular

points of F{p). In practice c is usually positive, but c can be negative as long as

for f(t) satisfying Dirichlet's conditions in any finite positive interval the integral

J e-c'f(t)dt is absolutely convergent (H. S. Carslaw and J. C. Jaeger [1]).

A note by the referee follows this paper and indicates relations between the

present work and work published elsewhere.

The examples treated in most textbooks on operational calculus and Laplace

transforms contain such functions F(p) that their poles and branch points (and

residues also) are obtainable without too much difficulty, and the inversion

integral in (1) is evaluated by suitable deformation of the path of integration,

and the use of Cauchy's theorem. But there are countless other examples where

F(p) might be too complicated to yield explicit information about the location

and nature of its singularities without a prohibitive amount of labor. For instance,

one will recall that in most textbook examples treating the solution of ordinary

and partial differential equations by operational means, the original system of

differential equations is transformed into a system whose solution F{p) is usually

some known elementary function or a very extensively tabulated function of a

simple differential equation (like a Bessel function), so that its analytic character

and singularities are well known. But in actual practice one might not be fortunate

enough to obtain such a comparatively simple F(p). Thus the transformed differ-

ential equations might not yield a known function. Instead it might be amenable



only to solution in series or numerical integration, with an F(p) that is given as

a tabulated function of p. Also, the solution F(p) might be given explicitly in

closed form as a combination of integrals of such complicated analytic expressions

that it might be easier to evaluate it for different numerical values of p than to

find its poles, residues, and branch points.

The purpose of this present article is to discuss the properties of a new set

of orthogonal polynomials which can be the basis for convenient formulas for

approximating f(t) in (1) for different positive values of t when one has an F(p)

that is too complicated to show its analytic character, but which can be calculated

for any p.

All further discussion will now be for F(p) assumed to be exactly of the

2» ar ....form 2~Z — , i-e-i a polynomial in l/p without a constant term.

To obtain a definite integral without a parameter / in the exponential term,

which is the "weight function," let pt = u in (1), so that we obtain

II. Use of orthogonal polynomials. At this point one may recall the application

of the theory of orthogonal polynomials to quadrature formulas for definite

integrals where the integrand is the product of a preassigned weight function and

a polynomial P(t). There it is possible to employ the value of Pit) at n fixed

irregularly spaced points f,-, i = 1, 2, such that the resulting quadrature

formula is exact when P{t) is any arbitrary polynomial of (In — l)-th degree.

Thus for the direct Laplace transform of P(t), namely I e~p'P(t)dt, which is

essentially I e~'Q(t)dt for polynomial Q(t), the points t{ are taken equal to the

zeros of the Laguerre polynomials, which have been tabulated extensively (H. E.

Salzer and R. Zucker [2]). In the present case, even though we are not dealing

with a polynomial in p, we can still solve the problem of finding a Gaussian-type

quadrature formula for (1') of approximately double the degree of accuracy of an

ordinary quadrature formula based upon the same number of equally spaced

points.

Thus let p-2.n(\/p) be any arbitrary (2w)-th degree polynomial in the variable

l/p, which vanishes at l/p = 0. Consider n distinct points I/pi, i = 1,2, • ■ •, n

other than l/p = 0 and construct the in + l)-point Lagrangian polynomial

approximation (of the nth degree in l/p), to p2re(l/£), based upon the points l/pi,

i = 1, 2, • • •, n and l/p = 0. The (n + l)th point l/p = 0 is needed in order to

provide for the property that p2„(l/p) vanishes at p = °o. We have for this

polynomial approximation L(n+1)(l/p) the explicit expression

where Fl — I is still a polynomial in l/u, without a constant term.

(2)



where on the right hand side of (2)

(3> wi).ff(i-i)/ff(i.i).\P' k-l\P Pk/' *_l \pi Pk)

the TT' denoting the absence of k — i. In (2), pn+i is oo, so that there is actually

no (w + l)-th term in (2) and Lnn^{\/p) is not used. (The summation in (2)

is written with n + 1 instead of n to avoid confusion with the (n — l)-th degree

coefficients L/n) (1/p) which differ from the Lfn+l)(l/p) by not having the

factor pi/p.)

Following the method in G. Szegö [3], we consider the (2w)-th degree poly-

nomial in 1/p, namely, pln(l/p) — Z,("+1)(l//>) which vanishes at 1/p = 0, 1 /pi,

i = 1,2, •••,», and thus has

as a factor. Writing

-G)-*~G)+M;Mi)-it follows that

1 r<s+in ( 1 \ 1 fe+j» / 1 \

<5) üjL "-(->)*-üiL ■"""(?)*+sj£T*"i<'*(;)f-'(;)*-27TJ p \

Thus if the second term in the right member of (5) always vanishes, (5) will be

an ra-point quadrature formula that is exact for any (2w)-th degree polynomial in

1/p without a constant term, namely,

where the "Christoffel numbers" A/n) are given by [6]

(7) di™ = f+ " e"L/"+» (-) dp.2trjJo~ix \p/

A sufficient condition for (6) to hold is obviously the "orthogonality" of

(l/p)pn(l/p) with respect to any arbitrary p„_i(l/p), namely,

(8) üjL» ePppAp)\p)dp = 0' •• L

The necessity of (8) is also obvious from (6) by choosing

P1»(i//o = (i/p)pn(i/p)Pn-xa/p)

where p„_i(l//>) is any arbitrary polynomial in 1/p of the (n — l)-th degree.



Hence the points l/pi, now denoted by l/p/n), are the zeros of a certain set

of orthogonal polynomials in the variable 1 /p.

The condition of orthogonality (8) is also mathematically equivalent, in

terms of actual polynomials (by setting x = 1/p), to having a polynomial of the

nth degree q„(x) which is orthogonal to any p„_i(x), with weight function ellx/x,

where the path of integration is a circle of radius l/2c whose center is at (l/2c, 0).

If the polynomial pn(l/p) is written as

0)"+"-' or+sr+•"+0)+the determination of bi, i = 0, 1, 1, to satisfy the conditions of orthogo-

nality (8), making use of

i_ Cc+i"C+j'OO gp

2Tj.

is in the solution of this system of linear equations:

(9')

»! (»-l)! (m-2)! 1! 0!

1 +bn-l _b==±_ ... +h+h=Q(n + 1)! n\ (n - 1)! 2! 1!

I /rt f->\ I 1 //-» 1 \ I ' ' I[(2«-l)! ' (2«-2)! ' (2w-3)! ' '«!'(»-1)!

For numerical work it is somewhat easier to solve (9') in the form

(9)

1 + m&„_! + n(n - 1)&„_2 + • • • + «!&i + n\b0 = 0

1 + (n + l)&n_! + (n + l)nbn-2 + • • •

+ (n + 1) ■ ■ ■ 3h + (n + 1) ■ ■ ■ 2b0 = 0

.

1 + (2« - l)&„_i + (In - l)(2n - 2)&n_2 + • • •

+ (2n - 1) • • • (n + l)öi + (2n - 1) • • • nb0 = 0.

III. Explicit expression for orthogonal polynomials. It is convenient to nor-

malize the polynomials pn(x), where x = 1/p, by multiplying pn(x), for n > 2,

by (4w — 2) (An — 6) • • • 6. This normalization produces polynomials with all

coefficients integral (proven below) and it is not the usual normalization by

multiplication by

[2^jZeV-p\P\p)\dp\-

Denoting (An - 2) (An - 6) •■■ 6pn(l/p) by Pn(l/p) for n > 2, and pi(l/p) by

Pi(l/p), one can avoid the labor of solving (9') or (9) directly by showing that



P„(l/p) has the following more elegant definition:

That (10) yields the leading coefficient of l/pn in P„ (1/p), namely

1, for n = 1,

(4n - 2) (4n - 6) • • • 6, for n > 2,

is obvious by induction. To prove the orthogonality property, or (8), it suffices

to prove the vanishing of

\ rc+jf if &n / ep \ "I f" dm /ep\l(A) —I ep- \ (-l)»e-*p«— (— I \\ (-l)™e-ppm — (— ) \dp

2irjJc-j«, plK e dpn\pn/ JL dpm\pmJ J

for m < n. This last expression is written as

(—1)»+» fc+'°° d" (ep\ dm (ep\ ,(B) I—^t— I e-j>^,™+»-i —- ( — I — ( — ) dp,

2tj Jc-ix e dpn \pn/ dpm \pmJ

and after integrating by parts m times, noting that the integrated parts always

vanish, we have

(-l)m(-l)m+n Cc+'x dm \ d* (ep\~\ep ,(C)- „ . — I — e^p^"-1 -— ( — ) — dp,

2irj Jc-ix dpmI dpn \p"/ \pm

which by Leibnitz's rule is expressible as

(-l)m(-l)",+B Cc+1'x ™{m\ dT , . ^m-r+n / £,\ gp

Application of Leibnitz's rule a second time to

in the above and cancellation of ep/pm, yields

(e) t^f-Z (-)[ £ (;) («+: -

dpm-r+n \pn

Now we integrate by parts (m — r + n) times each term of the above double

summation. The integrated part will always vanish since it will have a factor of

1/p to at least the first power. Furthermore, at some stage in the partial integra-

tion of each term, that stage varying with the term, the integral part will also

vanish if m < n. This last follows because the lowest power of pn~l~a is positive

or zero, since 5 can equal at the most r which can equal at the most m < n — 1.

Then in the integration by parts the positive or zero power pn~l~s, for each value


170 ORTHOGONAL POLYNOMIALS IN INVERSE LAPLACE TRANSFORMS

of s between 0 and r, would always be eventually annulled because the initially

occurring differential operator dm~r+n/dpm~T+n is of order m — r + n > n — 1 — s

even for the highest value of n — 1 — s when 5 = 0 (due to m — r -\- n > n — 1

for every r between 0 and m). Thus (E) vanishes, which proves (10), and estab-

lishes at the same time that this normalization yields all integral coefficients

forP»(l//>).IV. Normalization factor. To obtain the normalization factor, which turns out

to be given by

we repeat the preceding argument for m = n and now notice that in the final

integral (E) the lowest power of will survive the integration by parts,

because it is equal to I/p. Retaining in the double summation in (E) only the

single non-vanishing term 5 = r = m = n, we get

(F) t^ru2n~i)^-^(-)^' 2-wj Jc-j«, \ n ) pdpn\pnJ

which is integrated by parts n times, the integrated part always vanishing, to give

(-D--2-3 ■■■ (n-l)n ^ep-dp.

2irj \ n I J*-to pin+1(G) (-D^V2»-A ,

2-kj \ n / Jc-jx,

But (G) is

( 1; V n )n-n-{2n)\ ~ «! (2»)!

which reduces to ( — \)nn/2n or f (— 1)", thus proving (11).

From (10), the explicit formula for Pn(l/p) is seen to be [7]

nln

(12) P„(i)-(-l)-

<-'"(;)(2::,2)"--"'

<-'-(;)(2::/)"-^+ pn-2

( n\(2n - r - l \ ,(-1

+-" ' ' .". '-:- +


ORTHOGONAL POLYNOMIALS IN INVERSE LAPLACE TRANSFORMS 171

V. Recurrence formula. It is easy to obtain the recurrence relation for the

polynomials Pn(x) by employing a fundamental theorem about the existence of

a recurrence formula connecting any three successive orthogonal polynomials

(G. Szegö [8]), namely,

(13) Pn{x) = (anx + bn)Pn-i(x) + cP_t(*).

Thus an is immediately seen to be An — 2. Equating constant terms in (13), one

finds c.= b„ + I* and after substitution into the equation derived from the

coefficients of x, one obtains

2 _ In - 1

2w — 3 2w — 3

so that the recurrence formula satisfied by Pn(x) is seen to be [9]

(14) (In - 3)P»(») = \_(An - 2)(2n - 3)* + 2~]Pn^(x) + (2n - l)Pn-s(x),

for n > 3.

From (14) and (8) only, without making use of (10), one can again find the

normalization factor given in (11), through the following inductive argument:

Multiply (14) by P„_2(x) and then operate with

I Cc+jx i

— I c - • • • dp2tT1 Jc-jx p2*7

to obtain (making use of (8)):

(4, - 2)(2, - 3) .p^l / 1 VI / 1\

(2w - l) 1 r /1 vi2+ 0 +

Denoting the left member of (11) by Fn, still making use of (8) to replace in the

first of the above integrals (1 /p)Pn^2(l/p) by

a

one now obtains

n (4«~2)(2«-3) „0 =-7-Fn_! + (2» - l)Fn_2,An — 6

or F„_i = — F„_2. Since Fi = — i, (11) follows by induction.

The normalization given in (11) can be seen in a third way, directly from the



explicit formula for Pn(l/p) in (12). For, in view of (8), it suffices to consider only

(H) — I-— ( )n\Pn[-)dp,2ttj Jc-j«, p p» \ n ) \p)

or

, , til,-1\ lf'-"--(;)(2";:;1)<--^(I) (-DM -

V n / r=0 (In - r)!

or

/2k-1\ " n(n-i)---(n-r + l) (2n-r-1) • ■ • (n+l)n

Uj V n / 'r_0 H (2»-f)(2«-f-l) - (»+l)»!'

which, after cancellations, is written as

" (2»-l)(2w-2)-•-w n n(n-l)-■ ■ (n-r+i)(n-r)\

(K) r?0 ' (n-r)\n\ 2n-r r\

or

" , 1 (2n - l)(2n - 2) • • • n

and this, in turn, is expressible in the form

(M) K-D"- (2»-0)(2«-l)(2n-2) - ■ • (2«-[r-l])(2«-[r + l]) ■ ■ • (2n-»)

r=o (r-0)(r-l)- • • (f-[r-i])(r-Cr+l])- • • (f-n)

In (if), the |( — 1)" is multiplied by the sum of the coefficients of the Lagrangian

interpolation polynomial for the (n + 1) points 0,1, •••,«, for the variable

equal to 2n. But that sum is identically equal to 1, i.e., for any value, 2n or

otherwise. Thus we obtain once more §( — 1)" for the normalization.

VI. Integral coefficients. It may be of interest to show that (14) alone, without

any knowledge of (10), implies that Pn(x) has integral coefficients. We prove

this by noting that Pn+i(x) will have integral coefficients if Pm(x), m < n, has

integral coefficients and the following identical polynomial congruence holds

for m = n + 1:

(15) 2Pm-i(x) + (2m - l)Pm_2(x) = 0 (mod (2m - 3)).

Now the existence of integral coefficients of Pm(x) and congruence (15) can be

verified for the first few values of m. We then show that if (15) holds for some

particular m = n, it holds for m = n + 1, provided Pm(x), m < n — 1, has



integral coefficients, or, in other words, that

2Pn(x) + (In + l)P„_i(x) ■ 0 (mod (In - 1)).

This last congruence, by (14), is equivalent to

4 2(2« — 1)- Pn^(x) + -P„_2(x) + (2« + l)Pn_!(x) - 0 (mod (2n - 1)),2n — 3 2n — 3

or to

(2« — l)2 2(2« — 1)\-{- Pn_i(x) + ^-^ P„_2(x) = 0 (mod (2« - 1)),

2« — 3 2« — 3

which in turn is expressible as

n <J(2w- l)Pn-i(x) +2Pn_2(x)l(2« - 1) I -2« - 3- * 0 (mod (2« - 1)),

or

, I" (2 + (2« - 3)Pn_x(x) + (2« - 1 - (2« - 3))P„_2(x) ] ={ " ' L 2« - 3 J

(mod (2« - 1)).

But under the assumptions that (15) holds form = w, and that Pm(x), m < n — 1,

has integral coefficients, the last quantity in brackets is a polynomial with integral

coefficients, which shows that the last congruence is satisfied identically in x.

Thus (15) holds for m = n + 1 and P„+i(x) has integral coefficients. We proceed

in this way to every n. There is a slight subtlety in the argument of this induction

in the sense that the integral coefficients of Pm(x) up to m = n — 1 only are

needed to go from m = n to m = n + 1 in (15), but then use is made of the in-

tegral coefficients of P„(x) in using (14) with n + 1 in place of n.

VII. Differential equation. It is easy to show that P„(x) satisfies the differ-

ential equation

(16) x*Pn"(x) + (x- l)Pn'(x) - n*Pn(x) = 0.

Thus one merely expresses (12) in the form

(120 P.(») = (-1)4 1 + £ ~ P)(W2 -^■■■(^-r-l) x>]

and then observes that (12') is equivalent to the automatically terminating

"infinite series."

(12") Pn(x) = £ arx',r-0



where-2

a0 = (—1)", and ra, = — («2 — r — 1 )ar-i, for r > 0.

Working backwards from (12"), by equating coefficients of x'*""1, one sees that

(12") must arise from (16).

VIII. Explicit expressions for polynomials. Because these polynomials P„(x)

are of fundamental importance, and their role in the inverse Laplace transform

is comparable to the role of the Laguerre polynomials in the direct Laplace trans-

form, their explicit expressions are given below for n = 1(1)12:

Pi(x) = x - 1

P2(x) = 6x2 - 4x + 1

P3(x) = 60x3 - 36x2 + 9x - 1

P4(x) = 840x4 - 480x3 + 120x2 - 16x + 1

P6(x) = 15120xs - 8400x4 + 2100x3 - 300x2 + 25x - 1

P6(x) = 3 32640x6 - 1 81440x6 + 45360x4 - 6720x3 + 630x2 - 36x + 1

P7(x) = 86 48640x7 - 46 56960x6 + 11 64240xs - 1 76400x4 + 17640x3

- 1176x2 + 49x - 1

P8(x) = 2594 59200x8 - 1383 78240x7 + 345 94560x6 - 53 22240x5

+ 5 54400x4 - 40320x3 + 2016x2 - 64x + 1

P9(x) = 88216 12800x9 - 46702 65600x8 + 11675 66400x7 - 1816 21440x6

+ 194 59440x6 - 14 96880x4 + 83160x3 - 3240x2 + 81x - 1

P10(x) = 33 52212 86400x10 - 17 64322 56000x9 + 4 41080 64000x8 - 69189

12000x7 + 7567 56000x6 - 605 40480x5 + 36 03600x4 - 1 58400x3

+ 4950x2 - lOOx + 1

Pn(x) = 1407 92940 28800X11 - 737 48683 00800x10 + 184 37170 75200x9

- 29 11132 22400x8 + 3 23459 13600x7 - 26637 81120x6 + 1664

86320x6 - 79 27920x4 + 2 83140x3 - 7260x2 + 121x - 1

P12(x) = 64764 75253 24800x12 - 33790 30566 91200X11 + 8447 57641 72800x10

- 1340 88514 56000x9 + 150 84957 88800x8 - 12 70312 24320x7

+ 82335 05280x6 - 4151 34720xs + 162 16200x4 - 4 80480x3

+ 10296x2 - 144x + 1.

IX. Zeros and Christoffel numbers. In the numerical table below there are

given the values of the reciprocals of the zeros of Pn(x) or piin), the zeros of

P„(x), or i/piM, and the corresponding Christoffel numbers AiM, for n = 1(1)8.

Use of these quantities in the quadrature formula (6) above can give theoretically

exact accuracy for any polynomial in 1/p (with no constant term) up to the 16th

degree. However, the fact that these tabulated values of £»(n), l/£/n) and -4,(n)

are correct to only about a unit in the last significant figure that is given, must



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improvement of solutions of simultaneous linear equations 177

be taken into account in any practical example where some upper bound for the

error should be estimated.

Herbert E. SalzerOrdnance Corps

Diamond Ordnance Fuze Laboratories

Washington, D. C.note by referee

The function F{p) is subject to certain restrictions because it is a Laplace transform. In orderfor F(p) to be the Laplace transform of the function f{t) given by (1), it is sufficient that F(p) havethe form (cf. G. Doetsch [5]):

F(p) = a/p + F^/P1^,

where S > 0, a is a constant, and F\{p) is analytic and bounded in the half plane Re(p) > c.We assume that this condition is satisfied. Whether this condition is also sufficient for the con-vergence of the re-point quadrature formula to the true value of f{t) in (1), when re tends to infinity,has not been determined. The author makes use here of the fact that the convergence occurswhenever F(p) is a polynomial in 1/p without a constant term; in fact, the quadrature is exactfor polynomials of degree not greater than 2re. G. Szegö [10] has shown that under quite generalconditions a Gauss-Jacobi type quadrature formula which converges for polynomials also convergesfor a much wider class of functions. Unfortunately his theorems do not seem to apply directly tothe present case because the integral (1) involves a complex valued weight function which is notof bounded variation.

1. H. S. Carslaw & J. C. Jaeger, Operational Methods in Applied Mathematics, 2nd edition,Oxford University Press, 1949, p. 75.

2. H. E. Salzer & R. Zucker, "Table of the Zeros and Weight Factors of the First FifteenLaguerre Polynomials," Amer. Math. Soc, Bull., v. 55, 1949, p. 1004-1012.

3. G. Szegö, Orthogonal Polynomials, Amer. Math. Soc, Colloquium Pub., v. 23, 1939, p. 46-47.4. H. L. Krall & O. Frink, "A New Class of Orthogonal Polynomials: The Bessel Poly-

nomials," Amer. Math. Soc, Trans., v. 65, 1, 1949, p. 100-115.5. G. Doetsch, Theorie und Anwendung der Laplace-Transformation, Springer, Berlin, 1937,

p. 128.6. The shift in notation from (re + 1) to re in AiM will cause no confusion after the Ai's have

been computed and are ready for use in (6).7. It was called to the author's attention by H. L. Krall that P„(x) m ( —l)"y„(x, 1, — 1)

where y„(x, a, b) are "generalized Bessel polynomials" (see [4J).8. G. Szegö, op. cit., p. 41-42.9. Formula (14) holds for re = 2 if we define P0(x) = 1.

10. G. Szegö, op. cit., p. 341-342.

On the Improvement of the Solutions to a Set ofSimultaneous Linear Equations using the ILLIAC

The basic method used for solving simultaneous linear equations on the Uni-

versity of Illinois' electronic digital computer, the ILLIAC, has already been

described in detail by Wheeler and Nash [1]. The routine currently in use on

the ILLIAC, programmed by Wheeler [2], makes use of the method of elimina-

tion to solve the set of n simultaneous linear equations

n-l

(1) Z 9nm + ain = 0 i = 0, 1, 2, • • •, n - 13=0

in a manner very similar to that used by a human solving such a system.

In brief, the procedure used is as follows:

a) The augmented matrix

i = 0, 1,2, ■■■,n- 1(2) Otf

j = 0,1,2, ■..,»


orthogonal polynomials arising in the numerical evaluation of … · 2018. 11. 16. · orthogonal...

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