Tables For The Evaluation Of [ xße~xf(x) dx...ß = — \ the Gauss-Laguerre quadrature formula is...

$: Tables For The Evaluation Of [ xße~xf(x) dx...ß = — \ the Gauss-Laguerre quadrature formula is directly related to the Gauss-Hermite quadrature formula, since the Laguerre polynomial$
Tables For The Evaluation Of [ xße~xf(x) dx

By Gauss-Laguerre Quadrature

By P. Conçus, D. Cassatt, G. Jaehnig, and E. Melby

Abstract. Tables of abscissae and weight coefficients to fifteen places are pre-

xße~xf(x) dx ~ 23 Hkfiak)k=l

for ß = — \, —\, and —f and n = 1(1)15.

1. Introduction. The n-point Gauss quadrature formula for evaluating the

definite integral / xße~xf(x) dx is (see, for example, [1])Jo

f xße-xfix) dx = ¿ Hkfiak) + En,JO fc=l

where ak is the kth zero of the Laguerre polynomial Lnßix), Hk is the corresponding

weight coefficient, and En is the truncation error. If the normalization of the

Laguerre polynomials is chosen so that

L/ix) = ±(n + ß)m=o \n — m)

(~xTml

then the weight coefficients are given by

(1) Hk = —T-,-. ,u,—T—^To, (k = 1,2, ■■■ ,n)ni(n + l)Li+l(ak)}-

or the alternative formula

<o\ ti T(n + ß + 1) ,(2) Hk = -=--=-2 ik = 1,2, ■■■ ,n)

n\ak\ —L/(ak)

and the truncation error by

nlTin + ß + 1) f(2»)/,,.

K = -(2njl-; (l)'

For ß = 0, a 12-place table has been prepared by Salzer and Zucker [2] for values

of n ranging from 1 to 15. A short table has also been prepared by Burnett [3] for n,

ß = 2, 3, 4. Both of these tables have been reproduced in [1], More recently, Rabino-

witz and Weiss [4] have prepared tables to 18 significant digits for several integral

values of ß and n.

The tables presented here are designed to fill the need for integrands involving

certain fractional powers of ß. The roots and weight coefficients are given to 15

places for ß = — \, —|, and — f and for n = 1(1)15. It should be noted that for

Received October 4, 1962.

245

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246 CONÇUS ET AL.

ß = — \ the Gauss-Laguerre quadrature formula is directly related to the Gauss-

Hermite quadrature formula, since the Laguerre polynomial of order — § and

degree n in x is merely a multiple of the Hermite polynomial of degree 2n in x (see,

for example, [1]).

TABLE I. ABCISSAE AND WEIGHT COEFFICIENTSFOR THE GAUSS-LAGUERRE QUADRATURE FORMULA

T -1/4 -x N1 x l/ e xf(x)dx~k|1 Hkf(ak)

0,75000 00000 00000N=l

1.22541 67024 65178

Hk-exp(ak)

2.59420 71794 76230

0.42712 43444 67705

3.07287 56555 32295

N = 21.07587 23293 965450,1*954 43730 68633

1.64914 36384 641013.23074 75017 97830

0,29934 69776 68357

2,03993 89987 790685.91071 40235 52575

0,23055 42735 675461.54191 41073 683444.23305 69607 049018.99447 46583 59209

0.18751 15165 66305

1.24301 39853 183453.34155 40660 744816.75166 77467 61627

12.22625 26852 79242

N=30.92931 04968 569130.28658 93410 36511

0.(2)95168 64571 75359

N = 4

0.81598 32950 693120.37322 42412 00384

0.(1)35751 54927 853450.(3)45761 69169 47827

N = 5

0.72848 46471 500100,42348 79614 74704

0.(1)70352 68628 749160,(2)30727 12884 805100.(4)18694 66816 67895

1.25361 90505 660462.20391 21770 ¿>18073.51143 28986 92471

1.02756 59781 222731.74427 36982 350942.46426 91838 244403.68767 62186 46727

0,87873 03375 290351.46782 80537 799061.98837 90409 783662.62865 76456 552273.81515 19201 88150

0.15802 68336 99440

1.04245 2O3O3 148342.77254 78222 382645.48634 937Û5 047219.48223 30795 38669

15.55839 08637 04072

N=60.65938 18290 679550.45086 00686 648170,10607 62609 78039

0.(2)88885 64891 213360.(3)20929 37342 749770.(6)68512 88785 64549

0.77226 64300 961411.27871 25784 050211.69715 07607 604042.14547 22653 179262.74686 76167 187503.91467 37444 25026

0.136560.89815

2.373834.645347.86894

12.3633918.96375

0.120230.78921

2.077534.038086.76498

10.4267815.3578822.42527

13094 4954386048 1408276105 7058453499 6639697868 8666939179 2877534203 83951

3199746737628083826692788757O21226239436

5111212894551263467469790419393547298994

N = 70,60350 36343 916100.46409 82356 507590.13918 94937 12046

0.(1)17749 20341 990250,(3)86398 60993 516230,(4)12125 97533 515220.(7)23216 17328 41405

N=80,55736 92781 858130,46857 43009 095400.16824 66863 03240

0.(1)28898 18348 806820.(2)22580 83788 306310.(4)69546 10642 172010.(6)62294 26466 515780.(9)74114 08084 41323

0,69181 13979 526021,13939 745O4 24671

1.49468 96606 340011.84767 98338 830352.25916 64298 180232.83837 93377 503963.99617 13631 89639

0.62857 86736 214231.03164 44443 482231.34341 15012 673111.63903 43963 323501.95766 13503 648622.34730 28524 252162.91266 90027 086674.06512 08009 99067

Note:The number in parentheses designates the number of zeros between the decimal

point and the first significant figure.


EVALUATION OF jo Xß€~xf(x) dx BY GAUSS-LAGUERRE QUADRATURE 247

TABLE I. (continued)

H, Hk . exp(ak)

0.107390,70398

1.848103.576135.949909.07193

13.1198918.4412025.93145

0,(1)970340,635441.664923.211655.318738.05413

11.5265615.9210621.5963129.47412

0,(1)884970.579111,515142.916124.813417.25481

10.3128414.1010818.8108124.8108433.04728

0.(1)813410.53198

1.390322.671344.398636.607309.34863

12.6987016.7751521.7747128.0754936.64635

4450800035

13190649283050016344991642325608072

09237

1141409791988127032946553

57824185657294472839

42523240737985995590748277003921061722613689276736744O4

7468278171921191134837O9509190876756862864946857770779839779

4593760476080270496218857357234966402641

73712

949103834893740874497713970353808662564701876

45 092

9336237242775678502359961656593385102414201678031698465

89478877559685336481143877957055780O59O447187386066044248087

0.518590,467580,19297

0,(1)414690.(2)454900,(3)234660.(5)484950,(7)29145

0.(10)22569

0.485520.463180.21362

0,(1)547140.(2)776540.(3)581220.(4)208430.(6)301840.(8)12649

0,(12)66140

0.456940,456660,23065

0,(1)680620.(1)118390,(2)117770.(4)629880.(5)163180.(7)17127

0,(10)516060.(13)18773

0.431980,448880,24455

0.(1)811100.(1)166470,(2)207610.(3)150910,(5)597000,(6)115140,(9)89973

0.(11)199860,(15)51860

N»9

926304756140810995236499633195083317584650863

N»1055709262685661298209928562289202466

689957016261032

N = ll8572188117512092062656866853752018180536257267986704174

N = 12

47895080079786895088236213426377920046815505200086

7378769499

0669768817659060807971575744282216795145492O5

45590496829651805935350285214310583182533291238781

9250459281419878571264997096009837446445796200397117480

82901874219244204867419818079304223

8129548887470211635856289

0,577390,945351.224951.481931.745542.043312.418782.975004.12484

0,534990,874421.129061.358031.585131.828972.112922.478653.028594,17752

0.49923b,814901.049501.257081.458061.666451.897012.171262.529993.075534.22463

0,468590,764130.982191.172811.354081.537331.733011.954122.221282.574853.117254.26724

459305107923250901139642837570822985488394049

96261153096220195736833338069277990469968563647969

0681444236258046026438162330620041545099933278410352952

184605548241702147590518794271640057918580921234247217156763

206121079355226289778699772879411964O74314763

O4O34050082922270919796654642516324437921031518064

0058745476025274926391429108634854929840483398085925459

00142435724992977219473420776809255

9291488296561104900977220


248 CONÇUS ET AL.

TABLE I. (continued)

H,_ H . exPlak>

0.(1)752560.491971.284662.465064.051456.070458.55992

11.5738915.1913419.5334024.8017431.3830640,26775

0,(1)700190.45757

1.194022.288773.756255.617207.90052

10,6462413,9106117,7750822,3639027.8832434.7278843.90864

0,(1)654630.427681.115402.136293.501955.228897.339739.86515

12.8469416.3432220.4376225.2572031.0123238.1053847.56671

94975585612037499450766891433850625793360416537331279801290948739

13055098539861246952152341335209374777124293447340835767112032637

3275355271472264016556780713641852748263587514257848361596192576034054

437911269556095030954281869990884890421116773698766884454392335024169003144

5837439819588796871163075123568260911831484426880110144257942826445437

967188115625728538621408640331564648892416643665749229618028320817132222833

0,409960,440360.25581

0.(1)935980.(1)220400.(2)330660,(3)307090.(4)168320.(6)506050,(8)74440

0,(10)442730,(13)740420.(16)13996

0000

0.(10.(20.(30.(40.(50.(70.(9

0.(110.(140.(18

39039.43148.26483.10536J27870148790155403139431116705139019144657120594126394137016

0000

0.(10,(20,(30.(40.(50.(60.(8

0.(100.(130.(160.(20

.37285

.42246

.27198

.11633133996167854191201180484144908115004127728125109191216190971196166

N = 139579747508168788774568471847321416016309643 0274118821700199967043

N=14

3376501095118507648214767590034902844527254981622429308483300600197405

N=15

889004574806452341875837750385093978626543881569704056013570127801801858309

12328589906554947659692806106157276719860539072474149490322761178

8934994078340782727659414025636068328790567936581013467855669559595552

32600031163818638067746427193447528835886609903404

7582248455793849103690149

0.442010.720230.924361.101121.266921.431381.602491.789032.003152.264952.614613.154774,30614

0,418700,681840,874031.039221.192501.342321.495161.657471.837202.045962.303632.650273.188844.34192

0,398080.647930.829760.985101.128011.266071.404771.549151.704841.879302.083882.338282.682573.220044.37505

327296203618587573994665646334

20999953263652279457079866538632907

8182813053227954107054801640458607091597163598529707058389799118163599

417382721500113740634542460471939514020108279315353698510736908464728598374

2699449O3O6770382658690275055263391946957192467152995619504018488

18740353170287304988

73131643557368271874287251496404425611007952263870

789568215390182686028482056155530811281697837243856779925637026205775842590


EVALUATION OF jo xh~xf(x) dx BY GAUSS-LAGUERRE QUADRATURE 249

TABLE II. ABCISSAE AND WEIGHT COEFFICIENTSFOR THE GAUSS-LAGUERRE QUADRATURE FORMULA

■1/2 e^fMdx-jSj Hkf(ak)

0,50000 00000 00000

"k_-RîT

1.77245 38509 05516

Hk' exP(ak)

2.92228 23653 22278

0.27525 51286 084112.72474 48713 91589

N = 2

1.60982 81800 110260.16262 56708 94490

2.11992 89657 899382.48045 16353 91632

0,19016 35091 934881.78449 27485 432525.52534 37422 63260

N = 3

1.44925 91904 487850.31413 46406 45713

0.(2)90600 19811 01769

1.75280 26688 724611.87116 11152 623612.27381 66653 49049

0,14530 35215 03317

1,33909 72881 263613.92696 35013 582878.58863 56890 12034

N=41.32229 40251 164830.41560 46516 29784

0.(1)34155 96601 482700,(3)39920 81444 22735

1.52908 82573 034581.58578 00967 728031.73350 52131 267622.14386 02884 95960

0.11758 13202 117781.07456 20124 369043.08593 74437 175506.41472 97336 62030

11.80718 94899 71737

N = 5

1.22172 52674 706520,48027 72221 64629

0,(1)67748 78891 096210,(2)26872 91493 562470,(4)15280 86571 04652

1.37416 37079 025471.40659 26462 098121.48288 38638 871301.64133 22528 096332.05090 33827 31474

0,(1)98747 014Û6 848120,89830 28345 696182.55258 98026 681715.19615 25300 544669.12424 80375 31179

15.12995 97811 08085

0.(1)85115 44299 759400.77213 79200 42777

2.18059 18884 504594.38979 2B867 310147.55409 13261 01784

11.98999 30398 2387918.52827 74958 52492

0.(1)747910.67724

1.905113.809476.48314

10,0933214.9726221.98427

8825990876363506361454286367527088428409

6818349289314288490727170213432639362651

N = 61.14027 04725 249590.52098 46205 283220.10321 59712 31768

0.(2)78107 81169 258120,(3)17147 37408 717570,(6)53171 03368 71260

N = 7

1,07281 18194 241800,54621 12181 284930,13701 10684 46930

0,(1)15700 10945 291590,(3)71018 52271 038470,(5)94329 68710 037830,(7)17257 18233 62503

1.015850.561290.16762

0.(1)257600.(2)186450.(4)542370.(6)464190,(9)53096

N = 8

8958049170008276230768017201856168914948

3322757067979721019924836075767304202236

1,25861 57487 389861.27924 24640 405131.32532 55465 337431,41044 07322 244401.57328 78789 266451.97939 80941 84596

1.16812 33810 439921.18221 33340 863251.21275 94782 521951.26580 12129 446661.35541 35183 847941.51997 41747 951321.92175 74060 51319

1.09475 04100 756881.10488 39147 349191.12643 56581 764001.16249 45508 017281.21947 39165 119951.31151 13457 522351.47649 12445 553631.87374 89857 68139


250 CONÇUS ET AL.

TABLE II. (continued)

0.(1)667020,60323

1,692393.369175.694428.76975

12.7718218.0465025.48597

2309563570507976270233429673Û2535485467791660

819448174993179432695775568602691942898099077

0.966990.569610.19460

0,(1)372800.(2)377700,(3)183620,(5)362130,(7)20934

0,(10)15656

N*9

138944571334952084774526025373089624115939954

509113995982631508935368458588186861584242318

Hk • exp(aR)

1.033691.041261.057171.083151.122551.181901.275261.439981.83278

167299893388858735738091060069034406766270751

632435212937602242659960926216123201062803831

0.(1)601920.543861.522943.022515.084907.77743

11.2081315.5611621.1938929.02495

063147500241054337647750092315020433332120963034Û2

958799464604444

515749852425445486638935001541

36226

0.924480,573350,21803

0,(1)496210,(2)648750,(3)456670,(4)156050.(6)217210.(9)87986

0.(12)44587

N»1073392101074412004177466847272011295387418198487291

01220256684004749272475723270870641538565463606830

0,981840.987680,99984

1.019351.048161.089701.150521.244551.408661.79718

30013677051742680542070183484748657739235922339229

33492441067258134916971152904005006828255388506383

0,(1)548390,49517

1,384652.741914.597736.99939

10.0189013.7693018.4411124.4019632.59498

0.(1)503610.45450

1.269582.509844.19841

6.369979.07543

12.3904416.4321921.3967527.6611036.19136

8695741233574009940177004746958275958661968091242300914

8891166815994018097256448

53880423Û97963850876593618779803606

8818550356846000670285711288365723401691781928704340815

72940

6378003961

3212878413306356120309471

75313661094609015602

0.887090.573940.23820

0.(1)622800,(2)995670,(3)929770.(4)473100.(5)117680.(7)11933

0.(10)348860,(13)12334

0.853860.572350,25547

0,(1)748900,(1)140960.(2)164730.(3)113770.(5)431640.(7)803790.(9)60925

0.(11)131690,(15)33287

N = ll045282866447219741769867001017

2571057512981977801536684

N»12

232779070692435941007116284965383279140942349085392404836992

6991993814175658847710329685045020966020211930959988081

3739892886691186461501453

37683280888046788286975136156397822

0.937090.941710,951250,966390,988301.019001.061961.123711.218041.381331.76578

0.897960.901690,909350.921380.938560,962140,994151,038081.100411.194781.357141.73775

702256222388570455926924425582911678268509659

1779810531

569062201609542781929771244056338577735150947340089324511301

9198511077966252833119434152132089781661398526159477138

524523999370243594945 2978

66107315046606695685800888021583380


EVALUATION OF jo Xße~xf(x) dx BY GAUSS-LAGUERRE QUADRATURE 251

ak

0.(1)46560 08324 502480,42002 74064 012141,17231 07732 777802.31454 08643 494343.86458 50382 281595.84873 48113 063438.30455 34899 85900

11.28575 09935 1763814.87096 O3775 2540119.18091 94856 1045624.41669 23330 5651730.96393 82747 4679539.81042 60687 49338

0,(1)43292 03573 977350,39042 09260 42032

1.08896 58675 692702.14779 947O5 822323.58102 82499 917715.4O9II 23306 164607.66069 11156 10085

10,37556 3OO97 7OO5213,60971 14293 9023617,44429 44757 0418922,00319 67669 1492327.49204 15048 4385234,30462 O5O93 7308343.44926 23078 52043

0,(1)40452 70430 457530,36472 06450 51408

1,01674 60688 574962,00371 89531 339233.33698 32057 345105.03280 52776 251167.11359 37697 298759.60981 72843 04444

12.56308 23699 4849816.03128 41080 7397520.09778 53347 5592724.88931 24751 5655030,61571 74008 9949237.67847 17842 0529947.IO55O 86182 18914

TABLE II. (continued)

Hk

N = 13

0.82408 73011 807390.56926 44823 535690.27022 66558 23576

0.(1)87196 45443 450160.(1)18795 80258 231920.(2)26381 29444 647710,(3)23245 94032 062190.(4)12206 58343 479200.(6)35402 12674 794710,(8)50489 88068 98107

0,(10)29219 99867 963220.(13)47662 97318 744320,(17)87938 32189 50780

N=140,79720 94356 529030,56512 27825 187770,28278 92195 73910

0,(1)99029 77857 979630,(1)23936 84642 870960,(2)39146 62588 817980,(3)42123 62000 480650,(4)28691 00845 942880,(5)11715 43944 198600,(7)26513 65003 083420,(9)29517 06336 55538

0.(11)13278 87342 981930,(14)16631 87590 241370,(18)22802 78695 80735

N=150,77278 97790 836280,56026 18616 784250.29347 16950 817800.11028 83537 40469

0.(1)29407 65940 965340.(2)54758 44946 135320.(3)69662 02486 373710.(4)58774 50457 845980.(5)31581 89774 649420,(6)10217 04490 155190,(8)18357 16084 87571

0.(10)16212 37259 492610.(13)57572 14161 097410.(16)56206 67205 501810.(20)58165 09400 26245

Hk.exp(ak)

0,86336 41458 628960,86642 24023 158660.87268 25391 630630.88245 21216 809230.89624 93048 011490,91489 10010 414840,93965 59624 897790.97259 82965 528981.01719 77366 999411.07987 08919 421391.17412 66307 795731.33551 46629 086031.71248 40599 42642

0,83248 02184 914610,83502 69065 723990,84022 32921 886560,84828 78884 060870,85958 28499 066050,87466 54515 275000,89437 89423 397970,92001 64132 472480,95363 27504 402620,99868 91944 940021,06154 91030 647811.15558 83454 505431.31597 79936 612311.68951 78822 56733

0.80469 21334 038060,80683 96338 496080.81121 02466 51369

0,81796 31500 070&30.82735 87272 222780,83979 00074 736480.85583 61258 654870.87635 40453 053670.90264 20719 823770.93674 96251 294580.98211 59916 657651.04505 13786 627091.13880 53838 992811.29819 59631 085331.66849 49420 25524


252 CONÇUS ET AL.

TABLE III. ABCISSAE AND WEIGHT COEFFICIENTSFOR THE GAUSS-LAGUERRE QUADRATURE FORMULA

r 3 /4 Nx-^VftxJdx-jZjïy^)

0.25000 00000 00000

0.13196 60112 501052.36803 39887 49895

0.(1189682 15690 63772

1.52744 70442 329445.13287 07988 60679

0.(1)67925 85575 574471.13717 85166 857843.61792 70673 492768.17696 85602 09195

0.(1)54666 24082 215390.90803 85112 16728

2.82938 55747 822846.07468 94151 36615

11.38322 02580 42220

0,(1)45738 46887 456520,75652 09910 189562.33277 47212 405854.90438 29130 172398.76328 12419 57814

14.69730 16638 90640

0,(1)39317 63548 387350,64865 48953 46562

1,98810 35172 018984,13359 89124 270367.23743 52581 17768

11.61379 87096 6950218.08909 10717 53362

0.(1)344770.56786

1.733803.580826.200289.75800

14,5847721.53996

73946 4094006512 1097197161 6238629874 0606833693 1881954150 9908840368 8980160844 48773

,Hk-RTT-

3.62560 99082 21908

N = 2

3.43422 69970 471460.19138 29111 74762

N = 33.24393 87634 476530,37265 78935 90914

0,(2)90132 51183 34119

N = 4

3.08969 94954 258300,50145 73316 17305

0,(1)34094 06814 870930,(3)35901 30300 64269

N=52.96406 33525 364790,59092 56178 90062

0.(1)68185 62440 391500.(2)24225 29604 499100.(4)12783 78695 27085

N=62.85950 04515 808450.65387 47519 623330.10501 50577 52041

0.(2)70754 47946 514610.(3)14377 85241 339900.(6)42045 60410 57221

N = 7

2.77067 03312 628860.69892 31918 255460,14109 25065 29910

0.(1)14318 94278 219930.(3)59744 60275 167630.(5)74767 62941 455160.(7)13030 90851 92778

N»8

2.69388 23811 283690.73163 28384 647610.17479 22836 49657

0.(1)23683 09161 980730.(2)15758 63871 000590.(4)43097 71850 899280.(6)35138 42384 639590,(9)38556 56065 39643

Hk- exp(ak)

4.65537 52731 51840

3.91869 18028 834992.04327 70202 76164

3.54830 63889 476621.71661 20134 448111.52777 33797 59560

3.30686 19596 575821.56352 32369 869301.27035 34933 248591.27738 71281 81308

3.13060 82727 959611.46517 30370 775091.15472 71156 639801.05310 86629 39Û561.12286 94566 54552

2.99332 68004 646881.39330 90568 348981.08233 76462 712510.95433 39556 612570.91947 24224 776841.01549 45763 88438

2.88177 64394 669831.33701 62733 96839

1.03021 13631 747350.89353 20848 886440.83076 24788 814120.82703 19824 815700.93533 48819 61205

2.78838 10453 654021.29095 74773 448520,98970 61041 184000,85029 58495 062360,77672 54803 174440,74525 03461 249000,75834 90745 323940.87254 27164 86626


EVALUATION OF jo Xß6~xf(x) dx BY GAUSS-LAGUERRE QUADRATURE

TABLE III. (continued)

ak Hk Hk- exp(ak)

0,(1)30698 86435 381970,50504 32964 741271,53802 20591 179573.16256 36390 737615.43845 97639 224218.46637 97776 07115

12.42191 36357 4474317.64939 17104 4532425.03752 72532 60732

0.(1)27666 55867 079720.45478 44226 05949

1.38242 57611 585992.83398 00120 926974.85097 14487 649147.50001 09426 42825

10.86840 80238 3440415.19947 80442 3760320,78921 46210 7010728,57306 01649 22106

0,(1)25179 46133 088970,41365 01260 335111.25569 13731 971232.56850 48683 608414.38222 21869 103016.74359 09253 553789.72409 81589 96743

13.43620 72707 4337318.06970 34327 2726323.99096 96631 3071032.14018 25332 13868

0.(1)23102 65376 351390.37935 66724 223531.15041 19328 481112.34927 40135 885903.99856 00495 716896.13252 25283 388138.80166 58896 42439

12.08123 15274 6367416.06791 60473 2106321.01714 03815 8O8O527.24475 27410 9831935.73406 55621 60630

N=9

2.62653 93174 586590,75563 32293 042990,20550 49875 62283

0.(1)34572 91461 079660.(2)32102 91256 107440,(3)14640 57805 069600.(5)27469 96771 970040.(7)15241 49374 21216

0,(10)10990 41120 78605

N = 102.56676 55577 907720.77334 79703 443410,23313 28349 73219

0,(1)46436 74708 956700,(2)55491 23502 036250.(3)36564 66626 776380.(4)11868 79857 102450.(6)15844 10942 056780.(9)61932 66726 79684

0.(12)30377 59926 51750

N=ll2.51317 19430 878500.78643 57877 875160,25781 87298 84105

0,(1)58823 92300 575470,(2)85745 36376 854800.(3)74801 95764 975440.(4)36099 77825 528700.(6)86028 63687 142740.(8)84148 62506 45160

0,(10)23835 16025 782150,(14)81815 15671 71140

N = 122.46470 56914 014170.79605 87051 085530.27980 63187 19897

0,(1)71390 73605 831630.(1)12225 76741 131500.(2)13323 27324 016080.(4)87141 39386 072270.(5)31636 04948 317390.(7)56781 75865 577600.(9)41694 96761 00274

0.(12)87605 37829 778610.(15)21551 77086 21477

2.70842 15062 704041.25212 75312 699660.95670 23921 851100,81699 60007 957200.73864 79577 615180.69576 12596 247730.68175 24318 327250.70479 19920 299250.82162 54223 10382

2.63877 06006 673541.21866 77411 442790.92893 07183 602180.79003 01592 710160.70953 61427 726810.66111 19090 973860.63559 76470 800180.63229 14008 156460,66154 74158 116830,77924 82029 39204

2.57725 56699 587641.18934 87914 084490,90501 19190 490310,76743 47767 079490,68609 70648 462620.63477 13122 214480.60342 97339 293830.58872 66789 824200,59240 60903 630250,62569 66308 951240,74325 47410 56718

2,52230 97751 444891,16331 57428 837020,88404 64246 331020,74802 92843 688360.66654 38020 032830.61366 53880 652880.57908 11522 448710.55846 30362 168630.55093 70815 550760.55938 64499 534020.59535 57406 410930.71218 11106 77831


CONÇUS ET AL.

TABLE III. (continued)

Hk Hk exP<ak)

0.(1)21342 34275 846640.35032 52295 437891.06153 06703 466922.16501 56086 410613.67822 11595 444825.62708 86089 596718.04886 00306 83873

10.99692 74644 7724514.54957 95697 3492018.82713 43854 3417824.O3OO5 84247 7159430.54295 81792 2792639.35095 83258 76102

0.(1)19831 3OOO3 516570.32542 89264 556600.98547 25901 579272.00788 56746 413483.40641 81936 785665.20124 01688 3O5727.42072 82152 96688

10.10442 73IO3 0871713.30805 87811 0110317.11248 45724 2270421.64121 43114 1218727.09932 09059 5176533.87960 60056 6291742.98788 30440 44682

0.(1)18520 O79O1 007340.30384 16926 176800.91963 63045 601251.87224 52173 434863.17271 03197 159594.83705 43317 026956.88746 68596 046859.35420 88904 05215

12.27867 88974 7981315.71854 86393 9062119.75691 95767 4194324.52017 68467 9169430.21765 93557 5617237.24990 09353 6824946.64243 20535 11588

N=13

2.42055 25676 886030.80304 82073 885690.29936 95808 67135

0.(1)83886 78142 334040.(1)16419 26539 633760,(2)21454 89695 314520,(3)17878 81714 216700,(5)89734 80851 082760,(6)25062 98260 658820,(6)34610 09164 64354

0,(10)19470 01505 878740,(13)30950 21290 105340,(17)55690 70926 31006

N=14

2.38007 14721 580480.80801 06466 618540.31677 86929 18489

0.(1)96136 40563 501460.(1)21063 17077 546870.(2)32020 73881 697800.(3)32543 30123 381280.(4)21163 29905 461330.(6)83147 37759 670360.(7)18208 32941 192560.(9)19697 33497 60737

0.(12)86368 19556 754610.114)10564 40502 504580.(18)14150 18667 34415

N=152.34274 96154 187580.81139 57220 935960.33228 46565 180900.10802 11707 58834

0,(1)26066 57355 435960,(2)45056 12328 440130.(3)54072 48503 187600,(4)43513 39705 O47OO0.(5)22477 5O554 547170.(7)70314 31690 102550.(8)12269 93807 80246

0.(10)10559 52348 157060,(13)36629 02248 519010,(16)34984 59074 228580,(20)35421 03787 93840

2.47276 80436 203561.13995 03363 067630.86541 53255 645350.73105 x3168 188720,64980 77333 065200,59612 86094 581320,55964 71281 956970,53563 11874 082160,52219 76028 486350.51966 48268 968990,53148 13887 172220,56924 84267 849310,68499 40632 29722

2.42774 25106 3Û5961.11879 12735 325860.84867 47291 955970,71598 10854 401750,63520 27556 672330,58116 74101 541610,54355 42825 740320,51746 42994 125150,50057 02672 710150,49218 42625 925580,49325 03791 848350,50750 38942 077200.54647 58577 729120.66094 08033 36074

2.38654 17889 593451.09948 54439 565000.83349 55016 792790.70244 87542 090560,62226 37924 869130.56814 58338 421830,52986 49355 508760.50246 32386 159270.48340 44051 587730,47154 45172 594190,46683 51221 701510,47056 73819 435690,48661 78584 172140,52638 46611 827230.63945 81778 13751


EVALUATION OF ja Xße"xf(x) dx BY GAUSS-LAGUERRE QUADRATURE 255

In fact the integral

(3) /= i" x-ll2e-xf(x) dxJo

can be transformed by the substitution u = x112 into the integral

(4) /= f e~u2f(u2) du,V— oo

which is of a form suitable for Gauss-Hermite quadrature. However, the tables

presented here for ß = — § allow more accuracy to be obtained in evaluating inte-

grals of the form (3) than could be obtained by transforming them to type (4)

and utilizing the existing tables for Gauss-Hermite quadrature [5] (duplicated in

[I])-

2. Method of Calculation. The Laguerre polynomials and their derivatives

were calculated by using the recursion relations [6]

Loß(x) = 1, lf(x) = 1 + ß - x,

T a, \ 2n + ß — x — 1 xa / x n + ß — 1 r/3 / -, / 00 sLrT(x) =-L„-iix)-L„-2(x) in = 2, 3, • • • )

and

A Lj(x) = - inLj(x) - (n + ß)LßUx)] (n = 1, 2, • • • ).dx x

The roots and weight coefficients were calculated on the IBM-7090 computer by

using a combination of Muller's method, as programmed in the SHARE code

B4 RW GRT, and Newton's method. The high degree of accuracy was obtained

through the use of revised versions of the SHARE double-precision arithmetic

package and input-output routines Al NR NPRE, Al NR DICV, and Al NR

DOCV.The sums and products of the roots were checked by means of the formulas

^2 ak = n(n + ß),k=i

and

and the weight coefficients checked by comparison of the calculations from (1)

and (2). In all cases, the calculations proved accurate to at least one more signifi-

cant digit than the rounded values given in the tables. The required gamma func-

tions were calculated with the aid of Gauss' original table reproduced in [7].*

* Added in proof: A recently published book [8], which is reviewed in Math. Comp. v. 17,

January, 1963, p. 93, contains tables to eight significant digits f or ß = — 0.90(0.02)0.00, ß =

k0.55(0.05)3.00, ß = -%,-M, and ß = n + - , where n = -1(1)2, k = 1, 2. The values given

here are in agreement with the e'ght-digit ones given there.


256 CONÇUS ET AL.

3. Acknowledgments. A large share of the work of programming the problem

for the computer was done by Messrs. Cassatt, Jaehnig, and Melby under the

National Science Foundation program of research participation for high school

teachers at the Lawrence Radiation Laboratory. This work was carried out in part

under the auspices of the U. S. Atomic Energy Commission.

Lawrence Radiation Laboratory

University of California

Berkeley, California

1. Z. Kopal, Numerical Analysis, John Wiley and Sons, New York, 1961.2. H. E. Salzbr & R. Zucker, "Table of the zeros and weight factors of the first fifteen

Laguerre polynomials," Bull. Amer. Math. Soc. v. 55, 1949, p. 1004-1012.

3. D. Burnett, "The numerical calculation of / xme"xf{x) dx," Proc. Cambridge Philos.Jo

Soc. v. 33, 1937, p. 359-362.4. P. Rabinowitz & G. Weiss, "Tables of abscissas and weights for numerical evaluation

of integrals of the form / e-xxnf{x) dx." MTAC v. 13, 1959, p. 285-294.Jo

5. H. E. Salzer, R. Zucker, & R. Capuano, "Table of the zeros and weight factors ofthe first twenty Hermite polynomials", J. Res. Nat. Bur. Standards v. 48, 1952, p. 111-116.

6. H. Bateman, Higher Transcendental Functions, v. 2, McGraw-Hill Book Co., NewYork, 1953, p. 188.

7. H. T. Davis, Tables of the Higher Mathematical Functions v. 1, Principia Press, Bloom-ington, Ind., 1933, p. 366-367.

8. V. S. Aizenshtat, V. I. Krylov, & A. S. Metleskii, Tables for Calculating Laplace

rTransforms and Integrals of the form / x"e x f(x) dx, Izdatel' stov Akad. Nauk SSSR, Minsk,

Jo1962 (Russian).


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