coefficients and roots of the polynomials which define the … · 2018. 11. 16. · coefficients...

Coefficients and Roots of the Polynomials whichDefine the Derivatives of the Exponential

of (-e/T)

By Edwin S. Campbell, E. M. Fishbach,and J. O. Hirschfelder

The w-th derivative of exp (—t/T) can be shown to be of the form

(1) d" exp 1-e/rydT* = T~" exp l-e/T]WJ[t/T]

Wnlt/r\ = ¿ bn,i[e/Ty.

The b„,i are constant coefficients in a homogeneous polynomial of the w-th order in

[e/7"]. These polynomials attain added importance from their relation to the La-

guerre polynomials [1]

(2) WJMT] = (-mL.it/T2 - «Ln-i[e/r]}

L„[e/r]: the w-th order Laguerre polynomial.

The zeros of these polynomials are of interest since they locate the extreme values

of the derivatives of exp (—t/T). Furthermore, an accurate, simple computation

of the Wn for values of (t/T) which occur in physical problems sometimes requires

use of the polynomial roots. This occurs whenever the direct evaluation of Wn[t/T~\

by synthetic division introduces at intermediate steps of the calculation numbers

which are larger than the value of the polynomial. In such cases the subtraction of

these larger numbers can require the use of considerably more than the usual guard

figures which allow for the effect of ordinary round-off error. Fortunately the occur-

rence of this disastrous subtraction in synthetic division does not imply the occur-

rence when the polynomial is calculated using its roots

(3) WJL</r\ = fi &/Ï - r..,]

r„,j: the Z-th root of Wn-

The polynomial coefficients, bn,i, are integers. Table I lists all of their non-zero

digits which were computed from the recursion relations

(4) bn,i = i-iy-H\

K.i = -in + l- l)bn^i,i + Bb-t.t-i(2 < I < n - 1)

b»,n = 1.

The values in the Table were checked numerically by the distinct relations

(5)_ in - l)bn,i = -»(« - l)bn^,i.

Received 16 September 1957.

1

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

coefficients and roots of polynomials

Table I. Exact Values of the Coefficients, bn_i

l ßn.l Pn.t n l ßn,, i>„,,

1+1 0 11 1 +3.9916 8 72 -1.9958 4 8

, _, n 3 +2-9937 6 82-4-1 n 4 -1-9958 4 8¿ +1 "5 +6.9854 4 7

6 -1.3970 88 71+6 0 7 +1.6632 62-6 0 8-1.188 53+1 0 9 +4.95 3

10 -1.1 2

1 -2.4 111 +1 0

2 +3-6 1 12 l -4.7900 16 83 -I-2 1 2 +2.6345 088 94+l 0 3 -4.3908 48 9

4 +3.2931 36 9

1+12 ? S -1.3172 544 92 _2'4 % 6 +3.0735 936 83 j-l'î 7 7 -4.3908 48 74_2' 7 8 +3.9204 65+1 0 9 ~2178 5T U 10 +7.26 3

H -1-32 21 -7.2 2 12 +1 02 +1.8 33 -1.2 3 13 1 +6.2270 208 94+3 2 2 -3.7362 1248 105-3 1 3 +6.8497 2288 106+1 0 4 -5.7081 024 10

5 +2.5686 4608 10

1 +5.04 32 -1.512 4

6 -6.8497 2288 97 +1.1416 2048 9

, , -, 8 -1.2231 648 8

4 I42 f 9 +8.4942 65 +6^3 26 -4.2 17 +1 0

10 -3.7752 511 +1.0296 412 -1.56 213 +1 0

8 1 -4.032 4 14 1 -8.7178 2912 102 +1.4112 S 2 +5.6665 88928 113 -1.4112 5 3 -1.1333 17785 6 124 +5.88 4 4 +1.0388 74636 8 125 -1.176 4 5 -5.1943 73184 116 +1.176 3 6 +1.5583 11955 2 117 -5.6 1 7 -2.9682 13248 108+1 O« +3.7102 6656 9

9 -3.0918 888 89 1 +3.6288 S 10 +1-7177 16 7

2 -1.4515 2 6 » -6.2462 4 53 +1.6934 4 6 12 +1-4196 44 -8.4672 S 13 -1.82 25 +2.1168 5 14 +1 0

7 I2 0164 1 1S ' +1-3076 74368 12ó _7,° 1 2 -9.1537 20576 12o ., 9 3 +1.9833 06124 8 13"•" 4 4 -1.9833 06124 8 13

10 1 < «8« , S +10908 18368 64 13J. T3-6288 , 6 6 -3.6360 61228 8 122 +í-6^ £ 7 7 +7.7915 59776 11

, 7. ill2 8 7 8 -1.1130 79968 114 +1.2700 8 7 9 +1.08216108 10

^ TH!0.2 4 6 10 -7.2144 072 86 +6.3504 S n +3.2792 76 77 -6°48 4 12 -9.9372 S5

4

10 +1 Ô ¡5 +Ï" 0

8 +3-24 3 13 +1.911

.?. -? 1 14 -2.1


coefficients and roots of polynomials 3

Table I. Exact Values of the Coefficients, &„,r—Continued

A.,1 Pn.i n I /3..i

123456789

10111213141516

123456789

1011121314151617

123456789

101112131415161718

-2

+1-3

+3-2.

+8-2

+3-3

+2-1

+5-1

+2,-2.

+1

0922569266149666379972650777339271028857574096235288524

4676

+3-2

+7-8

+5-2

+5-1

+1-1

+7-3

+ 1-2

+3-2

+1

-6

+5-1

+ 1-1

+5-1

+3-4

+4-3

+ 1-6

+1-3

+4-3

+1

.5568

.8454

.1137

.2993394515786514009126141213135724350395284826472

.4023

.4420

.4512

.8140

.26985024572108812890289011936541362274783292161606

78988 809241 688230122496734946949 1249273 6399046656628852482

74280 9699424 76848561 9273322 2492659 45637063 782478024 19292504 3290630 425004 804576020884

737051764904706058830411884512384322906368143681432228686060540872

72886886316828963027264512184322192363608

13141414141313121110976420

1415151515151414131210986420

151617171716161514131211986420

19

20

21

123456789

10111213141516171819

123456789

1011121314151617181920

123456789

101112131415161718192021

+ 1.2164-1.0948

+3.1019-4.1359

+3.1019-1.4475

+4.4805-9.6012

+ 1.4668-1.6298

+ 1.3335-8.0818

+3.6264-1.1955

+2.8465-4.7442

+5.2326-3.42

+ 1

-2.4329

+2.3112-6.9337

+9.8228-7.8582

+3.9291-1.3097

+3.0404-5.0673

+6.1934-5.6303

+3.8388-1.9686

+7.5717-2.1633

+4.5070-6.6279

+6.498-3.8

+ 1

+5.1090-5.1090

+ 1.6178-2.4268

+2.0627-1.1001

+3.9291-9.8228

+ 1.7735-2.3647

+2.3647-1.7914

+ 1.0335-4.5430

+ 1.5143-3.7858

+6.9593-9.0972

+ 7.98-4.2

+1

510040590350060334135006076694945317399661305458941027780468848254444834424

0883267948 842521 690028 842521 686510 0872531 2554240291247682992486

0200856907707234185873486367431224703432390531439976726932226319181504661441286

9421794217798351975396790582883674341858686685822558225835034817568902563009075258

1766476780 830342 401318 441054 7220527 3673509 1224217 67369601184374452804

17094 417094 443746 5615619 8418276 86409747 660820527 3601318 480793 607724 807724 8846429648

171818181818171616151412111086420

18191919191919181716151413111086420

1920212121212019191817161513121086420



Table I. Exact Values of the Coefficients, b„,¡—Continued

I ßn.l Pn.l

1 -1.1240 00727 77760 768 212 +1.1802 00764 16648 8064 223 -3.9340 02547 22162 688 224 +6.2288 37366 43424 256 225 -5.6059 53629 79081 8304 226 +3.1767 07056 88146 37056 227 -1.210174116 90722 42688 228 +3.2415 37813 14435 072 219 -6.3029 90192 22512 64 20

10 +9.1043 19166 54740 48 1911 -9.9319 84545 32444 16 1812 +8.2766 53787 77036 8 1713 -5.3055 47299 8528 1614 +2.6236 22291 136 1515 -9.9947 51585 28 1316 +2.9151 35879 04 1217 -6.4304 46792 1018 +1.0507 266 919 -1.2289 2 720 +9.702 421 -4.62 222 +1 0

1 +2.5852 01673 88849 7664 222 -2.8437 21841 27734 74304 233 +9.9530 26444 47071 60064 234 -1.6588 37740 74511 93344 245 +1.5758 95853 70786 33676 8 246 -9.4553 75122 24718 02060 8 237 +3.8271 75644 71909 67500 8 238 -1.0934 78755 63402 76428 8 239 +2.2780 80740 90422 4256 22

10 -3.5436 81152 51768 2176 2111 +4.1879 86816 61180 6208 2012 -3.8072 60742 37436 928 1913 +2.6846 06933 72551 68 1814 -1.4750 58754 79424 1715 +6.3216 80377 6896 1516 -2.1072 26792 5632 1417 +5.4230 10127 92 1218 -1.0633 35319 2 1119 +1.5545 838 920 -1.6364 04 721 +1.1688 6 522 -5.06 223 +1 0

1 -6.2044 84017 33239 43936 232 +7.1351 56619 93225 35526 4 243 -2.6162 24093 97515 96359 68 254 +4.5783 92164 45652 93629 44 255 -4.5783 92164 45652 93629 44 256 +2.8996 48370 82246 85965 312 257 -1.2427 06444 63820 08270 848 258 +3.7725 01706 93739 53679 36 249 -8.3833 37126 52754 52620 8 23

10 +1.3972 22854 42125 75436 8 23II -1.7782 83632 89978 23283 2 2212 +1.7513 39941 49220 98688 2113 -1.3471 84570 37862 2976 2014 +8.1423 24326 46420 48 1815 -3.8772 97298 31628 8 1716 +1.4539 86486 86860 8 1617 -4.2764 30843 7312 1418 +9.7826 84936 64 1219 -1.7162 60515 2 1120 +2.2582 3752 9



Table I. Exact Values of the Coefficients, bn,t—Continued

« l ßn.l Pn.l

24 21 -2.1507 024 722 +1.3965 6 523 -5.52 224 +1 0

25 1 +1.5511 21004 33309 85984 252 -1.8613 45205 19971 83180 8 263 +7.1351 56619 93225 35526 4 264 -1.308112046 98757 98179 84 275 +1.3735 17649 33695 88088 832 276 -9.1567 84328 91305 87258 88 267 +4.1423 54815 46066 94236 16 268 -1.3314 71190 68378 66004 48 269 +3.1437 51422 44782 94732 8 25

10 -5.5888 91417 68503 01747 2 2411 +7.6212 15569 57049 56928 2312 -8.0831 07422 27173 7856 2213 +6.7359 22851 89311 488 2114 -4.4412 67814 43502 08 2015 +2.3263 78378 98977 28 1916 -9.6932 43245 79072 1717 +3.2073 23132 7984 1618 -8.3851 58517 12 1419 +1.7162 60515 2 1320 -2.7098 85024 1121 +3.2260 536 922 -2.7931 2 723 +1.656 524 -6 225 +1 0

26 1 -4.0329 14611 26605 63558 4 262 +5.0411 43264 08257 04448 273 -2.0164 57305 63302 81779 2 284 +3.8648 76502 46330 40076 8 285 -4.2513 64152 70963 44084 48 286 +2.9759 54906 89674 40859 136 287 -1.4171 21384 23654 48028 16 288 +4.8080 90410 80256 27238 4 279 -1.2020 22602 70064 06809 6 27

10 +2.2704 87138 43454 35084 8 2611 -3.3025 26746 81388 14668 8 2512 +3.7528 71303 19759 2576 2413 -3.3679 61425 94655 744 2314 +2.4056 86732 81896 96 2215 -1.3746 78133 03941 12 2116 +6.3006 08109 76396 8 1917 -2.3164 00040 3544 1818 +6.8129 41295 16 1619 -1.5936 70478 4 1520 +2.9357 08776 1321 -4.1938 6968 1122 +4.5388 2 923 -3.588 724 +1.95 525 -6.5 226 +1 0

27 1 +1.0888 86945 04183 52160 768 282 -1.4155 53028 55438 57808 9984 293 +5.8981 37618 97660 74204 16 294 -1.1796 27523 79532 14840 832 305 +1.3565 71652 36461 97066 9568 306 -9.9481 92117 34054 45157 6832 297 +4.9740 96058 67027 22578 8416 298 -1.7764 62878 09652 58063 872 299 +4.6878 88150 53249 86557 44 28



Table I. Exact Values of the Coefficients, 6„,¡—Continued

I ßn.l Pn.l

10 -9.3757 76301 06499 73114 88 2711 +1.4489 83610 16459 04935 936 2712 -1.7563 43769 89647 33255 68 2613 +1.6887 92086 43891 66592 2514 -1.2990 70835 72224 3584 2415 +8.0418 67078 28055 552 2216 -4.0209 33539 14027 776 2117 +1.6261 12828 32878 88 2018 -5.3140 94210 2248 1819 +1.3984 45844 796 1720 -2.9440 96515 36 1521 +4.9068 27525 6 1322 -6.3725 0328 1123 +6.2969 4 924 -4.563 725 +2.2815 526 -7.02 227 +1 0

1 -3.0488 83446 11713 86050 1504 292 +4.1159 92652 25813 71167 70304 303 -1.7835 96815 97852 60839 33798 4 314 +3.7158 26699 95526 26748 6208 315 -4.4589 92039 94631 52098 34496 316 +3.4185 60563 95884 16608 73113 6 317 -1.7906 74581 12129 80128 38297 6 318 +6.7150 29679 20486 75481 43616 309 -1.8652 86022 00135 20967 0656 30

10 +3.9378 26046 44729 88708 2496 2911 -6.4437 15348 73194 36068 0448 2812 +8.2987 24312 76083 64633 088 2713 -8.5115 12115 65213 99623 68 2614 +7.0149 82512 90011 53536 2515 -4.6766 55008 60007 69024 2416 +2.5331 88129 65837 49888 2317 -1.1175 82998 37869 4848 2218 +4.0174 55222 92994 88 2019 -1.1746 94509 62864 1920 +2.7821 71207 0152 1721 -5.2993 73727 648 1522 +8.0293 54132 8 1323 -9.5209 7328 1124 +8.6240 7 925 -5.7493 8 726 +2.6535 6 527 -7.56 228 +1 0

1 +8.8417 61993 73970 19545 43616 302 -1.2378 46679 12355 82736 36106 24 323 +5.5703 10056 05601 22313 62478 08 324 -1.2069 00512 14546 93167 95203 584 335 +1.5086 25640 18183 66459 94004 48 336 -1.2069 00512 14546 93167 95203 584 337 +6.6092 17090 32042 72110 21352 96 328 -2.5964 78142 62588 21186 15531 52 329 +7.5730 61249 32548 95126 28633 6 31

10 -1.6829 02499 85010 87805 84140 8 31II +2.9068 31590 65018 78937 36243 2 3012 -3.9638 61259 97752 89460 03968 2913 +4.3195 92398 69346 10309 0176 2814 -3.7974 43866 98326 24447 488 2715 +2.7124 59904 98804 46033 92 2616 -1.5822 68277 90969 26853 12 2517 +7.5623 11622 36250 18048 2318 -2.9656 12400 92647 1296 2219 +9.5385 19418 18455 68 20



Table I. Exact Values of the Coefficients, bn,i—Continued

» / ßn.i P..I

29 20 -2.5101 36688 99593 6 1921 +5.3788 64333 56272 1722 -9.3140 50794 048 1523 +1.2885 05050 56 1424 -1.4005 48968 1225 +1.1671 2414 1026 -7.1823 024 727 +3.0693 6 528 -8.12 229 +1 0

30 1 -2.6525 28598 12191 05863 63084 8 322 +3.8461 66467 27677 03502 26472 96 333 -1.7948 77684 72915 94967 72354 048 344 +4.0384 74790 64060 88677 37796 608 345 -5.2500 17227 83279 15280 59135 5904 346 +4.3750 14356 52732 62733 82612 992 347 -2.5000 08203 72990 07276 47207 424 348 +1.0267 89083 67478 06559 97960 192 349 -3.1374 11089 00627 42266 60433 92 33

10 +7.3206 25874 34797 31955 41012 48 3211 -1.3310 22886 24508 60355 52911 36 3212 +1.9158 66275 65580 56572 35251 2 3113 -2.2106 14933 44900 65275 79136 3014 +2.0648 60102 67214 89543 3216 2915 -1.5732 26744 89306 58699 6736 2816 +9.8326 67155 58166 16872 96 2617 -5.0609 31624 19644 35155 2 2518 +2.1500 68990 67169 16896 2419 -7.5441 01721 65505 856 2220 +2.1838 18919 42646 432 2121 -5.1995 68855 77729 6 1922 +1.0129 03023 85272 1823 -1.6014 27705 696 1624 +2.0307 96003 6 1425 -2.0307 96003 6 1226 +1.5621 50772 1027 -8.9011 44 728 +3.5322 529 -8.7 230 +1 0

Legend: Wn(x) = 2 b„,, xli-i

b..t = ßn,I X ÎO*"".', 1 g ft.,1 < 10

Table II. Roots of Wn (t/T)

n I rn_i C e„,i

2 1 2.00000 00000 0000 0 0

3 1 1.26794 91924 3112 0 53 2 4.73205 08075 6888 0 5

4 1 0.93582 22275 2409 0 54 2 3.30540 72893 3227 0 54 3 7.75877 04831 4363 0 5


8 coefficients and roots of polynomials

Table II. Roots of Wn (t/T)—Continued

» / r„,i C e„,i

5 1 0.74329 19279 8143 0 55 2 2.57163 50076 4627 0 55 3 5.73117 87516 8905 0 505 4 10.95389 43126 8321 1 6

6 1 0.61703 08532 7825 0 506 2 2.11296 59585 7838 0 506 3 4.61083 31510 1760 0 506 4 8.39906 69712 0486 0 506 5 14.26010 30659 2081 1 10

1 0.52766 81217 1117 0 502 1.79629 98096 4345 0 503 3.87664 15204 7699 0 504 6.91881 65667 0471 0 505 11.23461 04290 8311 1 66 17.64596 35523 8068 1 16

8 1 0.46102 42198 0496 0 508 2 1.56358 61896 5431 0 508 3 3.35205 05025 3674 0 508 4 5.91629 72490 2042 0 508 5 9.42069 93830 2156 0 508 6 14.19416 55480 0748 1 108 7 21.09217 69079 5447 1 22

9 1 0.40938 35732 0319 0 59 2 1.38496 31848 0312 0 59 3 2.95625 45561 6887 0 59 4 5.18194 31010 4007 0 59 5 8.16170 96881 4582 0 59 6 12.07005 51268 3715 1 79 7 17.24973 55261 4898 1 159 8 24.58595 52436 5281 1 30

10 1 0.36817 84529 4174 0 510 2 1.24335 79621 4047 0 510 3 2.64603 38413 8420 0 510 4 4.61688 25146 3485 0 5010 5 7.22178 65393 9663 0 5010 6 10.56732 08077 4184 1 5610 7 14.83591 45152 6107 1 11010 8 20.38218 19854 4899 1 20710 9 28.11834 33810 4993 1 39

11 1 0.33452 86763 2476 0 511 2 1.12825 33558 7666 0 511 3 2.39586 99247 4731 0 511 4 4.16684 09879 2878 0 511 5 6.48735 30313 8081 0 511 6 9.42835 48133 3561 0 511 7 13.10172 35803 6780 1 911 8 17.69648 75668 4621 1 1611 9 23.57778 70883 6019 1 2811 10 31.68280 09748 3192 1 50




' r».i C

1212121212121212121212

131313131313131313131313

123456789

1011

123456789

101112

0.306521.032792.189613.799045.894918.52729

11.7710015.7422620.6358026.8263435.27439

0.282850.952322.016493.492355.405497.79281

10.7073814.2271518.4719923.6417830.1200538.88928

6702173987194197606011715200806565402870566869949307009

834826041321385406971020039404868892363766342375248626143760

30057972684054970275702790682615412970826510

399264627771779915721242890989962982012206509550

000000

000000

5555557

122133662

55

5050556

10171279455

76

14141414141414141414141414

123456789

10111213

0.262580.883551.869033.232414.993577.180619.83280

13.0056216.7796121.2779126.7050333.4527842.52444

83981030733815186994560700494182510240023641117554409894965775660

7108877499795120741900792000586575854831277367211840

0000000

55

5050555

8514122635755991

1515151515151515151515151515

123456789

1011121314

0.245030.824081.741873.009134.641636.661349.09830

11.9937015.4049819.4149924.1497529.8181036.8196246.17743

3015022200240072459040665149662703734331803102828179520511964256900169

93104784644917929064271770538574514566152928713297218345

0000000

550505050557

11918902920446067601060



Table II. Roots of Wn (t/T)~

rn.l

161616161616161616161616161616

123456789

101112131415

0.229680.772141.631052.815144.337166.214648.47116

11.1383314.2589117.8920522.1226227.0793132.9749740.2165849.84622

050544910330990459004077327645398131965700216343810174814990355243711417085

251375396745122537565924467150802446695326684742016285685745

-Continued

C

0000000

e.,i

55

50505556

102160024503680545080601240

123456789

10111213141516

0.216140.726381.533592.644974.070975.825857.92850

10.4038013.2846616.6151720.4560024.8938430.0598636.1706943.6403653.52915

03052824323160309986816085515141853828991070732168602007025329201454365184111602

3945518373531195801805630666510406976805337135756784321650015465

0000000

5505050505055

8813802090

3090045000656009540

14300

1818181818181818181818181818181818

123456789

1011121314151617

0.204100.685761.447192.494433.836155.484117.453729.76498

12.4438615.5243419.0517123.0880027.7215533.0858539.4011547.0882057.22481

9108575894867930885960319542082949543667105193001287593070086346298280379339654618319

79339453804962489932404781285180839514578568717478246407086101411492

00000000

55

505050555

7712001810

26700385005480077500

1 1100016300




191919191919191919191919191919191919

I

123456789

101112131415161718

?n,l

0.193340.649461.370072.360273.627375.181177.034439.20350

11.7093214.5787617.8467521.5596525.7807030.5998136.1526542.6628850.5577860.93200

776861810754867762575865649562062306684247926113069827487534100504486502790777973397777265

790196551706750855602015656682147879928318820514524019184671856417780073

000000001112222222

en,i

55555555

691060

1590012172333456493

20202020202020202020202020202020202020

123456789

10111213141516171819

0.183660.616811.300792.239943.440474.910646.661168.70559

11.0611013.7493916.7981220.2430824.1315628.5279433.5236139.2562945.9529554.0470964.64971

51730638219402993990194285879606120717718683541200300594963372676797137617242907064863024724378

9616267366770886815074735821280275758341530002460267141488908263383401941605

0000000011122222222

5050

500500500505050

6129450

1410010152028395373

105

2121212121212121212121212121

123456789

1011121314

0.174900.587301.238222.131393.272134.667496.326538.26067

10.4841613.0148415.8750819.0932522.7058926.76117

6752330806510186260031335446586197609520738128772170127190763888102293

86613812342077121677883773621371082958281499062473217952

00000000

5050

5005005005005005055

84701 26000

1826

357



Table II. Roots of Wn (t/T)-

212121212121

222222222222222222222222222222222222222222

23232323232323232323232323232323232323232323

151617181920

123456789

101112131415161718192021

123456789

10111213141516171819202122

fn.l

-Continued

C

31.3245136.4887042.3934249.2688157.5544268.37703

0.166940.560491.181422.032953.119694.447696.024717.860439.96688

12.3589115.0549318.0778821.4567825.2288729.4431134.1659339.4914045.5611252.6082861.0782772.11320

0.159670.536021.129621.943272.980974.247985.750987.498299.50013

11.7690414.3203717.1730420.3505423.8824427.8066132.1727937.0483242.5285548.7568955.9694664.6173575.85754

613703346127457384980971378145

613900861207583745202040250426216817758688709869880217577154846626349242999029748253699787924355757196312

90124445070322826194423426097594501710428740861305809237097365523314299634815883561288450107697933673106284527

072948247640684915555231

197154065136590414901758150909334385968592764042634348930910601128338368304772127984

6359253342969229998036458811472518573152680262040563712764177417114998394619795996345272

000000000

000000000

490666898

12101650235

5050

500500500500505050

7651130063000

230318

431058307810

10400139001860259

5050

500500500500500505

691030

1470010

28438505170

68600906001900015600208002870



Table II. Roots of Wn (t/T)— Continued

I r..i C

2424242424242424242424242424242424242424242424

123456789

1011121314151617181920212223

0.153010.513601.082181.861212.854184.065695.501547.168919.07661

11.2353113.6579916.3604619.3620922.6868326.3647930.4344434.9462039.9683345.5973851.9784959.3506868.1705079.60945

8489583707809426410883431479662391089382446605487192756867130188157294431502691346462922664313453322023605181044050

99420681375398973952604724831722197459786470706753877187029094038518739530941339349002408424

000000000

55

505050505055

63933

134001 88000

25803470

463006110080000

1 040001 360001 77000

231003140

252525252525252525252525252525252525252525252525

123456789

101112131415161718192021222324

0.146890.492991.038591.785842.737843.898605.273226.867948.69039

10.7497713.0571515.6259118.4722421.6159225.0813628.8991333.1082637.7598642.9230248.6954555.2239962.7504471.7367183.36839

162574846120353954674532282901154035638261931142413675413581241442873520017240317305110435518643221177787417545963749267

814621828078396905074203064476674098089782166414427583419402567783896086737068035012715483612194

000000000

55

5050505050505

58852

122001 71000

23403140

21273646597698

129174

26 1 0.14123 67262 5809 026 2 0.47397 45378 8442 026 3 0.99838 34056 2151 026 4 1.71638 16871 9240 0

55

5050




» I r„,i C e»,i

26 5 2.63069 31145 8470 0 5026 6 3.74487 77262 0273 0 5026 7 5.06340 83123 3855 0 5026 8 6.59177 56068 7320 0 5026 9 8.33662 63598 0514 0 526 10 10.30594 30256 1368 1 526 11 12.50927 80113 1609 1 7826 12 14.95806 12826 7794 1 1120026 13 17.66600 89930 4484 1 1 5600026 14 20.64967 47456 6110 1 214026 15 23.92920 78044 9273 2 1426 16 27.52942 09021 3584 2 1926 17 31.48133 78942 1101 2 2526 18 35.82451 67628 4751 2 3226 19 40.61069 00156 5943 2 41426 20 45.90978 68582 2976 2 52626 21 51.82061 58754 0514 2 67226 22 58.49167 48142 7643 2 85426 23 66.16744 93598 1048 2 11026 24 75.31508 13581 0590 2 14126 25 87.13389 48199 8148 2 189

2727272727272727272727272727272727272727272727272727

123456789

1011121314151617181920212223242526

0.136000.456360.961181.652142.531673.602944.869906.337408.011279.89850

12.0073814.3477816.9314419.7723922.8875026.2972730.0268834.1077838.5800343.4959748.9263254.9709661.7799969.6005278.9047990.90552

1257093581102732892976794557806082631557625433483061381788107620441164439621002918438471425190658235265883025323077545732954780995

44443000092183759454890823762642804876088993550627626035785271237846172146291797533906083237475164981289

00000000001111222222222222

55

50505050505055

7210300

1431950

131723

292374472598758952

1210156206

28 1 0.13114 02048 4340 028 2 0.44002 68410 2122 028 3 0.92665 89979 7975 028 4 1.59256 14353 9651 0

5555




2828282828282828282828282828282828282828282828

I

56789

101112131415161718192021222324252627

r».i

2.439893.471484.690856.102297.710979.52302

11.5457113.7876216.2588918.9715521.9398925.1811028.7159932.5700936.7752641.3720246.4133051.9705858.1447965.0875773.0486282.5051294.68291

8859898767736868066327209142140798326083714830819161709697434022465076283372670939270205380245679811351084675612582

82148727210759261486826426830963233493275973269281034767641405632954364079046239162159399295

00000011112222222222222

e».i

5555557

10132

1800121621

266338430

538067608480

106001330171224

29292929292929292929292929292929292929292929292929292929

123456789

10111213141516171819202122232425262728

0.126610.424810.894531.537142.354583.349384.524685.884317.432869.17577

11.1194413.2713815.6404318.2369821.0733024.1639527.5263931.1817235.1558239.4809144.1979849.3605155.0407261.3405668.4131976.5107986.1154298.46569

47772568946996763597600740894300068365747384757820122764445565527621111427146279399717433150255565242543471053955779022189167933571699276629

6314280329826714036247310930590733635631951895700260430147528314527946749695102369700558584966743268689744013931

0000000000111122222222222222

555555555569

1231670

111519

244308

39004880610075809440

11700147001860242


16 coefficients and roots of polynomials

Table II.

I

Roots of Wn (t/T)-

r*,i

3030303030303030303030303030303030303030303030303030303030

123456789

1011121314151617181920212223242526272829

0.122390.410620.864571.485472.275073.235644.370015.681617.174538.85361

10.7245212.7939015.0694917.5603520.2770923.2322126.4405229.9197333.6912237.7811742.2221847.0556752.3356758.1351164.5569271.7557479.9861989.73509

102.25357

1363622202256603650948206712567377756855651835745973673506125611739915387553332046096141940722259363887418225193797503962825328866969891211528564

82324781978411745174649871741702225080894938065932804328251260064200915570267412506592964686878312619527150988207799

-Continued

C

00000000001111222222222222222

555555555568

111540

101418

224284

35804440

552006840084800

1 0400012900160002020260

Legend:r»,i: the Ith root of the n-th order polynomial, W„. The estimated error bound is

| r„., - root | g e„,, X 10-»

The significance of the numbers in column C is:0: the root was computed explicitly1: the root was computed as rn,i = 10/V„,i2: the root was computed as rn.i = 20/f„j

The formulae of equation (4) can be verified by induction based on the actual

formal construction of successive derivatives of exp (—t/T) [1]. After these rela-

tions have been established they can be used with a simple induction argument to

establish the correctness of explicit formulae which are convenient for other pur-

poses, albeit not for calculations

,,, r (-!)-'/(/+ l)t ... (n- l)'n

(6) *-•=-prnn-•The roots were located by the method of false position [2], which depends upon

evaluation of the polynomials for successive approximations to the roots. The evalu-

ations were computed by synthetic division using double precision arithmetic on

the ORDVAC computer at the Aberdeen Proving Ground. The polynomials were

calculated in one of three forms as described in the legend to Table II. This was

necessary since the original form of the polynomial was relatively insensitive to

changes in approximations for certain roots; that is, the remainder was too small


COEFFICIENTS AND ROOTS OF POLYNOMIALS 17

Table III. Symmetric Function Check on Roots

n in.n-l dn

2 2 03 6 04 12 - 15 20 - 46 30 - 107 42 +118 56 - 69 72 +1

10 90 - 2811 110 + 512 132 + 2013 156 + 4714 182 + 9415 210 - 37116 240 - 126317 272 -1305918 306 +2128919 342 + 92720 380 + 2721 420 - 626722 462 +3909623 506 - 241124 552 -5526725 600 - 591526 650 +2301027 702 + 176228 756 + 22329 812 - 432930 870 -18880

Legend: fc„.„_i - 2 r„,¡ = dn X 10"»i=i

compared with the largest term in the synthetic division schema. When this oc-

curred, one of the other forms of the polynomial was used in order to define the root

more accurately.

The error estimates of Table II were based upon locating the root between two

approximate values which gave remainders of opposite sign. Although this method

of locating roots is inherently self-checking, a further numerical test was made by

computing the symmetric function summations of Table III.

Edwin S. CampbellNew York University,New York

E. M. FischbachBallistic Research Laboratories,Aberdeen, Maryland

J. O. HirschfelderUniversity of WisconsinMilwaukee, Wisconsin

1. J. O. Hirschfelder & Edwin S. Campbell, Analytical (power series) Solutions of FlamePropagation (CM-784), University of Wisconsin Naval Research Laboratory, May 27, 1953, p.17 ff.

2. F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill Book Co., New York,1956, p. 446.


coefficients and roots of the polynomials which define the … · 2018. 11. 16. · coefficients...

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