1828:
1343:
3617:
1332:
1823:{\displaystyle {\begin{aligned}\ H&~=~-{\frac {\operatorname {d} ^{2}}{\operatorname {d} x^{2}}}\ln {\Bigl |}\ p(x)\ {\Bigr |}~=~{\frac {1}{~~(x-x_{1})^{2}\ }}+{\frac {1}{~~(x-x_{2})^{2}\ }}+\ \cdots \ +{\frac {1}{~~(x-x_{n})^{2}\ }}\\&~=~-{\frac {\ p''(x)\ }{\ {\bigl |}\ p(x)\ {\bigr |}\ }}\ +\ \left({\frac {\ p'(x)\ }{\ p(x)\ }}\right)^{2}\cdot \ \operatorname {sgn} \!{\Bigl (}\ p(x)\ {\Bigr )}~.\end{aligned}}}
3231:
1030:
1015:
3942:
The algorithm is fairly simple to use, compared to other "sure-fire" methods, and simple enough for hand calculation, aided by a pocket calculator, if a computer is not available. The speed at which the method converges means that one is only very rarely required to compute more than a few iterations
3934:
Empirical evidence shows that convergence failure is extremely rare, making this a good candidate for a general purpose polynomial root finding algorithm. However, given the fairly limited theoretical understanding of the algorithm, many numerical analysts are hesitant to use it as a default, and
2857:
1327:{\displaystyle {\begin{aligned}\ G&~=~{\frac {\operatorname {d} }{\operatorname {d} x}}\ln {\Bigl |}\ p(x)\ {\Bigr |}~=~{\frac {1}{\ x-x_{1}\ }}+{\frac {1}{\ x-x_{2}\ }}+\ \cdots \ +{\frac {1}{\ x-x_{n}\ }}\\&~=~{\frac {\ p'(x)\ }{\ {\bigl |}\ p(x)\ {\bigr |}\ }}\ ,\end{aligned}}}
2308:
756:
2967:
3901:, which notoriously fail to converge for poorly chosen initial guesses. Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction,
3446:, enough times to make the smaller roots significantly smaller than the largest root (and so, clustered comparatively nearer to zero). The approximate root from Graeffe's method, can then be used to start the new iteration for Laguerre's method on
2687:
2135:
667:
3357:
3930:
given above â manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability depending on the application to which the method is being used.
2728:
492:
can be found. Note however that deflation can lead to approximate factors that differ significantly from the corresponding exact factors. This error is least if the roots are found in the order of increasing magnitude.
2536:
327:
1010:{\displaystyle \ln {\bigl |}\ p(x)\ {\bigr |}~=~\ln {\bigl |}\ C\ {\bigr |}\ +\ \ln {\bigl |}\ x-x_{1}\ {\bigr |}\ +\ \ln {\bigl |}\ x-x_{2}\ {\bigr |}\ +\ \cdots \ +\ \ln {\bigl |}\ x-x_{n}\ {\bigr |}~.}
411:
2146:
1348:
1035:
3226:{\displaystyle a={\frac {\ p(x)\ }{p'(x)}}\cdot {\Biggl \{}\ {\frac {\ 1\ }{n}}+{\frac {\ n-1\ }{n}}\ {\sqrt {1-{\frac {n}{\ n-1\ }}\ {\frac {\ p(x)\ p''(x)\ }{~p'(x)^{2}\ }}\ }}\ {\Biggr \}}^{-1}\ ,}
2408:
57:. One of the most useful properties of this method is that it is, from extensive empirical study, very close to being a "sure-fire" method, meaning that it is almost guaranteed to always converge to
238:
3566:
744:
3716:
2013:
2897:
2930:
2576:
470:
3807:
4134:
3602:
3876:
3843:
3761:
2024:
1872:
541:
3272:
161:
3928:
2959:
2568:
1930:
1901:
3492:
2720:
2437:
2337:
1959:
533:
4353:
3882:
convergence is merely linear, with the penalty of calculating values for the polynomial and its first and second derivatives at each stage of the iteration.
3281:
2852:{\displaystyle \operatorname {\mathcal {R_{e}}} {\biggl \{}\ {\overline {G}}{\sqrt {\left(n-1\right)\left(n\ H-G^{2}\right)\ }}\ {\biggr \}}>0\ ,}
4127:
4083:
3992:
2445:
4332:
4258:
2303:{\displaystyle \ b\ \equiv \ \operatorname {\mathsf {harmonic\ mean}} {\Bigl \{}\ x-x_{2},\ x-x_{3},\ \ldots \ x-x_{n}\ {\Bigr \}}\ }
4102:
3980:
334:
4120:
4322:
2345:
3424:, that give distinct roots clearly distinct magnitudes, if necessary (which it will be if some roots are complex conjugates).
502:
245:
3497:
675:
4286:
4022:
3936:
488:. This deflation step reduces the degree of the polynomial by one, so that eventually, approximations for all roots of
3623:
4291:
169:
414:
4276:
4220:
1967:
4301:
4235:
4179:
4151:
4143:
2865:
4225:
3898:
3397:
for which the assumptions are viable; e.g. by first shifting the origin towards a suitable complex number
1021:
66:
28:
2902:
4317:
2682:{\displaystyle a={\frac {n}{\ G\pm {\sqrt {{\bigl (}n-1{\bigr )}{\bigl (}n\ H-G^{2}{\bigr )}\ }}\ }}\ ,}
4271:
4047:
4001:
423:
3616:
4281:
4210:
4202:
3764:
3443:
70:
3770:
2130:{\displaystyle \ b\ \approx \ x-x_{2}\ \approx \ x-x_{3}\ \approx \ \ldots \ \approx \ x-x_{n}\ ,}
662:{\displaystyle \ p(x)=C\left(x-x_{1}\right)\left(x-x_{2}\right)\ \cdots \ \left(x-x_{n}\right)\ ,}
65:
computation, more efficient methods are known, with which it is guaranteed to find all roots (see
20:
4327:
4248:
4192:
4187:
3894:
3605:
3571:
3275:
3848:
3812:
3728:
1844:
4243:
4098:
4079:
4073:
3976:
3242:
747:
130:
4159:
4055:
4009:
3904:
2935:
2544:
1906:
1877:
413:, where the sign is chosen to give the denominator with the larger absolute value, to avoid
3471:
2699:
2416:
2316:
1935:
512:
3954:
77:
4051:
4005:
2693:
3885:
A major advantage of
Laguerre's method is that it is almost guaranteed to converge to
3236:
where the square root of a complex number is chosen to have a non-negative real part.
4347:
4266:
4215:
4036:
Pan, V.Y. (1997). "Solving a polynomial equation: Some history and recent progress".
3959:
3879:
3352:{\displaystyle \ \operatorname {\mathcal {O}} {\bigl \{}\ (p(x))^{3}\ {\bigr \}}\ ,}
4164:
4068:
4028:
35:. In other words, Laguerre's method can be used to numerically solve the equation
4038:
4027:(Master's thesis). Mathematics. Oxford, UK: University of Oxford. Archived from
4169:
4059:
484:
If a root has been found, the corresponding linear factor can be removed from
32:
3939:, for which more solid theory has been developed and whose limits are known.
1932:
and all the other roots are all clustered together, at some further distance
61:
root of the polynomial, no matter what initial guess is chosen. However, for
3990:
Goedecker, S. (1994). "Remark on algorithms to find roots of polynomials".
4112:
62:
2696:
is chosen to produce largest absolute value of the denominator and make
480:
is small enough or if the maximum number of iterations has been reached.
4067:
Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007).
88:
The algorithm of the
Laguerre method to find one root of a polynomial
2531:{\displaystyle H={\frac {1}{~a^{2}\ }}+{\frac {\ n-1\ }{~b^{2}\ }}~.}
4013:
3615:
4116:
3468:
If we make the even more extreme assumption that the terms in
3442:
by repeatedly applying the root squaring transformation from
406:{\displaystyle a={\frac {n}{G\pm {\sqrt {(n-1)(nH-G^{2})}}}}}
3290:
2879:
2875:
2739:
2735:
76:
This method is named in honour of the French mathematician,
4078:(3rd ed.). New York, NY: Cambridge University Press.
3620:
Attraction zones of
Laguerre's method for the polynomial
2403:{\displaystyle G={\frac {\ 1\ }{a}}+{\frac {\ n-1\ }{b}}}
3274:
this formula differs from the offset of the third order
3359:
so convergence close to a root will be cubic as well.
2692:
where in this case, the square root of the (possibly)
322:{\displaystyle H=G^{2}-{\frac {p''(x_{k})}{p(x_{k})}}}
3907:
3851:
3815:
3773:
3731:
3626:
3574:
3500:
3474:
3461:
may then be obtained straightforwardly from that for
3284:
3245:
2970:
2938:
2905:
2868:
2731:
2702:
2579:
2547:
2448:
2419:
2348:
2319:
2149:
2027:
1970:
1938:
1909:
1880:
1847:
1346:
1033:
759:
678:
544:
515:
426:
337:
248:
172:
133:
4310:
4257:
4234:
4201:
4178:
4150:
3893:. This is in contrast to other methods such as the
3891:
no matter where the initial approximation is chosen
3561:{\displaystyle \ x_{2},\ x_{3},\ \ldots ,\ x_{n}\ }
3958:
3922:
3870:
3837:
3801:
3755:
3710:
3596:
3560:
3486:
3371:does not work well for some particular polynomial
3351:
3266:
3225:
2953:
2924:
2891:
2851:
2722:as small as possible; equivalently, it satisfies:
2714:
2681:
2562:
2530:
2431:
2402:
2331:
2302:
2129:
2007:
1953:
1924:
1895:
1866:
1822:
1326:
1009:
739:{\displaystyle \ x_{1},x_{2},\ \ldots ,\ x_{n}\ ,}
738:
661:
527:
464:
405:
321:
232:
155:
67:Root-finding algorithm § Roots of polynomials
3203:
3025:
2832:
2751:
2292:
2213:
1805:
1780:
1777:
1436:
1411:
1106:
1081:
3711:{\displaystyle ~p(x)=x^{4}+2x^{3}+3x^{2}+4x+1~.}
746:are the roots of the polynomial. If we take the
233:{\displaystyle G={\frac {p'(x_{k})}{p(x_{k})}}}
4069:"Section 9.5.3 Laguerre's method"
4128:
4093:Ralston, Anthony; Rabinowitz, Philip (1978).
3935:prefer better understood methods such as the
3393:can be transformed into a related polynomial
3338:
3300:
2657:
2628:
2621:
2605:
1684:
1659:
1303:
1278:
996:
967:
933:
904:
882:
853:
831:
815:
793:
768:
8:
3985:– via Internet Archive (archive.org).
2899:denotes real part of a complex number, and
4135:
4121:
4113:
4024:Iterative methods for roots of polynomials
3568:are negligibly small compared to the root
3906:
3859:
3850:
3823:
3814:
3781:
3772:
3730:
3681:
3665:
3649:
3625:
3582:
3573:
3549:
3524:
3508:
3499:
3473:
3337:
3336:
3327:
3299:
3298:
3289:
3288:
3283:
3244:
3208:
3202:
3201:
3180:
3116:
3089:
3081:
3054:
3033:
3024:
3023:
2977:
2969:
2937:
2909:
2904:
2878:
2873:
2872:
2867:
2831:
2830:
2811:
2769:
2759:
2750:
2749:
2738:
2733:
2732:
2730:
2701:
2656:
2655:
2649:
2627:
2626:
2620:
2619:
2604:
2603:
2601:
2586:
2578:
2546:
2510:
2483:
2468:
2455:
2447:
2418:
2376:
2355:
2347:
2318:
2291:
2290:
2281:
2253:
2231:
2212:
2211:
2191:
2166:
2165:
2148:
2115:
2078:
2053:
2026:
1996:
1969:
1937:
1908:
1879:
1855:
1846:
1841:, that the root we are looking for, say,
1804:
1803:
1779:
1778:
1762:
1709:
1683:
1682:
1658:
1657:
1626:
1597:
1587:
1562:
1535:
1525:
1500:
1485:
1475:
1450:
1435:
1434:
1410:
1409:
1394:
1377:
1371:
1347:
1345:
1302:
1301:
1277:
1276:
1245:
1219:
1200:
1173:
1154:
1139:
1120:
1105:
1104:
1080:
1079:
1055:
1034:
1032:
995:
994:
985:
966:
965:
932:
931:
922:
903:
902:
881:
880:
871:
852:
851:
830:
829:
814:
813:
792:
791:
767:
766:
758:
724:
699:
686:
677:
642:
607:
581:
543:
514:
450:
431:
425:
389:
356:
344:
336:
307:
286:
268:
259:
247:
218:
197:
179:
171:
144:
132:
3427:After that, getting a third polynomial
2008:{\displaystyle \ a\ \equiv \ x-x_{1}\ }
2892:{\displaystyle \ {\mathcal {R_{e}}}\ }
2203:
2200:
2197:
2194:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
1834:
1337:and the negated second derivative by
7:
4095:A First Course in Numerical Analysis
3993:SIAM Journal on Scientific Computing
4354:Polynomial factorization algorithms
3725:is a simple root of the polynomial
2925:{\displaystyle \ {\overline {G}}\ }
1384:
1374:
1061:
1057:
14:
4076:: The art of scientific computing
3763:then Laguerre's method converges
16:Polynomial root-finding algorithm
4333:Sidi's generalized secant method
1961:If we denote these distances by
4323:Inverse quadratic interpolation
465:{\displaystyle x_{k+1}=x_{k}-a}
3788:
3782:
3744:
3738:
3639:
3633:
3324:
3320:
3314:
3308:
3258:
3252:
3177:
3170:
3151:
3145:
3131:
3125:
3014:
3008:
2992:
2986:
1797:
1791:
1749:
1743:
1729:
1723:
1676:
1670:
1646:
1640:
1594:
1574:
1532:
1512:
1482:
1462:
1428:
1422:
1295:
1289:
1265:
1259:
1098:
1092:
785:
779:
557:
551:
503:fundamental theorem of algebra
395:
373:
370:
358:
313:
300:
292:
279:
224:
211:
203:
190:
150:
137:
1:
3401:, giving a second polynomial
3809:is close enough to the root
3802:{\displaystyle \ x^{(0)}\ ,}
3767:whenever the initial guess,
3369:'drastic set of assumptions'
2932:is the complex conjugate of
2914:
2764:
2541:Solving these equations for
1839:'drastic set of assumptions'
750:of both sides, we find that
163:is very small, exit the loop
3494:corresponding to the roots
535:can be written in the form
4370:
4152:Bracketing (no derivative)
4021:Mekwi, Wankere R. (2001).
3597:{\displaystyle \ x_{1}\ ,}
3450:. An approximate root for
4060:10.1137/S0036144595288554
3871:{\displaystyle \ x_{1}\ }
3838:{\displaystyle \ x_{1}~.}
3756:{\displaystyle \ p(x)\ ,}
1867:{\displaystyle \ x_{1}\ }
415:catastrophic cancellation
69:) or all real roots (see
3845:On the other hand, when
3267:{\displaystyle \ p(x)\ }
156:{\displaystyle p(x_{k})}
107:Choose an initial guess
4302:Splitting circle method
4287:JenkinsâTraub algorithm
4144:Root-finding algorithms
3961:Numerical Methods that
3937:JenkinsâTraub algorithm
3889:root of the polynomial
2413:and the expression for
46:for a given polynomial
4292:LehmerâSchur algorithm
3943:to get high accuracy.
3924:
3923:{\displaystyle \ a\ ,}
3872:
3839:
3803:
3757:
3718:
3712:
3598:
3562:
3488:
3353:
3268:
3227:
2955:
2954:{\displaystyle \ G\ ;}
2926:
2893:
2853:
2716:
2683:
2564:
2563:{\displaystyle \ a\ ,}
2532:
2433:
2404:
2333:
2313:then our equation for
2304:
2131:
2009:
1955:
1926:
1925:{\displaystyle \ x\ ,}
1897:
1896:{\displaystyle \ a\ ,}
1868:
1824:
1328:
1022:logarithmic derivative
1011:
740:
663:
529:
466:
407:
323:
234:
157:
29:root-finding algorithm
4318:Fixed-point iteration
3925:
3895:NewtonâRaphson method
3873:
3840:
3804:
3758:
3713:
3619:
3599:
3563:
3489:
3487:{\displaystyle \ G\ }
3354:
3269:
3228:
2956:
2927:
2894:
2854:
2717:
2715:{\displaystyle \ a\ }
2684:
2565:
2533:
2434:
2432:{\displaystyle \ H\ }
2405:
2334:
2332:{\displaystyle \ G\ }
2305:
2132:
2010:
1956:
1954:{\displaystyle \ b~.}
1927:
1898:
1869:
1825:
1329:
1012:
741:
664:
530:
528:{\displaystyle \ p\ }
509:th degree polynomial
467:
408:
324:
235:
158:
4277:DurandâKerner method
4221:NewtonâKrylov method
4031:on 23 December 2012.
3975:. Harper & Row.
3905:
3849:
3813:
3771:
3729:
3624:
3572:
3498:
3472:
3282:
3243:
3239:For small values of
2968:
2936:
2903:
2866:
2729:
2700:
2577:
2545:
2446:
2417:
2346:
2317:
2147:
2025:
1968:
1936:
1907:
1903:away from our guess
1878:
1874:is a short distance,
1845:
1344:
1031:
757:
676:
542:
513:
424:
335:
246:
170:
131:
4226:Steffensen's method
4052:1997SIAMR..39..187P
4006:1994SJSC...15.1059G
3899:Stephensen's method
71:Real-root isolation
4259:Polynomial methods
3920:
3868:
3835:
3799:
3753:
3719:
3708:
3594:
3558:
3484:
3349:
3264:
3223:
2951:
2922:
2889:
2849:
2712:
2679:
2560:
2528:
2429:
2400:
2339:may be written as
2329:
2300:
2127:
2005:
1951:
1922:
1893:
1864:
1833:We then make what
1820:
1818:
1324:
1322:
1007:
736:
659:
525:
505:states that every
462:
403:
319:
230:
153:
21:numerical analysis
4341:
4340:
4297:Laguerre's method
4272:Bairstow's method
4085:978-0-521-88068-8
4074:Numerical Recipes
3916:
3910:
3867:
3854:
3831:
3818:
3795:
3776:
3749:
3734:
3704:
3629:
3590:
3577:
3557:
3544:
3535:
3519:
3503:
3483:
3477:
3345:
3335:
3307:
3287:
3263:
3248:
3219:
3199:
3195:
3194:
3190:
3188:
3161:
3156:
3136:
3121:
3115:
3111:
3109:
3097:
3080:
3076:
3071:
3059:
3049:
3044:
3038:
3032:
3018:
2997:
2982:
2947:
2941:
2921:
2917:
2908:
2888:
2871:
2845:
2829:
2825:
2824:
2800:
2767:
2758:
2711:
2705:
2675:
2671:
2669:
2665:
2664:
2638:
2594:
2556:
2550:
2524:
2520:
2518:
2505:
2500:
2488:
2478:
2476:
2463:
2428:
2422:
2398:
2393:
2381:
2371:
2366:
2360:
2328:
2322:
2299:
2289:
2270:
2264:
2242:
2220:
2193:
2164:
2158:
2152:
2123:
2104:
2098:
2092:
2086:
2067:
2061:
2042:
2036:
2030:
2004:
1985:
1979:
1973:
1947:
1941:
1918:
1912:
1889:
1883:
1863:
1850:
1812:
1802:
1787:
1773:
1756:
1754:
1739:
1734:
1714:
1703:
1697:
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1656:
1651:
1631:
1622:
1616:
1607:
1605:
1573:
1570:
1558:
1552:
1545:
1543:
1511:
1508:
1495:
1493:
1461:
1458:
1449:
1443:
1433:
1418:
1401:
1367:
1361:
1353:
1316:
1312:
1310:
1300:
1285:
1275:
1270:
1250:
1244:
1238:
1229:
1227:
1208:
1196:
1190:
1183:
1181:
1162:
1149:
1147:
1128:
1119:
1113:
1103:
1088:
1071:
1054:
1048:
1040:
1003:
993:
974:
958:
952:
946:
940:
930:
911:
895:
889:
879:
860:
844:
838:
828:
822:
806:
800:
790:
775:
748:natural logarithm
732:
719:
710:
681:
655:
626:
620:
547:
524:
518:
401:
398:
317:
228:
25:Laguerre's method
4361:
4282:Graeffe's method
4211:Broyden's method
4160:Bisection method
4137:
4130:
4123:
4114:
4108:
4089:
4063:
4032:
4017:
4000:(5): 1059â1063.
3986:
3974:
3970:
3968:
3967:
3955:Acton, Forman S.
3929:
3927:
3926:
3921:
3914:
3908:
3877:
3875:
3874:
3869:
3865:
3864:
3863:
3852:
3844:
3842:
3841:
3836:
3829:
3828:
3827:
3816:
3808:
3806:
3805:
3800:
3793:
3792:
3791:
3774:
3762:
3760:
3759:
3754:
3747:
3732:
3724:
3717:
3715:
3714:
3709:
3702:
3686:
3685:
3670:
3669:
3654:
3653:
3627:
3603:
3601:
3600:
3595:
3588:
3587:
3586:
3575:
3567:
3565:
3564:
3559:
3555:
3554:
3553:
3542:
3533:
3529:
3528:
3517:
3513:
3512:
3501:
3493:
3491:
3490:
3485:
3481:
3475:
3464:
3460:
3449:
3444:Graeffe's method
3441:
3430:
3423:
3400:
3396:
3392:
3381:
3358:
3356:
3355:
3350:
3343:
3342:
3341:
3333:
3332:
3331:
3305:
3304:
3303:
3294:
3293:
3285:
3273:
3271:
3270:
3265:
3261:
3246:
3232:
3230:
3229:
3224:
3217:
3216:
3215:
3207:
3206:
3197:
3196:
3192:
3191:
3189:
3186:
3185:
3184:
3169:
3159:
3157:
3154:
3144:
3134:
3119:
3117:
3113:
3112:
3110:
3107:
3095:
3090:
3082:
3078:
3077:
3072:
3069:
3057:
3055:
3050:
3045:
3042:
3036:
3034:
3030:
3029:
3028:
3019:
3017:
3007:
2998:
2995:
2980:
2978:
2960:
2958:
2957:
2952:
2945:
2939:
2931:
2929:
2928:
2923:
2919:
2918:
2910:
2906:
2898:
2896:
2895:
2890:
2886:
2885:
2884:
2883:
2882:
2869:
2858:
2856:
2855:
2850:
2843:
2836:
2835:
2827:
2826:
2822:
2821:
2817:
2816:
2815:
2798:
2789:
2785:
2770:
2768:
2760:
2756:
2755:
2754:
2745:
2744:
2743:
2742:
2721:
2719:
2718:
2713:
2709:
2703:
2688:
2686:
2685:
2680:
2673:
2672:
2670:
2667:
2666:
2662:
2661:
2660:
2654:
2653:
2636:
2632:
2631:
2625:
2624:
2609:
2608:
2602:
2592:
2587:
2569:
2567:
2566:
2561:
2554:
2548:
2537:
2535:
2534:
2529:
2522:
2521:
2519:
2516:
2515:
2514:
2503:
2501:
2498:
2486:
2484:
2479:
2477:
2474:
2473:
2472:
2461:
2456:
2438:
2436:
2435:
2430:
2426:
2420:
2409:
2407:
2406:
2401:
2399:
2394:
2391:
2379:
2377:
2372:
2367:
2364:
2358:
2356:
2338:
2336:
2335:
2330:
2326:
2320:
2309:
2307:
2306:
2301:
2297:
2296:
2295:
2287:
2286:
2285:
2268:
2262:
2258:
2257:
2240:
2236:
2235:
2218:
2217:
2216:
2207:
2206:
2162:
2156:
2150:
2136:
2134:
2133:
2128:
2121:
2120:
2119:
2102:
2096:
2090:
2084:
2083:
2082:
2065:
2059:
2058:
2057:
2040:
2034:
2028:
2014:
2012:
2011:
2006:
2002:
2001:
2000:
1983:
1977:
1971:
1960:
1958:
1957:
1952:
1945:
1939:
1931:
1929:
1928:
1923:
1916:
1910:
1902:
1900:
1899:
1894:
1887:
1881:
1873:
1871:
1870:
1865:
1861:
1860:
1859:
1848:
1829:
1827:
1826:
1821:
1819:
1810:
1809:
1808:
1800:
1785:
1784:
1783:
1771:
1767:
1766:
1761:
1757:
1755:
1752:
1737:
1735:
1732:
1722:
1712:
1710:
1701:
1695:
1694:
1692:
1689:
1688:
1687:
1679:
1664:
1663:
1662:
1654:
1652:
1649:
1639:
1629:
1627:
1620:
1614:
1612:
1608:
1606:
1603:
1602:
1601:
1592:
1591:
1571:
1568:
1563:
1556:
1550:
1546:
1544:
1541:
1540:
1539:
1530:
1529:
1509:
1506:
1501:
1496:
1494:
1491:
1490:
1489:
1480:
1479:
1459:
1456:
1451:
1447:
1441:
1440:
1439:
1431:
1416:
1415:
1414:
1402:
1400:
1399:
1398:
1382:
1381:
1372:
1365:
1359:
1351:
1333:
1331:
1330:
1325:
1323:
1314:
1313:
1311:
1308:
1307:
1306:
1298:
1283:
1282:
1281:
1273:
1271:
1268:
1258:
1248:
1246:
1242:
1236:
1234:
1230:
1228:
1225:
1224:
1223:
1206:
1201:
1194:
1188:
1184:
1182:
1179:
1178:
1177:
1160:
1155:
1150:
1148:
1145:
1144:
1143:
1126:
1121:
1117:
1111:
1110:
1109:
1101:
1086:
1085:
1084:
1072:
1070:
1056:
1052:
1046:
1038:
1016:
1014:
1013:
1008:
1001:
1000:
999:
991:
990:
989:
972:
971:
970:
956:
950:
944:
938:
937:
936:
928:
927:
926:
909:
908:
907:
893:
887:
886:
885:
877:
876:
875:
858:
857:
856:
842:
836:
835:
834:
826:
820:
819:
818:
804:
798:
797:
796:
788:
773:
772:
771:
745:
743:
742:
737:
730:
729:
728:
717:
708:
704:
703:
691:
690:
679:
668:
666:
665:
660:
653:
652:
648:
647:
646:
624:
618:
617:
613:
612:
611:
591:
587:
586:
585:
545:
534:
532:
531:
526:
522:
516:
508:
471:
469:
468:
463:
455:
454:
442:
441:
412:
410:
409:
404:
402:
400:
399:
394:
393:
357:
345:
328:
326:
325:
320:
318:
316:
312:
311:
295:
291:
290:
278:
269:
264:
263:
239:
237:
236:
231:
229:
227:
223:
222:
206:
202:
201:
189:
180:
162:
160:
159:
154:
149:
148:
124:
115:
102:
98:
56:
45:
4369:
4368:
4364:
4363:
4362:
4360:
4359:
4358:
4344:
4343:
4342:
4337:
4328:Muller's method
4306:
4253:
4249:Ridders' method
4230:
4197:
4193:Halley's method
4188:Newton's method
4174:
4146:
4141:
4111:
4105:
4097:. McGraw-Hill.
4092:
4086:
4066:
4035:
4020:
4014:10.1137/0915064
3989:
3983:
3965:
3964:
3962:
3953:
3949:
3903:
3902:
3855:
3847:
3846:
3819:
3811:
3810:
3777:
3769:
3768:
3727:
3726:
3722:
3677:
3661:
3645:
3622:
3621:
3614:
3606:Newton's method
3578:
3570:
3569:
3545:
3520:
3504:
3496:
3495:
3470:
3469:
3462:
3451:
3447:
3432:
3428:
3402:
3398:
3394:
3383:
3372:
3365:
3323:
3280:
3279:
3278:by an error of
3276:Halley's method
3241:
3240:
3200:
3176:
3162:
3158:
3137:
3118:
3094:
3056:
3035:
3000:
2999:
2979:
2966:
2965:
2934:
2933:
2901:
2900:
2874:
2864:
2863:
2807:
2794:
2790:
2775:
2771:
2734:
2727:
2726:
2698:
2697:
2645:
2591:
2575:
2574:
2543:
2542:
2506:
2502:
2485:
2464:
2460:
2444:
2443:
2415:
2414:
2378:
2357:
2344:
2343:
2315:
2314:
2277:
2249:
2227:
2145:
2144:
2111:
2074:
2049:
2023:
2022:
1992:
1966:
1965:
1934:
1933:
1905:
1904:
1876:
1875:
1851:
1843:
1842:
1817:
1816:
1736:
1715:
1711:
1705:
1704:
1653:
1632:
1628:
1610:
1609:
1593:
1583:
1567:
1531:
1521:
1505:
1481:
1471:
1455:
1390:
1383:
1373:
1357:
1342:
1341:
1321:
1320:
1272:
1251:
1247:
1232:
1231:
1215:
1205:
1169:
1159:
1135:
1125:
1060:
1044:
1029:
1028:
981:
918:
867:
755:
754:
720:
695:
682:
674:
673:
638:
631:
627:
603:
596:
592:
577:
570:
566:
540:
539:
511:
510:
506:
499:
446:
427:
422:
421:
385:
349:
333:
332:
303:
296:
282:
271:
270:
255:
244:
243:
214:
207:
193:
182:
181:
168:
167:
140:
129:
128:
119:
114:
108:
100:
89:
86:
78:Edmond Laguerre
47:
36:
17:
12:
11:
5:
4367:
4365:
4357:
4356:
4346:
4345:
4339:
4338:
4336:
4335:
4330:
4325:
4320:
4314:
4312:
4308:
4307:
4305:
4304:
4299:
4294:
4289:
4284:
4279:
4274:
4269:
4263:
4261:
4255:
4254:
4252:
4251:
4246:
4244:Brent's method
4240:
4238:
4236:Hybrid methods
4232:
4231:
4229:
4228:
4223:
4218:
4213:
4207:
4205:
4199:
4198:
4196:
4195:
4190:
4184:
4182:
4176:
4175:
4173:
4172:
4167:
4162:
4156:
4154:
4148:
4147:
4142:
4140:
4139:
4132:
4125:
4117:
4110:
4109:
4103:
4090:
4084:
4064:
4046:(2): 187â220.
4033:
4018:
3987:
3981:
3950:
3948:
3945:
3919:
3913:
3862:
3858:
3834:
3826:
3822:
3798:
3790:
3787:
3784:
3780:
3752:
3746:
3743:
3740:
3737:
3707:
3701:
3698:
3695:
3692:
3689:
3684:
3680:
3676:
3673:
3668:
3664:
3660:
3657:
3652:
3648:
3644:
3641:
3638:
3635:
3632:
3613:
3610:
3604:this leads to
3593:
3585:
3581:
3552:
3548:
3541:
3538:
3532:
3527:
3523:
3516:
3511:
3507:
3480:
3364:
3361:
3348:
3340:
3330:
3326:
3322:
3319:
3316:
3313:
3310:
3302:
3297:
3292:
3260:
3257:
3254:
3251:
3234:
3233:
3222:
3214:
3211:
3205:
3183:
3179:
3175:
3172:
3168:
3165:
3153:
3150:
3147:
3143:
3140:
3133:
3130:
3127:
3124:
3106:
3103:
3100:
3093:
3088:
3085:
3075:
3068:
3065:
3062:
3053:
3048:
3041:
3027:
3022:
3016:
3013:
3010:
3006:
3003:
2994:
2991:
2988:
2985:
2976:
2973:
2950:
2944:
2916:
2913:
2881:
2877:
2860:
2859:
2848:
2842:
2839:
2834:
2820:
2814:
2810:
2806:
2803:
2797:
2793:
2788:
2784:
2781:
2778:
2774:
2766:
2763:
2753:
2748:
2741:
2737:
2708:
2694:complex number
2690:
2689:
2678:
2659:
2652:
2648:
2644:
2641:
2635:
2630:
2623:
2618:
2615:
2612:
2607:
2600:
2597:
2590:
2585:
2582:
2559:
2553:
2539:
2538:
2527:
2513:
2509:
2497:
2494:
2491:
2482:
2471:
2467:
2459:
2454:
2451:
2425:
2411:
2410:
2397:
2390:
2387:
2384:
2375:
2370:
2363:
2354:
2351:
2325:
2311:
2310:
2294:
2284:
2280:
2276:
2273:
2267:
2261:
2256:
2252:
2248:
2245:
2239:
2234:
2230:
2226:
2223:
2215:
2210:
2205:
2202:
2199:
2196:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2161:
2155:
2138:
2137:
2126:
2118:
2114:
2110:
2107:
2101:
2095:
2089:
2081:
2077:
2073:
2070:
2064:
2056:
2052:
2048:
2045:
2039:
2033:
2016:
2015:
1999:
1995:
1991:
1988:
1982:
1976:
1950:
1944:
1921:
1915:
1892:
1886:
1858:
1854:
1831:
1830:
1815:
1807:
1799:
1796:
1793:
1790:
1782:
1776:
1770:
1765:
1760:
1751:
1748:
1745:
1742:
1731:
1728:
1725:
1721:
1718:
1708:
1700:
1686:
1678:
1675:
1672:
1669:
1661:
1648:
1645:
1642:
1638:
1635:
1625:
1619:
1613:
1611:
1600:
1596:
1590:
1586:
1582:
1579:
1576:
1566:
1561:
1555:
1549:
1538:
1534:
1528:
1524:
1520:
1517:
1514:
1504:
1499:
1488:
1484:
1478:
1474:
1470:
1467:
1464:
1454:
1446:
1438:
1430:
1427:
1424:
1421:
1413:
1408:
1405:
1397:
1393:
1389:
1386:
1380:
1376:
1370:
1364:
1358:
1356:
1350:
1349:
1335:
1334:
1319:
1305:
1297:
1294:
1291:
1288:
1280:
1267:
1264:
1261:
1257:
1254:
1241:
1235:
1233:
1222:
1218:
1214:
1211:
1204:
1199:
1193:
1187:
1176:
1172:
1168:
1165:
1158:
1153:
1142:
1138:
1134:
1131:
1124:
1116:
1108:
1100:
1097:
1094:
1091:
1083:
1078:
1075:
1069:
1066:
1063:
1059:
1051:
1045:
1043:
1037:
1036:
1018:
1017:
1006:
998:
988:
984:
980:
977:
969:
964:
961:
955:
949:
943:
935:
925:
921:
917:
914:
906:
901:
898:
892:
884:
874:
870:
866:
863:
855:
850:
847:
841:
833:
825:
817:
812:
809:
803:
795:
787:
784:
781:
778:
770:
765:
762:
735:
727:
723:
716:
713:
707:
702:
698:
694:
689:
685:
670:
669:
658:
651:
645:
641:
637:
634:
630:
623:
616:
610:
606:
602:
599:
595:
590:
584:
580:
576:
573:
569:
565:
562:
559:
556:
553:
550:
521:
498:
495:
482:
481:
474:
473:
472:
461:
458:
453:
449:
445:
440:
437:
434:
430:
418:
397:
392:
388:
384:
381:
378:
375:
372:
369:
366:
363:
360:
355:
352:
348:
343:
340:
329:
315:
310:
306:
302:
299:
294:
289:
285:
281:
277:
274:
267:
262:
258:
254:
251:
240:
226:
221:
217:
213:
210:
205:
200:
196:
192:
188:
185:
178:
175:
164:
152:
147:
143:
139:
136:
123:= 0, 1, 2, ...
116:
112:
85:
82:
15:
13:
10:
9:
6:
4:
3:
2:
4366:
4355:
4352:
4351:
4349:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4315:
4313:
4311:Other methods
4309:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4267:Aberth method
4265:
4264:
4262:
4260:
4256:
4250:
4247:
4245:
4242:
4241:
4239:
4237:
4233:
4227:
4224:
4222:
4219:
4217:
4216:Secant method
4214:
4212:
4209:
4208:
4206:
4204:
4200:
4194:
4191:
4189:
4186:
4185:
4183:
4181:
4177:
4171:
4168:
4166:
4163:
4161:
4158:
4157:
4155:
4153:
4149:
4145:
4138:
4133:
4131:
4126:
4124:
4119:
4118:
4115:
4106:
4104:0-07-051158-6
4100:
4096:
4091:
4087:
4081:
4077:
4075:
4070:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4040:
4034:
4030:
4026:
4025:
4019:
4015:
4011:
4007:
4003:
3999:
3995:
3994:
3988:
3984:
3982:0-88385-450-3
3978:
3973:
3972:
3969:
3956:
3952:
3951:
3946:
3944:
3940:
3938:
3932:
3917:
3911:
3900:
3896:
3892:
3888:
3883:
3881:
3880:multiple root
3860:
3856:
3832:
3824:
3820:
3796:
3785:
3778:
3766:
3750:
3741:
3735:
3705:
3699:
3696:
3693:
3690:
3687:
3682:
3678:
3674:
3671:
3666:
3662:
3658:
3655:
3650:
3646:
3642:
3636:
3630:
3618:
3611:
3609:
3607:
3591:
3583:
3579:
3550:
3546:
3539:
3536:
3530:
3525:
3521:
3514:
3509:
3505:
3478:
3466:
3458:
3454:
3445:
3439:
3435:
3425:
3421:
3417:
3413:
3409:
3405:
3390:
3386:
3379:
3375:
3370:
3362:
3360:
3346:
3328:
3317:
3311:
3295:
3277:
3255:
3249:
3237:
3220:
3212:
3209:
3181:
3173:
3166:
3163:
3148:
3141:
3138:
3128:
3122:
3104:
3101:
3098:
3091:
3086:
3083:
3073:
3066:
3063:
3060:
3051:
3046:
3039:
3020:
3011:
3004:
3001:
2989:
2983:
2974:
2971:
2964:
2963:
2962:
2948:
2942:
2911:
2846:
2840:
2837:
2818:
2812:
2808:
2804:
2801:
2795:
2791:
2786:
2782:
2779:
2776:
2772:
2761:
2746:
2725:
2724:
2723:
2706:
2695:
2676:
2650:
2646:
2642:
2639:
2633:
2616:
2613:
2610:
2598:
2595:
2588:
2583:
2580:
2573:
2572:
2571:
2570:we find that
2557:
2551:
2525:
2511:
2507:
2495:
2492:
2489:
2480:
2469:
2465:
2457:
2452:
2449:
2442:
2441:
2440:
2423:
2395:
2388:
2385:
2382:
2373:
2368:
2361:
2352:
2349:
2342:
2341:
2340:
2323:
2282:
2278:
2274:
2271:
2265:
2259:
2254:
2250:
2246:
2243:
2237:
2232:
2228:
2224:
2221:
2208:
2159:
2153:
2143:
2142:
2141:
2124:
2116:
2112:
2108:
2105:
2099:
2093:
2087:
2079:
2075:
2071:
2068:
2062:
2054:
2050:
2046:
2043:
2037:
2031:
2021:
2020:
2019:
1997:
1993:
1989:
1986:
1980:
1974:
1964:
1963:
1962:
1948:
1942:
1919:
1913:
1890:
1884:
1856:
1852:
1840:
1836:
1813:
1794:
1788:
1774:
1768:
1763:
1758:
1746:
1740:
1726:
1719:
1716:
1706:
1698:
1673:
1667:
1643:
1636:
1633:
1623:
1617:
1598:
1588:
1584:
1580:
1577:
1564:
1559:
1553:
1547:
1536:
1526:
1522:
1518:
1515:
1502:
1497:
1486:
1476:
1472:
1468:
1465:
1452:
1444:
1425:
1419:
1406:
1403:
1395:
1391:
1387:
1378:
1368:
1362:
1354:
1340:
1339:
1338:
1317:
1292:
1286:
1262:
1255:
1252:
1239:
1220:
1216:
1212:
1209:
1202:
1197:
1191:
1185:
1174:
1170:
1166:
1163:
1156:
1151:
1140:
1136:
1132:
1129:
1122:
1114:
1095:
1089:
1076:
1073:
1067:
1064:
1049:
1041:
1027:
1026:
1025:
1023:
1004:
986:
982:
978:
975:
962:
959:
953:
947:
941:
923:
919:
915:
912:
899:
896:
890:
872:
868:
864:
861:
848:
845:
839:
823:
810:
807:
801:
782:
776:
763:
760:
753:
752:
751:
749:
733:
725:
721:
714:
711:
705:
700:
696:
692:
687:
683:
656:
649:
643:
639:
635:
632:
628:
621:
614:
608:
604:
600:
597:
593:
588:
582:
578:
574:
571:
567:
563:
560:
554:
548:
538:
537:
536:
519:
504:
496:
494:
491:
487:
479:
476:Repeat until
475:
459:
456:
451:
447:
443:
438:
435:
432:
428:
419:
416:
390:
386:
382:
379:
376:
367:
364:
361:
353:
350:
346:
341:
338:
330:
308:
304:
297:
287:
283:
275:
272:
265:
260:
256:
252:
249:
241:
219:
215:
208:
198:
194:
186:
183:
176:
173:
165:
145:
141:
134:
126:
125:
122:
117:
111:
106:
105:
104:
96:
92:
83:
81:
79:
74:
72:
68:
64:
60:
54:
50:
43:
39:
34:
30:
26:
22:
4296:
4203:Quasi-Newton
4165:Regula falsi
4094:
4072:
4043:
4037:
4029:the original
4023:
3997:
3991:
3963:
3960:
3941:
3933:
3890:
3886:
3884:
3720:
3467:
3456:
3452:
3437:
3433:
3426:
3419:
3415:
3411:
3407:
3403:
3388:
3384:
3377:
3373:
3368:
3367:Even if the
3366:
3238:
3235:
2861:
2691:
2540:
2412:
2312:
2140:or exactly,
2139:
2017:
1838:
1835:Acton (1970)
1832:
1336:
1019:
671:
500:
489:
485:
483:
477:
120:
109:
94:
90:
87:
75:
58:
52:
48:
41:
37:
31:tailored to
24:
18:
4180:Householder
4039:SIAM Review
1020:Denote the
33:polynomials
4170:ITP method
3971:Work
3947:References
3612:Properties
497:Derivation
331:Calculate
242:Calculate
166:Calculate
99:of degree
84:Definition
3765:cubically
3537:…
3296:
3210:−
3102:−
3087:−
3064:−
3021:⋅
2915:¯
2805:−
2780:−
2765:¯
2747:
2643:−
2614:−
2599:±
2493:−
2386:−
2275:−
2266:…
2247:−
2225:−
2209:
2160:≡
2109:−
2100:≈
2094:…
2088:≈
2072:−
2063:≈
2047:−
2038:≈
1990:−
1981:≡
1769:⋅
1624:−
1581:−
1554:⋯
1519:−
1469:−
1407:
1388:
1369:−
1213:−
1192:⋯
1167:−
1133:−
1077:
1065:
979:−
963:
948:⋯
916:−
900:
865:−
849:
811:
764:
712:…
636:−
622:⋯
601:−
575:−
457:−
383:−
365:−
354:±
266:−
4348:Category
3957:(1970).
3363:Fallback
3167:′
3142:″
3005:′
2439:becomes
1837:calls a
1720:′
1637:″
1256:′
672:so that
276:″
187:′
63:computer
4048:Bibcode
4002:Bibcode
3966:USUALLY
3382:, then
4101:
4082:
3979:
3915:
3909:
3866:
3853:
3830:
3817:
3794:
3775:
3748:
3733:
3703:
3628:
3589:
3576:
3556:
3543:
3534:
3518:
3502:
3482:
3476:
3344:
3334:
3306:
3286:
3262:
3247:
3218:
3198:
3193:
3187:
3160:
3155:
3135:
3120:
3114:
3108:
3096:
3079:
3070:
3058:
3043:
3037:
3031:
2996:
2981:
2946:
2940:
2920:
2907:
2887:
2870:
2862:where
2844:
2828:
2823:
2799:
2757:
2710:
2704:
2674:
2668:
2663:
2637:
2593:
2555:
2549:
2523:
2517:
2504:
2499:
2487:
2475:
2462:
2427:
2421:
2392:
2380:
2365:
2359:
2327:
2321:
2298:
2288:
2269:
2263:
2241:
2219:
2192:
2163:
2157:
2151:
2122:
2103:
2097:
2091:
2085:
2066:
2060:
2041:
2035:
2029:
2003:
1984:
1978:
1972:
1946:
1940:
1917:
1911:
1888:
1882:
1862:
1849:
1811:
1801:
1786:
1772:
1753:
1738:
1733:
1713:
1702:
1696:
1690:
1680:
1665:
1655:
1650:
1630:
1621:
1615:
1604:
1572:
1569:
1557:
1551:
1542:
1510:
1507:
1492:
1460:
1457:
1448:
1442:
1432:
1417:
1366:
1360:
1352:
1315:
1309:
1299:
1284:
1274:
1269:
1249:
1243:
1237:
1226:
1207:
1195:
1189:
1180:
1161:
1146:
1127:
1118:
1112:
1102:
1087:
1053:
1047:
1039:
1002:
992:
973:
957:
951:
945:
939:
929:
910:
894:
888:
878:
859:
843:
837:
827:
821:
805:
799:
789:
774:
731:
718:
709:
680:
654:
625:
619:
546:
523:
517:
3878:is a
3431:from
44:) = 0
27:is a
4099:ISBN
4080:ISBN
3977:ISBN
3897:and
3887:some
3410:) =
2838:>
2018:and
501:The
420:Set
118:For
103:is:
59:some
4056:doi
4010:doi
3721:If
2961:or
1775:sgn
1024:by
127:If
73:).
19:In
4350::
4071:.
4054:.
4044:39
4042:.
4008:.
3998:15
3996:.
3608:.
3465:.
3418:â
1404:ln
1074:ln
960:ln
897:ln
846:ln
808:ln
761:ln
80:.
23:,
4136:e
4129:t
4122:v
4107:.
4088:.
4062:.
4058::
4050::
4016:.
4012::
4004::
3918:,
3912:a
3861:1
3857:x
3833:.
3825:1
3821:x
3797:,
3789:)
3786:0
3783:(
3779:x
3751:,
3745:)
3742:x
3739:(
3736:p
3723:x
3706:.
3700:1
3697:+
3694:x
3691:4
3688:+
3683:2
3679:x
3675:3
3672:+
3667:3
3663:x
3659:2
3656:+
3651:4
3647:x
3643:=
3640:)
3637:x
3634:(
3631:p
3592:,
3584:1
3580:x
3551:n
3547:x
3540:,
3531:,
3526:3
3522:x
3515:,
3510:2
3506:x
3479:G
3463:r
3459:)
3457:x
3455:(
3453:p
3448:r
3440:)
3438:x
3436:(
3434:q
3429:r
3422:)
3420:w
3416:x
3414:(
3412:p
3408:x
3406:(
3404:q
3399:w
3395:r
3391:)
3389:x
3387:(
3385:p
3380:)
3378:x
3376:(
3374:p
3347:,
3339:}
3329:3
3325:)
3321:)
3318:x
3315:(
3312:p
3309:(
3301:{
3291:O
3259:)
3256:x
3253:(
3250:p
3221:,
3213:1
3204:}
3182:2
3178:)
3174:x
3171:(
3164:p
3152:)
3149:x
3146:(
3139:p
3132:)
3129:x
3126:(
3123:p
3105:1
3099:n
3092:n
3084:1
3074:n
3067:1
3061:n
3052:+
3047:n
3040:1
3026:{
3015:)
3012:x
3009:(
3002:p
2993:)
2990:x
2987:(
2984:p
2975:=
2972:a
2949:;
2943:G
2912:G
2880:e
2876:R
2847:,
2841:0
2833:}
2819:)
2813:2
2809:G
2802:H
2796:n
2792:(
2787:)
2783:1
2777:n
2773:(
2762:G
2752:{
2740:e
2736:R
2707:a
2677:,
2658:)
2651:2
2647:G
2640:H
2634:n
2629:(
2622:)
2617:1
2611:n
2606:(
2596:G
2589:n
2584:=
2581:a
2558:,
2552:a
2526:.
2512:2
2508:b
2496:1
2490:n
2481:+
2470:2
2466:a
2458:1
2453:=
2450:H
2424:H
2396:b
2389:1
2383:n
2374:+
2369:a
2362:1
2353:=
2350:G
2324:G
2293:}
2283:n
2279:x
2272:x
2260:,
2255:3
2251:x
2244:x
2238:,
2233:2
2229:x
2222:x
2214:{
2204:n
2201:a
2198:e
2195:m
2189:c
2186:i
2183:n
2180:o
2177:m
2174:r
2171:a
2168:h
2154:b
2125:,
2117:n
2113:x
2106:x
2080:3
2076:x
2069:x
2055:2
2051:x
2044:x
2032:b
1998:1
1994:x
1987:x
1975:a
1949:.
1943:b
1920:,
1914:x
1891:,
1885:a
1857:1
1853:x
1814:.
1806:)
1798:)
1795:x
1792:(
1789:p
1781:(
1764:2
1759:)
1750:)
1747:x
1744:(
1741:p
1730:)
1727:x
1724:(
1717:p
1707:(
1699:+
1685:|
1677:)
1674:x
1671:(
1668:p
1660:|
1647:)
1644:x
1641:(
1634:p
1618:=
1599:2
1595:)
1589:n
1585:x
1578:x
1575:(
1565:1
1560:+
1548:+
1537:2
1533:)
1527:2
1523:x
1516:x
1513:(
1503:1
1498:+
1487:2
1483:)
1477:1
1473:x
1466:x
1463:(
1453:1
1445:=
1437:|
1429:)
1426:x
1423:(
1420:p
1412:|
1396:2
1392:x
1385:d
1379:2
1375:d
1363:=
1355:H
1318:,
1304:|
1296:)
1293:x
1290:(
1287:p
1279:|
1266:)
1263:x
1260:(
1253:p
1240:=
1221:n
1217:x
1210:x
1203:1
1198:+
1186:+
1175:2
1171:x
1164:x
1157:1
1152:+
1141:1
1137:x
1130:x
1123:1
1115:=
1107:|
1099:)
1096:x
1093:(
1090:p
1082:|
1068:x
1062:d
1058:d
1050:=
1042:G
1005:.
997:|
987:n
983:x
976:x
968:|
954:+
942:+
934:|
924:2
920:x
913:x
905:|
891:+
883:|
873:1
869:x
862:x
854:|
840:+
832:|
824:C
816:|
802:=
794:|
786:)
783:x
780:(
777:p
769:|
734:,
726:n
722:x
715:,
706:,
701:2
697:x
693:,
688:1
684:x
657:,
650:)
644:n
640:x
633:x
629:(
615:)
609:2
605:x
598:x
594:(
589:)
583:1
579:x
572:x
568:(
564:C
561:=
558:)
555:x
552:(
549:p
520:p
507:n
490:p
486:p
478:a
460:a
452:k
448:x
444:=
439:1
436:+
433:k
429:x
417:.
396:)
391:2
387:G
380:H
377:n
374:(
371:)
368:1
362:n
359:(
351:G
347:n
342:=
339:a
314:)
309:k
305:x
301:(
298:p
293:)
288:k
284:x
280:(
273:p
261:2
257:G
253:=
250:H
225:)
220:k
216:x
212:(
209:p
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199:k
195:x
191:(
184:p
177:=
174:G
151:)
146:k
142:x
138:(
135:p
121:k
113:0
110:x
101:n
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95:x
93:(
91:p
55:)
53:x
51:(
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