42:
2220:
77:. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the
2811:
2591:
4133:
Vrahatis, M.N.; Perdiou, A.E.; Kalantonis, V.S.; Perdios, E.A.; Papadakis, K.; Prosmiti, R.; Farantos, S.C. (July 2001). "Application of the
Characteristic Bisection Method for locating and computing periodic orbits in molecular systems".
3257:
of a characteristic polygon is a edge between a pair of vertices, such that the sign vector differs by only a single sign. In the above example, the proper edges of the characteristic quadrilateral are AB, AC, BD and CD. A
245:) have opposite signs and bracket a root. The method selects the subinterval that is guaranteed to be a bracket as the new interval to be used in the next step. In this way an interval that contains a zero of
3390:
3330:
2480:
2115:
2430:
1355:
2120:
This formula can be used to determine, in advance, an upper bound on the number of iterations that the bisection method needs to converge to a root to within a certain tolerance. The number
3231:
3158:
3085:
3012:
2130:
2696:
2308:
475:
When implementing the method on a computer, there can be problems with finite precision, so there are often additional convergence tests or limits to the number of iterations. Although
1270:
3450:
1141:
2862:
1056:
2704:
2272:
825:
3456:. If the topological degree of the initial polyhedron is not zero, then there is a procedure that can choose an edge such that the next polyhedron also has nonzero degree.
4314:
2336:
2488:
1669:
1391:
2886:
2369:
to the root. And, a strict improvement to the bisection method can be achieved with a higher order of convergence without trading-off worst case performance with the
1213:
1180:
2627:
1632:
1604:
1576:
1443:
1501:
1472:
926:
897:
1417:
978:
952:
1541:
1521:
868:
848:
2345:
However, despite the bisection method being optimal with respect to worst case performance under absolute error criteria it is sub-optimal with respect to
2310:
The main motivation to use the bisection method is that over the set of continuous functions, no other method can guarantee to produce an estimate c
3507:
2342:
iterations. This is also true under several common assumptions on function f and the behaviour of the function in the neighbourhood of the root.
4283:
4307:
3953:
Polymilis, C.; Servizi, G.; Turchetti, G.; Skokos, Ch.; Vrahatis, M. N. (May 2003). "LOCATING PERIODIC ORBITS BY TOPOLOGICAL DEGREE THEORY".
3929:
4244:
356:). The function values are of opposite sign (there is at least one zero crossing within the interval). Each iteration performs these steps:
4512:
4438:
3342:
3282:
4199:
2435:
4161:
Vrahatis, Michael N. (December 1988). "Solving systems of nonlinear equations using the nonzero value of the topological degree".
4533:
4300:
35:
4502:
3567:
If the function has the same sign at the endpoints of an interval, the endpoints may or may not bracket roots of the function.
3591:, p. 29. This version recomputes the function values at each iteration rather than carrying them to the next iterations.
2937:
73:
defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a
2036:
4466:
2400:
1278:
520:
decreases, at some point the midpoint of will be numerically identical to (within floating point precision of) either
110:). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see
2215:{\displaystyle n\leq n_{1/2}\equiv \left\lceil \log _{2}\left({\frac {\epsilon _{0}}{\epsilon }}\right)\right\rceil ,}
4471:
3470:
2124:
of iterations needed to achieve a required tolerance ε (that is, an error guaranteed to be at most ε), is bounded by
99:
45:
A few steps of the bisection method applied over the starting range . The bigger red dot is the root of the function.
3910:"Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions"
3183:
3110:
3037:
2964:
213:
itself is a root then the process has succeeded and stops. Otherwise, there are now only two possibilities: either
171:
4287:
2636:
2277:
3916:. Lecture Notes in Computer Science. Vol. 11974. Cham: Springer International Publishing. pp. 223–238.
4456:
4400:
1931:
After 13 iterations, it becomes apparent that there is a convergence to about 1.521: a root for the polynomial.
1221:
3408:
1543:
will become increasingly smaller, converging on the root of the function. See this happen in the table below.
1067:
3269:
At each iteration, the algorithm picks a proper edge of the polyhedron (say, A—B), and computes the signs of
4481:
4415:
4359:
4331:
4323:
3465:
2806:{\displaystyle \operatorname {sgn}(x)={\begin{cases}1,&x>0\\0,&x=0\\-1,&x<0\\\end{cases}}}
249:
is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.
83:
31:
2823:
986:
4405:
70:
58:
2365:(amongst others), typically perform better since they trade-off worst case performance to achieve higher
2228:
4497:
3511:
30:
This article is about searching zeros of continuous functions. For searching a finite sorted array, see
772:
4034:
4476:
4451:
4046:
479:
is continuous, finite precision may preclude a function value ever being zero. For example, consider
3401:
Suppose the diameter (= length of longest proper edge) of the original characteristic polyhedron is
2731:
4461:
4390:
4382:
2366:
1972:
1968:
1948:
143:
111:
62:
2381:
The bisection method has been generalized to multi-dimensional functions. Such methods are called
4250:
4112:
4015:
3958:
3935:
3909:
3849:
3818:
3727:
3645:
2586:{\displaystyle \deg(f,\Omega ):=\sum _{y\in f^{-1}(\mathbf {0} )}\operatorname {sgn} \det(Df(y))}
2394:
74:
4507:
4428:
4372:
4367:
3544:
2358:
107:
103:
3869:"On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree"
3452:
bisections of edges are required so that the diameter of the remaining polygon will be at most
2321:
4423:
4266:
4232:
4195:
4104:
4062:
4007:
3925:
3890:
3810:
3768:
3719:
3680:
3637:
2362:
1638:
1360:
2871:
1185:
1152:
4224:
4170:
4143:
4096:
4054:
3999:
3968:
3917:
3880:
3841:
3802:
3758:
3711:
3672:
3629:
3476:
2865:
2600:
1610:
1582:
1554:
1422:
189:
At each step the method divides the interval in two parts/halves by computing the midpoint
2630:
1477:
1448:
902:
873:
3791:"An Enhancement of the Bisection Method Average Performance Preserving Minmax Optimality"
1396:
957:
931:
4050:
2944:
is not zero (a necessary criterion to ensure the existence of a root). For example, for
1526:
1506:
853:
833:
4147:
4035:"An Efficient Method for Locating and Computing Periodic Orbits of Nonlinear Mappings"
65:
for which one knows two values with opposite signs. The method consists of repeatedly
34:. For the method of determining what software change caused a change in behavior, see
4527:
4446:
4395:
4116:
4085:"A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations"
4019:
3939:
3853:
3822:
3763:
3746:
3731:
3676:
3649:
2953:
2817:
2354:
4188:
1149:
In the first iteration, the end points of the interval which brackets the root are
4344:
98:, more elaborate methods exist for testing the existence of a root in an interval (
17:
3921:
131:
50:
4269:
3972:
1146:
Because the function is continuous, there must be a root within the interval .
41:
4349:
4186:
Burden, Richard L.; Faires, J. Douglas (1985), "2.1 The
Bisection Algorithm",
3988:"An efficient degree-computation method for a generalized method of bisection"
2370:
537:
95:
4236:
4108:
4066:
4011:
3894:
3814:
3772:
3723:
3684:
3641:
766:
Suppose that the bisection method is used to find a root of the polynomial
328:) have opposite signs, so the method is applicable to this smaller interval.
4274:
512:
is limited by the floating point precision; i.e., as the difference between
66:
4058:
3885:
3868:
4292:
3394:, then B is replaced by M, and we get a smaller characteristic polyhedron.
3334:, then A is replaced by M, and we get a smaller characteristic polyhedron.
4215:
Corliss, George (1977), "Which root does the bisection algorithm find?",
4174:
3837:
2921:
4100:
4003:
3715:
3633:
4084:
3987:
3963:
3699:
3698:
Graf, Siegfried; Novak, Erich; Papageorgiou, Anargyros (1989-07-01).
3663:
Sikorski, K (1985-12-01). "Optimal solution of nonlinear equations".
3617:
4228:
3845:
3806:
3790:
264:
may be taken as the solution and the process stops. Otherwise, if
3262:
is a pair of vertices, such that the sign vector differs by all
4296:
3867:
Mourrain, B.; Vrahatis, M. N.; Yakoubsohn, J. C. (2002-06-01).
3385:{\displaystyle \operatorname {sgn} f(M)=\operatorname {sgn}(B)}
3325:{\displaystyle \operatorname {sgn} f(M)=\operatorname {sgn}(A)}
2475:{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}
1503:
have opposite signs. As this continues, the interval between
504:
that gives exactly zero. Additionally, the difference between
122:
The method is applicable for numerically solving the equation
3473:, generalization of the bisection method in the complex plane
2353:. Popular alternatives to the bisection method, such as the
471:)) so that there is a zero crossing within the new interval.
2900:
uses only the signs of a function in different points. Lef
2799:
3266:
signs. In the above example, the diagonals are AD and BC.
3273:
in its mid-point (say, M). Then it proceeds as follows:
3912:. In Sergeyev, Yaroslav D.; Kvasov, Dmitri E. (eds.).
3789:
Oliveira, I. F. D.; Takahashi, R. H. C. (2020-12-06).
2820:. In order for a root to exist, it is sufficient that
4286:
Notes, PPT, Mathcad, Maple, Matlab, Mathematica from
3411:
3345:
3285:
3186:
3113:
3040:
2967:
2874:
2826:
2707:
2639:
2603:
2491:
2438:
2403:
2324:
2280:
2231:
2133:
2110:{\displaystyle |c_{n}-c|\leq {\frac {|b-a|}{2^{n}}}.}
2039:
1641:
1613:
1585:
1557:
1529:
1509:
1480:
1451:
1425:
1399:
1363:
1281:
1224:
1188:
1155:
1070:
989:
960:
934:
905:
876:
856:
836:
775:
201:) / 2 of the interval and the value of the function
4490:
4437:
4414:
4381:
4358:
4330:
4187:
3444:
3384:
3324:
3225:
3152:
3079:
3006:
2924:in R, having 2 vertices, such that in each vertex
2880:
2856:
2805:
2690:
2621:
2585:
2474:
2424:
2330:
2302:
2266:
2214:
2109:
1939:The method is guaranteed to converge to a root of
1663:
1626:
1598:
1570:
1535:
1515:
1495:
1466:
1437:
1411:
1385:
1349:
1264:
1207:
1174:
1135:
1050:
972:
946:
920:
891:
862:
842:
819:
336:The input for the method is a continuous function
4083:Vrahatis, M. N.; Iordanidis, K. I. (1986-03-01).
2425:{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}
2393:Some of these methods are based on computing the
1350:{\displaystyle f(c_{1})=(1.5)^{3}-(1.5)-2=-0.125}
493:; there is no floating-point value approximating
3545:"Dichotomy method - Encyclopedia of Mathematics"
2559:
3397:Else, we pick a new proper edge and try again.
3226:{\displaystyle \operatorname {sgn} f(D)=(+,+)}
3153:{\displaystyle \operatorname {sgn} f(C)=(+,-)}
3080:{\displaystyle \operatorname {sgn} f(B)=(-,+)}
3007:{\displaystyle \operatorname {sgn} f(A)=(-,-)}
393:Calculate the function value at the midpoint,
4308:
3914:Numerical Computations: Theory and Algorithms
2004:is the midpoint of the initial interval, and
928:have opposite signs. For the above function,
229:) have opposite signs and bracket a root, or
178:must have at least one root in the interval (
8:
3600:
3588:
3576:
3531:
3494:
2904:be a function from R to R, for some integer
2691:{\displaystyle \mathbf {0} =(0,0,...,0)^{T}}
2303:{\displaystyle \epsilon \leq \epsilon _{0}.}
642:// limit iterations to prevent infinite loop
280:) have opposite signs, then the method sets
304:) have opposite signs then the method sets
4315:
4301:
4293:
1265:{\displaystyle c_{1}={\frac {2+1}{2}}=1.5}
27:Algorithm for finding a zero of a function
4163:ACM Transactions on Mathematical Software
3962:
3884:
3795:ACM Transactions on Mathematical Software
3762:
3700:"Bisection is not optimal on the average"
3445:{\displaystyle \log _{2}(D/\varepsilon )}
3431:
3416:
3410:
3344:
3284:
3185:
3112:
3039:
2966:
2873:
2825:
2726:
2706:
2682:
2640:
2638:
2602:
2543:
2531:
2520:
2490:
2466:
2462:
2461:
2451:
2447:
2446:
2437:
2416:
2412:
2411:
2402:
2323:
2291:
2279:
2259:
2245:
2236:
2230:
2189:
2183:
2170:
2148:
2144:
2132:
2096:
2086:
2072:
2069:
2061:
2049:
2040:
2038:
1652:
1640:
1618:
1612:
1590:
1584:
1562:
1556:
1528:
1508:
1479:
1450:
1424:
1398:
1374:
1362:
1314:
1292:
1280:
1238:
1229:
1223:
1193:
1187:
1160:
1154:
1129:
1096:
1069:
1015:
988:
959:
933:
904:
875:
855:
835:
813:
795:
774:
762:Example: Finding the root of a polynomial
404:If convergence is satisfactory (that is,
170:are said to bracket a root since, by the
2956:with vertices (say) A,B,C,D, such that:
1545:
1136:{\displaystyle f(2)=(2)^{3}-(2)-2=+4\,.}
340:, an interval , and the function values
40:
3955:Libration Point Orbits and Applications
3747:"Average-case results for zero finding"
3487:
2349:under standard assumptions as well as
2013:is the midpoint of the interval in the
1445:for the next iteration to ensure that
1275:The function value at the midpoint is
4128:
4126:
4078:
4076:
3784:
3782:
2857:{\displaystyle \deg(f,\Omega )\neq 0}
2017:th step, then the difference between
1971:is halved at each step so the method
1051:{\displaystyle f(1)=(1)^{3}-(1)-2=-2}
7:
4288:Holistic Numerical Methods Institute
3611:
3609:
2482:is defined as a sum over its roots:
162:) have opposite signs. In this case
4246:Numerical Methods with Applications
4033:Vrahatis, Michael N. (1995-06-01).
2948:=2, a characteristic polyhedron of
2864:, and this can be verified using a
2389:Methods based on degree computation
2377:Generalization to higher dimensions
2267:{\displaystyle \epsilon _{0}=|b-a|}
612:value which differs from a root of
2875:
2842:
2507:
2404:
420:)| is sufficiently small), return
146:defined on an interval and where
25:
820:{\displaystyle f(x)=x^{3}-x-2\,.}
4513:Sidi's generalized secant method
4194:(3rd ed.), PWS Publishers,
4039:Journal of Computational Physics
2641:
2544:
2338:absolute error with less than n
364:, the midpoint of the interval,
36:Bisection (software engineering)
4503:Inverse quadratic interpolation
4243:Kaw, Autar; Kalu, Egwu (2008),
4136:Computer Physics Communications
2898:characteristic bisection method
2892:Characteristic bisection method
2225:where the initial bracket size
757:// max number of steps exceeded
4249:(1st ed.), archived from
3986:Kearfott, Baker (1979-06-01).
3838:"An Improved Bisection Method"
3508:"Interval Halving (Bisection)"
3439:
3425:
3379:
3373:
3361:
3355:
3319:
3313:
3301:
3295:
3220:
3208:
3202:
3196:
3147:
3135:
3129:
3123:
3074:
3062:
3056:
3050:
3001:
2989:
2983:
2977:
2928:, the combination of signs of
2845:
2833:
2720:
2714:
2679:
2648:
2616:
2610:
2580:
2577:
2571:
2562:
2548:
2540:
2510:
2498:
2457:
2432:and a differentiable function
2314:to the solution c that in the
2274:and the required bracket size
2260:
2246:
2087:
2073:
2062:
2041:
1658:
1645:
1490:
1484:
1461:
1455:
1380:
1367:
1329:
1323:
1311:
1304:
1298:
1285:
1111:
1105:
1093:
1086:
1080:
1074:
1030:
1024:
1012:
1005:
999:
993:
915:
909:
886:
880:
785:
779:
1:
4148:10.1016/S0010-4655(01)00190-4
3908:Vrahatis, Michael N. (2020).
2397:, which for a bounded region
2383:generalized bisection methods
562:, maximum iterations
536:The method may be written in
3922:10.1007/978-3-030-40616-5_17
3836:Ivo, Oliveira (2020-12-14).
3764:10.1016/0885-064X(89)90022-8
3677:10.1016/0885-064X(85)90011-1
1967:) have opposite signs. The
980:satisfy this criterion, as
3745:Novak, Erich (1989-12-01).
3616:Sikorski, K. (1982-02-01).
870:have to be found such that
412:is sufficiently small, or |
4550:
4332:Bracketing (no derivative)
3973:10.1142/9789812704849_0031
3549:www.encyclopediaofmath.org
550:, endpoint values
174:, the continuous function
172:intermediate value theorem
29:
3603:, p. 31, Theorem 2.1
2910:characteristic polyhedron
2331:{\displaystyle \epsilon }
755:Output("Method failed.")
707:// increment step counter
312:. In both cases, the new
3601:Burden & Faires 1985
3589:Burden & Faires 1985
3579:, p. 28 for section
3577:Burden & Faires 1985
3532:Burden & Faires 1985
3495:Burden & Faires 1985
1664:{\displaystyle f(c_{n})}
1386:{\displaystyle f(c_{1})}
130:) = 0 for the
100:Descartes' rule of signs
4534:Root-finding algorithms
4482:Splitting circle method
4467:Jenkins–Traub algorithm
4324:Root-finding algorithms
3466:Binary search algorithm
2881:{\displaystyle \Omega }
1208:{\displaystyle b_{1}=2}
1175:{\displaystyle a_{1}=1}
32:binary search algorithm
4472:Lehmer–Schur algorithm
4059:10.1006/jcph.1995.1119
3886:10.1006/jcom.2001.0636
3618:"Bisection is optimal"
3471:Lehmer–Schur algorithm
3446:
3386:
3326:
3227:
3154:
3081:
3008:
2938:topological degree of
2882:
2858:
2807:
2692:
2623:
2587:
2476:
2426:
2351:asymptotic performance
2332:
2304:
2268:
2216:
2111:
1665:
1628:
1600:
1572:
1537:
1517:
1497:
1468:
1439:
1413:
1387:
1351:
1266:
1209:
1176:
1137:
1052:
974:
948:
922:
893:
864:
844:
821:
576:, either
435:) and replace either (
46:
4498:Fixed-point iteration
4089:Numerische Mathematik
3992:Numerische Mathematik
3873:Journal of Complexity
3751:Journal of Complexity
3704:Numerische Mathematik
3665:Journal of Complexity
3622:Numerische Mathematik
3447:
3387:
3327:
3228:
3155:
3082:
3009:
2883:
2868:over the boundary of
2859:
2808:
2693:
2624:
2622:{\displaystyle Df(y)}
2588:
2477:
2427:
2367:orders of convergence
2333:
2305:
2269:
2217:
2112:
1951:on the interval and
1666:
1629:
1627:{\displaystyle c_{n}}
1601:
1599:{\displaystyle b_{n}}
1573:
1571:{\displaystyle a_{n}}
1538:
1518:
1498:
1469:
1440:
1438:{\displaystyle a=1.5}
1414:
1388:
1352:
1267:
1215:, so the midpoint is
1210:
1177:
1138:
1053:
975:
949:
923:
894:
865:
845:
822:
284:as the new value for
44:
4457:Durand–Kerner method
4401:Newton–Krylov method
4175:10.1145/50063.214384
3409:
3343:
3283:
3184:
3111:
3038:
2965:
2936:) is unique and the
2872:
2824:
2705:
2637:
2601:
2489:
2436:
2401:
2322:
2278:
2229:
2131:
2037:
1639:
1611:
1583:
1555:
1527:
1507:
1496:{\displaystyle f(b)}
1478:
1467:{\displaystyle f(a)}
1449:
1423:
1397:
1361:
1279:
1222:
1186:
1153:
1068:
987:
958:
932:
921:{\displaystyle f(b)}
903:
892:{\displaystyle f(a)}
874:
854:
834:
773:
558:, tolerance
427:Examine the sign of
209:) at that point. If
84:binary search method
61:that applies to any
4406:Steffensen's method
4051:1995JCoPh.119..105V
2347:average performance
1975:. Specifically, if
1949:continuous function
1412:{\displaystyle a=1}
973:{\displaystyle b=2}
947:{\displaystyle a=1}
830:First, two numbers
620:) = 0 by less than
424:and stop iterating.
144:continuous function
112:Real-root isolation
63:continuous function
59:root-finding method
18:Bisection algorithm
4439:Polynomial methods
4267:Weisstein, Eric W.
4190:Numerical Analysis
4101:10.1007/BF01389620
4004:10.1007/BF01404868
3716:10.1007/BF01396051
3634:10.1007/BF01459080
3442:
3382:
3322:
3223:
3150:
3077:
3004:
2914:admissible polygon
2878:
2854:
2803:
2798:
2688:
2619:
2583:
2552:
2472:
2422:
2395:topological degree
2328:
2300:
2264:
2212:
2107:
1973:converges linearly
1661:
1624:
1596:
1568:
1533:
1513:
1493:
1464:
1435:
1409:
1383:
1347:
1262:
1205:
1172:
1133:
1048:
970:
944:
918:
889:
860:
840:
817:
47:
4521:
4520:
4477:Laguerre's method
4452:Bairstow's method
3931:978-3-030-40616-5
3405:. Then, at least
2516:
2198:
2102:
1929:
1928:
1536:{\displaystyle b}
1516:{\displaystyle a}
1419:is replaced with
1254:
863:{\displaystyle b}
843:{\displaystyle a}
685:// solution found
16:(Redirected from
4541:
4462:Graeffe's method
4391:Broyden's method
4340:Bisection method
4317:
4310:
4303:
4294:
4284:Bisection Method
4280:
4279:
4254:
4239:
4204:
4193:
4179:
4178:
4158:
4152:
4151:
4130:
4121:
4120:
4080:
4071:
4070:
4030:
4024:
4023:
3983:
3977:
3976:
3966:
3950:
3944:
3943:
3905:
3899:
3898:
3888:
3864:
3858:
3857:
3833:
3827:
3826:
3786:
3777:
3776:
3766:
3742:
3736:
3735:
3695:
3689:
3688:
3660:
3654:
3653:
3613:
3604:
3598:
3592:
3586:
3580:
3574:
3568:
3565:
3559:
3558:
3556:
3555:
3541:
3535:
3529:
3523:
3522:
3520:
3519:
3510:. Archived from
3504:
3498:
3492:
3477:Nested intervals
3455:
3451:
3449:
3448:
3443:
3435:
3421:
3420:
3404:
3393:
3391:
3389:
3388:
3383:
3333:
3331:
3329:
3328:
3323:
3234:
3232:
3230:
3229:
3224:
3161:
3159:
3157:
3156:
3151:
3088:
3086:
3084:
3083:
3078:
3015:
3013:
3011:
3010:
3005:
2912:(also called an
2887:
2885:
2884:
2879:
2866:surface integral
2863:
2861:
2860:
2855:
2812:
2810:
2809:
2804:
2802:
2801:
2697:
2695:
2694:
2689:
2687:
2686:
2644:
2628:
2626:
2625:
2620:
2592:
2590:
2589:
2584:
2551:
2547:
2539:
2538:
2481:
2479:
2478:
2473:
2471:
2470:
2465:
2456:
2455:
2450:
2431:
2429:
2428:
2423:
2421:
2420:
2415:
2337:
2335:
2334:
2329:
2309:
2307:
2306:
2301:
2296:
2295:
2273:
2271:
2270:
2265:
2263:
2249:
2241:
2240:
2221:
2219:
2218:
2213:
2208:
2204:
2203:
2199:
2194:
2193:
2184:
2175:
2174:
2157:
2156:
2152:
2116:
2114:
2113:
2108:
2103:
2101:
2100:
2091:
2090:
2076:
2070:
2065:
2054:
2053:
2044:
2003:
2001:
2000:
1997:
1994:
1670:
1668:
1667:
1662:
1657:
1656:
1633:
1631:
1630:
1625:
1623:
1622:
1605:
1603:
1602:
1597:
1595:
1594:
1577:
1575:
1574:
1569:
1567:
1566:
1546:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1473:
1471:
1470:
1465:
1444:
1442:
1441:
1436:
1418:
1416:
1415:
1410:
1392:
1390:
1389:
1384:
1379:
1378:
1356:
1354:
1353:
1348:
1319:
1318:
1297:
1296:
1271:
1269:
1268:
1263:
1255:
1250:
1239:
1234:
1233:
1214:
1212:
1211:
1206:
1198:
1197:
1181:
1179:
1178:
1173:
1165:
1164:
1142:
1140:
1139:
1134:
1101:
1100:
1057:
1055:
1054:
1049:
1020:
1019:
979:
977:
976:
971:
953:
951:
950:
945:
927:
925:
924:
919:
898:
896:
895:
890:
869:
867:
866:
861:
849:
847:
846:
841:
826:
824:
823:
818:
800:
799:
503:
501:
492:
389:
387:
386:
383:
380:
89:dichotomy method
79:interval halving
55:bisection method
21:
4549:
4548:
4544:
4543:
4542:
4540:
4539:
4538:
4524:
4523:
4522:
4517:
4508:Muller's method
4486:
4433:
4429:Ridders' method
4410:
4377:
4373:Halley's method
4368:Newton's method
4354:
4326:
4321:
4265:
4264:
4261:
4242:
4229:10.1137/1019044
4214:
4211:
4209:Further reading
4202:
4185:
4182:
4160:
4159:
4155:
4132:
4131:
4124:
4082:
4081:
4074:
4032:
4031:
4027:
3985:
3984:
3980:
3952:
3951:
3947:
3932:
3907:
3906:
3902:
3866:
3865:
3861:
3846:10.1145/3423597
3835:
3834:
3830:
3807:10.1145/3423597
3801:(1): 5:1–5:24.
3788:
3787:
3780:
3744:
3743:
3739:
3697:
3696:
3692:
3662:
3661:
3657:
3615:
3614:
3607:
3599:
3595:
3587:
3583:
3575:
3571:
3566:
3562:
3553:
3551:
3543:
3542:
3538:
3530:
3526:
3517:
3515:
3506:
3505:
3501:
3493:
3489:
3485:
3462:
3453:
3412:
3407:
3406:
3402:
3341:
3340:
3338:
3281:
3280:
3278:
3248:
3241:
3182:
3181:
3179:
3175:
3168:
3109:
3108:
3106:
3102:
3095:
3036:
3035:
3033:
3029:
3022:
2963:
2962:
2960:
2942:on its interior
2894:
2870:
2869:
2822:
2821:
2797:
2796:
2785:
2773:
2772:
2761:
2752:
2751:
2740:
2727:
2703:
2702:
2678:
2635:
2634:
2631:Jacobian matrix
2599:
2598:
2527:
2487:
2486:
2460:
2445:
2434:
2433:
2410:
2399:
2398:
2391:
2379:
2359:Ridders' method
2341:
2320:
2319:
2313:
2287:
2276:
2275:
2232:
2227:
2226:
2185:
2179:
2166:
2165:
2161:
2140:
2129:
2128:
2092:
2071:
2045:
2035:
2034:
2026:and a solution
2025:
2012:
1998:
1995:
1986:
1985:
1983:
1981:
1937:
1648:
1637:
1636:
1614:
1609:
1608:
1586:
1581:
1580:
1558:
1553:
1552:
1525:
1524:
1505:
1504:
1476:
1475:
1447:
1446:
1421:
1420:
1395:
1394:
1370:
1359:
1358:
1310:
1288:
1277:
1276:
1240:
1225:
1220:
1219:
1189:
1184:
1183:
1156:
1151:
1150:
1092:
1066:
1065:
1011:
985:
984:
956:
955:
930:
929:
901:
900:
872:
871:
852:
851:
832:
831:
791:
771:
770:
764:
759:
750:// new interval
657:// new midpoint
534:
499:
494:
480:
384:
381:
372:
371:
369:
334:
332:Iteration tasks
252:Explicitly, if
120:
108:Budan's theorem
104:Sturm's theorem
39:
28:
23:
22:
15:
12:
11:
5:
4547:
4545:
4537:
4536:
4526:
4525:
4519:
4518:
4516:
4515:
4510:
4505:
4500:
4494:
4492:
4488:
4487:
4485:
4484:
4479:
4474:
4469:
4464:
4459:
4454:
4449:
4443:
4441:
4435:
4434:
4432:
4431:
4426:
4424:Brent's method
4420:
4418:
4416:Hybrid methods
4412:
4411:
4409:
4408:
4403:
4398:
4393:
4387:
4385:
4379:
4378:
4376:
4375:
4370:
4364:
4362:
4356:
4355:
4353:
4352:
4347:
4342:
4336:
4334:
4328:
4327:
4322:
4320:
4319:
4312:
4305:
4297:
4291:
4290:
4281:
4260:
4259:External links
4257:
4256:
4255:
4240:
4223:(2): 325–327,
4210:
4207:
4206:
4205:
4200:
4181:
4180:
4169:(4): 312–329.
4153:
4122:
4095:(2): 123–138.
4072:
4045:(1): 105–119.
4025:
3998:(2): 109–127.
3978:
3945:
3930:
3900:
3879:(2): 612–640.
3859:
3828:
3778:
3757:(4): 489–501.
3737:
3710:(4): 481–491.
3690:
3671:(2): 197–209.
3655:
3628:(1): 111–117.
3605:
3593:
3581:
3569:
3560:
3536:
3524:
3499:
3486:
3484:
3481:
3480:
3479:
3474:
3468:
3461:
3458:
3441:
3438:
3434:
3430:
3427:
3424:
3419:
3415:
3399:
3398:
3395:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3335:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3251:
3250:
3246:
3239:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3177:
3173:
3166:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3104:
3100:
3093:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3031:
3027:
3020:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2893:
2890:
2877:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2814:
2813:
2800:
2795:
2792:
2789:
2786:
2784:
2781:
2778:
2775:
2774:
2771:
2768:
2765:
2762:
2760:
2757:
2754:
2753:
2750:
2747:
2744:
2741:
2739:
2736:
2733:
2732:
2730:
2725:
2722:
2719:
2716:
2713:
2710:
2685:
2681:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2643:
2618:
2615:
2612:
2609:
2606:
2595:
2594:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2550:
2546:
2542:
2537:
2534:
2530:
2526:
2523:
2519:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2469:
2464:
2459:
2454:
2449:
2444:
2441:
2419:
2414:
2409:
2406:
2390:
2387:
2378:
2375:
2363:Brent's method
2339:
2327:
2311:
2299:
2294:
2290:
2286:
2283:
2262:
2258:
2255:
2252:
2248:
2244:
2239:
2235:
2223:
2222:
2211:
2207:
2202:
2197:
2192:
2188:
2182:
2178:
2173:
2169:
2164:
2160:
2155:
2151:
2147:
2143:
2139:
2136:
2118:
2117:
2106:
2099:
2095:
2089:
2085:
2082:
2079:
2075:
2068:
2064:
2060:
2057:
2052:
2048:
2043:
2030:is bounded by
2021:
2008:
1979:
1969:absolute error
1936:
1933:
1927:
1926:
1923:
1920:
1917:
1914:
1910:
1909:
1906:
1903:
1900:
1897:
1893:
1892:
1889:
1886:
1883:
1880:
1876:
1875:
1872:
1869:
1866:
1863:
1859:
1858:
1855:
1852:
1849:
1846:
1842:
1841:
1838:
1835:
1832:
1829:
1825:
1824:
1821:
1818:
1815:
1812:
1808:
1807:
1804:
1801:
1798:
1795:
1791:
1790:
1787:
1784:
1781:
1778:
1774:
1773:
1770:
1767:
1764:
1761:
1757:
1756:
1753:
1750:
1747:
1744:
1740:
1739:
1736:
1733:
1730:
1727:
1723:
1722:
1719:
1716:
1713:
1710:
1706:
1705:
1702:
1699:
1696:
1693:
1689:
1688:
1685:
1682:
1679:
1676:
1672:
1671:
1660:
1655:
1651:
1647:
1644:
1634:
1621:
1617:
1606:
1593:
1589:
1578:
1565:
1561:
1550:
1532:
1512:
1492:
1489:
1486:
1483:
1463:
1460:
1457:
1454:
1434:
1431:
1428:
1408:
1405:
1402:
1393:is negative,
1382:
1377:
1373:
1369:
1366:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1317:
1313:
1309:
1306:
1303:
1300:
1295:
1291:
1287:
1284:
1273:
1272:
1261:
1258:
1253:
1249:
1246:
1243:
1237:
1232:
1228:
1204:
1201:
1196:
1192:
1171:
1168:
1163:
1159:
1144:
1143:
1132:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1099:
1095:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1059:
1058:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1018:
1014:
1010:
1007:
1004:
1001:
998:
995:
992:
969:
966:
963:
943:
940:
937:
917:
914:
911:
908:
888:
885:
882:
879:
859:
839:
828:
827:
816:
812:
809:
806:
803:
798:
794:
790:
787:
784:
781:
778:
763:
760:
542:
533:
530:
473:
472:
425:
402:
391:
333:
330:
119:
116:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4546:
4535:
4532:
4531:
4529:
4514:
4511:
4509:
4506:
4504:
4501:
4499:
4496:
4495:
4493:
4491:Other methods
4489:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4447:Aberth method
4445:
4444:
4442:
4440:
4436:
4430:
4427:
4425:
4422:
4421:
4419:
4417:
4413:
4407:
4404:
4402:
4399:
4397:
4396:Secant method
4394:
4392:
4389:
4388:
4386:
4384:
4380:
4374:
4371:
4369:
4366:
4365:
4363:
4361:
4357:
4351:
4348:
4346:
4343:
4341:
4338:
4337:
4335:
4333:
4329:
4325:
4318:
4313:
4311:
4306:
4304:
4299:
4298:
4295:
4289:
4285:
4282:
4277:
4276:
4271:
4268:
4263:
4262:
4258:
4253:on 2009-04-13
4252:
4248:
4247:
4241:
4238:
4234:
4230:
4226:
4222:
4218:
4213:
4212:
4208:
4203:
4201:0-87150-857-5
4197:
4192:
4191:
4184:
4183:
4176:
4172:
4168:
4164:
4157:
4154:
4149:
4145:
4141:
4137:
4129:
4127:
4123:
4118:
4114:
4110:
4106:
4102:
4098:
4094:
4090:
4086:
4079:
4077:
4073:
4068:
4064:
4060:
4056:
4052:
4048:
4044:
4040:
4036:
4029:
4026:
4021:
4017:
4013:
4009:
4005:
4001:
3997:
3993:
3989:
3982:
3979:
3974:
3970:
3965:
3960:
3956:
3949:
3946:
3941:
3937:
3933:
3927:
3923:
3919:
3915:
3911:
3904:
3901:
3896:
3892:
3887:
3882:
3878:
3874:
3870:
3863:
3860:
3855:
3851:
3847:
3843:
3839:
3832:
3829:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3796:
3792:
3785:
3783:
3779:
3774:
3770:
3765:
3760:
3756:
3752:
3748:
3741:
3738:
3733:
3729:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3694:
3691:
3686:
3682:
3678:
3674:
3670:
3666:
3659:
3656:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3623:
3619:
3612:
3610:
3606:
3602:
3597:
3594:
3590:
3585:
3582:
3578:
3573:
3570:
3564:
3561:
3550:
3546:
3540:
3537:
3533:
3528:
3525:
3514:on 2013-05-19
3513:
3509:
3503:
3500:
3496:
3491:
3488:
3482:
3478:
3475:
3472:
3469:
3467:
3464:
3463:
3459:
3457:
3436:
3432:
3428:
3422:
3417:
3413:
3396:
3376:
3370:
3367:
3364:
3358:
3352:
3349:
3346:
3336:
3316:
3310:
3307:
3304:
3298:
3292:
3289:
3286:
3276:
3275:
3274:
3272:
3267:
3265:
3261:
3256:
3245:
3238:
3217:
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3211:
3205:
3199:
3193:
3190:
3187:
3178:
3172:
3165:
3144:
3141:
3138:
3132:
3126:
3120:
3117:
3114:
3105:
3099:
3092:
3071:
3068:
3065:
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3053:
3047:
3044:
3041:
3032:
3026:
3019:
2998:
2995:
2992:
2986:
2980:
2974:
2971:
2968:
2959:
2958:
2957:
2955:
2954:quadrilateral
2951:
2947:
2943:
2941:
2935:
2931:
2927:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2891:
2889:
2867:
2851:
2848:
2839:
2836:
2830:
2827:
2819:
2818:sign function
2793:
2790:
2787:
2782:
2779:
2776:
2769:
2766:
2763:
2758:
2755:
2748:
2745:
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2734:
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2711:
2708:
2701:
2700:
2699:
2683:
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2669:
2666:
2663:
2660:
2657:
2654:
2651:
2645:
2632:
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2574:
2568:
2565:
2556:
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2535:
2532:
2528:
2524:
2521:
2517:
2513:
2504:
2501:
2495:
2492:
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2484:
2483:
2467:
2452:
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2439:
2417:
2407:
2396:
2388:
2386:
2384:
2376:
2374:
2372:
2368:
2364:
2360:
2356:
2355:secant method
2352:
2348:
2343:
2325:
2317:
2297:
2292:
2288:
2284:
2281:
2256:
2253:
2250:
2242:
2237:
2233:
2209:
2205:
2200:
2195:
2190:
2186:
2180:
2176:
2171:
2167:
2162:
2158:
2153:
2149:
2145:
2141:
2137:
2134:
2127:
2126:
2125:
2123:
2104:
2097:
2093:
2083:
2080:
2077:
2066:
2058:
2055:
2050:
2046:
2033:
2032:
2031:
2029:
2024:
2020:
2016:
2011:
2007:
1993:
1989:
1978:
1974:
1970:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1934:
1932:
1924:
1921:
1918:
1915:
1912:
1911:
1907:
1904:
1901:
1898:
1895:
1894:
1890:
1887:
1884:
1881:
1878:
1877:
1873:
1870:
1867:
1864:
1861:
1860:
1856:
1853:
1850:
1847:
1844:
1843:
1839:
1836:
1833:
1830:
1827:
1826:
1822:
1819:
1816:
1813:
1810:
1809:
1805:
1802:
1799:
1796:
1793:
1792:
1788:
1785:
1782:
1779:
1776:
1775:
1771:
1768:
1765:
1762:
1759:
1758:
1754:
1751:
1748:
1745:
1742:
1741:
1737:
1734:
1731:
1728:
1725:
1724:
1720:
1717:
1714:
1711:
1708:
1707:
1703:
1700:
1697:
1694:
1691:
1690:
1686:
1683:
1680:
1677:
1674:
1673:
1653:
1649:
1642:
1635:
1619:
1615:
1607:
1591:
1587:
1579:
1563:
1559:
1551:
1548:
1547:
1544:
1530:
1510:
1487:
1481:
1458:
1452:
1432:
1429:
1426:
1406:
1403:
1400:
1375:
1371:
1364:
1344:
1341:
1338:
1335:
1332:
1326:
1320:
1315:
1307:
1301:
1293:
1289:
1282:
1259:
1256:
1251:
1247:
1244:
1241:
1235:
1230:
1226:
1218:
1217:
1216:
1202:
1199:
1194:
1190:
1169:
1166:
1161:
1157:
1147:
1130:
1126:
1123:
1120:
1117:
1114:
1108:
1102:
1097:
1089:
1083:
1077:
1071:
1064:
1063:
1062:
1045:
1042:
1039:
1036:
1033:
1027:
1021:
1016:
1008:
1002:
996:
990:
983:
982:
981:
967:
964:
961:
941:
938:
935:
912:
906:
883:
877:
857:
837:
814:
810:
807:
804:
801:
796:
792:
788:
782:
776:
769:
768:
767:
761:
758:
754:
751:
748:
744:
741:
738:
734:
731:
727:
723:
719:
715:
711:
708:
704:
700:
697:
694:
690:
686:
683:
680:
676:
672:
668:
664:
661:
658:
654:
650:
646:
643:
640:
637:
633:
630:
626:
623:
619:
615:
611:
607:
603:
600:) > 0 and
599:
595:
591:
587:
584:) < 0 and
583:
579:
575:
571:
568:
565:
561:
557:
553:
549:
545:
541:
539:
531:
529:
527:
523:
519:
515:
511:
507:
497:
491:
487:
483:
478:
470:
466:
462:
458:
454:
450:
446:
442:
438:
434:
430:
426:
423:
419:
415:
411:
407:
403:
400:
396:
392:
379:
375:
367:
363:
359:
358:
357:
355:
351:
347:
343:
339:
331:
329:
327:
323:
319:
315:
311:
307:
303:
299:
295:
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
250:
248:
244:
240:
236:
232:
228:
224:
220:
216:
212:
208:
204:
200:
196:
192:
187:
185:
181:
177:
173:
169:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
117:
115:
113:
109:
105:
101:
97:
92:
90:
86:
85:
80:
76:
72:
68:
64:
60:
56:
52:
43:
37:
33:
19:
4383:Quasi-Newton
4345:Regula falsi
4339:
4273:
4251:the original
4245:
4220:
4216:
4189:
4166:
4162:
4156:
4142:(1): 53–68.
4139:
4135:
4092:
4088:
4042:
4038:
4028:
3995:
3991:
3981:
3964:nlin/0211044
3954:
3948:
3913:
3903:
3876:
3872:
3862:
3831:
3798:
3794:
3754:
3750:
3740:
3707:
3703:
3693:
3668:
3664:
3658:
3625:
3621:
3596:
3584:
3572:
3563:
3552:. Retrieved
3548:
3539:
3534:, p. 28
3527:
3516:. Retrieved
3512:the original
3502:
3497:, p. 31
3490:
3400:
3270:
3268:
3263:
3259:
3254:
3252:
3243:
3236:
3235:, that is,
3170:
3163:
3162:, that is,
3097:
3090:
3089:, that is,
3024:
3017:
2949:
2945:
2939:
2933:
2929:
2925:
2917:
2913:
2909:
2905:
2901:
2897:
2895:
2815:
2596:
2392:
2382:
2380:
2350:
2346:
2344:
2315:
2224:
2121:
2119:
2027:
2022:
2018:
2014:
2009:
2005:
1991:
1987:
1976:
1964:
1960:
1956:
1952:
1944:
1940:
1938:
1930:
1274:
1148:
1145:
1060:
829:
765:
756:
752:
749:
746:
742:
739:
736:
732:
729:
725:
721:
717:
713:
709:
706:
702:
698:
695:
692:
688:
684:
681:
678:
674:
670:
666:
662:
659:
656:
652:
648:
644:
641:
638:
635:
631:
628:
624:
621:
617:
613:
609:
605:
601:
597:
593:
592:) > 0 or
589:
585:
581:
577:
573:
569:
566:
563:
559:
555:
551:
547:
543:
540:as follows:
535:
525:
521:
517:
513:
509:
505:
495:
489:
485:
481:
476:
474:
468:
464:
460:
456:
452:
448:
444:
440:
436:
432:
428:
421:
417:
413:
409:
405:
398:
394:
377:
373:
365:
361:
353:
349:
345:
341:
337:
335:
325:
321:
317:
313:
309:
305:
301:
297:
293:
289:
285:
281:
277:
273:
269:
265:
261:
257:
253:
251:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
188:
183:
179:
175:
167:
163:
159:
155:
151:
147:
139:
135:
127:
123:
121:
93:
88:
82:
81:method, the
78:
54:
48:
4360:Householder
4270:"Bisection"
4217:SIAM Review
3957:: 665–676.
3255:proper edge
3016:, that is,
1891:−0.0001034
1874:−0.0008289
1857:−0.0022794
1840:−0.0051789
1806:−0.0109712
1772:−0.0340538
567:conditions:
308:as the new
96:polynomials
51:mathematics
4350:ITP method
3554:2015-12-21
3518:2013-11-07
3483:References
3242:(D)>0,
3169:(C)>0,
3096:(B)<0,
3023:(A)<0,
2371:ITP Method
2316:worst case
1925:0.0000780
1908:0.0002594
1823:0.0006222
1789:0.0122504
1755:0.0591125
1738:0.2521973
1721:0.6660156
1704:1.6093750
1357:. Because
720:)) = sign(
691:)
669:) = 0 or (
538:pseudocode
360:Calculate
118:The method
4275:MathWorld
4237:1095-7200
4117:121771945
4109:0945-3245
4067:0021-9991
4020:122058552
4012:0945-3245
3940:211160947
3895:0885-064X
3854:230586635
3823:230586635
3815:0098-3500
3773:0885-064X
3732:119546369
3724:0945-3245
3685:0885-064X
3650:119952605
3642:0945-3245
3437:ε
3423:
3371:
3350:
3311:
3290:
3249:(D)>0.
3191:
3176:(C)<0.
3145:−
3118:
3103:(B)>0.
3066:−
3045:
3030:(A)<0.
2999:−
2993:−
2972:
2876:Ω
2849:≠
2843:Ω
2831:
2777:−
2712:
2557:
2533:−
2525:∈
2518:∑
2508:Ω
2496:
2458:→
2408:⊆
2405:Ω
2326:ϵ
2289:ϵ
2285:≤
2282:ϵ
2254:−
2234:ϵ
2196:ϵ
2187:ϵ
2177:
2159:≡
2138:≤
2081:−
2067:≤
2056:−
1922:1.5213928
1919:1.5214233
1916:1.5213623
1905:1.5214233
1902:1.5214844
1899:1.5213623
1888:1.5213623
1885:1.5214844
1882:1.5212402
1871:1.5212402
1868:1.5214844
1865:1.5209961
1854:1.5209961
1851:1.5214844
1848:1.5205078
1837:1.5205078
1834:1.5214844
1831:1.5195313
1820:1.5214844
1817:1.5234375
1814:1.5195313
1803:1.5195313
1800:1.5234375
1797:1.5156250
1786:1.5234375
1783:1.5312500
1780:1.5156250
1769:1.5156250
1766:1.5312500
1752:1.5312500
1549:Iteration
1342:−
1333:−
1321:−
1115:−
1103:−
1043:−
1034:−
1022:−
808:−
802:−
753:end while
677:)/2 <
608:) < 0
546:Function
532:Algorithm
459:)) with (
288:, and if
260:)=0 then
134:variable
87:, or the
67:bisecting
4528:Category
3460:See also
3260:diagonal
2922:polytope
2206:⌉
2163:⌈
1935:Analysis
488:) = cos
408:−
138:, where
71:interval
4047:Bibcode
3392:
3339:
3332:
3279:
3233:
3180:
3160:
3107:
3087:
3034:
3014:
2961:
2908:≥ 2. A
2816:is the
2629:is the
2318:has an
2002:
1984:
1687:−0.125
687:Output(
610:output:
447:)) or (
388:
370:
4235:
4198:
4115:
4107:
4065:
4018:
4010:
3938:
3928:
3893:
3852:
3821:
3813:
3771:
3730:
3722:
3683:
3648:
3640:
3454:ε
2698:, and
2597:where
1959:) and
1749:1.5625
1735:1.5625
696:end if
544:input:
348:) and
320:) and
296:) and
272:) and
237:) and
221:) and
154:) and
53:, the
4113:S2CID
4016:S2CID
3959:arXiv
3936:S2CID
3850:S2CID
3819:S2CID
3728:S2CID
3646:S2CID
2952:is a
2920:is a
2916:) of
1947:is a
1732:1.625
1718:1.625
1345:0.125
712:sign(
629:while
572:<
142:is a
57:is a
4233:ISSN
4196:ISBN
4105:ISSN
4063:ISSN
4008:ISSN
3926:ISBN
3891:ISSN
3811:ISSN
3769:ISSN
3720:ISSN
3681:ISSN
3638:ISSN
2896:The
2791:<
2746:>
2385:.
1715:1.75
1701:1.75
1523:and
1474:and
1182:and
1061:and
954:and
899:and
850:and
740:else
730:then
705:+ 1
693:Stop
682:then
655:)/2
636:NMAX
627:← 1
564:NMAX
516:and
508:and
166:and
132:real
94:For
75:root
69:the
4225:doi
4171:doi
4144:doi
4140:138
4097:doi
4055:doi
4043:119
4000:doi
3969:doi
3918:doi
3881:doi
3842:doi
3803:doi
3759:doi
3712:doi
3673:doi
3630:doi
3414:log
3368:sgn
3347:sgn
3337:If
3308:sgn
3287:sgn
3277:If
3188:sgn
3115:sgn
3042:sgn
2969:sgn
2828:deg
2709:sgn
2560:det
2554:sgn
2493:deg
2361:or
2340:1/2
2168:log
1943:if
1763:1.5
1746:1.5
1729:1.5
1712:1.5
1695:1.5
1684:1.5
1433:1.5
1327:1.5
1308:1.5
1260:1.5
728:))
679:TOL
647:← (
622:TOL
560:TOL
524:or
193:= (
186:).
49:In
4530::
4272:.
4231:,
4221:19
4219:,
4167:14
4165:.
4138:.
4125:^
4111:.
4103:.
4093:49
4091:.
4087:.
4075:^
4061:.
4053:.
4041:.
4037:.
4014:.
4006:.
3996:32
3994:.
3990:.
3967:.
3934:.
3924:.
3889:.
3877:18
3875:.
3871:.
3848:.
3840:.
3817:.
3809:.
3799:47
3797:.
3793:.
3781:^
3767:.
3753:.
3749:.
3726:.
3718:.
3708:55
3706:.
3702:.
3679:.
3667:.
3644:.
3636:.
3626:40
3624:.
3620:.
3608:^
3547:.
3253:A
2888:.
2633:,
2514::=
2373:.
2357:,
1982:=
1913:15
1896:14
1879:13
1862:12
1845:11
1828:10
745:←
735:←
710:if
701:←
673:–
660:if
651:+
639:do
634:≤
554:,
528:.
502:/2
498:=
463:,
451:,
439:,
401:).
376:+
374:a
368:=
182:,
114:.
106:,
102:,
91:.
4316:e
4309:t
4302:v
4278:.
4227::
4177:.
4173::
4150:.
4146::
4119:.
4099::
4069:.
4057::
4049::
4022:.
4002::
3975:.
3971::
3961::
3942:.
3920::
3897:.
3883::
3856:.
3844::
3825:.
3805::
3775:.
3761::
3755:5
3734:.
3714::
3687:.
3675::
3669:1
3652:.
3632::
3557:.
3521:.
3440:)
3433:/
3429:D
3426:(
3418:2
3403:D
3380:)
3377:B
3374:(
3365:=
3362:)
3359:M
3356:(
3353:f
3320:)
3317:A
3314:(
3305:=
3302:)
3299:M
3296:(
3293:f
3271:f
3264:d
3247:2
3244:f
3240:1
3237:f
3221:)
3218:+
3215:,
3212:+
3209:(
3206:=
3203:)
3200:D
3197:(
3194:f
3174:2
3171:f
3167:1
3164:f
3148:)
3142:,
3139:+
3136:(
3133:=
3130:)
3127:C
3124:(
3121:f
3101:2
3098:f
3094:1
3091:f
3075:)
3072:+
3069:,
3063:(
3060:=
3057:)
3054:B
3051:(
3048:f
3028:2
3025:f
3021:1
3018:f
3002:)
2996:,
2990:(
2987:=
2984:)
2981:A
2978:(
2975:f
2950:f
2946:d
2940:f
2934:v
2932:(
2930:f
2926:v
2918:f
2906:d
2902:f
2852:0
2846:)
2840:,
2837:f
2834:(
2794:0
2788:x
2783:,
2780:1
2770:0
2767:=
2764:x
2759:,
2756:0
2749:0
2743:x
2738:,
2735:1
2729:{
2724:=
2721:)
2718:x
2715:(
2684:T
2680:)
2676:0
2673:,
2670:.
2667:.
2664:.
2661:,
2658:0
2655:,
2652:0
2649:(
2646:=
2642:0
2617:)
2614:y
2611:(
2608:f
2605:D
2593:,
2581:)
2578:)
2575:y
2572:(
2569:f
2566:D
2563:(
2549:)
2545:0
2541:(
2536:1
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2522:y
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2453:n
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2443::
2440:f
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2413:R
2312:n
2298:.
2293:0
2261:|
2257:a
2251:b
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2243:=
2238:0
2210:,
2201:)
2191:0
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2172:2
2154:2
2150:/
2146:1
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2122:n
2105:.
2098:n
2094:2
2088:|
2084:a
2078:b
2074:|
2063:|
2059:c
2051:n
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2042:|
2028:c
2023:n
2019:c
2015:n
2010:n
2006:c
1999:2
1996:/
1992:b
1990:+
1988:a
1980:1
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1965:b
1963:(
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1955:(
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1941:f
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1482:f
1462:)
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1453:f
1430:=
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1407:1
1404:=
1401:a
1381:)
1376:1
1372:c
1368:(
1365:f
1339:=
1336:2
1330:)
1324:(
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1312:)
1305:(
1302:=
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1283:f
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1242:2
1236:=
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1227:c
1203:2
1200:=
1195:1
1191:b
1170:1
1167:=
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1158:a
1131:.
1127:4
1124:+
1121:=
1118:2
1112:)
1109:2
1106:(
1098:3
1094:)
1090:2
1087:(
1084:=
1081:)
1078:2
1075:(
1072:f
1046:2
1040:=
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1031:)
1028:1
1025:(
1017:3
1013:)
1009:1
1006:(
1003:=
1000:)
997:1
994:(
991:f
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965:=
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942:1
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916:)
913:b
910:(
907:f
887:)
884:a
881:(
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858:b
838:a
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805:x
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793:x
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780:(
777:f
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737:c
733:a
726:a
724:(
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718:c
716:(
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699:N
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671:b
667:c
665:(
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653:b
649:a
645:c
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625:N
618:x
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604:(
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233:(
231:f
227:c
225:(
223:f
219:a
217:(
215:f
211:c
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205:(
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199:b
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195:a
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160:b
158:(
156:f
152:a
150:(
148:f
140:f
136:x
128:x
126:(
124:f
38:.
20:)
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