4570:
1302:
2470:, starting with the interval ). It is mathematically possible with discontinuous functions for the method to fail to converge to a zero limit or sign change, but this is not a problem in practice since it would require an infinite sequence of coincidences for both endpoints to get stuck converging to discontinuities where the sign does not change, for example at
3331:
5075:
4376:
For manual calculation, by calculator, one tends to want to use faster methods, and they usually, but not always, converge faster than bisection. But a computer, even using bisection, will solve an equation, to the desired accuracy, so rapidly that there's no need to try to save time by using a less
4363:
on this point is queried, and the interval is then reduced to bracket the root by keeping the sub-interval with function values of opposite sign on each end. This three step procedure guarantees that the minmax properties of the bisection method are enjoyed by the estimate as well as the superlinear
5823:
900:
Many equations, including most of the more complicated ones, can be solved only by iterative numerical approximation. This consists of trial and error, in which various values of the unknown quantity are tried. That trial-and-error may be guided by calculating, at each step of the procedure, a new
4372:
When solving one equation, or just a few, using a computer, the bisection method is an adequate choice. Although bisection isn't as fast as the other methods—when they're at their best and don't have a problem—bisection nevertheless is guaranteed to converge at a useful rate, roughly halving the
3052:
used above looks arbitrary, but it guarantees superlinear convergence (asymptotically, the algorithm will perform two regular steps after any modified step, and has order of convergence 1.442). There are other ways to pick the rescaling which give even better superlinear convergence rates.
867:
translates the
Latinized version of Pacioli's term into the vernacular "false positions" in 1556. Pacioli's term nearly disappeared in the 16th century European works and the technique went by various names such as "Rule of False", "Rule of Position" and "Rule of False Position".
2013:
1331:
joining the endpoints of the function on the current bracketing interval. Essentially, the root is being approximated by replacing the actual function by a line segment on the bracketing interval and then using the classical double false position formula on that line segment.
3159:
2636:
failure mode is easy to detect: The same end-point is retained twice in a row. The problem is easily remedied by picking instead a modified false position, chosen to avoid slowdowns due to those relatively unusual unfavorable situations. A number of such improvements to
879:
The Rule of falsehoode is so named not for that it teacheth anye deceyte or falsehoode, but that by fayned numbers taken at all aduentures, it teacheth to finde out the true number that is demaunded, and this of all the vulgar Rules which are in practise) is y most
735:(九章算術), dated from 200 BC to AD 100, most of Chapter 7 was devoted to the algorithm. There, the procedure was justified by concrete arithmetical arguments, then applied creatively to a wide variety of story problems, including one involving what we would call
4402:
is also changing very little, then Newton's method most likely will not run into trouble, and will converge. So, under those favorable conditions, one could switch to Newton's method if one wanted the error to be very small and wanted very fast convergence.
4320:
1162:
Preserving the bracketing and ensuring that the solution estimates lie in the interior of the bracketing intervals guarantees that the solution estimates will converge toward the solution, a guarantee not available with other root finding methods such as
1288:
Since the bracketing interval's length is halved at each step, the bisection method's error is, on average, halved with each iteration. Hence, every 3 iterations, the method gains approximately a factor of 2, i.e. roughly a decimal place, in accuracy.
3020:
2851:
143:
Modern versions of the technique employ systematic ways of choosing new test values and are concerned with the questions of whether or not an approximation to a solution can be obtained, and if it can, how fast can the approximation be found.
4873:
4387:, maybe in one of its improved versions, such as the Illinois or Anderson–Björck versions. Or, if even that isn't converging as well as bisection would, switch to bisection, which always converges at a useful, if not spectacular, rate.
2625:
is one of the best methods, and even in its original un-improved version would often be the best choice; for example, when Newton's isn't used because the derivative is prohibitively time-consuming to evaluate, or when Newton's and
5218:
4096:
5603:
1748:
1114:
is guaranteed to lie between these two values, that is to say, these values "bracket" the root. A point strictly between these two values is then selected and used to create a smaller interval that still brackets a root. If
3433:
1570:
41:
technique of using test ("false") values for the variable and then adjusting the test value according to the outcome. This is sometimes also referred to as "guess and check". Versions of the method predate the advent of
2431:). In addition to sign changes, it is also possible for the method to converge to a point where the limit of the function is zero, even if the function is undefined (or has another value) at that point (for example at
4681:
3517:
2202:
The above formula is also used in the secant method, but the secant method always retains the last two computed points, and so, while it is slightly faster, it does not preserve bracketing and may not converge.
2578:
747:
Now an item is purchased jointly; everyone contributes 8 , the excess is 3; everyone contributes 7, the deficit is 4. Tell: The number of people, the item price, what is each? Answer: 7 people, item price 53.
552:
5387:
4165:
1768:
3855:
3975:
4759:
109:. This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply
438:
4551:
2404:
is negative on the interval) and thus the width of the bracket never falls below 1. Hence, the right endpoint approaches 0 at a linear rate (the number of accurate digits grows linearly, with a
1159:
The point selected in any current interval can be thought of as an estimate of the solution. The different variations of this method involve different ways of calculating this solution estimate.
3164:
2210:
always converges, and has versions that do well at avoiding slowdowns, makes it a good choice when speed is needed. However, its rate of convergence can drop below that of the bisection method.
3908:
1259:
3756:
3630:
3326:{\displaystyle {\begin{aligned}m'&=1-{\frac {f(c_{k})}{f(b_{k})}},\\m&={\begin{cases}m'&{\text{if }}m'>0,\\{\frac {1}{2}}&{\text{otherwise.}}\end{cases}}\end{aligned}}}
4170:
2390:
3562:
2602:
always converges, usually considerably faster than bisection, there are situations that can slow its convergence – sometimes to a prohibitive degree. That problem isn't unique to
4496:
875:
Several 16th century
European authors felt the need to apologize for the name of the method in a science that seeks to find the truth. For instance, in 1568 Humphrey Baker says:
2862:
5588:
5486:
4361:
2696:
5281:
4857:
3695:
1156:, a root has been found and the algorithm stops. Otherwise, the procedure is repeated as often as necessary to obtain an approximation to the root to any desired accuracy.
889:
The method of false position provides an exact solution for linear functions, but more direct algebraic techniques have supplanted its use for these functions. However, in
5070:{\displaystyle \ B(n)=\sum _{k=0}^{n-1}3\cdot {\frac {1}{2^{k}}}=3\left({\frac {1-({\tfrac {1}{2}})^{n-1+1}}{1-{\tfrac {1}{2}}}}\right)=6\left(1-{\frac {1}{2^{n}}}\right)}
344:
5531:
5429:
7229:
3668:
671:
4804:
2610:
of the numerical equation-solving methods can have a slow-convergence or no-convergence problem under some conditions. Sometimes, Newton's method and the secant method
863:, probably taking the term from Fibonacci. Other European writers would follow Pacioli and sometimes provided a translation into Latin or the vernacular. For instance,
800:("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of
4380:
An exception would be if the computer program had to solve equations very many times during its run. Then the time saved by the faster methods could be significant.
3582:
205:
2067:
are always of opposite sign the “subtraction” in the numerator of the improved formula is effectively an addition (as is the subtraction in the denominator too).
5818:{\displaystyle \ {\hat {x}}~=~{\frac {~x_{1}F(x_{2})-x_{2}F(x_{1})~}{F(x_{2})-F(x_{1})}}~=~{\frac {~2\times 1.75+3\times 1.5~}{1.75+1.5}}~\approx ~2.4615\ }
5081:
3982:
6871:
4364:
convergence of the secant method. And, is observed to outperform both bisection and interpolation based methods under smooth and non-smooth functions.
1607:
4412:
3352:
731:
1047:, but it can fail to find a root under certain circumstances and it may be computationally costly since it requires a computation of the function's
1433:
37:
is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the
2286:
also has the same sign) will remain fixed for all subsequent iterations while the converging endpoint becomes updated. As a result, unlike the
7222:
6809:
6758:
6728:
6700:
6669:
4603:
4448:
A club-rush grew 1 unit on its first day. At the end of each day, the plant has grown by 2 times as much as the previous day's growth.
3438:
2483:
7427:
2395:
on the initial bracket . The left end, −1, is never replaced (it does not change at first and after the first three iterations,
7353:
6942:
2008:{\displaystyle c_{k}=b_{k}-f(b_{k}){\frac {b_{k}-a_{k}}{f(b_{k})-f(a_{k})}}={\frac {a_{k}f(b_{k})-b_{k}f(a_{k})}{f(b_{k})-f(a_{k})}}.}
457:
7164:
7145:
5288:
801:
4569:
4106:
3064:
by some scholars. Ford (1995) summarizes and analyzes this and other similar superlinear variants of the method of false position.
3762:
7453:
7448:
7215:
3915:
7417:
4688:
2411:
For discontinuous functions, this method can only be expected to find a point where the function changes sign (for example at
3107:
and the left-hand endpoint has been retained. (So far, that's the same as ordinary Regula Falsi and the
Illinois algorithm.)
166:
is aimed at solving problems involving direct proportion. Such problems can be written algebraically in the form: determine
355:
4504:
793:
2035:
will be very close together, and nearly always of the same sign. Such a subtraction can lose significant digits. Because
864:
7381:
3864:
7386:
6840:
1199:
3705:
3587:
1009:
4315:{\displaystyle \rho _{k}\equiv \min \left\{\epsilon 2^{n_{1/2}+n_{0}-j}-{\frac {b-a}{2}},|x_{t}-x_{1/2}|\right\}}
7371:
7315:
7176:"Mathematical Philology in the Treatise on Double False Position in an Arabic Manuscript at Columbia University"
1313:
The convergence rate of the bisection method could possibly be improved by using a different solution estimate.
5834:
2641:
have been proposed; two of them, the
Illinois algorithm and the Anderson–Björk algorithm, are described below.
7396:
7330:
7274:
7246:
7238:
6774:
4373:
error with each iteration – gaining roughly a decimal place of accuracy with every 3 iterations.
3015:{\displaystyle c_{k}={\frac {f(b_{k})a_{k}-{\frac {1}{2}}f(a_{k})b_{k}}{f(b_{k})-{\frac {1}{2}}f(a_{k})}},}
2325:
7320:
6966:
3522:
2846:{\displaystyle c_{k}={\frac {{\frac {1}{2}}f(b_{k})a_{k}-f(a_{k})b_{k}}{{\frac {1}{2}}f(b_{k})-f(a_{k})}}}
2420:
1040:
894:
707:
4462:
2290:, the width of the bracket does not tend to zero (unless the zero is at an inflection point around which
7412:
6986:
Dowell, M.; Jarratt, P. (1971). "A modified regula falsi method for computing the root of an equation".
5536:
5434:
4330:
3076:-th iteration the bracketing interval is and that the functional value of the new calculated estimate
743:. A more typical example is this "joint purchase" problem involving an "excess and deficit" condition:
7057:
7391:
7366:
5230:
4809:
859:
715:
444:
284:
is aimed at solving more difficult problems that can be written algebraically in the form: determine
3673:
3256:
7376:
7305:
7297:
6971:
4429:
grew 3 units on its first day. At the end of each day, the plant is observed to have grown by
2584:
2405:
959:
726:
296:
7195:
7115:
7038:
7003:
6830:
Issues in the Origin and
Development of Hisab al-Khata'ayn (Calculation by Double False Position)
5497:
5395:
890:
789:
7422:
7343:
7287:
7282:
7204:(On a previously unpublished treatise on Double False Position in a medieval Arabic manuscript.)
6624:
3635:
1164:
1044:
837:(AD 1202) to explaining and demonstrating the uses of double false position, terming the method
629:
4768:
4383:
Then, a program could start with Newton's method, and, if Newton's isn't converging, switch to
7338:
7160:
7141:
7107:
6938:
6932:
6928:
6907:
6865:
6805:
6754:
6748:
6724:
6696:
6665:
6630:
804:), as well as purely recreational problems. The algorithm was often memorized with the aid of
6753:. Science and Civilisation in China. Vol. 3. Cambridge University Press. pp. 147–.
6716:
6661:
6654:
7254:
7187:
7099:
7030:
6995:
6835:
4864:
4594:
2287:
2277:
1175:
691:
677:
3567:
447:. By using a pair of test inputs and the corresponding pair of outputs, the result of this
178:
821:
620:
38:
7088:"An Enhancement of the Bisection Method Average Performance Preserving Minmax Optimality"
4564:
Suppose it is day 3. The club-rush is taller than the bulrush by 1.75 units. ∎
2264:
are of opposite signs, at each step, one of the end-points will get closer to a root of
1305:
The first two iterations of the false position method. The red curve shows the function
6744:
809:
773:
722:
558:
6963:
Improved
Algorithms of Illinois-type for the Numerical Solution of Nonlinear Equations
6890:
Regola
Helcataym (vocabulo Arabo) che in nostra lingua vuol dire delle false Positioni
4561:
Suppose it is day 2. The club-rush is shorter than the bulrush by 1.5 units.
1301:
901:
estimate for the solution. There are many ways to arrive at a calculated-estimate and
7442:
7361:
7310:
7199:
7119:
2428:
2316:, which is used to pick the false position, does not improve as rapidly as possible.
1414:
1168:
740:
687:
54:
7007:
5213:{\displaystyle \ C(n)=\sum _{k=0}^{n-1}2^{k}={\frac {~~1-2^{n}}{\ 1-2\ }}=2^{n}-1\ }
4091:{\displaystyle \delta \equiv \min\{\kappa _{1}|b-a|^{\kappa _{2}},|x_{1/2}-x_{f}|\}}
1043:
that can be used to obtain approximations to such a root. One of the most common is
1328:
850:
817:
6832:. Eighth North African Meeting on the History of Arab Mathematics. Radès, Tunisia.
6799:
1743:{\displaystyle f(b_{k})+{\frac {f(b_{k})-f(a_{k})}{b_{k}-a_{k}}}(c_{k}-b_{k})=0.}
3428:{\displaystyle \kappa _{1}\in (0,\infty ),\kappa _{2}\in \left[1,1+\phi \right)}
1321:
1055:. These methods proceed by producing a sequence of shrinking intervals , at the
833:
769:
736:
20:
2655:-value of the retained end point in the next estimate computation when the new
7264:
7191:
6911:
6618:
6609:
After running this code, the final answer is approximately 0.865474033101614.
3344:
1048:
908:
Given an equation, move all of its terms to one side so that it has the form,
7111:
2690:)), meaning that the end point of the previous step will be retained. Hence:
2199:. This process is repeated until the root is approximated sufficiently well.
1565:{\displaystyle y-f(b_{k})={\frac {f(b_{k})-f(a_{k})}{b_{k}-a_{k}}}(x-b_{k}).}
2614:
instead of converging – and often do so under the same conditions that slow
828:
813:
805:
760:
wrote a now-lost treatise on the use of double false position, known as the
757:
753:
703:
448:
152:
Two basic types of false position method can be distinguished historically,
7207:
6847:
4417:, a root finding problem can be translated to modern language as follows:
7021:
King, Richard F. (October 1983). "Anderson-Bjorck for Linear
Sequences".
1179:
557:
would be memorized and carried out by rote. Indeed, the rule as given by
57:, which asks for a solution of (written in modern notation) the equation
47:
7042:
6999:
5837:, is an example of the Illinois algorithm. To find the positive number
4426:
785:
781:
711:
43:
1186:, the current bracketing interval is , then the new solution estimate
16:
Numerical method used to approximate solutions of univariate equations
4676:{\displaystyle B(n)=\sum _{i=1}^{n}3\cdot {\frac {1}{2^{i-1}}}\quad }
7175:
7103:
7034:
4585:
To understand this, we shall model the heights of the plants on day
3512:{\displaystyle n_{1/2}\equiv \lceil (b_{0}-a_{0})/2\epsilon \rceil }
1051:. Other methods are needed and one general class of methods are the
872:
appears as the
Latinized version of Rule of False as early as 1690.
6801:
The Nine
Chapters on the Mathematical Art: Companion and Commentary
6798:
Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999).
2573:{\displaystyle f(x)={\frac {1}{(x-1)^{2}}}+{\frac {1}{(x+1)^{2}}}.}
7087:
4568:
4377:
reliable method—and every method is less reliable than bisection.
3335:
For simple roots, Anderson–Björck performs very well in practice.
1424:
1300:
768:). The oldest surviving writing on double false position from the
7062:
Proceedings of 2011 World Congress on Engineering and Technology
6627:, another root-finding method based on the false position method
6621:, a variation with guaranteed minmax and superlinear convergence
846:
777:
7211:
1110:
has opposite signs. Under the continuity assumption, a root of
676:
double false position provides the exact solution, while for a
547:{\displaystyle x={\frac {b_{1}x_{2}-b_{2}x_{1}}{b_{1}-b_{2}}},}
247:. The correct answer is then found by proportional adjustment,
7157:
Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculation
5382:{\displaystyle \ F(n):=C(n)-B(n)={\frac {6}{2^{n}}}+2^{n}-7\ }
4160:{\displaystyle x_{\text{ITP}}\equiv x_{1/2}-\sigma \rho _{k}}
3850:{\displaystyle x_{f}\equiv {\frac {bf(a)-af(b)}{f(a)-f(b)}}}
3025:
down-weighting one of the endpoint values to force the next
3315:
1119:
is the point selected, then the smaller interval goes from
3970:{\displaystyle \sigma \equiv {\text{sign}}(x_{1/2}-x_{f})}
2018:
This last symmetrical form has a computational advantage:
6836:
http://facstaff.uindy.edu/~oaks/Biblio/COMHISMA8paper.doc
6750:
Mathematics and the Sciences of the Heavens and the Earth
6721:
A History of Algorithms: From the Pebble to the Microchip
222:
are known. The method begins by using a test input value
4754:{\displaystyle C(n)=\sum _{i=1}^{n}1\cdot 2^{i-1}\quad }
1285:, thereby guaranteeing convergence toward the solution.
5854:, the equation is transformed into a root-finding form
3702:
Calculate the bisection and the regula falsi points:
7086:
Oliveira, I. F. D.; Takahashi, R. H. C. (2020-12-06).
6904:
Elementary Numerical Analysis: an algorithmic approach
5010:
4968:
3592:
1099:-values, initially found by trial-and-error, at which
897:
used in iterative numerical approximation techniques.
443:
Double false position is mathematically equivalent to
113:(currently set to 4) by 3 and substitute again to get
82:. This is solved by false position. First, guess that
5606:
5539:
5500:
5437:
5398:
5291:
5233:
5084:
4876:
4812:
4771:
4691:
4606:
4507:
4465:
4333:
4173:
4109:
3985:
3918:
3867:
3765:
3708:
3676:
3638:
3590:
3570:
3525:
3441:
3355:
3162:
2865:
2699:
2486:
2328:
1771:
1610:
1436:
1202:
632:
460:
433:{\displaystyle f(x_{1})=b_{1},\qquad f(x_{2})=b_{2}.}
358:
299:
181:
4546:{\displaystyle (4+{\frac {8}{10}}+{\frac {6}{130}})}
2280:) in the interval, then one endpoint (the one where
7405:
7352:
7329:
7296:
7273:
7245:
3110:But, whereas the Illinois algorithm would multiply
1320:method calculates the new solution estimate as the
725:as a purely arithmetical algorithm. In the ancient
6888:, vol. I, Venice, 1556, p. fol. 238, v,
6653:
5817:
5582:
5525:
5480:
5423:
5381:
5275:
5212:
5069:
4851:
4798:
4753:
4675:
4545:
4490:
4454:that the club-rush becomes as tall as the bulrush.
4355:
4314:
4159:
4090:
3969:
3902:
3849:
3750:
3689:
3662:
3624:
3576:
3556:
3511:
3427:
3325:
3036:to occur on that side of the function. The factor
3014:
2845:
2572:
2384:
2305:). As a consequence, the linear approximation to
2007:
1742:
1564:
1253:
665:
546:
432:
338:
199:
6660:(2nd ed.), Addison Wesley Longman, p.
5950:/* a,b: endpoints of an interval where we search
4187:
3992:
3903:{\displaystyle x_{t}\equiv x_{f}+\sigma \delta }
2319:One example of this phenomenon is the function
1182:of the bracketing interval. That is, if at step
702:The simple false position technique is found in
7136:Burden, Richard L.; Faires, J. Douglas (2000).
7058:"A family of regula falsi root-finding methods"
6965:, Technical Report, University of Essex Press,
1590:-intercept of this line, that is, the value of
1254:{\displaystyle c_{k}={\frac {a_{k}+b_{k}}{2}}.}
954:and is a solution of the original equation. If
877:
6067:/* starting values at endpoints of interval */
3751:{\displaystyle x_{1/2}\equiv {\frac {a+b}{2}}}
3625:{\displaystyle {\tfrac {1}{2}}(1+{\sqrt {5}})}
2587:avoids this hypothetical convergence problem.
7223:
3142:, Anderson–Björck algorithm multiplies it by
229:, and finding the corresponding output value
8:
6923:
6921:
4085:
3995:
3506:
3463:
3103:. In this case, the new bracketing interval
6409:/* fc and fa have same sign, copy c to a */
6307:/* fc and fb have same sign, copy c to b */
3861:Perturb the estimator towards the center:
7230:
7216:
7208:
6823:
6821:
6493:/* fc * f_ very small (looks like zero) */
4103:Project the estimator to minmax interval:
2674:)) has the same sign as the previous one (
1178:, calculates the solution estimate as the
788:. He justified the technique by a formal,
53:As an example, consider problem 26 in the
7092:ACM Transactions on Mathematical Software
6970:
6906:(2nd ed.). McGraw-Hill. p. 40.
6695:, vol. II, Dover, pp. 437–441,
5953:e: half of upper bound for relative error
5756:
5735:
5713:
5689:
5673:
5657:
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5611:
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2870:
2864:
2831:
2809:
2789:
2781:
2768:
2749:
2736:
2716:
2713:
2704:
2698:
2558:
2536:
2524:
2502:
2485:
2367:
2351:
2327:
1990:
1968:
1947:
1931:
1915:
1899:
1892:
1877:
1855:
1837:
1824:
1817:
1808:
1789:
1776:
1770:
1725:
1712:
1696:
1683:
1668:
1646:
1633:
1621:
1609:
1550:
1528:
1515:
1500:
1478:
1465:
1453:
1435:
1339:-th iteration the bracketing interval is
1236:
1223:
1216:
1207:
1201:
923:is some function of the unknown variable
732:The Nine Chapters on the Mathematical Art
631:
570:Gesse at this woorke as happe doth leade.
532:
519:
507:
497:
484:
474:
467:
459:
421:
405:
385:
369:
357:
298:
180:
6804:. Oxford University Press. p. 358.
1601:, and substitute these values to obtain
1361:. Construct the line through the points
812:and balance-scale diagrams explained by
752:Between the 9th and 10th centuries, the
610:In crossewaies multiplye contrary kinde,
6956:
6954:
6686:
6684:
6682:
6680:
6641:
931:that satisfies this equation, that is,
6870:: CS1 maint: archived copy as title (
6863:
6647:
6645:
4765:For the sake of better notations, let
1417:or chord of the graph of the function
614:All truthe by falsehode for to fynde.
590:That truth by it will soone be founde.
5833:This example program, written in the
3154:has one of the two following values:
2385:{\displaystyle f(x)=2x^{3}-4x^{2}+3x}
796:, double false position was known as
690:that can be successively improved by
586:Suche falsehode is so good a grounde,
574:By chaunce to truthe you may procede.
7:
3670:the ITP method calculates the point
3557:{\displaystyle n_{0}\in [0,\infty )}
2276:is of constant sign (so there is no
2083:is calculated as above and then, if
1335:More precisely, suppose that in the
1008:are of opposite signs, then, by the
820:, all three being mathematicians of
6902:Conte, S.D.; Boor, Carl de (1965).
4491:{\displaystyle (2+{\frac {6}{13}})}
1309:and the blue lines are the secants.
1174:The simplest variation, called the
3548:
3378:
2651:The Illinois algorithm halves the
602:With to much ioyne to fewe againe,
582:Although no truthe therein be don.
578:And firste woorke by the question,
14:
5583:{\displaystyle \ F(x_{2})=F(3)\ }
5481:{\displaystyle \ F(x_{1})=F(2)\ }
4356:{\displaystyle f(x_{\text{ITP}})}
1413:, as illustrated. This line is a
893:, double false position became a
831:) devoted Chapter 13 of his book
133:, verifying that the solution is
7428:Sidi's generalized secant method
6715:Chabert, Jean-Luc, ed. (2012) .
5597:Estimated root (1st iteration):
4806:Rewrite the plant height series
4421:Excess And Deficit Problem #11:
845:method that he had learned from
808:, such as a verse attributed to
606:To to fewe adde to manye plaine.
7418:Inverse quadratic interpolation
5276:{\displaystyle \ (C(n)-B(n))\ }
4852:{\displaystyle \ B(n),\ C(n)\ }
4750:
4672:
790:Euclidean-style geometric proof
721:Double false position arose in
598:From to fewe take to fewe also.
394:
6717:"3. Methods of False Position"
5956:m: maximal number of iteration
5741:
5728:
5719:
5706:
5695:
5682:
5663:
5650:
5616:
5574:
5568:
5559:
5546:
5472:
5466:
5457:
5444:
5334:
5328:
5319:
5313:
5304:
5298:
5267:
5264:
5258:
5249:
5243:
5237:
5097:
5091:
4980:
4964:
4889:
4883:
4843:
4837:
4825:
4819:
4701:
4695:
4616:
4610:
4540:
4508:
4485:
4466:
4350:
4337:
4303:
4267:
4081:
4045:
4024:
4009:
3964:
3930:
3841:
3835:
3826:
3820:
3812:
3806:
3794:
3788:
3690:{\displaystyle x_{\text{ITP}}}
3619:
3603:
3551:
3539:
3492:
3466:
3381:
3369:
3228:
3215:
3207:
3194:
3003:
2990:
2971:
2958:
2940:
2927:
2898:
2885:
2837:
2824:
2815:
2802:
2774:
2761:
2742:
2729:
2555:
2542:
2521:
2508:
2496:
2490:
2338:
2332:
2270:. If the second derivative of
1996:
1983:
1974:
1961:
1953:
1940:
1921:
1908:
1883:
1870:
1861:
1848:
1814:
1801:
1731:
1705:
1674:
1661:
1652:
1639:
1627:
1614:
1556:
1537:
1506:
1493:
1484:
1471:
1459:
1446:
1145:. In the improbable case that
1134:has the sign opposite that of
642:
636:
411:
398:
375:
362:
309:
303:
1:
7140:(7th ed.). Brooks/Cole.
4445:of the previous day's growth.
2218:Since the initial end-points
2021:As a solution is approached,
1095:These methods start with two
6723:. Springer. pp. 86–91.
4581:), and the approximated root
4407:Example: Growth of a bulrush
1091:Two-point bracketing methods
1053:two-point bracketing methods
339:{\displaystyle f(x)=ax+c=0,}
6937:. Dover. pp. 231–232.
6779:www-groups.dcs.st-and.ac.uk
5526:{\displaystyle \ x_{2}=3\ }
5424:{\displaystyle \ x_{1}=2\ }
4396:has become very small, and
1427:, its equation is given by
1018:has a root in the interval
962:and there exist two points
794:medieval Muslim mathematics
7470:
7247:Bracketing (no derivative)
7023:Mathematics of Computation
4327:The value of the function
3663:{\displaystyle j=0,1,2...}
3342:
2438:for the function given by
1753:Solving this equation for
1010:intermediate value theorem
792:. Within the tradition of
666:{\displaystyle f(x)=ax+c,}
594:From many bate to many mo,
7192:10.1163/24519197-BJA10007
4799:{\displaystyle \ k=i-1~.}
3068:Anderson–Björck algorithm
2630:have failed to converge.
7056:Galdino, Sérgio (2011).
6828:Schwartz, R. K. (2004).
6656:A History of Mathematics
6652:Katz, Victor J. (1998),
5875:
4577:, its exact root (point
4368:Practical considerations
3056:The above adjustment to
2628:Successive-Substitutions
2606:: Other than bisection,
2115:have the same sign, set
89:to obtain, on the left,
31:method of false position
7454:Latin words and phrases
7449:Root-finding algorithms
7397:Splitting circle method
7382:Jenkins–Traub algorithm
7239:Root-finding algorithms
7180:Philological Encounters
6931:; Björck, Åke (2003) .
3697:following three steps:
1297:(false position) method
1041:root-finding algorithms
905:provides one of these.
7387:Lehmer–Schur algorithm
7174:Roberts, A.M. (2020).
6834:Available online at:
6693:History of Mathematics
6691:Smith, D. E. (1958) ,
5835:C programming language
5819:
5584:
5527:
5482:
5425:
5383:
5277:
5214:
5129:
5071:
4921:
4853:
4800:
4755:
4727:
4677:
4642:
4593:= 1, 2, 3...) after a
4582:
4567:
4547:
4492:
4357:
4316:
4161:
4092:
3971:
3904:
3851:
3752:
3691:
3664:
3626:
3578:
3558:
3513:
3429:
3327:
3016:
2847:
2647:The Illinois algorithm
2574:
2386:
2009:
1744:
1566:
1310:
1255:
1123:to the endpoint where
895:root-finding algorithm
882:
762:Book of the Two Errors
750:
708:Babylonian mathematics
667:
548:
434:
340:
201:
7413:Fixed-point iteration
7155:Sigler, L.E. (2002).
5820:
5585:
5528:
5483:
5426:
5384:
5278:
5215:
5103:
5072:
4895:
4854:
4801:
4756:
4707:
4678:
4622:
4572:
4548:
4493:
4419:
4411:In chapter 7 of
4358:
4317:
4162:
4093:
3972:
3905:
3852:
3753:
3692:
3665:
3627:
3584:is the golden ration
3579:
3577:{\displaystyle \phi }
3559:
3514:
3430:
3328:
3087:has the same sign as
3017:
2848:
2575:
2387:
2236:are chosen such that
2010:
1745:
1567:
1304:
1256:
745:
706:tablets from ancient
668:
549:
435:
341:
282:Double false position
202:
200:{\displaystyle ax=b,}
164:Simple false position
158:double false position
154:simple false position
35:false position method
7372:Durand–Kerner method
7316:Newton–Krylov method
6961:Ford, J. A. (1995),
5604:
5537:
5498:
5435:
5396:
5289:
5231:
5227:to find the root of
5082:
4874:
4810:
4769:
4689:
4604:
4505:
4463:
4331:
4171:
4107:
3983:
3916:
3865:
3763:
3706:
3674:
3636:
3632:, in each iteration
3588:
3568:
3523:
3439:
3353:
3160:
3072:Suppose that in the
2863:
2697:
2484:
2326:
2070:At iteration number
1769:
1608:
1434:
1200:
860:Summa de arithmetica
776:(10th century), an
727:Chinese mathematical
716:Egyptian mathematics
630:
458:
445:linear interpolation
356:
349:if it is known that
297:
236:by multiplication:
179:
148:Two historical types
7321:Steffensen's method
7159:. Springer-Verlag.
4390:When the change in
2585:method of bisection
2406:rate of convergence
1081:contains a root of
1059:th step, such that
960:continuous function
802:Quranic inheritance
780:mathematician from
7354:Polynomial methods
7138:Numerical Analysis
7000:10.1007/BF01934364
6929:Dahlquist, Germund
5815:
5580:
5523:
5478:
5421:
5379:
5273:
5210:
5067:
5019:
4977:
4849:
4796:
4751:
4673:
4583:
4543:
4488:
4353:
4312:
4157:
4088:
3967:
3900:
3847:
3748:
3687:
3660:
3622:
3601:
3574:
3554:
3509:
3425:
3323:
3321:
3314:
3062:Illinois algorithm
3012:
2843:
2570:
2382:
2005:
1740:
1562:
1311:
1264:This ensures that
1251:
891:numerical analysis
885:Numerical analysis
849:sources. In 1494,
827:Leonardo of Pisa (
798:hisāb al-khaṭāʾayn
766:Kitāb al-khaṭāʾayn
663:
544:
430:
336:
197:
7436:
7435:
7392:Laguerre's method
7367:Bairstow's method
6934:Numerical Methods
6811:978-7-03-006101-0
6760:978-0-521-05801-8
6730:978-3-642-18192-4
6702:978-0-486-20430-7
6671:978-0-321-01618-8
5814:
5808:
5802:
5798:
5785:
5761:
5755:
5749:
5745:
5700:
5636:
5630:
5624:
5619:
5609:
5579:
5542:
5522:
5503:
5477:
5440:
5420:
5401:
5378:
5355:
5294:
5272:
5236:
5209:
5186:
5184:
5172:
5151:
5148:
5087:
5060:
5022:
5018:
4976:
4943:
4879:
4848:
4833:
4815:
4792:
4774:
4670:
4573:Plot of function
4538:
4525:
4483:
4414:The Nine Chapters
4347:
4261:
4117:
3928:
3845:
3746:
3684:
3617:
3600:
3310:
3303:
3272:
3232:
3007:
2985:
2922:
2841:
2797:
2724:
2659:-value (that is,
2565:
2531:
2296: ) = −sign(
2000:
1887:
1703:
1535:
1246:
1193:is obtained by,
839:regulis elchatayn
539:
7461:
7377:Graeffe's method
7306:Broyden's method
7255:Bisection method
7232:
7225:
7218:
7209:
7203:
7170:
7151:
7124:
7123:
7083:
7077:
7076:
7074:
7072:
7053:
7047:
7046:
7029:(164): 591–596.
7018:
7012:
7011:
6983:
6977:
6976:
6974:
6958:
6949:
6948:
6925:
6916:
6915:
6899:
6893:
6892:
6886:General Trattato
6882:
6876:
6875:
6869:
6861:
6859:
6858:
6852:
6846:. Archived from
6845:
6833:
6825:
6816:
6815:
6795:
6789:
6788:
6786:
6785:
6771:
6765:
6764:
6741:
6735:
6734:
6712:
6706:
6705:
6688:
6675:
6674:
6659:
6649:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6551:
6548:
6545:
6542:
6539:
6536:
6533:
6530:
6527:
6524:
6521:
6518:
6515:
6512:
6509:
6506:
6503:
6500:
6497:
6494:
6491:
6488:
6485:
6482:
6479:
6476:
6473:
6470:
6467:
6464:
6461:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6437:
6434:
6431:
6428:
6425:
6422:
6419:
6416:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6392:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6359:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6326:
6323:
6320:
6317:
6314:
6311:
6308:
6305:
6302:
6299:
6296:
6293:
6290:
6287:
6284:
6281:
6278:
6275:
6272:
6269:
6266:
6263:
6260:
6257:
6254:
6251:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6209:
6206:
6203:
6200:
6197:
6194:
6191:
6188:
6185:
6182:
6179:
6176:
6173:
6170:
6167:
6164:
6161:
6158:
6155:
6152:
6149:
6146:
6143:
6140:
6137:
6134:
6131:
6128:
6125:
6122:
6119:
6116:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6086:
6083:
6080:
6077:
6074:
6071:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6026:
6023:
6020:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5963:
5960:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5903:
5900:
5897:
5894:
5891:
5888:
5885:
5882:
5879:
5872:
5853:
5842:
5824:
5822:
5821:
5816:
5812:
5806:
5800:
5799:
5797:
5786:
5783:
5759:
5757:
5753:
5747:
5746:
5744:
5740:
5739:
5718:
5717:
5701:
5698:
5694:
5693:
5678:
5677:
5662:
5661:
5646:
5645:
5634:
5632:
5628:
5622:
5621:
5620:
5612:
5607:
5594:(the "excess").
5593:
5589:
5587:
5586:
5581:
5577:
5558:
5557:
5540:
5532:
5530:
5529:
5524:
5520:
5513:
5512:
5501:
5492:(the "deficit").
5491:
5487:
5485:
5484:
5479:
5475:
5456:
5455:
5438:
5430:
5428:
5427:
5422:
5418:
5411:
5410:
5399:
5388:
5386:
5385:
5380:
5376:
5369:
5368:
5356:
5354:
5353:
5341:
5292:
5282:
5280:
5279:
5274:
5270:
5234:
5219:
5217:
5216:
5211:
5207:
5200:
5199:
5187:
5185:
5182:
5170:
5168:
5167:
5166:
5149:
5146:
5144:
5139:
5138:
5128:
5117:
5085:
5076:
5074:
5073:
5068:
5066:
5062:
5061:
5059:
5058:
5046:
5027:
5023:
5021:
5020:
5011:
5001:
5000:
4999:
4978:
4969:
4956:
4944:
4942:
4941:
4929:
4920:
4909:
4877:
4862:
4858:
4856:
4855:
4850:
4846:
4831:
4813:
4805:
4803:
4802:
4797:
4790:
4772:
4760:
4758:
4757:
4752:
4749:
4748:
4726:
4721:
4682:
4680:
4679:
4674:
4671:
4669:
4668:
4650:
4641:
4636:
4595:geometric series
4592:
4588:
4580:
4576:
4554:
4552:
4550:
4549:
4544:
4539:
4531:
4526:
4518:
4499:
4497:
4495:
4494:
4489:
4484:
4476:
4444:
4442:
4441:
4438:
4435:
4401:
4395:
4362:
4360:
4359:
4354:
4349:
4348:
4345:
4321:
4319:
4318:
4313:
4311:
4307:
4306:
4301:
4300:
4296:
4280:
4279:
4270:
4262:
4257:
4246:
4241:
4240:
4233:
4232:
4220:
4219:
4215:
4183:
4182:
4166:
4164:
4163:
4158:
4156:
4155:
4140:
4139:
4135:
4119:
4118:
4115:
4097:
4095:
4094:
4089:
4084:
4079:
4078:
4066:
4065:
4061:
4048:
4040:
4039:
4038:
4037:
4027:
4012:
4007:
4006:
3976:
3974:
3973:
3968:
3963:
3962:
3950:
3949:
3945:
3929:
3926:
3909:
3907:
3906:
3901:
3890:
3889:
3877:
3876:
3856:
3854:
3853:
3848:
3846:
3844:
3815:
3780:
3775:
3774:
3757:
3755:
3754:
3749:
3747:
3742:
3731:
3726:
3725:
3721:
3696:
3694:
3693:
3688:
3686:
3685:
3682:
3669:
3667:
3666:
3661:
3631:
3629:
3628:
3623:
3618:
3613:
3602:
3593:
3583:
3581:
3580:
3575:
3563:
3561:
3560:
3555:
3535:
3534:
3518:
3516:
3515:
3510:
3499:
3491:
3490:
3478:
3477:
3459:
3458:
3454:
3434:
3432:
3431:
3426:
3424:
3420:
3396:
3395:
3365:
3364:
3332:
3330:
3329:
3324:
3322:
3318:
3317:
3311:
3308:
3304:
3296:
3281:
3273:
3270:
3266:
3233:
3231:
3227:
3226:
3210:
3206:
3205:
3189:
3174:
3153:
3147:
3141:
3139:
3138:
3135:
3132:
3125:
3106:
3102:
3086:
3075:
3051:
3049:
3048:
3045:
3042:
3035:
3021:
3019:
3018:
3013:
3008:
3006:
3002:
3001:
2986:
2978:
2970:
2969:
2953:
2952:
2951:
2939:
2938:
2923:
2915:
2910:
2909:
2897:
2896:
2880:
2875:
2874:
2852:
2850:
2849:
2844:
2842:
2840:
2836:
2835:
2814:
2813:
2798:
2790:
2787:
2786:
2785:
2773:
2772:
2754:
2753:
2741:
2740:
2725:
2717:
2714:
2709:
2708:
2689:
2673:
2658:
2654:
2591:Improvements in
2579:
2577:
2576:
2571:
2566:
2564:
2563:
2562:
2537:
2532:
2530:
2529:
2528:
2503:
2476:
2469:
2462:
2455:
2437:
2426:
2417:
2403:
2402:
2391:
2389:
2388:
2383:
2372:
2371:
2356:
2355:
2315:
2304:
2302:
2288:bisection method
2285:
2278:inflection point
2275:
2269:
2263:
2249:
2235:
2226:
2198:
2177:
2157:, otherwise set
2156:
2135:
2114:
2098:
2082:
2075:
2066:
2050:
2034:
2027:
2014:
2012:
2011:
2006:
2001:
1999:
1995:
1994:
1973:
1972:
1956:
1952:
1951:
1936:
1935:
1920:
1919:
1904:
1903:
1893:
1888:
1886:
1882:
1881:
1860:
1859:
1843:
1842:
1841:
1829:
1828:
1818:
1813:
1812:
1794:
1793:
1781:
1780:
1749:
1747:
1746:
1741:
1730:
1729:
1717:
1716:
1704:
1702:
1701:
1700:
1688:
1687:
1677:
1673:
1672:
1651:
1650:
1634:
1626:
1625:
1600:
1593:
1589:
1585:
1571:
1569:
1568:
1563:
1555:
1554:
1536:
1534:
1533:
1532:
1520:
1519:
1509:
1505:
1504:
1483:
1482:
1466:
1458:
1457:
1425:point-slope form
1422:
1412:
1386:
1360:
1338:
1324:
1308:
1284:
1277:
1270:
1260:
1258:
1257:
1252:
1247:
1242:
1241:
1240:
1228:
1227:
1217:
1212:
1211:
1192:
1185:
1176:bisection method
1155:
1144:
1133:
1122:
1118:
1113:
1109:
1098:
1086:
1080:
1058:
1035:
1017:
1007:
993:
979:
970:
957:
953:
950:of the function
941:
930:
926:
922:
918:
685:
672:
670:
669:
664:
615:
611:
607:
603:
599:
595:
591:
587:
583:
579:
575:
571:
553:
551:
550:
545:
540:
538:
537:
536:
524:
523:
513:
512:
511:
502:
501:
489:
488:
479:
478:
468:
439:
437:
436:
431:
426:
425:
410:
409:
390:
389:
374:
373:
345:
343:
342:
337:
289:
277:
272:
270:
269:
263:
260:
246:
235:
228:
221:
215:
206:
204:
203:
198:
171:
139:
132:
130:
128:
127:
124:
121:
112:
108:
106:
104:
103:
100:
97:
88:
81:
79:
77:
76:
73:
70:
7469:
7468:
7464:
7463:
7462:
7460:
7459:
7458:
7439:
7438:
7437:
7432:
7423:Muller's method
7401:
7348:
7344:Ridders' method
7325:
7292:
7288:Halley's method
7283:Newton's method
7269:
7241:
7236:
7173:
7167:
7154:
7148:
7135:
7132:
7130:Further reading
7127:
7104:10.1145/3423597
7098:(1): 5:1–5:24.
7085:
7084:
7080:
7070:
7068:
7055:
7054:
7050:
7035:10.2307/2007695
7020:
7019:
7015:
6985:
6984:
6980:
6960:
6959:
6952:
6945:
6927:
6926:
6919:
6901:
6900:
6896:
6884:
6883:
6879:
6862:
6856:
6854:
6850:
6843:
6841:"Archived copy"
6839:
6827:
6826:
6819:
6812:
6797:
6796:
6792:
6783:
6781:
6775:"Nine chapters"
6773:
6772:
6768:
6761:
6745:Needham, Joseph
6743:
6742:
6738:
6731:
6714:
6713:
6709:
6703:
6690:
6689:
6678:
6672:
6651:
6650:
6643:
6639:
6625:Ridders' method
6615:
6607:
6606:
6603:
6600:
6597:
6594:
6591:
6588:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6555:
6552:
6549:
6546:
6543:
6540:
6537:
6534:
6531:
6528:
6525:
6522:
6519:
6516:
6513:
6510:
6507:
6504:
6501:
6498:
6495:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6468:
6465:
6462:
6459:
6456:
6453:
6450:
6447:
6444:
6441:
6438:
6435:
6432:
6429:
6426:
6423:
6420:
6417:
6414:
6411:
6408:
6405:
6402:
6399:
6396:
6393:
6390:
6387:
6384:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6339:
6336:
6333:
6330:
6327:
6324:
6321:
6318:
6315:
6312:
6309:
6306:
6303:
6300:
6297:
6294:
6291:
6288:
6285:
6282:
6279:
6276:
6273:
6270:
6267:
6264:
6261:
6258:
6255:
6252:
6249:
6246:
6243:
6240:
6237:
6234:
6231:
6228:
6225:
6222:
6219:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6186:
6183:
6180:
6177:
6174:
6171:
6168:
6165:
6162:
6159:
6156:
6153:
6150:
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6120:
6117:
6114:
6111:
6108:
6105:
6102:
6099:
6096:
6093:
6090:
6087:
6084:
6081:
6078:
6075:
6072:
6069:
6066:
6063:
6060:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6027:
6024:
6021:
6018:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5994:
5991:
5988:
5985:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5904:
5901:
5898:
5895:
5892:
5889:
5886:
5883:
5881:<stdio.h>
5880:
5877:
5855:
5844:
5838:
5831:
5787:
5758:
5731:
5709:
5702:
5685:
5669:
5653:
5637:
5633:
5602:
5601:
5591:
5549:
5535:
5534:
5504:
5496:
5495:
5493:
5489:
5447:
5433:
5432:
5402:
5394:
5393:
5360:
5345:
5287:
5286:
5229:
5228:
5191:
5169:
5158:
5145:
5130:
5080:
5079:
5050:
5038:
5034:
5002:
4979:
4957:
4951:
4933:
4872:
4871:
4863:and invoke the
4860:
4808:
4807:
4767:
4766:
4734:
4687:
4686:
4654:
4602:
4601:
4590:
4586:
4578:
4574:
4503:
4502:
4501:
4461:
4460:
4459:
4439:
4436:
4433:
4432:
4430:
4409:
4397:
4391:
4370:
4340:
4329:
4328:
4284:
4271:
4247:
4224:
4203:
4198:
4194:
4190:
4174:
4169:
4168:
4147:
4123:
4110:
4105:
4104:
4070:
4049:
4029:
4022:
3998:
3981:
3980:
3954:
3933:
3914:
3913:
3881:
3868:
3863:
3862:
3816:
3781:
3766:
3761:
3760:
3732:
3709:
3704:
3703:
3677:
3672:
3671:
3634:
3633:
3586:
3585:
3566:
3565:
3526:
3521:
3520:
3482:
3469:
3442:
3437:
3436:
3404:
3400:
3387:
3356:
3351:
3350:
3347:
3341:
3320:
3319:
3313:
3312:
3305:
3292:
3291:
3274:
3267:
3259:
3252:
3244:
3238:
3237:
3218:
3211:
3197:
3190:
3175:
3167:
3158:
3157:
3149:
3143:
3136:
3133:
3130:
3129:
3127:
3123:
3111:
3104:
3100:
3088:
3085:
3077:
3073:
3070:
3046:
3043:
3040:
3039:
3037:
3034:
3026:
2993:
2961:
2954:
2943:
2930:
2901:
2888:
2881:
2866:
2861:
2860:
2827:
2805:
2788:
2777:
2764:
2745:
2732:
2715:
2700:
2695:
2694:
2688:
2675:
2672:
2660:
2656:
2652:
2649:
2596:
2554:
2541:
2520:
2507:
2482:
2481:
2471:
2464:
2457:
2439:
2432:
2419:
2412:
2400:
2396:
2363:
2347:
2324:
2323:
2306:
2300:
2291:
2281:
2271:
2265:
2261:
2251:
2247:
2237:
2234:
2228:
2225:
2219:
2216:
2197:
2188:
2179:
2176:
2167:
2158:
2155:
2146:
2137:
2134:
2125:
2116:
2112:
2100:
2096:
2084:
2081:
2077:
2071:
2064:
2052:
2048:
2036:
2033:
2029:
2026:
2022:
1986:
1964:
1957:
1943:
1927:
1911:
1895:
1894:
1873:
1851:
1844:
1833:
1820:
1819:
1804:
1785:
1772:
1767:
1766:
1761:
1721:
1708:
1692:
1679:
1678:
1664:
1642:
1635:
1617:
1606:
1605:
1595:
1591:
1587:
1584:
1576:
1546:
1524:
1511:
1510:
1496:
1474:
1467:
1449:
1432:
1431:
1418:
1410:
1397:
1388:
1384:
1371:
1362:
1358:
1349:
1340:
1336:
1322:
1306:
1299:
1283:
1279:
1276:
1272:
1269:
1265:
1232:
1219:
1218:
1203:
1198:
1197:
1191:
1187:
1183:
1165:Newton's method
1146:
1135:
1124:
1120:
1116:
1111:
1100:
1096:
1093:
1082:
1078:
1069:
1060:
1056:
1045:Newton's method
1039:There are many
1033:
1026:
1019:
1013:
1012:, the function
1005:
995:
991:
981:
978:
972:
969:
963:
955:
951:
932:
928:
924:
920:
909:
887:
700:
686:it provides an
681:
628:
627:
621:linear function
617:
613:
612:
609:
608:
605:
604:
601:
600:
597:
596:
593:
592:
589:
588:
585:
584:
581:
580:
577:
576:
573:
572:
569:
563:Ground of Artes
528:
515:
514:
503:
493:
480:
470:
469:
456:
455:
417:
401:
381:
365:
354:
353:
295:
294:
285:
264:
261:
256:
255:
253:
248:
237:
230:
223:
217:
211:
177:
176:
167:
150:
134:
125:
122:
119:
118:
116:
114:
110:
101:
98:
95:
94:
92:
90:
83:
74:
71:
66:
65:
63:
58:
46:and the use of
39:trial and error
17:
12:
11:
5:
7467:
7465:
7457:
7456:
7451:
7441:
7440:
7434:
7433:
7431:
7430:
7425:
7420:
7415:
7409:
7407:
7403:
7402:
7400:
7399:
7394:
7389:
7384:
7379:
7374:
7369:
7364:
7358:
7356:
7350:
7349:
7347:
7346:
7341:
7339:Brent's method
7335:
7333:
7331:Hybrid methods
7327:
7326:
7324:
7323:
7318:
7313:
7308:
7302:
7300:
7294:
7293:
7291:
7290:
7285:
7279:
7277:
7271:
7270:
7268:
7267:
7262:
7257:
7251:
7249:
7243:
7242:
7237:
7235:
7234:
7227:
7220:
7212:
7206:
7205:
7171:
7165:
7152:
7146:
7131:
7128:
7126:
7125:
7078:
7048:
7013:
6994:(2): 168–174.
6978:
6972:10.1.1.53.8676
6950:
6944:978-0486428079
6943:
6917:
6894:
6877:
6817:
6810:
6790:
6766:
6759:
6736:
6729:
6707:
6701:
6676:
6670:
6640:
6638:
6635:
6634:
6633:
6631:Brent's method
6628:
6622:
6614:
6611:
5887:<math.h>
5876:
5830:
5827:
5826:
5825:
5811:
5805:
5796:
5793:
5790:
5782:
5779:
5776:
5773:
5770:
5767:
5764:
5752:
5743:
5738:
5734:
5730:
5727:
5724:
5721:
5716:
5712:
5708:
5705:
5697:
5692:
5688:
5684:
5681:
5676:
5672:
5668:
5665:
5660:
5656:
5652:
5649:
5644:
5640:
5627:
5618:
5615:
5576:
5573:
5570:
5567:
5564:
5561:
5556:
5552:
5548:
5545:
5519:
5516:
5511:
5507:
5474:
5471:
5468:
5465:
5462:
5459:
5454:
5450:
5446:
5443:
5417:
5414:
5409:
5405:
5390:
5389:
5375:
5372:
5367:
5363:
5359:
5352:
5348:
5344:
5339:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5306:
5303:
5300:
5297:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5221:
5220:
5206:
5203:
5198:
5194:
5190:
5181:
5178:
5175:
5165:
5161:
5157:
5154:
5142:
5137:
5133:
5127:
5124:
5121:
5116:
5113:
5110:
5106:
5102:
5099:
5096:
5093:
5090:
5077:
5065:
5057:
5053:
5049:
5044:
5041:
5037:
5033:
5030:
5026:
5017:
5014:
5008:
5005:
4998:
4995:
4992:
4989:
4986:
4982:
4975:
4972:
4966:
4963:
4960:
4954:
4950:
4947:
4940:
4936:
4932:
4927:
4924:
4919:
4916:
4913:
4908:
4905:
4902:
4898:
4894:
4891:
4888:
4885:
4882:
4845:
4842:
4839:
4836:
4830:
4827:
4824:
4821:
4818:
4795:
4789:
4786:
4783:
4780:
4777:
4763:
4762:
4747:
4744:
4741:
4737:
4733:
4730:
4725:
4720:
4717:
4714:
4710:
4706:
4703:
4700:
4697:
4694:
4684:
4667:
4664:
4661:
4657:
4653:
4648:
4645:
4640:
4635:
4632:
4629:
4625:
4621:
4618:
4615:
4612:
4609:
4566:
4565:
4562:
4542:
4537:
4534:
4529:
4524:
4521:
4516:
4513:
4510:
4500:the height is
4487:
4482:
4479:
4474:
4471:
4468:
4456:
4455:
4451:Find the time
4449:
4446:
4408:
4405:
4369:
4366:
4352:
4343:
4339:
4336:
4325:
4324:
4310:
4305:
4299:
4295:
4291:
4287:
4283:
4278:
4274:
4269:
4265:
4260:
4256:
4253:
4250:
4244:
4239:
4236:
4231:
4227:
4223:
4218:
4214:
4210:
4206:
4201:
4197:
4193:
4189:
4186:
4181:
4177:
4154:
4150:
4146:
4143:
4138:
4134:
4130:
4126:
4122:
4113:
4100:
4087:
4083:
4077:
4073:
4069:
4064:
4060:
4056:
4052:
4047:
4043:
4036:
4032:
4026:
4021:
4018:
4015:
4011:
4005:
4001:
3997:
3994:
3991:
3988:
3966:
3961:
3957:
3953:
3948:
3944:
3940:
3936:
3932:
3924:
3921:
3899:
3896:
3893:
3888:
3884:
3880:
3875:
3871:
3858:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3778:
3773:
3769:
3745:
3741:
3738:
3735:
3729:
3724:
3720:
3716:
3712:
3680:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3621:
3616:
3611:
3608:
3605:
3599:
3596:
3573:
3553:
3550:
3547:
3544:
3541:
3538:
3533:
3529:
3508:
3505:
3502:
3498:
3494:
3489:
3485:
3481:
3476:
3472:
3468:
3465:
3462:
3457:
3453:
3449:
3445:
3423:
3419:
3416:
3413:
3410:
3407:
3403:
3399:
3394:
3390:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3363:
3359:
3343:Main article:
3340:
3337:
3316:
3306:
3302:
3299:
3294:
3293:
3290:
3287:
3284:
3280:
3277:
3268:
3265:
3262:
3258:
3257:
3255:
3250:
3247:
3245:
3243:
3240:
3239:
3236:
3230:
3225:
3221:
3217:
3214:
3209:
3204:
3200:
3196:
3193:
3187:
3184:
3181:
3178:
3176:
3173:
3170:
3166:
3165:
3119:
3096:
3081:
3069:
3066:
3060:is called the
3030:
3023:
3022:
3011:
3005:
3000:
2996:
2992:
2989:
2984:
2981:
2976:
2973:
2968:
2964:
2960:
2957:
2950:
2946:
2942:
2937:
2933:
2929:
2926:
2921:
2918:
2913:
2908:
2904:
2900:
2895:
2891:
2887:
2884:
2878:
2873:
2869:
2854:
2853:
2839:
2834:
2830:
2826:
2823:
2820:
2817:
2812:
2808:
2804:
2801:
2796:
2793:
2784:
2780:
2776:
2771:
2767:
2763:
2760:
2757:
2752:
2748:
2744:
2739:
2735:
2731:
2728:
2723:
2720:
2712:
2707:
2703:
2683:
2668:
2648:
2645:
2634:Regula falsi's
2616:regula falsi's
2595:
2589:
2581:
2580:
2569:
2561:
2557:
2553:
2550:
2547:
2544:
2540:
2535:
2527:
2523:
2519:
2516:
2513:
2510:
2506:
2501:
2498:
2495:
2492:
2489:
2468: (0) = 5
2393:
2392:
2381:
2378:
2375:
2370:
2366:
2362:
2359:
2354:
2350:
2346:
2343:
2340:
2337:
2334:
2331:
2259:
2245:
2232:
2223:
2215:
2212:
2206:The fact that
2193:
2183:
2172:
2162:
2151:
2141:
2130:
2120:
2108:
2092:
2079:
2060:
2044:
2031:
2024:
2016:
2015:
2004:
1998:
1993:
1989:
1985:
1982:
1979:
1976:
1971:
1967:
1963:
1960:
1955:
1950:
1946:
1942:
1939:
1934:
1930:
1926:
1923:
1918:
1914:
1910:
1907:
1902:
1898:
1891:
1885:
1880:
1876:
1872:
1869:
1866:
1863:
1858:
1854:
1850:
1847:
1840:
1836:
1832:
1827:
1823:
1816:
1811:
1807:
1803:
1800:
1797:
1792:
1788:
1784:
1779:
1775:
1757:
1751:
1750:
1739:
1736:
1733:
1728:
1724:
1720:
1715:
1711:
1707:
1699:
1695:
1691:
1686:
1682:
1676:
1671:
1667:
1663:
1660:
1657:
1654:
1649:
1645:
1641:
1638:
1632:
1629:
1624:
1620:
1616:
1613:
1580:
1573:
1572:
1561:
1558:
1553:
1549:
1545:
1542:
1539:
1531:
1527:
1523:
1518:
1514:
1508:
1503:
1499:
1495:
1492:
1489:
1486:
1481:
1477:
1473:
1470:
1464:
1461:
1456:
1452:
1448:
1445:
1442:
1439:
1406:
1393:
1380:
1367:
1354:
1345:
1298:
1291:
1281:
1274:
1267:
1262:
1261:
1250:
1245:
1239:
1235:
1231:
1226:
1222:
1215:
1210:
1206:
1189:
1092:
1089:
1074:
1065:
1031:
1024:
1003:
989:
976:
967:
942:, is called a
886:
883:
853:used the term
810:Ibn al-Yasamin
774:Qusta ibn Luqa
756:mathematician
723:late antiquity
699:
696:
674:
673:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
619:For an affine
567:
565:(c. 1542) is:
559:Robert Recorde
555:
554:
543:
535:
531:
527:
522:
518:
510:
506:
500:
496:
492:
487:
483:
477:
473:
466:
463:
441:
440:
429:
424:
420:
416:
413:
408:
404:
400:
397:
393:
388:
384:
380:
377:
372:
368:
364:
361:
347:
346:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
208:
207:
196:
193:
190:
187:
184:
149:
146:
15:
13:
10:
9:
6:
4:
3:
2:
7466:
7455:
7452:
7450:
7447:
7446:
7444:
7429:
7426:
7424:
7421:
7419:
7416:
7414:
7411:
7410:
7408:
7406:Other methods
7404:
7398:
7395:
7393:
7390:
7388:
7385:
7383:
7380:
7378:
7375:
7373:
7370:
7368:
7365:
7363:
7362:Aberth method
7360:
7359:
7357:
7355:
7351:
7345:
7342:
7340:
7337:
7336:
7334:
7332:
7328:
7322:
7319:
7317:
7314:
7312:
7311:Secant method
7309:
7307:
7304:
7303:
7301:
7299:
7295:
7289:
7286:
7284:
7281:
7280:
7278:
7276:
7272:
7266:
7263:
7261:
7258:
7256:
7253:
7252:
7250:
7248:
7244:
7240:
7233:
7228:
7226:
7221:
7219:
7214:
7213:
7210:
7201:
7197:
7193:
7189:
7185:
7181:
7177:
7172:
7168:
7166:0-387-40737-5
7162:
7158:
7153:
7149:
7147:0-534-38216-9
7143:
7139:
7134:
7133:
7129:
7121:
7117:
7113:
7109:
7105:
7101:
7097:
7093:
7089:
7082:
7079:
7067:
7063:
7059:
7052:
7049:
7044:
7040:
7036:
7032:
7028:
7024:
7017:
7014:
7009:
7005:
7001:
6997:
6993:
6989:
6982:
6979:
6973:
6968:
6964:
6957:
6955:
6951:
6946:
6940:
6936:
6935:
6930:
6924:
6922:
6918:
6913:
6909:
6905:
6898:
6895:
6891:
6887:
6881:
6878:
6873:
6867:
6853:on 2014-05-16
6849:
6842:
6837:
6831:
6824:
6822:
6818:
6813:
6807:
6803:
6802:
6794:
6791:
6780:
6776:
6770:
6767:
6762:
6756:
6752:
6751:
6746:
6740:
6737:
6732:
6726:
6722:
6718:
6711:
6708:
6704:
6698:
6694:
6687:
6685:
6683:
6681:
6677:
6673:
6667:
6663:
6658:
6657:
6648:
6646:
6642:
6636:
6632:
6629:
6626:
6623:
6620:
6617:
6616:
6612:
6610:
5874:
5870:
5866:
5862:
5858:
5852:
5848:
5841:
5836:
5828:
5809:
5803:
5794:
5791:
5788:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5750:
5736:
5732:
5725:
5722:
5714:
5710:
5703:
5690:
5686:
5679:
5674:
5670:
5666:
5658:
5654:
5647:
5642:
5638:
5625:
5613:
5600:
5599:
5598:
5595:
5590:which equals
5571:
5565:
5562:
5554:
5550:
5543:
5517:
5514:
5509:
5505:
5488:which equals
5469:
5463:
5460:
5452:
5448:
5441:
5415:
5412:
5407:
5403:
5373:
5370:
5365:
5361:
5357:
5350:
5346:
5342:
5337:
5331:
5325:
5322:
5316:
5310:
5307:
5301:
5295:
5285:
5284:
5283:
5261:
5255:
5252:
5246:
5240:
5226:
5204:
5201:
5196:
5192:
5188:
5179:
5176:
5173:
5163:
5159:
5155:
5152:
5140:
5135:
5131:
5125:
5122:
5119:
5114:
5111:
5108:
5104:
5100:
5094:
5088:
5078:
5063:
5055:
5051:
5047:
5042:
5039:
5035:
5031:
5028:
5024:
5015:
5012:
5006:
5003:
4996:
4993:
4990:
4987:
4984:
4973:
4970:
4961:
4958:
4952:
4948:
4945:
4938:
4934:
4930:
4925:
4922:
4917:
4914:
4911:
4906:
4903:
4900:
4896:
4892:
4886:
4880:
4870:
4869:
4868:
4866:
4840:
4834:
4828:
4822:
4816:
4793:
4787:
4784:
4781:
4778:
4775:
4745:
4742:
4739:
4735:
4731:
4728:
4723:
4718:
4715:
4712:
4708:
4704:
4698:
4692:
4685:
4665:
4662:
4659:
4655:
4651:
4646:
4643:
4638:
4633:
4630:
4627:
4623:
4619:
4613:
4607:
4600:
4599:
4598:
4596:
4571:
4563:
4560:
4559:
4558:
4557:Explanation:
4555:
4535:
4532:
4527:
4522:
4519:
4514:
4511:
4480:
4477:
4472:
4469:
4453:
4450:
4447:
4428:
4424:
4423:
4422:
4418:
4416:
4415:
4406:
4404:
4400:
4394:
4388:
4386:
4381:
4378:
4374:
4367:
4365:
4341:
4334:
4323:
4308:
4297:
4293:
4289:
4285:
4281:
4276:
4272:
4263:
4258:
4254:
4251:
4248:
4242:
4237:
4234:
4229:
4225:
4221:
4216:
4212:
4208:
4204:
4199:
4195:
4191:
4184:
4179:
4175:
4152:
4148:
4144:
4141:
4136:
4132:
4128:
4124:
4120:
4111:
4101:
4098:
4075:
4071:
4067:
4062:
4058:
4054:
4050:
4041:
4034:
4030:
4019:
4016:
4013:
4003:
3999:
3989:
3986:
3977:
3959:
3955:
3951:
3946:
3942:
3938:
3934:
3922:
3919:
3910:
3897:
3894:
3891:
3886:
3882:
3878:
3873:
3869:
3859:
3838:
3832:
3829:
3823:
3817:
3809:
3803:
3800:
3797:
3791:
3785:
3782:
3776:
3771:
3767:
3758:
3743:
3739:
3736:
3733:
3727:
3722:
3718:
3714:
3710:
3700:
3699:
3698:
3678:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3614:
3609:
3606:
3597:
3594:
3571:
3545:
3542:
3536:
3531:
3527:
3503:
3500:
3496:
3487:
3483:
3479:
3474:
3470:
3460:
3455:
3451:
3447:
3443:
3421:
3417:
3414:
3411:
3408:
3405:
3401:
3397:
3392:
3388:
3384:
3375:
3372:
3366:
3361:
3357:
3346:
3338:
3336:
3333:
3300:
3297:
3288:
3285:
3282:
3278:
3275:
3263:
3260:
3253:
3248:
3246:
3241:
3234:
3223:
3219:
3212:
3202:
3198:
3191:
3185:
3182:
3179:
3177:
3171:
3168:
3155:
3152:
3146:
3122:
3118:
3114:
3108:
3099:
3095:
3091:
3084:
3080:
3067:
3065:
3063:
3059:
3054:
3033:
3029:
3009:
2998:
2994:
2987:
2982:
2979:
2974:
2966:
2962:
2955:
2948:
2944:
2935:
2931:
2924:
2919:
2916:
2911:
2906:
2902:
2893:
2889:
2882:
2876:
2871:
2867:
2859:
2858:
2857:
2832:
2828:
2821:
2818:
2810:
2806:
2799:
2794:
2791:
2782:
2778:
2769:
2765:
2758:
2755:
2750:
2746:
2737:
2733:
2726:
2721:
2718:
2710:
2705:
2701:
2693:
2692:
2691:
2686:
2682:
2678:
2671:
2667:
2663:
2646:
2644:
2642:
2640:
2635:
2631:
2629:
2624:
2619:
2618:convergence.
2617:
2613:
2609:
2605:
2601:
2594:
2590:
2588:
2586:
2567:
2559:
2551:
2548:
2545:
2538:
2533:
2525:
2517:
2514:
2511:
2504:
2499:
2493:
2487:
2480:
2479:
2478:
2474:
2467:
2460:
2454:
2450:
2446:
2442:
2435:
2430:
2429:sign function
2425:
2424:
2415:
2409:
2407:
2399:
2379:
2376:
2373:
2368:
2364:
2360:
2357:
2352:
2348:
2344:
2341:
2335:
2329:
2322:
2321:
2320:
2317:
2313:
2309:
2299:
2295:
2289:
2284:
2279:
2274:
2268:
2258:
2254:
2244:
2240:
2231:
2222:
2213:
2211:
2209:
2204:
2200:
2196:
2192:
2186:
2182:
2175:
2171:
2165:
2161:
2154:
2150:
2144:
2140:
2133:
2129:
2123:
2119:
2111:
2107:
2103:
2095:
2091:
2087:
2076:, the number
2074:
2068:
2063:
2059:
2055:
2047:
2043:
2039:
2019:
2002:
1991:
1987:
1980:
1977:
1969:
1965:
1958:
1948:
1944:
1937:
1932:
1928:
1924:
1916:
1912:
1905:
1900:
1896:
1889:
1878:
1874:
1867:
1864:
1856:
1852:
1845:
1838:
1834:
1830:
1825:
1821:
1809:
1805:
1798:
1795:
1790:
1786:
1782:
1777:
1773:
1765:
1764:
1763:
1760:
1756:
1737:
1734:
1726:
1722:
1718:
1713:
1709:
1697:
1693:
1689:
1684:
1680:
1669:
1665:
1658:
1655:
1647:
1643:
1636:
1630:
1622:
1618:
1611:
1604:
1603:
1602:
1598:
1583:
1579:
1559:
1551:
1547:
1543:
1540:
1529:
1525:
1521:
1516:
1512:
1501:
1497:
1490:
1487:
1479:
1475:
1468:
1462:
1454:
1450:
1443:
1440:
1437:
1430:
1429:
1428:
1426:
1421:
1416:
1409:
1405:
1401:
1396:
1392:
1383:
1379:
1375:
1370:
1366:
1357:
1353:
1348:
1344:
1333:
1330:
1326:
1319:
1314:
1303:
1296:
1292:
1290:
1286:
1248:
1243:
1237:
1233:
1229:
1224:
1220:
1213:
1208:
1204:
1196:
1195:
1194:
1181:
1177:
1172:
1170:
1169:secant method
1166:
1160:
1157:
1153:
1149:
1142:
1138:
1131:
1127:
1107:
1103:
1090:
1088:
1085:
1077:
1073:
1068:
1064:
1054:
1050:
1046:
1042:
1037:
1030:
1023:
1016:
1011:
1002:
998:
988:
984:
975:
966:
961:
949:
945:
939:
935:
916:
912:
906:
904:
898:
896:
892:
884:
881:
876:
873:
871:
866:
862:
861:
856:
852:
848:
844:
840:
836:
835:
830:
825:
823:
819:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
767:
763:
759:
755:
749:
744:
742:
741:conic section
738:
734:
733:
728:
724:
719:
717:
714:from ancient
713:
709:
705:
697:
695:
693:
689:
688:approximation
684:
679:
660:
657:
654:
651:
648:
645:
639:
633:
626:
625:
624:
622:
616:
566:
564:
560:
541:
533:
529:
525:
520:
516:
508:
504:
498:
494:
490:
485:
481:
475:
471:
464:
461:
454:
453:
452:
450:
446:
427:
422:
418:
414:
406:
402:
395:
391:
386:
382:
378:
370:
366:
359:
352:
351:
350:
333:
330:
327:
324:
321:
318:
315:
312:
306:
300:
293:
292:
291:
288:
283:
279:
275:
267:
259:
251:
244:
240:
233:
226:
220:
214:
194:
191:
188:
185:
182:
175:
174:
173:
170:
165:
161:
159:
155:
147:
145:
141:
137:
86:
69:
61:
56:
55:Rhind papyrus
51:
49:
45:
40:
36:
32:
28:
27:
22:
7298:Quasi-Newton
7260:Regula falsi
7259:
7186:(3–4): 3–4.
7183:
7179:
7156:
7137:
7095:
7091:
7081:
7069:. Retrieved
7065:
7061:
7051:
7026:
7022:
7016:
6991:
6987:
6981:
6962:
6933:
6903:
6897:
6889:
6885:
6880:
6855:. Retrieved
6848:the original
6829:
6800:
6793:
6782:. Retrieved
6778:
6769:
6749:
6739:
6720:
6710:
6692:
6655:
6608:
6544:"%0.15f
5868:
5864:
5860:
5856:
5850:
5846:
5839:
5832:
5829:Example code
5596:
5533:and compute
5431:and compute
5391:
5225:regula falsi
5224:
5222:
4859:in terms of
4764:
4584:
4556:
4457:
4452:
4420:
4413:
4410:
4398:
4392:
4389:
4385:regula falsi
4384:
4382:
4379:
4375:
4371:
4326:
4102:
3979:
3912:
3860:
3701:
3348:
3334:
3156:
3150:
3144:
3120:
3116:
3112:
3109:
3097:
3093:
3089:
3082:
3078:
3071:
3061:
3058:regula falsi
3057:
3055:
3031:
3027:
3024:
2855:
2684:
2680:
2676:
2669:
2665:
2661:
2650:
2643:
2639:regula falsi
2638:
2633:
2632:
2627:
2623:regula falsi
2622:
2621:But, though
2620:
2615:
2611:
2607:
2604:regula falsi
2603:
2600:regula falsi
2599:
2597:
2593:regula falsi
2592:
2582:
2472:
2465:
2458:
2452:
2448:
2444:
2440:
2433:
2422:
2413:
2410:
2397:
2394:
2318:
2311:
2307:
2297:
2293:
2282:
2272:
2266:
2256:
2252:
2242:
2238:
2229:
2220:
2217:
2208:regula falsi
2207:
2205:
2201:
2194:
2190:
2184:
2180:
2173:
2169:
2163:
2159:
2152:
2148:
2142:
2138:
2131:
2127:
2121:
2117:
2109:
2105:
2101:
2093:
2089:
2085:
2072:
2069:
2061:
2057:
2053:
2045:
2041:
2037:
2020:
2017:
1758:
1754:
1752:
1596:
1581:
1577:
1574:
1419:
1407:
1403:
1399:
1394:
1390:
1381:
1377:
1373:
1368:
1364:
1355:
1351:
1346:
1342:
1334:
1329:line segment
1318:regula falsi
1317:
1315:
1312:
1295:regula falsi
1294:
1287:
1263:
1173:
1161:
1158:
1151:
1147:
1140:
1136:
1129:
1125:
1105:
1101:
1094:
1083:
1075:
1071:
1066:
1062:
1052:
1038:
1028:
1021:
1014:
1000:
996:
986:
982:
973:
964:
947:
943:
937:
933:
914:
910:
907:
903:regula falsi
902:
899:
888:
878:
874:
870:Regula Falsi
869:
858:
857:in his book
854:
843:al-khaṭāʾayn
842:
838:
832:
826:
818:Ibn al-Banna
797:
765:
761:
751:
746:
737:secant lines
730:
729:text called
720:
701:
682:
675:
618:
568:
562:
556:
442:
348:
286:
281:
280:
273:
265:
257:
249:
242:
238:
231:
224:
218:
212:
209:
168:
163:
162:
157:
153:
151:
142:
135:
84:
67:
59:
52:
34:
30:
26:regula falsi
25:
24:
18:
7275:Householder
7071:9 September
6556:FalsiMethod
5965:FalsiMethod
4865:sum formula
1575:Now choose
1271:is between
880:excellence.
834:Liber Abaci
772:is that of
770:Middle East
21:mathematics
7443:Categories
7265:ITP method
6912:1088854304
6857:2012-06-08
6784:2019-02-16
6637:References
6619:ITP method
3345:ITP method
3339:ITP method
3309:otherwise.
1594:for which
1586:to be the
1325:-intercept
1049:derivative
980:such that
927:. A value
841:after the
451:given by,
290:such that
172:such that
7200:229538951
7120:230586635
7112:0098-3500
6975:, CSM-257
6967:CiteSeerX
5804:≈
5778:×
5766:×
5723:−
5667:−
5617:^
5371:−
5323:−
5253:−
5223:Now, use
5202:−
5177:−
5156:−
5123:−
5105:∑
5043:−
5007:−
4988:−
4962:−
4926:⋅
4915:−
4897:∑
4785:−
4761:Club-rush
4743:−
4732:⋅
4709:∑
4663:−
4647:⋅
4624:∑
4282:−
4252:−
4243:−
4235:−
4196:ϵ
4185:≡
4176:ρ
4149:ρ
4145:σ
4142:−
4121:≡
4068:−
4031:κ
4017:−
4000:κ
3990:≡
3987:δ
3952:−
3923:≡
3920:σ
3898:δ
3895:σ
3879:≡
3830:−
3798:−
3777:≡
3728:≡
3572:ϕ
3549:∞
3537:∈
3507:⌉
3504:ϵ
3480:−
3464:⌈
3461:≡
3418:ϕ
3398:∈
3389:κ
3379:∞
3367:∈
3358:κ
3186:−
2975:−
2912:−
2819:−
2756:−
2515:−
2408:of 2/3).
2358:−
1978:−
1925:−
1865:−
1831:−
1796:−
1719:−
1690:−
1656:−
1544:−
1522:−
1488:−
1441:−
865:Tartaglia
855:el cataym
829:Fibonacci
814:al-Hassar
806:mnemonics
758:Abu Kamil
710:, and in
704:cuneiform
692:iteration
680:function
678:nonlinear
526:−
491:−
449:algorithm
48:equations
7008:50473598
6866:cite web
6747:(1959).
6613:See also
5884:#include
5878:#include
5863:) = cos(
5859: (
4458:Answer:
3279:′
3271:if
3264:′
3172:′
3148:, where
3115: (
3092: (
2679: (
2664: (
2447:) = abs(
2443: (
2310: (
2255: (
2241: (
2214:Analysis
2104: (
2088: (
2056: (
2040: (
1402: (
1376: (
1180:midpoint
1150: (
1139: (
1128: (
1104: (
999: (
985: (
936: (
919:, where
913: (
824:origin.
822:Moroccan
754:Egyptian
7043:2007695
4683:Bulrush
4443:
4431:
4427:bulrush
4099: ;
3911:where
3857: ;
3140:
3128:
3050:
3038:
2612:diverge
2598:Though
2463:and by
2427:or the
1762:gives:
1327:of the
1167:or the
851:Pacioli
786:Lebanon
782:Baalbek
698:History
561:in his
271:
254:
129:
117:
105:
93:
78:
64:
44:algebra
7198:
7163:
7144:
7118:
7110:
7041:
7006:
6969:
6941:
6910:
6808:
6757:
6727:
6699:
6668:
6595:return
6550:"
6538:printf
6508:return
6091:double
6070:double
6031:double
6010:double
6001:double
5992:double
5986:double
5971:double
5962:double
5911:return
5899:double
5890:double
5843:where
5813:
5810:2.4615
5807:
5801:
5784:
5760:
5754:
5748:
5699:
5635:
5629:
5623:
5608:
5578:
5541:
5521:
5502:
5476:
5439:
5419:
5400:
5377:
5293:
5271:
5235:
5208:
5183:
5171:
5150:
5147:
5086:
4878:
4847:
4832:
4814:
4791:
4773:
4553:units.
4167:where
3759:and
3564:where
3349:Given
1415:secant
712:papyri
23:, the
7196:S2CID
7116:S2CID
7039:JSTOR
7004:S2CID
6851:(PDF)
6844:(PDF)
6583:5E-15
6562:&
6496:break
6256:break
5867:) --
4498:days;
2456:when
2292:sign(
1423:. In
1154:) = 0
958:is a
940:) = 0
917:) = 0
739:on a
241:′ =
115:12 +
33:, or
7161:ISBN
7142:ISBN
7108:ISSN
7073:2016
6939:ISBN
6908:OCLC
6872:link
6838:and
6806:ISBN
6755:ISBN
6725:ISBN
6697:ISBN
6666:ISBN
6529:void
6523:main
6487:else
6469:side
6442:side
6397:>
6379:else
6364:side
6340:side
6295:>
6238:fabs
6229:<
6211:fabs
6133:<
6055:side
5849:) =
5845:cos(
5789:1.75
5769:1.75
5592:1.75
5494:Set
5490:−1.5
5392:Set
3978:and
3927:sign
3658:2...
3519:and
3283:>
2583:The
2477:in
2475:= ±1
2451:) −
2418:for
2250:and
2227:and
2178:and
2136:and
2099:and
2051:and
2028:and
1387:and
1316:The
1293:The
1278:and
994:and
971:and
948:zero
944:root
847:Arab
816:and
778:Arab
216:and
156:and
138:= 12
131:= 15
91:4 +
80:= 15
7188:doi
7100:doi
7031:doi
6996:doi
6988:BIT
6592:));
6589:100
6520:int
6112:for
6046:int
6019:int
5914:cos
5871:= 0
5795:1.5
5781:1.5
4536:130
4346:ITP
4188:min
4116:ITP
3993:min
3683:ITP
3126:by
2856:or
2687:− 1
2608:all
2461:≠ 0
2436:= 0
2416:= 0
2187:+ 1
2166:+ 1
2145:+ 1
2124:+ 1
1599:= 0
1350:,
946:or
252:=
210:if
107:= 5
87:= 4
19:In
7445::
7194:.
7182:.
7178:.
7114:.
7106:.
7096:47
7094:.
7090:.
7064:.
7060:.
7037:.
7027:41
7025:.
7002:.
6992:11
6990:.
6953:^
6920:^
6868:}}
6864:{{
6820:^
6777:.
6719:.
6679:^
6664:,
6662:15
6644:^
6547:\n
6460:/=
6457:fb
6445:==
6436:if
6430:fc
6424:fa
6394:fc
6388:fa
6382:if
6370:-1
6355:/=
6352:fa
6346:-1
6343:==
6334:if
6328:fc
6322:fb
6292:fb
6286:fc
6280:if
6277:);
6262:fc
6253:))
6205:if
6202:);
6199:fb
6193:fa
6175:fb
6163:fa
6145:++
6109:);
6094:fb
6088:);
6073:fa
6040:fc
5989:),
5983:)(
5959:*/
5873:.
5308::=
4867:.
4597:.
4523:10
4481:13
4434:1
4425:A
3435:,
3105:=
2421:1/
2189:=
2168:=
2147:=
2126:=
1738:0.
1411:))
1398:,
1385:))
1372:,
1171:.
1087:.
1070:,
1036:.
1027:,
784:,
718:.
694:.
623:,
278:.
239:ax
160:.
140:.
120:12
62:+
50:.
29:,
7231:e
7224:t
7217:v
7202:.
7190::
7184:5
7169:.
7150:.
7122:.
7102::
7075:.
7066:1
7045:.
7033::
7010:.
6998::
6947:.
6914:.
6874:)
6860:.
6814:.
6787:.
6763:.
6733:.
6604:}
6601:;
6598:0
6586:,
6580:,
6577:1
6574:,
6571:0
6568:,
6565:f
6559:(
6553:,
6541:(
6535:{
6532:)
6526:(
6517:}
6514:;
6511:c
6505:}
6502:}
6499:;
6490:{
6484:}
6481:;
6478:1
6475:+
6472:=
6466:;
6463:2
6454:)
6451:1
6448:+
6439:(
6433:;
6427:=
6421:;
6418:c
6415:=
6412:a
6406:{
6403:)
6400:0
6391:*
6385:(
6376:}
6373:;
6367:=
6361:;
6358:2
6349:)
6337:(
6331:;
6325:=
6319:;
6316:c
6313:=
6310:b
6304:{
6301:)
6298:0
6289:*
6283:(
6274:c
6271:(
6268:f
6265:=
6259:;
6250:a
6247:+
6244:b
6241:(
6235:*
6232:e
6226:)
6223:a
6220:-
6217:b
6214:(
6208:(
6196:-
6190:(
6187:/
6184:)
6181:a
6178:*
6172:-
6169:b
6166:*
6160:(
6157:=
6154:c
6151:{
6148:)
6142:n
6139:;
6136:m
6130:n
6127:;
6124:0
6121:=
6118:n
6115:(
6106:b
6103:(
6100:f
6097:=
6085:a
6082:(
6079:f
6076:=
6064:;
6061:0
6058:=
6052:,
6049:n
6043:;
6037:,
6034:c
6028:{
6025:)
6022:m
6016:,
6013:e
6007:,
6004:b
5998:,
5995:a
5980:f
5977:*
5974:(
5968:(
5947:}
5944:;
5941:x
5938:*
5935:x
5932:*
5929:x
5926:-
5923:)
5920:x
5917:(
5908:{
5905:)
5902:x
5896:(
5893:f
5869:x
5865:x
5861:x
5857:f
5851:x
5847:x
5840:x
5792:+
5775:3
5772:+
5763:2
5751:=
5742:)
5737:1
5733:x
5729:(
5726:F
5720:)
5715:2
5711:x
5707:(
5704:F
5696:)
5691:1
5687:x
5683:(
5680:F
5675:2
5671:x
5664:)
5659:2
5655:x
5651:(
5648:F
5643:1
5639:x
5626:=
5614:x
5575:)
5572:3
5569:(
5566:F
5563:=
5560:)
5555:2
5551:x
5547:(
5544:F
5518:3
5515:=
5510:2
5506:x
5473:)
5470:2
5467:(
5464:F
5461:=
5458:)
5453:1
5449:x
5445:(
5442:F
5416:2
5413:=
5408:1
5404:x
5374:7
5366:n
5362:2
5358:+
5351:n
5347:2
5343:6
5338:=
5335:)
5332:n
5329:(
5326:B
5320:)
5317:n
5314:(
5311:C
5305:)
5302:n
5299:(
5296:F
5268:)
5265:)
5262:n
5259:(
5256:B
5250:)
5247:n
5244:(
5241:C
5238:(
5205:1
5197:n
5193:2
5189:=
5180:2
5174:1
5164:n
5160:2
5153:1
5141:=
5136:k
5132:2
5126:1
5120:n
5115:0
5112:=
5109:k
5101:=
5098:)
5095:n
5092:(
5089:C
5064:)
5056:n
5052:2
5048:1
5040:1
5036:(
5032:6
5029:=
5025:)
5016:2
5013:1
5004:1
4997:1
4994:+
4991:1
4985:n
4981:)
4974:2
4971:1
4965:(
4959:1
4953:(
4949:3
4946:=
4939:k
4935:2
4931:1
4923:3
4918:1
4912:n
4907:0
4904:=
4901:k
4893:=
4890:)
4887:n
4884:(
4881:B
4861:k
4844:)
4841:n
4838:(
4835:C
4829:,
4826:)
4823:n
4820:(
4817:B
4794:.
4788:1
4782:i
4779:=
4776:k
4746:1
4740:i
4736:2
4729:1
4724:n
4719:1
4716:=
4713:i
4705:=
4702:)
4699:n
4696:(
4693:C
4666:1
4660:i
4656:2
4652:1
4644:3
4639:n
4634:1
4631:=
4628:i
4620:=
4617:)
4614:n
4611:(
4608:B
4591:n
4589:(
4587:n
4579:K
4575:F
4541:)
4533:6
4528:+
4520:8
4515:+
4512:4
4509:(
4486:)
4478:6
4473:+
4470:2
4467:(
4440:2
4437:/
4399:x
4393:y
4351:)
4342:x
4338:(
4335:f
4322:.
4309:}
4304:|
4298:2
4294:/
4290:1
4286:x
4277:t
4273:x
4268:|
4264:,
4259:2
4255:a
4249:b
4238:j
4230:0
4226:n
4222:+
4217:2
4213:/
4209:1
4205:n
4200:2
4192:{
4180:k
4153:k
4137:2
4133:/
4129:1
4125:x
4112:x
4086:}
4082:|
4076:f
4072:x
4063:2
4059:/
4055:1
4051:x
4046:|
4042:,
4035:2
4025:|
4020:a
4014:b
4010:|
4004:1
3996:{
3965:)
3960:f
3956:x
3947:2
3943:/
3939:1
3935:x
3931:(
3892:+
3887:f
3883:x
3874:t
3870:x
3842:)
3839:b
3836:(
3833:f
3827:)
3824:a
3821:(
3818:f
3813:)
3810:b
3807:(
3804:f
3801:a
3795:)
3792:a
3789:(
3786:f
3783:b
3772:f
3768:x
3744:2
3740:b
3737:+
3734:a
3723:2
3719:/
3715:1
3711:x
3679:x
3655:,
3652:1
3649:,
3646:0
3643:=
3640:j
3620:)
3615:5
3610:+
3607:1
3604:(
3598:2
3595:1
3552:)
3546:,
3543:0
3540:[
3532:0
3528:n
3501:2
3497:/
3493:)
3488:0
3484:a
3475:0
3471:b
3467:(
3456:2
3452:/
3448:1
3444:n
3422:)
3415:+
3412:1
3409:,
3406:1
3402:[
3393:2
3385:,
3382:)
3376:,
3373:0
3370:(
3362:1
3301:2
3298:1
3289:,
3286:0
3276:m
3261:m
3254:{
3249:=
3242:m
3235:,
3229:)
3224:k
3220:b
3216:(
3213:f
3208:)
3203:k
3199:c
3195:(
3192:f
3183:1
3180:=
3169:m
3151:m
3145:m
3137:2
3134:/
3131:1
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3117:a
3113:f
3101:)
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3044:/
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3010:,
3004:)
2999:k
2995:a
2991:(
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2983:2
2980:1
2972:)
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2963:b
2959:(
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2949:k
2945:b
2941:)
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2928:(
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2920:2
2917:1
2907:k
2903:a
2899:)
2894:k
2890:b
2886:(
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2877:=
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2868:c
2838:)
2833:k
2829:a
2825:(
2822:f
2816:)
2811:k
2807:b
2803:(
2800:f
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2792:1
2783:k
2779:b
2775:)
2770:k
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2762:(
2759:f
2751:k
2747:a
2743:)
2738:k
2734:b
2730:(
2727:f
2722:2
2719:1
2711:=
2706:k
2702:c
2685:k
2681:c
2677:f
2670:k
2666:c
2662:f
2657:y
2653:y
2568:.
2560:2
2556:)
2552:1
2549:+
2546:x
2543:(
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2534:+
2526:2
2522:)
2518:1
2512:x
2509:(
2505:1
2500:=
2497:)
2494:x
2491:(
2488:f
2473:x
2466:f
2459:x
2453:x
2449:x
2445:x
2441:f
2434:x
2423:x
2414:x
2401:"
2398:f
2380:x
2377:3
2374:+
2369:2
2365:x
2361:4
2353:3
2349:x
2345:2
2342:=
2339:)
2336:x
2333:(
2330:f
2314:)
2312:x
2308:f
2303:)
2301:"
2298:f
2294:f
2283:f
2273:f
2267:f
2262:)
2260:0
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2248:)
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2224:0
2221:a
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2164:k
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2132:k
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2122:k
2118:a
2113:)
2110:k
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2097:)
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2090:a
2086:f
2080:k
2078:c
2073:k
2065:)
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2058:a
2054:f
2049:)
2046:k
2042:b
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2032:k
2030:b
2025:k
2023:a
2003:.
1997:)
1992:k
1988:a
1984:(
1981:f
1975:)
1970:k
1966:b
1962:(
1959:f
1954:)
1949:k
1945:a
1941:(
1938:f
1933:k
1929:b
1922:)
1917:k
1913:b
1909:(
1906:f
1901:k
1897:a
1890:=
1884:)
1879:k
1875:a
1871:(
1868:f
1862:)
1857:k
1853:b
1849:(
1846:f
1839:k
1835:a
1826:k
1822:b
1815:)
1810:k
1806:b
1802:(
1799:f
1791:k
1787:b
1783:=
1778:k
1774:c
1759:k
1755:c
1735:=
1732:)
1727:k
1723:b
1714:k
1710:c
1706:(
1698:k
1694:a
1685:k
1681:b
1675:)
1670:k
1666:a
1662:(
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1653:)
1648:k
1644:b
1640:(
1637:f
1631:+
1628:)
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1615:(
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1597:y
1592:x
1588:x
1582:k
1578:c
1560:.
1557:)
1552:k
1548:b
1541:x
1538:(
1530:k
1526:a
1517:k
1513:b
1507:)
1502:k
1498:a
1494:(
1491:f
1485:)
1480:k
1476:b
1472:(
1469:f
1463:=
1460:)
1455:k
1451:b
1447:(
1444:f
1438:y
1420:f
1408:k
1404:b
1400:f
1395:k
1391:b
1389:(
1382:k
1378:a
1374:f
1369:k
1365:a
1363:(
1359:)
1356:k
1352:b
1347:k
1343:a
1341:(
1337:k
1323:x
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1280:b
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1273:a
1268:k
1266:c
1249:.
1244:2
1238:k
1234:b
1230:+
1225:k
1221:a
1214:=
1209:k
1205:c
1190:k
1188:c
1184:k
1152:c
1148:f
1143:)
1141:c
1137:f
1132:)
1130:x
1126:f
1121:c
1117:c
1112:f
1108:)
1106:x
1102:f
1097:x
1084:f
1079:)
1076:k
1072:b
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1063:a
1061:(
1057:k
1034:)
1032:0
1029:b
1025:0
1022:a
1020:(
1015:f
1006:)
1004:0
1001:b
997:f
992:)
990:0
987:a
983:f
977:0
974:b
968:0
965:a
956:f
952:f
938:c
934:f
929:c
925:x
921:f
915:x
911:f
764:(
683:f
661:,
658:c
655:+
652:x
649:a
646:=
643:)
640:x
637:(
634:f
542:,
534:2
530:b
521:1
517:b
509:1
505:x
499:2
495:b
486:2
482:x
476:1
472:b
465:=
462:x
428:.
423:2
419:b
415:=
412:)
407:2
403:x
399:(
396:f
392:,
387:1
383:b
379:=
376:)
371:1
367:x
363:(
360:f
334:,
331:0
328:=
325:c
322:+
319:x
316:a
313:=
310:)
307:x
304:(
301:f
287:x
276:′
274:x
268:′
266:b
262:/
258:b
250:x
245:′
243:b
234:′
232:b
227:′
225:x
219:b
213:a
195:,
192:b
189:=
186:x
183:a
169:x
136:x
126:4
123:/
111:x
102:4
99:/
96:4
85:x
75:4
72:/
68:x
60:x
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