Regula Falsi Method
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   The false position method or regula falsi method is a term for problem-solvingmethods in arithmetic, algebra, and calculus. In simple terms, these methods begin by attempting to evaluate a problem using test ( false ) values for the variables, and then adjust the values accordingly. Two basic types of false position method can be distinguished, simple false position and double false position. imple false position is aimed at solving problems involving direct proportion. uch problems can be written algebraically in the form! determine  such thata#b,if a and b are $nown. %ouble false position is aimed at solving more di&cult problems that can be written algebraically in the form! determine  such thatf()#b,if it is $nown thatf(')#b',f()#b.%ouble false position is mathematically euivalent to linear interpolation* for an a&ne linear function,f()#a+c,it provides the eact solution, while for a nonlinear function f it provides an approimation that can be successively improved by iteration.In problems involving arithmetic or algebra, the false position method or regula falsi is used to refer to basic trial and error methods of solving problems by substituting test values for the un$nown uantities. This is sometimes also referred to as guess and chec$ . ersions of this method predate the advent of algebra and the use of euations.or simple false position, the method of solving what we would now write as a # b begins by using a test input value , and /nding the corresponding output value b by multiplication! a # b. The correct answer is then found  by proportional adjustment,  #  0 b 1 b. This techniue is found in cuneiform tablets from ancient 2abylonian mathematics, and possibly in papyri from ancient 3gyptian mathematics.4'56i$ewise, double false position arose in late antiuity as a purely arithmetical algorithm. It was used mostly to solve what are now called a&ne linear problems by using a pair of test inputs and the corresponding pair of outputs. This algorithm would be memori7ed and carried out by rote. In the ancient 8hinese mathematical tet called The 9ine 8hapters on the :athematical ;rt ( 九章算術 ), dated from 200 BC to AD 100, most of Chapter 7 was devoted to the algorithm. There, the procedure was ustified ! co#crete arithmetical argume#ts, the# applied creativel to a wide variet of stor pro!lems, i#cludi#g o#e i#volvi#g what we would call seca#t li#es o# a $uadratic pol #omial. A more t pical e%ample is this &oi#t purchase& pro!lem'(ow a# item is purchased oi#tl  ever o#e co#tri!utes * +coi#s, the e%cess is - ever o#e co#tri!utes 7, the deficit is . Tell' The #um!er of people, the item price, what is each/ A#swer' 7 people, item price -.+2Betwee# the th a#d 10th ce#turies, the g ptia# 3uslim mathematicia# A!u 4amil wrote a #ow5lost treatise o# the use of dou!le false positio#, 6#ow# as the Boo6 of the Two rrors 4it <b al-$ha=<>ayn). The oldest surviving writing on double false position from the :iddle 3ast is that of ?usta ibn 6ua ('@th century), a 8hristian ;rab mathematician from 2aalbe$, 6ebanon. Ae justi/edthe techniue by a formal, 3uclidean-style geometric proof. Bithin the tradition of medieval :uslim mathematics, double false position was $nown as his<b al-$ha=<>ayn ( rec$oning by two errors ). It was used for centuries, especially in the :aghreb, to solve practical problems such as commercial and juridical uestions (estate partitions according to rules of ?uranic inheritance), as well as purely recreational problems. The algorithm was oftenmemori7ed with the aid of mnemonics, such as a verse attributed to Ibn al- Casamin and balance-scale diagrams eplained by al-Aassar and Ibn al-2anna, all three being mathematicians of :oroccan srcin.4D56eonardo of Eisa (ibonacci) devoted 8hapter 'D of his boo$ 6iber ;baci (;% '@) to eplaining and demonstrating the uses of double false position, terming the method regulis elchatayn after the al-$ha=<>ayn method that he had learned from ;rab sources.4D5  9umerical analysis4edit5In numerical analysis, double false position became a root-/nding algorithm that combines features from the bisection method and the secant method. The /rst two iterations of the false position method. The red curve shows the function f and the blue lines are the secants.6i$e the bisection method, the false position method starts with two points a@and b@ such that f(a@) and f(b@) are of opposite signs, which implies by the intermediate value theorem that the function f has a root in the interval 4a@, b@5, assuming continuity of the function f. The method proceeds by producinga seuence of shrin$ing intervals 4a$, b$5 that all contain a root of f.;t iteration number $, the numberc$#b$Ff(b$)(b$Fa$)f(b$)Ff(a$)is computed. ;s eplained below, c$ is the root of the secant line through (a$,f(a$)) and (b$, f(b$)). If f(a$) and f(c$) have the same sign, then we set a$+' # c$ and b$+' # b$, otherwise we set a$+' # a$ and b$+' # c$. This process is repeated until the root is approimated su&ciently well. The above formula is also used in the secant method, but the secant method always retains the last two computed points, while the false position method retains two points which certainly brac$et a root. Gn the other hand, the only diHerence between the false position method and the bisection method is that the latter uses c$ # (a$ + b$)  .inding the root of the secant4edit5Jiven a$ and b$, we construct the line through the points (a$, f(a$)) and (b$, f(b$)), as demonstrated in the picture immediately above. 9ote that this line is a secant or chord of the graph of the function f. In point-slope form, it can be de/ned as  yFf(b$)#f(b$)Ff(a$)b$Fa$(Fb$).Be now choose c$ to be the root of this line (substituting for ), and setting y#@ and see thatf(b$)+f(b$)Ff(a$)b$Fa$(c$Fb$)#@.olving this euation gives the above euation for c$.;nalysis4edit5If the initial end-points a@ and b@ are chosen such that f(a@) and f(b@) are of opposite signs, then at each step, one of the end-points will get closer to a root of f. If the second derivative of f is of constant sign (so there is no inKection point) in the interval, then one endpoint (the one where f also has the same sign) will remain /ed for all subseuent iterations while the converging endpoint becomes updated. ;s a result, unli$e the bisection method, the width of the brac$et does not tend to 7ero (unless the 7ero is at an inKection point around which sign(f)#-sign(fL)). ;s a conseuence, the linear approimation to f(), which is used to pic$ the false position, does not improve in its uality.Gne eample of this phenomenon is the functionf()#DFM+Don the initial brac$et 4F','5. The left end, F', is never replaced (after the /rstthree iterations, fL is negative on the interval) and thus the width of the brac$et never falls below '. Aence, the right endpoint approaches @ at a linear rate (the number of accurate digits grows linearly, with a rate of convergence of D).or discontinuous functions, this method can only be epected to /nd a point where the function changes sign (for eample at #@ for ' or the sign function). In addition to sign changes, it is also possible for the method to converge to a point where the limit of the function is 7ero, even if the functionis unde/ned (or has another value) at that point (for eample at #@ for the function given by f()#abs()-N when O@ and by f(@)#P, starting with the
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