Questions tagged [regula-falsi]

In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown, that, in modified form, is still in use. In simple terms, the method is the trial and error 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 algebra and the use of equations.

Two basic types of false position method can be distinguished historically, simple false position and double false position.

Simple false position is aimed at solving problems involving direct proportion. Such problems can be written algebraically in the form: determine $x$ such that $ax=b$, if $a$ and $b$ are known. The method begins by using a test input value $x′$, and finding the corresponding output value $b′$ by multiplication: $ax′ = b′$. The correct answer is then found by proportional adjustment, $x = \frac b {b′} x′$.

Double false position is aimed at solving more difficult problems that can be written algebraically in the form: determine $x$ such that $f ( x ) = a x + c = 0 $, if it is known that $ f ( x_1 ) = b_1 $ and $ f ( x_2 ) = b_2 $. Double false position is mathematically equivalent to linear interpolation. By using a pair of test inputs and the corresponding pair of outputs, the result of this algorithm given by $ x = \frac { b_1 x_2 - b_2 x_1 } { b_1 - b_2 } $. For an affine linear function $f(x)=ax+c$, double false position provides the exact solution, while for a nonlinear function $f$ it provides an approximation that can be successively improved by iteration.

Source: Wikipedia