False position method

False position method and regula falsi method are two early, and still current, names for a very old method for solving an equation in one unknown.

To solve an equation means to write, or determine the numerical value of, one of its quantities in terms of the other quantities mentioned in the equation.

Many equations, including most of the more complicated ones, can be solved only by iterative numerical approximation. That consists of trial and error, in which various values of the unknown quantity, referred to here as “x”, are tried. That trial-and-error may be informed by a calculated estimate for the solution. The iterative numerical approximation methods for solving equations, which use a calculated estimate for the solution, for use in calculating the next, improved, solution-estimate, differ only by how their calculated solution-estimates are made.

By moving all of an equation’s terms to one side, we can get an equation that says: f(x) = 0, where f(x) is some function of the unknown variable “x”.

That transforms the problem into one of finding the x-value at which f(x) = 0. That x-value is the equation’s solution.

In this section, the symbol “y” will be used interchangeably with f(x) when that improves brevity, clarity, and reduces clutter.

Here, “y” means “y(x)” means “f(x)”. The expressions “y” and “f(x)” will both be used here, and they mean the same thing. The symbol “y” is familiar, as the often-used name for the vertical co-ordinate on a graph, often a function of “x”, the horizontal co-ordinate.

Let's solve the equation x + x/4 = 15 by false position. Try with x = 4. We get 4 + 4/4 = 5, note 4 is not the solution. Let's now multiply with 3 on both sides to get 12 + 12/4 = 15, obtaining the solution x = 12. The example is problem 26 on the Rhind papyrus. A History of Mathematics, 3rd edition, by Victor J. Katz categorizes the problem as false position.

Many methods for the calculated-estimate are used. The oldest and simplest class of such methods, and the class that contains the most reliable method (Bisection), are the two-point bracketing methods.

Those methods start with two x-values, initially found by trial-and-error, at which f(x) has opposite signs. In other words: Two x-values such that, for one of them, f(x) is positive, and for the other, f(x) is negative. In that way, those two f(x) values (i.e. y-values) can be said to “bracket” zero, because they’re on opposite sides of zero.

That bracketing, along with the fact that the solution-estimate-calculation method (to be discussed later) always chooses an x-value between the two current bracketing values, guarantees that the solution-estimates will converge toward the solution. …a guarantee not available with such other methods as Newton’s method or the Secant method.

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