Väärustamismeetodile

Väärustamismeetodile, also known as the false position method or Regula Falsi, is a numerical method for finding the root of a continuous function. It is similar to the bisection method in that it requires a bracketed interval where the function changes sign, guaranteeing a root exists within that interval. However, instead of always bisecting the interval, the false position method uses a secant line connecting the function values at the interval endpoints to estimate the root's location.

The method begins with an interval [a, b] such that f(a) and f(b) have opposite signs. A new estimate for the root, c, is calculated using the formula: c = b - f(b) * (b - a) / (f(b) - f(a)). This formula finds the x-intercept of the line passing through (a, f(a)) and (b, f(b)).

After calculating c, the function value f(c) is evaluated. If f(c) is close enough to zero, then c is considered the root. Otherwise, the interval is updated. If f(a) and f(c) have opposite signs, the new interval becomes [a, c]. If f(c) and f(b) have opposite signs, the new interval becomes [c, b]. This process is repeated iteratively until the desired level of accuracy is achieved, meaning the interval becomes sufficiently small or the function value at the estimated root is close enough to zero.

While generally convergent, the false position method can sometimes converge slowly, especially if the function is highly curved within the interval. This can occur if one of the endpoints remains fixed for many iterations. Despite this potential drawback, it remains a robust and widely used technique for root-finding.