Sekanttiyhtälö

Sekanttiyhtälö, often referred to as the secant method, is a numerical technique used to find roots of a function. It is an iterative root-finding algorithm that approximates the root by using a sequence of secant lines. The method is similar to Newton's method, but instead of using the derivative of the function, it approximates the derivative using a finite difference from two previous points.

The secant method starts with two initial guesses, x0 and x1, that are reasonably close to the root. The formula for the next approximation, xn+1, is derived from the equation of the secant line passing through the points (xn-1, f(xn-1)) and (xn, f(xn)) and finding where this line intersects the x-axis. The iterative formula is given by:

xn+1 = xn - f(xn) * (xn - xn-1) / (f(xn) - f(xn-1))

The process continues until a desired level of accuracy is achieved, meaning the difference between successive approximations is sufficiently small, or the function value at an approximation is close to zero. The secant method typically converges faster than the bisection method but slower than Newton's method. It is a useful alternative when the derivative of the function is difficult or impossible to compute. However, convergence is not guaranteed, and the method can diverge if the initial guesses are not suitable or if the function behaves poorly in the interval of interest.