lõikemeetodi

Lõikemeetod, also known as the secant method, is a numerical method for finding successively better approximations to the roots of a real-valued function. It is similar to Newton's method but does not require the calculation of the derivative of the function. Instead, it uses a finite difference approximation of the derivative.

The method begins with two initial guesses, x0 and x1, which are typically chosen to be close to the expected root. The next approximation, x2, is found by calculating the point where the secant line connecting the points (x0, f(x0)) and (x1, f(x1)) intersects the x-axis. The formula for x2 is:

x2 = x1 - f(x1) * (x1 - x0) / (f(x1) - f(x0))

The process is then repeated. The new pair of approximations becomes x1 and x2, and the next approximation, x3, is calculated using the same formula. This iterative process continues until the difference between successive approximations is smaller than a predefined tolerance, or until a maximum number of iterations is reached.

The lõikemeetod converges linearly, which is slower than the quadratic convergence of Newton's method. However, it can be advantageous when the derivative of the function is difficult or computationally expensive to obtain. The method's convergence is not guaranteed for all functions and initial guesses. It may diverge if the initial guesses are too far from the root or if the function has certain characteristics, such as a derivative close to zero near the root.