lõigumeetod

The lõigumeetod, often translated as the secant method, is a numerical root-finding algorithm. It is an iterative method that approximates the root of a function by constructing a sequence of roots of secant lines. The secant method is similar to Newton's method, but it approximates the derivative of the function using a finite difference.

To use the lõigumeetod, two initial guesses, x0 and x1, are required. These guesses should be relatively close to the actual root. The algorithm then iteratively calculates the next approximation, xn+1, using the formula: xn+1 = xn - f(xn) * (xn - xn-1) / (f(xn) - f(xn-1)). This formula represents the x-intercept of the secant line passing through the points (xn-1, f(xn-1)) and (xn, f(xn)).

The process continues until a specified level of accuracy is reached, meaning the difference between successive approximations is below a certain tolerance, or the function value at the approximation is close enough to zero. The lõigumeetod generally converges faster than the bisection method but slower than Newton's method. It does not require the derivative of the function to be computed, which can be an advantage when the derivative is difficult or impossible to find analytically. However, it can diverge if the initial guesses are poor or if the function has certain characteristics, such as horizontal tangents near the root.