How do you choose between bisection and false position methods?

1 Bisection method

The bisection method is a simple and reliable bracketing method that works by repeatedly dividing an interval that contains a root into two subintervals and choosing the one that still contains a root. The bisection method is guaranteed to converge to a root, but it may be slow and inefficient, especially if the function is flat or oscillatory near the root. The bisection method requires only one function evaluation per iteration, but it does not use any information about the slope or shape of the function.

2 False position method

The false position method is another bracketing method that tries to improve on the bisection method by using a linear approximation of the function to estimate the root. The false position method draws a straight line between the endpoints of the interval that contains a root and finds the point where the line crosses the x-axis. This point is then used as the new endpoint of the interval, depending on whether it has the same sign as the other endpoint or not. The false position method can converge faster than the bisection method, but it may also fail or stagnate if the function is not well approximated by a line.

3 Newton's method

Newton's method is a classic example of an open method that uses the derivative of the function to find the root. Newton's method starts with an initial guess for the root and then iterates the following formula: x_(n+1) = x_n - f(x_n) / f'(x_n), where f' is the derivative of f. Newton's method can converge very quickly if the initial guess is close to the root and the function is smooth and well-behaved. However, Newton's method can also diverge or oscillate if the initial guess is far from the root or the function is steep or has multiple roots.

4 Secant method

The secant method is another open method that approximates the derivative of the function by using a finite difference. The secant method uses two previous points to draw a secant line that intersects the x-axis and then uses that point as the next approximation for the root. The secant method does not require the explicit calculation of the derivative, but it still uses some information about the slope of the function. The secant method can converge faster than the bisection method, but it may also be unstable or inaccurate if the function is not smooth or has multiple roots.