The Secant Method only requires one function input, f(x), but two initial guesses. The Newton method is in principle faster; its convergence is quadratic while the secant method’s convergence is of order (1+sqrt(5))/2 which is about 1.6.

The quick answer would be, because the Newton method is an higher order method, and thus builds better approximation of your function. Newton method typically exactly minimizes the second order approximation of a function f. That is, iteratively sets x←x−[∇2f(x)]−1∇f(x).

The Newton-Raphson method is a method for approximating the roots of polynomial equations of any order. In fact the method works for any equation, polynomial or not, as long as the function is differentiable in a desired interval. ‘s often become increasingly better approximations of the function’s root.

The results of numerical tests suggest that the simple iteration method implemented in Bowring’s algorithm executes approximately 30% faster than the Newton-Raphson method implemented in Borkowski’s algorithm. Only two iterations of each algorithm are considered.

The average rate of convergence of Newton-Raphson method has been found to be 0.217920.

Disadvantages of secant method

• It may not converge.
• There is no guaranteed error bound for the computed iterates.
• It is likely to have difficulty if f′(α) = 0.
• Newton’s method generalizes more easily to new methods for solving simultaneous systems of nonlinear equations.

Advantages of secant method: 1. It converges at faster than a linear rate, so that it is more rapidly convergent than the bisection method. 2. It does not require use of the derivative of the function, something that is not available in a number of applications.

What is the difference between Newton-Raphson and secant method on the basis of geometric interpretation? The two methods are almost the same, from a geometric perspective. The difference is that Newton’s Method uses a line that is tangent to one point, while the Secant Method uses a line that is secant to two points.

An algorithm for finding roots which retains that prior estimate for which the function value has opposite sign from the function value at the current best estimate of the root. In this way, the method of false position keeps the root bracketed (Press et al. 1992).

In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown, that, in modified form, is still in use. However, 4 is not the solution of the original equation, as it gives a value which is three times too small.

REGULA–FALSI METHOD

1. Find points a and b such that a < b and f(a) * f(b) < 0.
2. Take the interval [a, b] and determine the next value of x1.
3. If f(x1) = 0 then x1 is an exact root, else if f(x1) * f(b) < 0 then let a = x1, else if f(a) * f(x1) < 0 then let b = x1.

Order of convergence of false position method is the golden ratio.