Sidi's generalized secant method
Sidi's generalized secant method is a root-finding algorithm, that is, a numerical method for solving equations of the form . The method was published by Avram Sidi.
The method is a generalization of the secant method. Like the secant method, it is an iterative method which requires one evaluation of in each iteration and no derivatives of . The method can converge much faster though, with an order which approaches 2 provided that satisfies the regularity conditions described below.
We call the root of , that is, . Sidi's method is an iterative method which generates a sequence of approximations of . Starting with k + 1 initial approximations , the approximation is calculated in the first iteration, the approximation is calculated in the second iteration, etc. Each iteration takes as input the last k + 1 approximations and the value of at those approximations. Hence the nth iteration takes as input the approximations and the values .
The number k must be 1 or larger: k = 1, 2, 3, .... It remains fixed during the execution of the algorithm. In order to obtain the starting approximations one could carry out a few initializing iterations with a lower value of k.
The approximation is calculated as follows in the nth iteration. A polynomial of interpolation of degree k is fitted to the k + 1 points . With this polynomial, the next approximation of is calculated as
with the derivative of at . Having calculated one calculates and the algorithm can continue with the (n + 1)th iteration. Clearly, this method requires the function to be evaluated only once per iteration; it requires no derivatives of .
The iterative cycle is stopped if an appropriate stop-criterion is met. Typically the criterion is that the last calculated approximation is close enough to the sought-after root .
To execute the algorithm effectively, Sidi's method calculates the interpolating polynomial in its Newton form.
Sidi showed that if the function is (k + 1)-times continuously differentiable in an open interval containing (that is, ), is a simple root of (that is, ) and the initial approximations are chosen close enough to , then the sequence converges to , meaning that the following limit holds: .
Sidi furthermore showed that
and that the sequence converges to of order , i.e.
The order of convergence is the only positive root of the polynomial
We have e.g. ≈ 1.6180, ≈ 1.8393 and ≈ 1.9276. The order approaches 2 from below if k becomes large:  
Sidi's method reduces to the secant method if we take k = 1. In this case the polynomial is the linear approximation of around which is used in the nth iteration of the secant method.
We can expect that the larger we choose k, the better is an approximation of around . Also, the better is an approximation of around . If we replace with in (Template:EquationNote) we obtain that the next approximation in each iteration is calculated as
This is the Newton–Raphson method. It starts off with a single approximation so we can take k = 0 in (Template:EquationNote). It does not require an interpolating polynomial but instead one has to evaluate the derivative in each iteration. Depending on the nature of this may not be possible or practical.
Once the interpolating polynomial has been calculated, one can also calculate the next approximation as a solution of instead of using (Template:EquationNote). For k = 1 these two methods are identical: it is the secant method. For k = 2 this method is known as Muller's method. For k = 3 this approach involves finding the roots of a cubic function, which is unattractively complicated. This problem becomes worse for even larger values of k. An additional complication is that the equation will in general have multiple solutions and a prescription has to be given which of these solutions is the next approximation . Muller does this for the case k = 2 but no such prescriptions appear to exist for k > 2.
- ↑ Sidi, Avram, "Generalization Of The Secant Method For Nonlinear Equations", Applied Mathematics E-notes 8 (2008), 115–123, http://www.math.nthu.edu.tw/~amen/2008/070227-1.pdf
- ↑ The home page of Avram Sidi at the Israel Institute of Technology is at Avraham Sidi
- ↑ Traub, J.F., "Iterative Methods for the Solution of Equations", Prentice Hall, Englewood Cliffs, N.J. (1964)
- ↑ 4.0 4.1 Muller, David E., "A Method for Solving Algebraic Equations Using an Automatic Computer", Mathematical Tables and Other Aids to Computation 10 (1956), 208–215