Sidi's generalized secant method is a root-finding algorithm, that is, a numerical method for solving equations of the form ${\displaystyle f(x)=0}$ . 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 ${\displaystyle f}$ in each iteration and no derivatives of ${\displaystyle f}$. The method can converge much faster though, with an order which approaches 2 provided that ${\displaystyle f}$ satisfies the regularity conditions described below.

We call ${\displaystyle \alpha }$ the root of ${\displaystyle f}$, that is, ${\displaystyle f(\alpha )=0}$. Sidi's method is an iterative method which generates a sequence ${\displaystyle \{x_{i}\}}$ of approximations of ${\displaystyle \alpha }$. Starting with k + 1 initial approximations ${\displaystyle x_{1},\dots ,x_{k+1}}$, the approximation ${\displaystyle x_{k+2}}$ is calculated in the first iteration, the approximation ${\displaystyle x_{k+3}}$ is calculated in the second iteration, etc. Each iteration takes as input the last k + 1 approximations and the value of ${\displaystyle f}$ at those approximations. Hence the nth iteration takes as input the approximations ${\displaystyle x_{n},\dots ,x_{n+k}}$ and the values ${\displaystyle f(x_{n}),\dots ,f(x_{n+k})}$.

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 ${\displaystyle x_{1},\dots ,x_{k+1}}$ one could carry out a few initializing iterations with a lower value of k.

The approximation ${\displaystyle x_{n+k+1}}$ is calculated as follows in the nth iteration. A polynomial of interpolation ${\displaystyle p_{n,k}(x)}$ of degree k is fitted to the k + 1 points ${\displaystyle (x_{n},f(x_{n})),\dots (x_{n+k},f(x_{n+k}))}$. With this polynomial, the next approximation ${\displaystyle x_{n+k+1}}$ of ${\displaystyle \alpha }$ is calculated as

with ${\displaystyle p_{n,k}'(x_{n+k})}$ the derivative of ${\displaystyle p_{n,k}}$ at ${\displaystyle x_{n+k}}$. Having calculated ${\displaystyle x_{n+k+1}}$ one calculates ${\displaystyle f(x_{n+k+1})}$ and the algorithm can continue with the (n + 1)th iteration. Clearly, this method requires the function ${\displaystyle f}$ to be evaluated only once per iteration; it requires no derivatives of ${\displaystyle f}$.

The iterative cycle is stopped if an appropriate stopping criterion is met. Typically the criterion is that the last calculated approximation is close enough to the sought-after root ${\displaystyle \alpha }$.

To execute the algorithm effectively, Sidi's method calculates the interpolating polynomial ${\displaystyle p_{n,k}(x)}$ in its Newton form.