Sidi's generalized secant method is a root-finding algorithm, that is, a numerical method for solving equations of the form f ( x ) = 0 {\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 f {\displaystyle f} in each iteration and no derivatives of f {\displaystyle f} . The method can converge much faster though, with an order which approaches 2 provided that f {\displaystyle f} satisfies the regularity conditions described below.
Algorithm We call α {\displaystyle \alpha } the root of f {\displaystyle f} , that is, f ( α ) = 0 {\displaystyle f(\alpha )=0} . Sidi's method is an iterative method which generates a sequence { x i } {\displaystyle \{x_{i}\}} of approximations of α {\displaystyle \alpha } . Starting with k + 1 initial approximations x 1 , … , x k + 1 {\displaystyle x_{1},\dots ,x_{k+1}} , the approximation x k + 2 {\displaystyle x_{k+2}} is calculated in the first iteration, the approximation x k + 3 {\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 f {\displaystyle f} at those approximations. Hence the nth iteration takes as input the approximations x n , … , x n + k {\displaystyle x_{n},\dots ,x_{n+k}} and the values f ( x n ) , … , f ( x n + k ) {\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 x 1 , … , x k + 1 {\displaystyle x_{1},\dots ,x_{k+1}} one could carry out a few initializing iterations with a lower value of k. The approximation x n + k + 1 {\displaystyle x_{n+k+1}} is calculated as follows in the nth iteration. A polynomial of interpolation p n , k ( x ) {\displaystyle p_{n,k}(x)} of degree k is fitted to the k + 1 points ( x n , f ( x n ) ) , … ( x n + k , f ( x n + k ) ) {\displaystyle (x_{n},f(x_{n})),\dots (x_{n+k},f(x_{n+k}))} . With this polynomial, the next approximation x n + k + 1 {\displaystyle x_{n+k+1}} of α {\displaystyle \alpha } is calculated as
… excerpt ends here. Continue reading the full article.
