The secant method for solving an equation $F(x)=0$ in one variable is much older than Newton's one. Recall that given two iterates $x_{k-1}$ and $x_k$, it provides an update $x_k$ by taking the intersection of the chord joining the graph points $(x_{k-1},F(x_{k-1}))$ and $(x_k,F(x_k))$, with the horizontal axis $y=0$. The intersection is $(x_k,0)$. Students are taught that this method is convergent in the following sense, that if $F$ is ${\mathcal C}^2$ at $\hat x$, if $F(\hat x)=0$ and $F'(\hat x)\ne0$, and if $x_0,x_1$ are close enough to $\hat x$, then the sequence converges towards $\hat x$. In addition, the error $e_k:=|x_k-\hat x|$ satisfies $e_{k+1}\le{\rm cst}\cdot e_ke_{k-1}$ for some constant. This ensures that $e_k\le\rho^{\omega^k}$ for some $\rho< 1$ and $\omega=\frac{1+\sqrt{5}}{2}$ the golden ratio. This order is optimal, as seen on the example of a quadratic polynomial $F$.
The following is a natural generalization to $n\ge2$ variables. Now, $F:B(\hat x;r) \rightarrow {\mathbb R}^n$ is a vector field. Given $n+1$ iterates $x_{k-n},\ldots,x_{k-1},x_k$, we determine $x_{k+1}$ as the point $x\in{\mathbb R}^n$ such that $(x,0)$ is the intersection of the horizontal space ${\mathbb R}^n\times 0$, with the $n$-dimensional affine space spanned by the graph points $(x_j,F(x_j))$.
Question. Is this method classical ? Does there exist a reference ? Do we know convergence ? Order ?
Remark. Unlike the one-variable case, the algorithm can stop in finite time, when the affine space spanned by graph points does not intersect the horizontal space. Thus we expect convergence only for reasonnable initial data $(x_0,\ldots,x_n)$.
