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The secant method starts from two approximations $x_0,x_1$. Replace the function near its zero of a function by the secant line through $(x_{k-1},f(x_{k-1}))$ and $(x_k,f(x_k))$; the zero of that line is the next approximation:Stop immediately if $f(x_k)=0$; otherwise the denominator must be nonzero. Each further step needs one new function evaluation and no derivative. There is no automatic bracketing or global convergence guarantee.^[ref]

For iterates approaching ${\mathbf{x}}_{\ast }$, write $e_k=\Vert {\mathbf{x}}_k-{\mathbf{x}}_{\ast }\Vert$. Exact order of convergence $p$ means $e_{k+1}/e_k^p\to C$ for a finite constant $C>0$, with nonzero errors. Thus quadratic convergence has $p=2$, and cubic convergence has $p=3$. The weaker eventual bound $e_{k+1}\le C{e}_k^p$ means at least order $p$. Finite termination is a separate case.^[ref]

The assertion about being always faster is false. For example, the fixed-point iterations with errors $e_{k+1}=e_k^2$ and $d_{k+1}=50d_k^3$ have exact quadratic convergence and cubic convergence, respectively. Starting both at $0.1$, their next errors are $0.01$ and $0.05$: the cubic convergence method is worse on that step, although both converge. Error constants, starting points, and work per step all matter. The higher order of convergence describes asymptotic behavior within a convergence region, not a uniform comparison of all computations.

The secant method approximates a zero of a function by repeatedly taking the zero of a secant line through the two most recent sampled points. The formula follows by setting the line $f(x_k)+(x-x_k)(f(x_k)-f(x_{k-1}))/(x_k-x_{k-1})$ equal to zero. It requires distinct sampled positions and a nonzero difference of sampled values, unless a root has already been found. After initialization it needs one new function evaluation per step. Unlike bisection, it need not retain a bracket. For a sufficiently smooth function near a simple root, its usual local order of convergence is $(1+\sqrt{5})/2$.^[ref]