---
title: Secant Method Dynamical System
url: https://www.emergentmind.com/topics/secant-method-dynamical-system
type: topic
---

# Secant Method Dynamical System

to=arxiv_search  北京赛车前json
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to=arxiv_search 彩票总代json
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The secant method dynamical system is the study of secant iteration as a discrete dynamical system generated by successive root-finding updates rather than as a purely numerical recurrence. In its classical form, the method depends on two consecutive iterates and is therefore naturally represented as a map on pairs, typically on \(\mathbb{R}^2\) or \(\mathbb{C}^2\), with roots appearing as fixed points of the induced map. In the literature, this viewpoint has been developed in several directions: planar dynamics of the real secant map for polynomials, exact solvability for special functions, higher-order Newton–Secant-like rational maps, extensions to multiple roots and non-root cycles, Riemannian generalizations, and a recent potential-theoretic formulation near root-type fixed points [1812.10954] [1410.5097] [2508.05847].

## 1. Two-point formulation and phase space

For a real polynomial \(p\), the classical secant iteration
\[
x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}
\]
is represented as the planar map
\[
S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).
\]
This is the standard state-space formulation of the secant method as a discrete dynamical system on \(\mathbb{R}^2\) [1812.10954] [2006.01528].

A useful algebraic device is the symmetric polynomial \(q(x,y)\) determined by
\[
p(x)-p(y)=(x-y)\,q(x,y),
\]
which rewrites the map as
\[
S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).
\]
On the diagonal, \(q(x,x)=p'(x)\), and one obtains
\[
S(x,x)=(x,N_p(x)),
\]
so the second coordinate reduces to the Newton step. This establishes a direct structural link between the two-dimensional secant dynamics and the one-dimensional Newton map [1812.10954].

The natural singular set is
\[
\delta_S^1=\{(x,y)\in\mathbb{R}^2:\;p(x)=p(y),\ x\neq y\},\qquad
\delta_S^2=\{(x,x)\in\mathbb{R}^2:\;p'(x)=0\},
\]
with
\[
\delta_S=\delta_S^1\cup\delta_S^2.
\]
The maximal smooth domain for iteration is
\[
E_S=\mathbb{R}^2\setminus \bigcup_{n\ge 0}S^{-n}(\delta_S).
\]
Accordingly, the secant method is not globally a smooth self-map of the plane, but a rational plane map defined on the complement of a singular algebraic set and its preimages [2006.01528] [1812.10954].

In the holomorphic setting, the secant method may also be written in the coordinate order
\[
S_f(x,y)=\left(\frac{f(x)y-f(y)x}{f(x)-f(y)},\;x\right),
\]
which is conjugate to the standard ordering by coordinate permutation. This form is convenient for local series expansions near a simple root and for the construction of a Böttcher-type potential [2508.05847].

## 2. Fixed points, local convergence, and multiplicity effects

If \(r\) is a simple real root of \(p\), then \((r,r)\) is a fixed point of the secant map. The Jacobian on the diagonal is
\[
DS(x,x)=
\begin{pmatrix}
0 & 1\\[4pt]
\dfrac{p(x)\,p''(x)}{2[p'(x)]^2} &
\dfrac{p(x)\,p''(x)}{2[p'(x)]^2}
\end{pmatrix},
\]
and at a simple root this reduces to
\[
DS(r,r)=
\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}.
\]
Thus \((r,r)\) is a superattracting fixed point in the planar system [1812.10954].

The classical local order of convergence of the secant method at a simple root is the golden ratio
\[
\varphi=\frac{1+\sqrt{5}}{2}\approx 1.618.
\]
This is slower than Newton’s quadratic order, but it is achieved without derivative evaluation. In the exact potential-theoretic treatment of the holomorphic secant map, the same \(\varphi\) reappears through Fibonacci exponents in the iterates and in the functional equation \(h\circ S_f=h^\phi\) for the local potential [1410.5097] [2508.05847].

The local picture changes fundamentally at multiple roots. If \(\alpha\) is a real root of multiplicity \(d\ge 2\), then the parity of \(d\) determines the local dynamics. When \(d\) is odd, \((\alpha,\alpha)\) is locally attracting: there exists an open neighborhood \(U\) of \((\alpha,\alpha)\) such that \(S^n(x,y)\to(\alpha,\alpha)\) for all \((x,y)\in U\). When \(d\) is even, \((\alpha,\alpha)\) lies on the common boundary of all basins of attraction associated to simple real roots of \(p\), and also on \(\partial\mathcal A(\alpha)\) [1907.09323].

This parity effect is encoded in the quantity
\[
G_d(m)=1+m+m^2+\cdots+m^{d-1},
\]
which has the unique real zero \(m=-1\) precisely when \(d\) is even. In the even-multiplicity case, the direction \(m=-1\) generates a continuum of focal images along the prefocal line \(x=\alpha\), producing the boundary phenomenon. In the odd-multiplicity case, \(G_d(m)\neq 0\) for all real \(m\), forcing convergence back to \((\alpha,\alpha)\) [1907.09323].

A common misconception is that secant dynamics near a multiple root behaves like Newton’s method with merely slower linear contraction. The published analysis shows a more specific dichotomy: Newton’s method has a clean attracting fixed point irrespective of parity, whereas the secant map distinguishes odd and even multiplicity through its focal-point structure [1907.09323].

## 3. Basin topology, focal points, and boundary organization

For each simple real root \(\alpha\), the basin of attraction is
\[
\mathcal A(\alpha)=\{(x,y)\in E_S:\ S^m(x,y)\to (\alpha,\alpha)\text{ as }m\to\infty\},
\]
and the immediate basin \(\mathcal A^*(\alpha)\) is the connected component of \(\mathcal A(\alpha)\) containing \((\alpha,\alpha)\) [2006.01528].

The secant map has a distinguished set of focal points
\[
Q_{i,j}=(\alpha_i,\alpha_j),\qquad i\neq j,
\]
each associated with the prefocal vertical line
\[
L_j=\{(x,y)\in\mathbb{R}^2:\ x=\alpha_j\}.
\]
For a simple focal point \(Q_{i,j}\), the one-to-one correspondence between slopes \(m\) of arcs through \(Q_{i,j}\) and landing points on \(L_j\) is
\[
y(m)=\frac{\alpha_j p'(\alpha_i)-\alpha_i p'(\alpha_j)\,m}{p'(\alpha_i)-p'(\alpha_j)\,m}.
\]
This focal-line correspondence is central in the organization of basin boundaries [1812.10954] [2006.01528].

A global consequence is the Wada-type boundary property proved for the real secant map: each focal point lies on the common boundary of all root basins,
\[
Q_{i,j}\in \bigcap_{\ell=1}^n \partial A(\alpha_\ell).
\]
Hence every neighborhood of a focal point contains points from every basin [1812.10954].

For internal roots, the topology is more rigid. If \(\alpha_0<\alpha_1<\alpha_2\) are consecutive simple roots and \(\alpha_1\) is the internal root, then
\[
\mathcal A^*(\alpha_1)\subset
R=\{(x,y)\in\mathbb{R}^2:\ \alpha_0<x<\alpha_2,\ \alpha_0<y<\alpha_2\}.
\]
Under the hypothesis that the external boundary is piecewise smooth, the boundary \(\partial\mathcal A^*(\alpha_1)\) contains a smooth hexagon-like polygon with lobes whose six vertices are the focal points
\[
Q_{1,0},\ Q_{2,0},\ Q_{0,1},\ Q_{2,1},\ Q_{0,2},\ Q_{1,2},
\]
and countably many \(C^1\)-lobes are attached at \(Q_{1,0},Q_{2,0},Q_{0,2},Q_{1,2}\). Under the same hypothesis, there exists a \(4\)-cycle in \(\partial\mathcal A^*(\alpha_1)\) [2006.01528].

If \(p\) has exactly one inflection in \((\alpha_0,\alpha_2)\), then every point of the rectangle \(R\setminus\{(\alpha_1,\alpha_1)\}\) has at most two preimages in \(R\). This controls the folding geometry through the critical curves
\[
\Theta=\{(x,y)\in R:\ x\neq y,\ \partial q/\partial x(x,y)=0\}\cup\{(\gamma_0,\gamma_0)\}
\]
and \(\Gamma=S(\Theta)\), and it yields simple connectivity of the immediate basin:
\[
\mathcal A^*(\alpha_1)\ \text{is simply connected}.
\]
A degree-\(5\) counterexample with more than one change of convexity shows that multiply connected immediate basins can occur when the one-inflection hypothesis fails [2006.01528].

## 4. Exactly solvable and model secant dynamics

The real secant method is exactly solvable for \(f(x)=x^2+1\). In this case,
\[
x_n=\frac{x_{n-1}x_{n-2}-1}{x_{n-1}+x_{n-2}}.
\]
Introducing angle variables
\[
\theta_k=\arccot(x_k),\qquad x_k=\cot\theta_k,
\]
and using the cotangent addition formula gives
\[
x_n=\cot(\theta_{n-1}+\theta_{n-2}),\qquad
\theta_n=\theta_{n-1}+\theta_{n-2}\pmod{\pi}.
\]
Hence
\[
\theta_n=F_n\theta_1+F_{n-1}\theta_0,
\qquad
x_n=\cot\big(F_n\theta_1+F_{n-1}\theta_0\big),
\]
where \(F_n\) are the Fibonacci numbers [1808.03229].

On the torus
\[
\mathbb T^2=(\mathbb R/\pi\mathbb Z)^2,
\]
the angle pair evolves by the hyperbolic toral automorphism
\[
A=
\begin{pmatrix}
1&1\\
1&0
\end{pmatrix},
\]
whose eigenvalues are
\[
\lambda_\pm=\frac{1\pm\sqrt{5}}{2},\qquad
\lambda_+=\varphi,\qquad
\lambda_-=-\varphi^{-1}.
\]
This yields hyperbolicity, dense periodic points, ergodicity, mixing, and maximal Lyapunov exponent
\[
\chi_{\max}=\log\varphi.
\]
The erratic spikes in the real sequence \(x_n=\cot\theta_n\) are thus driven by mixing in angle space together with the singularities of the cotangent [1808.03229].

A different non-root mechanism appears at a nondegenerate local extremum of a polynomial. If \(p'(\alpha)=0\), \(p''(\alpha)\neq 0\), and \(p(\alpha)\neq 0\), the secant map exhibits the critical three-cycle
\[
(\alpha,\alpha)\longmapsto(\alpha,\infty)\longmapsto(\infty,\alpha)\longmapsto(\alpha,\alpha)
\]
in the projective extension. Near this cycle, the third iterate \(S_p^3\) is modeled by
\[
T_{a,d}(x,y)=\big(y-a(x+y)^d,\ y-2a(x+y)^d\big),
\]
where \(a=(-a_2)^d a_{d+1}\neq 0\) and \(\deg(p)=d+1\) [2405.08791].

The global basin geometry of this model depends sharply on parity and sign. If \(d\) is even, \(\mathcal A_{a,d}(0)\) is compact and homeomorphic to a closed topological disk, and its boundary is exactly the global stable manifold of the origin. If \(d\) is odd and \(a>0\), \(\mathcal A_{a,d}(0)\) is open, simply connected, and unbounded, and its boundary contains the stable manifold of the hyperbolic two-cycle \(\{(0,1),(0,-1)\}\). If \(d\) is odd and \(a<0\), \(\mathcal A_{a,d}(0)\) is equal to the global stable manifold of the origin and is unbounded [2405.08791].

## 5. Non-root attractors and higher-order secant families

For the classical real secant map on \(E_S\), there are no periodic orbits of minimal period \(2\) or \(3\). However, there exists a polynomial \(p^\star\) for which the secant map has an attracting periodic orbit of minimal period \(4\). The explicit construction yields a degree-\(7\) polynomial with cycle
\[
(a,b)\mapsto(b,d)\mapsto(d,c)\mapsto(c,a)\mapsto(a,b),
\]
whose monodromy has eigenvalues \(0\) and
\[
\frac14(3-\sqrt{5})^2\approx 0.1459<1.
\]
Thus open sets of initial conditions can converge to a non-root attractor [1812.10954].

After extension to the punctured torus \(\mathbb T^2_\infty\), a universal \(3\)-cycle appears at every critical point \(x_0\) of \(p\):
\[
(x_0,x_0)\mapsto(x_0,\infty)\mapsto(\infty,x_0)\mapsto(x_0,x_0).
\]
For \(k\ge 3\), the eigenvalues of \(D\hat S^3(x_0,x_0)\) are \(0\) and \(1\), and numerics show open regions of initial conditions attracted to this \(3\)-cycle [1812.10954].

The secant dynamical viewpoint also motivates higher-order one-point maps obtained by Newton–Secant compositions. Two optimal Newton–Secant-like methods without memory were constructed: a two-point method of order \(4\) using exactly \(3\) evaluations per iteration, and a three-point method of order \(8\) using exactly \(4\) evaluations per iteration. They attain the Kung–Traub bound, with efficiency indices
\[
4^{1/3}\approx 1.5874,\qquad
8^{1/4}\approx 1.6818.
\]
For the order-\(8\) family, the weight conditions are
\[
\phi(0)=0,\quad \phi'(0)=-\frac12,\quad \phi''(0)=-\frac52,\qquad
\psi(0)=1,\quad \psi'(0)=1,
\]
and a representative explicit choice is
\[
\phi(t)=-\frac12 t-\frac54 t^2,\qquad
\psi(s)=\frac{1+2s}{1+s}.
\]
In basin plots for
\[
f(z)=z^3-\frac1z=\frac{z^4-1}{z},
\]
sampled on a \(256\times 256\) grid in \([-3,3]\times[-3,3]\subset\mathbb C\) with \(100\) iterations and tolerance \(10^{-3}\), the proposed SLSS method produced basins larger than BCST, SS, CTV, TP, CL, BRW, and WL, while CFGT exhibited slightly larger stability basins than SLSS. No spurious non-root attractors or periodic cycles were reported for the proposed methods under the tested settings [1410.5097].

For multiple zeros, a parameterized Newton–Secant method modifies the classical predictor–corrector by
\[
y_n=x_n-\frac{f(x_n)}{f'(x_n)},\qquad
x_{n+1}=x_n-\frac{\theta f(x_n)}{\theta f(x_n)-f(y_n)}\cdot \frac{f(x_n)}{f'(x_n)},
\]
with
\[
\theta=\left(\frac{m-1}{m}\right)^{m-1}.
\]
At a root of multiplicity \(m\), this yields cubic convergence with efficiency index
\[
3^{1/3}\approx 1.44225,
\]
and the corresponding complex dynamical plots show basins of attraction typically larger than those of the compared Osada, Dong, and Chun methods [1507.03493].

A derivative-free three-point secant modification uses the three most recent iterates and induces a \(3\)-dimensional dynamical system
\[
X_{k+1}=T(X_k)=(x_{k-1},x_k,\Phi(x_{k-2},x_{k-1},x_k)).
\]
Its asymptotic error recurrence is
\[
e_{k+1}=K e_k e_{k-1} e_{k-2}+\text{higher-order terms},
\]
leading to the characteristic equation
\[
p^3-p^2-p-1=0
\]
and order
\[
p\approx 1.8392867552.
\]
At the fixed point \((r,r,r)\), the Jacobian is
\[
J_T(r,r,r)=
\begin{pmatrix}
0&1&0\\
0&0&1\\
0&0&0
\end{pmatrix},
\]
so all eigenvalues vanish [1902.09058].

## 6. Geometric and potential-theoretic extensions

The secant method has also been generalized from Euclidean space to complete Riemannian manifolds. For a vector field \(X:M\to TM\), the iteration is
\[
u_n=-[p_{n-1},p_n;X]^{-1}(X(p_n)),\qquad
p_{n+1}=\operatorname{Exp}_{p_n}(u_n),
\]
and the associated two-point dynamical system on \(M\times M\) is
\[
T(p_{n-1},p_n)=\Big(p_n,\operatorname{Exp}_{p_n}\big(-[p_{n-1},p_n;X]^{-1}X(p_n)\big)\Big).
\]
Its fixed points are exactly the pairs \((p^*,p^*)\) with \(X(p^*)=0\) [1712.02655].

Under the \(w\)-condition for divided differences and the semilocal hypotheses encoded in the quantities \(a(u)\), \(b(u)\), \(c(u)\), and the smallest positive root \(R\) of the scalar equation appearing in Theorem 6, the iteration is well defined, remains in \(B(p_0,R)\), and converges to the unique zero \(p^*\in B(p_0,R)\). The local contraction estimate is
\[
d(p_{n+1},p^*)\le c(R)\,d(p_n,p^*),\qquad c(R)<1.
\]
The paper establishes at least linear convergence locally; it does not establish the golden-ratio order known for the scalar secant method [1712.02655].

In the holomorphic setting, a recent development provides a Böttcher-type potential theory for the secant map near a simple root. If \(\alpha\) is a simple root of \(f\), then \((\alpha,\alpha)\) is a root-type fixed point, and the secant map admits the local factorization
\[
S_f(x,y)=\big(\alpha+G_{f,\alpha}(x,y)(x-\alpha)(y-\alpha),\ x\big),
\]
with
\[
G_{f,\alpha}(\alpha,\alpha)=\frac{f''(\alpha)}{2f'(\alpha)}.
\]
The iterates have the exact Fibonacci form
\[
S_f^{\circ n}(x,y)=\Big(\alpha+G_n(x,y)(x-\alpha)^{F_{n+1}}(y-\alpha)^{F_n},\ 
\alpha+G_{n-1}(x,y)(x-\alpha)^{F_n}(y-\alpha)^{F_{n-1}}\Big),
\]
which makes the classical golden-ratio order explicit in the two-dimensional holomorphic dynamics [2508.05847].

Assuming \(f''(\alpha)\neq 0\), there exists a Böttcher-type holomorphic germ \(H_{f,\alpha,w_0}\) on a forward-invariant bidisk \(V\subset A(f,\alpha)\), unique up to the normalization specified in the theorem, and a unique continuous modulus \(\widehat H_{f,\alpha}\) on the entire basin \(A(f,\alpha)\) satisfying
\[
\widehat H_{f,\alpha}(x,y)
=
\big[\widehat H_{f,\alpha}(S_f(x,y))\big]^{1/\phi}
|G_{f,\alpha}(x,y)|^{1/\sqrt{5}}.
\]
The associated potential
\[
h_{f,\alpha}(x,y)=\big[\widehat H_{f,\alpha}(x,y)\big]^{1/\phi}
|x-\alpha|^{1/\sqrt{5}}
|y-\alpha|^{\phi^{-1}/\sqrt{5}}
\]
is continuous on the whole basin and satisfies
\[
h_{f,\alpha}(S_f(x,y))=\big(h_{f,\alpha}(x,y)\big)^\phi.
\]
The Green’s function \(G_{f,\alpha}=\log h_{f,\alpha}\) is pluriharmonic wherever it is finite [2508.05847].

These constructions place the secant method dynamical system in a broader framework: a two-step root-finding algorithm can generate planar or higher-dimensional rational dynamics, noninvertible folding, focal geometry, exact toral models, non-root cycles, invariant manifolds, and a local potential theory analogous to one-dimensional Böttcher coordinates, but with Fibonacci scaling and golden-ratio asymptotics.

Source: https://www.emergentmind.com/topics/secant-method-dynamical-system