Secant Method Dynamical System

• The secant method dynamical system is a discrete two-point iteration for root finding, mapping pairs in ℝ² or ℂ² to reveal complex convergence behavior.
• It employs a symmetric polynomial formulation that links its dynamics with Newton’s method, achieving local golden-ratio convergence at simple roots.
• Extensions cover multiple roots, basin topology, and geometric generalizations to holomorphic and Riemannian settings, deepening insights into potential theory and higher-order iterations.

to=arxiv_search 北京赛车前json {"query":"secant method dynamical system basins attraction Newton-Secant like methods without memory dynamics", "max_results": 10, "sort_by": "relevance"} to=arxiv_search 彩票总代json {"query":"(Salimi et al., 2014, Li et al., 2018, Garijo et al., 2018, Gardini et al., 2020, Garijo et al., 2019, Fontic et al., 2024, Castro et al., 2017, Ferrara et al., 2015, Tiruneh, 2019, Freeman, 7 Aug 2025)", "max_results": 10, "sort_by": "relevance"} 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 (Garijo et al., 2018, Salimi et al., 2014, Freeman, 7 Aug 2025).

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 $\mathbb{R}^2$ (Garijo et al., 2018, Gardini et al., 2020).

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

$\mathbb{C}^2$0

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 (Garijo et al., 2018).

The natural singular set is

$\mathbb{C}^2$1

with

$\mathbb{C}^2$2

The maximal smooth domain for iteration is

$\mathbb{C}^2$3

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 (Gardini et al., 2020, Garijo et al., 2018).

In the holomorphic setting, the secant method may also be written in the coordinate order

$\mathbb{C}^2$4

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 (Freeman, 7 Aug 2025).

2. Fixed points, local convergence, and multiplicity effects

If $\mathbb{C}^2$5 is a simple real root of $\mathbb{C}^2$6, then $\mathbb{C}^2$7 is a fixed point of the secant map. The Jacobian on the diagonal is

$\mathbb{C}^2$8

and at a simple root this reduces to

$\mathbb{C}^2$9

Thus $p$0 is a superattracting fixed point in the planar system (Garijo et al., 2018).

The classical local order of convergence of the secant method at a simple root is the golden ratio

$p$1

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 $p$2 reappears through Fibonacci exponents in the iterates and in the functional equation $p$3 for the local potential (Salimi et al., 2014, Freeman, 7 Aug 2025).

The local picture changes fundamentally at multiple roots. If $p$4 is a real root of multiplicity $p$5, then the parity of $p$6 determines the local dynamics. When $p$7 is odd, $p$8 is locally attracting: there exists an open neighborhood $p$9 of $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$0 such that $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$1 for all $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$2. When $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$3 is even, $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$4 lies on the common boundary of all basins of attraction associated to simple real roots of $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$5, and also on $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$6 (Garijo et al., 2019).

This parity effect is encoded in the quantity

$$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$7

which has the unique real zero $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$8 precisely when $$x_{n+2}=x_{n+1}-p(x_{n+1})\frac{x_{n+1}-x_n}{p(x_{n+1})-p(x_n)}$$9 is even. In the even-multiplicity case, the direction $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$0 generates a continuum of focal images along the prefocal line $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$1, producing the boundary phenomenon. In the odd-multiplicity case, $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$2 for all real $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$3, forcing convergence back to $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$4 (Garijo et al., 2019).

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 (Garijo et al., 2019).

3. Basin topology, focal points, and boundary organization

For each simple real root $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$5, the basin of attraction is

$$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$6

and the immediate basin $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$7 is the connected component of $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$8 containing $$S_p(x,y)=\left(y,\;y-p(y)\frac{y-x}{p(y)-p(x)}\right).$$9 (Gardini et al., 2020).

The secant map has a distinguished set of focal points

$\mathbb{R}^2$0

each associated with the prefocal vertical line

$\mathbb{R}^2$1

For a simple focal point $\mathbb{R}^2$2, the one-to-one correspondence between slopes $\mathbb{R}^2$3 of arcs through $\mathbb{R}^2$4 and landing points on $\mathbb{R}^2$5 is

$\mathbb{R}^2$6

This focal-line correspondence is central in the organization of basin boundaries (Garijo et al., 2018, Gardini et al., 2020).

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,

$\mathbb{R}^2$7

Hence every neighborhood of a focal point contains points from every basin (Garijo et al., 2018).

For internal roots, the topology is more rigid. If $\mathbb{R}^2$8 are consecutive simple roots and $\mathbb{R}^2$9 is the internal root, then

$q(x,y)$0

Under the hypothesis that the external boundary is piecewise smooth, the boundary $q(x,y)$1 contains a smooth hexagon-like polygon with lobes whose six vertices are the focal points

$q(x,y)$2

and countably many $q(x,y)$3-lobes are attached at $q(x,y)$4. Under the same hypothesis, there exists a $q(x,y)$5-cycle in $q(x,y)$6 (Gardini et al., 2020).

If $q(x,y)$7 has exactly one inflection in $q(x,y)$8, then every point of the rectangle $q(x,y)$9 has at most two preimages in $p(x)-p(y)=(x-y)\,q(x,y),$0. This controls the folding geometry through the critical curves

$p(x)-p(y)=(x-y)\,q(x,y),$1

and $p(x)-p(y)=(x-y)\,q(x,y),$2, and it yields simple connectivity of the immediate basin: $p(x)-p(y)=(x-y)\,q(x,y),$3 A degree-$p(x)-p(y)=(x-y)\,q(x,y),$4 counterexample with more than one change of convexity shows that multiply connected immediate basins can occur when the one-inflection hypothesis fails (Gardini et al., 2020).

4. Exactly solvable and model secant dynamics

The real secant method is exactly solvable for $p(x)-p(y)=(x-y)\,q(x,y),$5. In this case,

$p(x)-p(y)=(x-y)\,q(x,y),$6

Introducing angle variables

$p(x)-p(y)=(x-y)\,q(x,y),$7

and using the cotangent addition formula gives

$p(x)-p(y)=(x-y)\,q(x,y),$8

Hence

$p(x)-p(y)=(x-y)\,q(x,y),$9

where $$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$0 are the Fibonacci numbers (Li et al., 2018).

On the torus

$$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$1

the angle pair evolves by the hyperbolic toral automorphism

$$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$2

whose eigenvalues are

$$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$3

This yields hyperbolicity, dense periodic points, ergodicity, mixing, and maximal Lyapunov exponent

$$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$4

The erratic spikes in the real sequence $$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$5 are thus driven by mixing in angle space together with the singularities of the cotangent (Li et al., 2018).

A different non-root mechanism appears at a nondegenerate local extremum of a polynomial. If $$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$6, $$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$7, and $$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$8, the secant map exhibits the critical three-cycle

$$S(x,y)=\left(y,\;\frac{y\,q(x,y)-p(y)}{q(x,y)}\right).$$9

in the projective extension. Near this cycle, the third iterate $q(x,x)=p'(x)$0 is modeled by

$q(x,x)=p'(x)$1

where $q(x,x)=p'(x)$2 and $q(x,x)=p'(x)$3 (Fontic et al., 2024).

The global basin geometry of this model depends sharply on parity and sign. If $q(x,x)=p'(x)$4 is even, $q(x,x)=p'(x)$5 is compact and homeomorphic to a closed topological disk, and its boundary is exactly the global stable manifold of the origin. If $q(x,x)=p'(x)$6 is odd and $q(x,x)=p'(x)$7, $q(x,x)=p'(x)$8 is open, simply connected, and unbounded, and its boundary contains the stable manifold of the hyperbolic two-cycle $q(x,x)=p'(x)$9. If $\mathbb{C}^2$00 is odd and $\mathbb{C}^2$01, $\mathbb{C}^2$02 is equal to the global stable manifold of the origin and is unbounded (Fontic et al., 2024).

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

For the classical real secant map on $\mathbb{C}^2$03, there are no periodic orbits of minimal period $\mathbb{C}^2$04 or $\mathbb{C}^2$05. However, there exists a polynomial $\mathbb{C}^2$06 for which the secant map has an attracting periodic orbit of minimal period $\mathbb{C}^2$07. The explicit construction yields a degree-$\mathbb{C}^2$08 polynomial with cycle

$\mathbb{C}^2$09

whose monodromy has eigenvalues $\mathbb{C}^2$10 and

$\mathbb{C}^2$11

Thus open sets of initial conditions can converge to a non-root attractor (Garijo et al., 2018).

After extension to the punctured torus $\mathbb{C}^2$12, a universal $\mathbb{C}^2$13-cycle appears at every critical point $\mathbb{C}^2$14 of $\mathbb{C}^2$15: $\mathbb{C}^2$16 For $\mathbb{C}^2$17, the eigenvalues of $\mathbb{C}^2$18 are $\mathbb{C}^2$19 and $\mathbb{C}^2$20, and numerics show open regions of initial conditions attracted to this $\mathbb{C}^2$21-cycle (Garijo et al., 2018).

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 $\mathbb{C}^2$22 using exactly $\mathbb{C}^2$23 evaluations per iteration, and a three-point method of order $\mathbb{C}^2$24 using exactly $\mathbb{C}^2$25 evaluations per iteration. They attain the Kung–Traub bound, with efficiency indices

$\mathbb{C}^2$26

For the order-$\mathbb{C}^2$27 family, the weight conditions are

$\mathbb{C}^2$28

and a representative explicit choice is

$\mathbb{C}^2$29

In basin plots for

$\mathbb{C}^2$30

sampled on a $\mathbb{C}^2$31 grid in $\mathbb{C}^2$32 with $\mathbb{C}^2$33 iterations and tolerance $\mathbb{C}^2$34, 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 (Salimi et al., 2014).

For multiple zeros, a parameterized Newton–Secant method modifies the classical predictor–corrector by

$\mathbb{C}^2$35

with

$\mathbb{C}^2$36

At a root of multiplicity $\mathbb{C}^2$37, this yields cubic convergence with efficiency index

$\mathbb{C}^2$38

and the corresponding complex dynamical plots show basins of attraction typically larger than those of the compared Osada, Dong, and Chun methods (Ferrara et al., 2015).

A derivative-free three-point secant modification uses the three most recent iterates and induces a $\mathbb{C}^2$39-dimensional dynamical system

$\mathbb{C}^2$40

Its asymptotic error recurrence is

$\mathbb{C}^2$41

leading to the characteristic equation

$\mathbb{C}^2$42

and order

$\mathbb{C}^2$43

At the fixed point $\mathbb{C}^2$44, the Jacobian is

$\mathbb{C}^2$45

so all eigenvalues vanish (Tiruneh, 2019).

6. Geometric and potential-theoretic extensions

The secant method has also been generalized from Euclidean space to complete Riemannian manifolds. For a vector field $\mathbb{C}^2$46, the iteration is

$\mathbb{C}^2$47

and the associated two-point dynamical system on $\mathbb{C}^2$48 is

$\mathbb{C}^2$49

Its fixed points are exactly the pairs $\mathbb{C}^2$50 with $\mathbb{C}^2$51 (Castro et al., 2017).

Under the $\mathbb{C}^2$52-condition for divided differences and the semilocal hypotheses encoded in the quantities $\mathbb{C}^2$53, $\mathbb{C}^2$54, $\mathbb{C}^2$55, and the smallest positive root $\mathbb{C}^2$56 of the scalar equation appearing in Theorem 6, the iteration is well defined, remains in $\mathbb{C}^2$57, and converges to the unique zero $\mathbb{C}^2$58. The local contraction estimate is

$\mathbb{C}^2$59

The paper establishes at least linear convergence locally; it does not establish the golden-ratio order known for the scalar secant method (Castro et al., 2017).

In the holomorphic setting, a recent development provides a Böttcher-type potential theory for the secant map near a simple root. If $\mathbb{C}^2$60 is a simple root of $\mathbb{C}^2$61, then $\mathbb{C}^2$62 is a root-type fixed point, and the secant map admits the local factorization

$\mathbb{C}^2$63

with

$\mathbb{C}^2$64

The iterates have the exact Fibonacci form

$\mathbb{C}^2$65

which makes the classical golden-ratio order explicit in the two-dimensional holomorphic dynamics (Freeman, 7 Aug 2025).

Assuming $\mathbb{C}^2$66, there exists a Böttcher-type holomorphic germ $\mathbb{C}^2$67 on a forward-invariant bidisk $\mathbb{C}^2$68, unique up to the normalization specified in the theorem, and a unique continuous modulus $\mathbb{C}^2$69 on the entire basin $\mathbb{C}^2$70 satisfying

$\mathbb{C}^2$71

The associated potential

$\mathbb{C}^2$72

is continuous on the whole basin and satisfies

$\mathbb{C}^2$73

The Green’s function $\mathbb{C}^2$74 is pluriharmonic wherever it is finite (Freeman, 7 Aug 2025).

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.