Iterative Function Solver

Two reference-backed routes to the functional square root \(f(f(x)) = F(x)\) — live, with a draggable cobweb grapher.

This calculator implements two complementary methods. Composita / Paper Mode builds the truncated power series of \(f\) from the triangular composita recurrence of Kruchinin & Kruchinin, extending naturally to the family \(A^{\circ 2^m}(x) = F(x)\). Analytic Fixed-Point Mode classifies a chosen real fixed point and, when the multiplier is suitable, constructs a local half-iterate through Schröder / Koenigs linearization (see the Theory page). Everything below updates live as you edit \(F\).

1. Choose a Solver Mode

2. Define the Target Function \(F(x)\)

Set the coefficients of \(F(x) = c_0 + c_1 x + c_2 x^2 + \cdots + c_6 x^6\) with the boxes or sliders. Paper mode requires \(c_0 = 0\) and \(c_1 > 0\); analytic mode accepts any constant, but the chosen \(x_0\) must satisfy \(F(x_0) = x_0\).

Live target

Paper depth \(m\) (solves \(A^{\circ 2^m}=F\))

Series order

Presets: \(x + x^2\) \(x + 16x^2\) \(x(1+2x)^2\) \(x + x^3\) \(A^{\circ 4}=x+16x^2\)

3. Computed Candidate \(f(x)\)

Mode:

Target \(F(x)\)

Candidate

Verification

Method note

Max residual / error

Classification

Series method

Chosen fixed point

n/a

Coefficient data

4. Composita Triangles

The composita \(F^{\Delta}(n,k) = [x^n]\,F(x)^k\) (Kruchinin & Kruchinin, 2013) is the coefficient of \(x^n\) in the \(k\)-th power of \(F\). Theorem 3 of the 2014 paper turns \(A(A(x)) = F(x)\) into the triangular recurrence below, from which \(a(n) = A^{\Delta}(n,1)\).

5. Interactive Cobweb Grapher

Compare \(F\), the candidate \(f\), and the recomposition \(f(f(x))\). Hover to read values, click to drop the cobweb start point, drag to pan, and scroll to zoom. The orange staircase is the orbit \(x_{n+1}=F(x_n)\).

— candidate \(f(x)\) — recomposed \(f(f(x))\) — target \(F(x)\) ● fixed point — orbit cobweb — \(y=x\)

Tip: the cobweb starts at \(x_0\) from the Orbit Explorer below — click the plot to move it.

6. Orbit Explorer

Iterate the target dynamics \(x_{n+1} = F(x_n)\). The start \(x_0\) is shared with the cobweb above, so the table and the staircase always agree.

\(x_0\)

Steps

\(n\)	\(x_n\)

7. Method & References

The two solver modes implement the following constructions:

[1] V. V. Kruchinin, D. V. Kruchinin (2013), Composita and its Properties, arXiv:1103.2582 — defines \(F^{\Delta}(n,k)=[x^n]F(x)^k\) and the composition law \(F^{\Delta}(n,k)=\sum_{m=k}^{n} A^{\Delta}(n,m)\,G^{\Delta}(m,k)\) used in Paper Mode.
[2] D. V. Kruchinin, V. V. Kruchinin (2013), Method for solving an iterative functional equation \(A^{2^n}(x)=F(x)\), arXiv:1302.1986 — Theorem 3's recurrence for \(A^{\Delta}(n,k)\) and the repeated-square-root scheme behind the paper depth \(m\).
[3] G. Koenigs (1884) — local analytic linearization \(\sigma(F(x))=\lambda\,\sigma(x)\) near a non-parabolic fixed point, giving the Analytic Mode half-iterate \(f=\sigma^{-1}(\sqrt{\lambda}\,\sigma)\).
[4] Kuczma, Choczewski, Ger (1990), Iterative Functional Equations — the structural background; see the Bibliography for the full list.

Integer footnote (Cor. 7 of [2]). If \(F\) has integer coefficients with \(f(1)=1\), then \(A(A(x))=\tfrac14 F(4x)\) has integer coefficients — which is why presets like \(x+16x^2\) yield clean integer series \(a(n)\).