Parallel Evaluation of Nonlinear Recursions

• Parallel evaluation of nonlinear recursions is defined as designing algorithms that decompose complex recursive relations into independent tasks, enabling simultaneous computation.
• The methodology uses tree-based decomposition, fixed-point iterations, and algebraic transformations to convert nonlinearity into more tractable, parallelizable structures.
• Applications span combinatorial counting, numerical simulation, and optimal control, highlighting the importance of scalable methods in modern computational systems.

Parallel evaluation of nonlinear recursions refers to the design, analysis, and implementation of algorithms that compute the terms or solutions of nonlinear recursive relations using multiple processing units or computational processes simultaneously. Nonlinear recursions, particularly those termed nested, meta-Fibonacci, or involving polynomial, rational, or transcendental nonlinearities, arise broadly in combinatorics, numerical analysis, dynamical systems, and scientific computing. General strategies for parallel evaluation exploit structural properties—such as independence in subtree computations, algebraic solvability, or measure preservation in chaotic maps—to decompose the overall computation into smaller concurrent tasks, accelerating solution and enabling scalability for large or complex systems.

1. Formalism and Model Problem Classes

Nonlinear recursions of interest have the general form

$$R(n) = \sum_{i=1}^k R\Big(n - a_i - \sum_{j=1}^p R(n - b_{ij})\Big)$$

where $a_i, b_{ij}$ are shift parameters, $k$ is the "arity," and $p$ the "order" (Isgur et al., 2013). Other families include systems of nonlinearly coupled first-order polynomial recursions with multiple dependent variables (Calogero et al., 2019, Calogero, 20 Jul 2024, Calogero, 8 Sep 2024), continued fraction recursions for constants (e.g., $x_{k} = 1 + 1/x_{k-1}$ for the Golden mean) (Finch, 17 Mar 2025), and nonlinear difference equations relevant to ODE/PDE discretization (Christlieb et al., 2020). Recursions may involve nested dependencies, explicit nonlinearity in argument or right-hand side, or be part of coupled systems derived from transformations of solvable linear recursions.

Key terms:

• Arity ($k$): Number of terms in the sum or number of recursive calls per iteration.
• Order ($p$): Number of inner summations or nesting depth.
• Polynomial vs. Rational vs. Other Nonlinearities: The right-hand side can be polynomial functions of the dependent variables, involve rational expressions, or non-polynomial functions.

2. Combinatorial and Tree-Based Solution Methods

The tree-based methodology provides a combinatorial framework for nested nonlinear recursions, particularly meta-Fibonacci types (Isgur et al., 2013). The approach associates each recursion with an infinite, labeled $k$-ary tree (skeleton), with labels assigned in preorder and leaves subdivided into "cells" or substructures. The n-th term of the recursion corresponds to a count—e.g., the number of nonempty cells in the leaves after $n$ labels have been distributed.

Key steps:

• Skeleton and labeling rules: The recurrence parameters (e.g., $s, j, m, p$) dictate both the recursion and labeling structure.
• Pruning operations: Multi-step processes that adjust the tree's label structure (initial correction, deletion, lifting, relabeling).
• Isomorphism: The pruned tree at $n$ labels is shown to be isomorphic (with shifted $n'$) to another labeled tree; this leads directly to a closed-form recursion for counting nonempty "cells."

For example, nested recursions such as $$R(n) = R(n - s - R(n-j)) + R(n - s - j - m - R(n-2j-m))$$ are mirrored by the pruning steps, allowing for derivation of recursions for split parts (e.g., left and right subtrees), which are then recombined.

This framework has been extended to general $k$-ary trees; higher arity allows for even finer-grained decomposition since each nonleaf node's children can be evaluated independently, which is ideal for parallel computation. Linear combinations of solution sequences are handled via tree "superpositions" (Isgur et al., 2013).

3. Structural and Algebraic Solvability

Several works exploit explicit solvability or reductions that enable direct, parallel computation:

• Change of variables to linear systems: Many nonlinear recursions or systems can be mapped via invertible (often rational or linear-fractional) transformations to linear systems whose solutions are exponential-polynomial forms (Calogero, 20 Jul 2024, Calogero, 8 Sep 2024). These explicit formulas allow evaluation of the nonlinear recurrence at any $n$ in parallel, given the initial data and transformation parameters.
• Symmetric function methods: For polynomially coupled systems in discrete time, the dynamics of the symmetric functions of the dependent variables can be solved independently and inverted to recover all variables as roots of an associated polynomial, feasible for systems of small degree (Calogero et al., 2019).
• Asymptotic expansion and reciprocal transformation: Theoretical analysis of nonlinear recursions often yields detailed asymptotic expansions, with constants and higher-order corrections extracted via reciprocal sequence analysis and dedicated algorithms (e.g., Mavecha–Laohakosol) (Finch, 16 Dec 2024). Terms of the expansion (and error corrections/products in continued fraction recursions) are separable and can be computed concurrently (Finch, 17 Mar 2025).

4. Parallelization Strategies

A variety of specific techniques have been developed to parallelize nonlinear recursion evaluation:

a) Divide-and-conquer via arity:

• For $k$-ary recursions (especially when $k$ is large), the natural tree decomposition splits the computation into $k$ independent subproblems at each step (Isgur et al., 2013).
• Extension to multidimensional or block recursions (e.g., systems with 2 or 3 dependent variables) admits parallel precomputation of the explicit terms (especially when closed-form solutions exist) (Calogero, 20 Jul 2024, Calogero, 8 Sep 2024).

b) Parallel-in-time schemes:

• ODE-based and parareal algorithms split the evolution "in time" (the recursion index) into intervals over which coarse and fine propagators can be applied independently, with finishes synchronized during correction steps (Chehab et al., 2015).
• Modified parareal methods efficiently accelerate convergence for nonlinear ODEs or matrix functions, with global projections and orthonormal basis management (Chehab et al., 2015).

c) Jacobi-type fixed-point iterations:

• For triangular (feedforward) nonlinear systems, Jacobi updates on all variables can be executed in parallel, with at most $T$ iterations required for exact convergence. Hybrid block Jacobi–Gauss-Seidel schemes further balance parallelism and convergence speed (Song et al., 2020).

d) Domain decomposition and convolution-based approaches:

• Successive convolution methods for PDEs (with nonlinear terms) use domain decomposition, assembling global solutions from local convolution integrals, with near-neighbor communication enforced by an artificial CFL constraint. Fast summation techniques yield $O(N)$ complexity, leveraging hierarchical parallelism in MPI + Kokkos implementations (Christlieb et al., 2020).

e) Tree superposition and linear combination:

• Composite solution sequences built as frequency linear combinations of base recursions are constructed by superposing building-block trees, allowing the entire solution sequence to be reconstructed in parallel after relabeling (Isgur et al., 2013).

f) Parallel shooting and multiple-trajectory methods:

• In nonlinear optimization, parallel shooting sequential quadratic programming (PS-SQP) schemes simultaneously evaluate multiple initial guesses or step size variants, accelerating solution of nonlinear recursions in optimal control contexts (Verheijen et al., 2023).

Table: Summary of Parallelization Techniques

Technique	Structure Exploited	Key Application Area
Arity-based divide & conquer	Tree arity, independence of subtrees	Combinatorial recursions
Jacobi/GS fixed-point iteration	Triangular nonlinear systems	ML (nets, autoregressive)
Parallel-in-time (Parareal)	Temporal decomposition	ODEs, matrix functions
Symmetric function/algebraic inversion	Explicit solvability/roots	Polynomial systems
Domain decomposition + convolution	Locality & analytic kernel decay	Nonlinear PDEs
Parallel shooting (multi-trajectory)	Multiple shooting directions/steps	Optimization/NMPC
Expansion/product parallelization	Infinite product or sum decomposition	Asymptotics/convergence

5. Applications and Computational Implications

Parallel evaluation of nonlinear recursions is central in:

• Enumerative combinatorics and meta-Fibonacci sequences: Structural recursions for counting configurations often allow combinatorial interpretations that translate directly to parallelizable algorithms (Isgur et al., 2013).
• Feedforward and autoregressive computation in machine learning: Triangular recurrence systems can be recast for parallel evaluation, with substantial wall-clock speedup in RNNs, DenseNets, and autoregressive samplers (Song et al., 2020).
• Numerical PDEs (matrix-free methods): Matrix-free successive convolution methods apply to nonlinear advection-diffusion and Hamilton–Jacobi equations, scaling efficiently across modern distributed architectures (Christlieb et al., 2020).
• Optimal control and nonlinear programming: Parallelization via multi-trajectory shooting and step-size search accelerates convergence in nonlinear MPC (Verheijen et al., 2023).
• Arithmetic dynamics and asymptotic expansions: Parallel computation of expansion terms and product corrections underpins high-precision evaluation in arithmetic and continued fraction dynamics (Finch, 16 Dec 2024, Finch, 17 Mar 2025).

6. Open Questions and Limitations

Substantial challenges and open problems remain specifically relevant to parallelization:

• Nonstandard parameter regimes: For certain parameter ranges (e.g., negative "shift" parameters or beyond canonical intervals), tree pruning or combinatorial constructions may fail, and alternative representations for parallel evaluation are sought (Isgur et al., 2013).
• Existence and structure of “ceiling function” and hybrid solutions: Characterizing existence and explicit constructions (especially for $k$-ary recursions and cases with nondivisible parameters) is incomplete, limiting automated parallel reduction (Isgur et al., 2013).
• Extension to higher dimensions and noncommuting variables: While explicit techniques scale for small $N$, high-degree systems and noncommuting variable recursions lack sufficiently robust frameworks for parallel inversion (Calogero et al., 2019, Calogero, 20 Jul 2024, Calogero, 8 Sep 2024).
• Numerical stability and symbolic overhead: Explicit but highly nonlinear transformations can introduce numerical instability (especially in polynomial denominators), and symbolic preprocessing may be a computational bottleneck.
• Summation with special denominators: Parallel summation algorithms for P-recursive sequences currently cannot fully predict "special" polynomial denominators, limiting completeness (Chen et al., 7 Feb 2024).

A plausible implication is that future progress in parallel evaluation may depend on (a) finding new combinatorial or algebraic reductions for recursions outside standard parameter regimes, (b) improving automated handling of singularities or parameter dependencies, and (c) developing general symbolic-numeric frameworks optimized for distributed architectures.

7. Outlook and Future Directions

The integration of combinatorial, algebraic, and computational insights has enabled robust parallel strategies for many classes of nonlinear recursions, but ongoing research aims to extend these methods to more complex and less structured systems. Notable directions include:

• Generalizing tree-based constructions and algebraic solution mapping to broader classes of non-polynomial or non-nested recursions.
• Extending parallel evaluation schemes to multidimensional and matrix-valued recursions, particularly for scientific computing and control contexts.
• Systematic implementation of hybrid symbolic-numeric parallel routines for high accuracy, especially where asymptotic expansions or infinite product representations play a critical role (Finch, 16 Dec 2024, Finch, 17 Mar 2025).
• Refining domain decomposition and communication patterns for large-scale simulation via matrix-free or explicit convolution approaches (Christlieb et al., 2020, Heinlein et al., 31 Oct 2024).
• Developing practical software libraries that synthesize these algorithmic advances and abstract parallel scheduling and synchronization burdens.

Continued progress will likely require close collaboration among researchers in combinatorics, dynamical systems, numerical analysis, and parallel computing, with a focus on bridging explicit model structure with scalable algorithmic implementations.