Compatible Functions in Analysis

• Compatible functions are a multifaceted concept describing a structural fit between functions and their formal frameworks across disciplines.
• They ensure that in numerical analysis the composite condition number controls subproblem conditioning, and in algebra they define square-root relations for means.
• This compatibility underpins modular numerical design, surrogate modeling in optimization, and unified controller synthesis in control theory.

Searching arXiv for the primary paper and closely related background on amenable/compatible functions in numerical analysis. Searching arXiv for (Beltrán, 23 Jul 2025) and related work on amenable/compatible functions. “Compatible functions” is a context-dependent technical term rather than a single invariant notion. In current arXiv usage it denotes formally distinct compatibility relations between functions and surrounding structures: in numerical analysis, pairs of functions whose composite condition number controls the product of the constituent condition numbers; in algebra, means satisfying Gauss’ functional equation with a binary law; in optimization and symbolic computation, surrogates or coefficient families that fit a prescribed representational calculus; and in control theory, Lyapunov and barrier functions that admit a common feasible control input (Beltrán, 23 Jul 2025, Padmanabhan et al., 2019, Karcher, 2021, Chen et al., 2013, Schneeberger et al., 2023, Dai et al., 2024). This suggests that the common theme is structural fit: compatibility identifies the cases in which composition, inference, or control synthesis can be carried out without violating the governing formalism.

1. Compatibility in numerical analysis

In the numerical-analysis sense developed for univariate real-analytic functions, compatibility is a property of a pair $g,h$ that complements the single-function notion of amenability. For a real-analytic univariate $f:(a,b)\to\mathbb{R}$, $f\not\equiv 0$, the condition number used in the paper is

$\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$

For analytic composition, the forward bound

$\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$

always holds, with equality in the analytic nonzero case via the chain rule. Compatibility asks for a reverse control, up to a constant factor: $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$ for all $x$ in the domain of $h$. Equivalently, the condition number of the composite should control the combined conditioning of the two stages, not merely be bounded by it (Beltrán, 23 Jul 2025).

This notion is designed to interact with amenability. Amenability is a local regularity property requiring relative perturbation balls of radius $1/(C\mu(f,x))$ to remain inside the domain and to keep $\mu(f,\cdot)$ uniformly controlled. The key theorem recalled in the univariate paper is that amenability turns backward stability into mixed stability and mixed stability into forward stability. A second theorem states that if $f:(a,b)\to\mathbb{R}$0 and $f:(a,b)\to\mathbb{R}$1 are both amenable and compatible, then $f:(a,b)\to\mathbb{R}$2 is amenable, and the composition of forward stable algorithms for $f:(a,b)\to\mathbb{R}$3 and $f:(a,b)\to\mathbb{R}$4 is forward stable for $f:(a,b)\to\mathbb{R}$5 (Beltrán, 23 Jul 2025).

In this setting, compatibility is therefore not an isolated analytic curiosity. It is the hypothesis that makes modular numerical design rigorous: a decomposition $f:(a,b)\to\mathbb{R}$6 is justified only when the conditioning of the whole problem controls the combined conditioning of the subproblems.

2. Univariate analytic tests, examples, and nonexamples

For univariate real-analytic functions, the theory becomes computationally usable because both amenability and compatibility admit simple tests. The practical compatibility criterion is Proposition 2.9 of the paper: if

$f:(a,b)\to\mathbb{R}$7

is bounded on the domain of $f:(a,b)\to\mathbb{R}$8, with the convention $f:(a,b)\to\mathbb{R}$9, then $f\not\equiv 0$0 and $f\not\equiv 0$1 are compatible. This replaces direct verification of

$f\not\equiv 0$2

by the boundedness of a simpler ratio (Beltrán, 23 Jul 2025).

Amenability is tested through the auxiliary quantity

$f\not\equiv 0$3

On bounded or semi-infinite intervals, the propositions in the paper reduce amenability to endpoint behavior of $f\not\equiv 0$4, boundedness of $f\not\equiv 0$5, and appropriate blow-up of $f\not\equiv 0$6 near zeros. On $f\not\equiv 0$7, for example, the sufficient test requires that any sequence approaching $f\not\equiv 0$8 or $f\not\equiv 0$9 while getting arbitrarily close to the zero set forces $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$0, and that $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$1 remain bounded at both ends (Beltrán, 23 Jul 2025).

The resulting tables identify many elementary amenable functions. Polynomials are amenable on $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$2, rational functions on their natural domains, $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$3 on $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$4, finite sums of monomials with real exponents on $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$5, $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$6 on $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$7, $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$8 on $\kappa(f,x)= \begin{cases} 0 & x=0,\[4pt] \infty & x\neq 0 \text{ and } f(x)=0,\[4pt] \displaystyle \frac{|x|\cdot |f'(x)|}{|f(x)|} & \text{otherwise}, \end{cases} \qquad \mu(f,x)=1+\kappa(f,x).$9, $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$0, $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$1, $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$2, $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$3, $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$4 on the stated domains, and even $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$5, $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$6, and $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$7 on $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$8. Trigonometric functions are amenable only on finite intervals avoiding the problematic global oscillatory geometry: $\kappa(g\circ h,x)\le \kappa(g,h(x))\,\kappa(h,x)$9 and $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$0 are not amenable on unbounded domains such as $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$1, and $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$2 on $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$3 is a non-amenable endpoint example because $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$4 is unbounded as $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$5 (Beltrán, 23 Jul 2025).

These examples matter for compatibility because amenable functions supply the admissible building blocks. The paper does not tabulate many concrete compatible pairs, but it does provide the analytic machinery needed to certify them once a decomposition is proposed.

3. Algebraic compatibility of means and binary laws

A different use of the term appears in the algebraic study of means. Padmanabhan and Shukla define a mean $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$6 by idempotence, symmetry, and left cancellation, and call it compatible with a binary operation $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$7 when Gauss’ functional equation holds: $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$8 In this setting, compatibility says that $\mu(g,h(x))\,\mu(h,x)\le C\,\mu(g\circ h,x)$9 acts as a $x$0-square root of $x$1 (Padmanabhan et al., 2019).

Classical examples fit this scheme exactly. The arithmetic mean is compatible with addition on $x$2, the geometric mean with multiplication on $x$3, and the harmonic mean with the law $x$4 on $x$5. Under mild additional hypotheses—a left identity for $x$6 and injectivity of squaring—the authors show that compatibility is highly restrictive: $x$7 must be commutative, and, more importantly,

$x$8

Thus any associative law compatible with a mean forces the mean to satisfy the medial identity. They further show that self-distributivity of the mean is equivalent to the Moufang property for $x$9 under the same framework (Padmanabhan et al., 2019).

The arithmetic–geometric mean provides the central obstruction. Tanimoto constructed a theta-function-based loop operation $h$0 on $h$1 satisfying

$h$2

but Padmanabhan and Shukla prove that no semigroup law, and hence no group law, can be compatible with $h$3. The reason is structural: $h$4 is not medial, so Theorem 3 rules out any associative compatible law. The same paper also rules out compatibility with Moufang loops by showing that $h$5 is not self-distributive (Padmanabhan et al., 2019).

Here compatibility is an algebraic square-root relation, and the principal content of the theory is a rigidity theorem: associative compatibility is possible for arithmetic, geometric, and harmonic means, but not for the arithmetic–geometric mean.

4. Representational compatibility in optimization, symbolic computation, and AD

In engineering optimization, “compatible functions” can mean surrogate models whose algebraic form fits a solver class. In signomial programming, a model is GP-compatible or SP-compatible when it can be expressed using monomials, posynomials, signomials, or posynomial ratios after log-log transformation. The Soft Difference of Max Affine class

$h$6

is explicitly DC in log-space, and after inverse transformation becomes a ratio of powers of two posynomials. That is the paper’s notion of compatibility: the fitted function can be embedded into a signomial program through a small set of auxiliary constraints. The reported examples include 2D, 3D, and NACA 24xx airfoil data, with RMS error driven to less than $h$7 in each case (Karcher, 2021).

In symbolic computation, the phrase refers to coefficient functions in a mixed first-order linear functional system

$h$8

whose rational certificates satisfy the commutativity conditions for derivations, shifts, and $h$9-shifts. The compatible conditions are the six families $1/(C\mu(f,x))$0, $1/(C\mu(f,x))$1, $1/(C\mu(f,x))$2, $1/(C\mu(f,x))$3, $1/(C\mu(f,x))$4, and $1/(C\mu(f,x))$5. The structure theorem states that any compatible family $1/(C\mu(f,x))$6 can be written using a rational factor $1/(C\mu(f,x))$7, symbolic-power data $1/(C\mu(f,x))$8, differential terms $1/(C\mu(f,x))$9, shift terms $\mu(f,\cdot)$0, and $\mu(f,\cdot)$1-shift terms $\mu(f,\cdot)$2, yielding decompositions of $\mu(f,\cdot)$3-solutions into a rational function, symbolic powers, a hyperexponential function, a hypergeometric term, and a $\mu(f,\cdot)$4-hypergeometric term (Chen et al., 2013).

A third representational meaning arises in global optimization with implicit functions. There, “AD-compatible” subgradients are subgradient propagation rules that allow convex relaxations of implicit functions—defined as optimal-value functions of convex problems—to be added to the elemental libraries used by forward AD modes. The paper derives LP-based formulas for directional derivatives and L-derivatives of these optimal-value relaxations from supplied AD-like information about the residual function, thereby extending forward-mode subgradient propagation beyond explicit factorable functions (Song et al., 30 Jan 2025).

Across these three literatures, compatibility denotes formal admissibility within a calculus: a surrogate compatible with SP syntax, coefficients compatible with operator commutativity, or relaxations compatible with AD propagation.

5. Compatibility in control theory

In control theory, compatibility frequently concerns the simultaneous satisfiability of stability and safety certificates. One line of work considers polynomial control-affine systems and seeks a Control Lyapunov Function $\mu(f,\cdot)$5 together with multiple Control Barrier Functions $\mu(f,\cdot)$6 that share a single rational feedback law

$\mu(f,\cdot)$7

The SOS conditions are written so that the same numerator $\mu(f,\cdot)$8 and denominator $\mu(f,\cdot)$9 satisfy the CLF inequality and every CBF inequality. In that sense, the CLF and CBFs are compatible precisely when the stability- and safety-induced feedback laws coincide, and feasibility of the unified SOS system certifies a single rational controller enforcing both properties (Schneeberger et al., 2023).

A more exact formulation appears in the verification and synthesis of compatible CLFs and CBFs independent of any nominal controller. For a control-affine polynomial system with input constraints $f:(a,b)\to\mathbb{R}$00, a CLF $f:(a,b)\to\mathbb{R}$01 and CBF $f:(a,b)\to\mathbb{R}$02 are called compatible when, for every state in

$f:(a,b)\to\mathbb{R}$03

there exists an input satisfying the CLF decrease condition, the CBF forward-invariance condition, and the input bounds simultaneously. The paper derives exact necessary and sufficient conditions for this pointwise feasibility by combining Farkas’ Lemma with Positivstellensatz, and then formulates SOS programs for verification and alternating synthesis of a larger compatible region (Dai et al., 2024).

A related but distinct notion is the extent-compatible control barrier function. There the issue is not CLF–CBF coexistence but compatibility between a barrier function and the physical extent of the system. Given an extent function $f:(a,b)\to\mathbb{R}$04 with extent set $f:(a,b)\to\mathbb{R}$05, an extent-compatible CBF is a safe function $f:(a,b)\to\mathbb{R}$06 for which there exist class-$f:(a,b)\to\mathbb{R}$07 functions $f:(a,b)\to\mathbb{R}$08 such that

$f:(a,b)\to\mathbb{R}$09

admits a feasible control uniformly over $f:(a,b)\to\mathbb{R}$10. This yields guarantees that the entire physical volume $f:(a,b)\to\mathbb{R}$11, not merely the state point $f:(a,b)\to\mathbb{R}$12, remains inside the safe set (Srinivasan et al., 2020).

In these control-theoretic usages, compatibility is fundamentally a feasibility property for shared control authority.

6. Other specialized usages and the broader pattern

Several additional literatures use the term in yet more specialized senses. In graph-flow theory, a vertex labeling $f:(a,b)\to\mathbb{R}$13 is compatible with a graph when its restriction to every connected component sums to zero; this $f:(a,b)\to\mathbb{R}$14-compatibility governs explicit expansions of assigning polynomials and the definition of $f:(a,b)\to\mathbb{R}$15-compatible broken bonds (Fu et al., 2024). In discrete mechanism design, “compatible” means dominant-strategy incentive compatible: an outcome function belongs to $f:(a,b)\to\mathbb{R}$16, and for two-player discrete type spaces any such outcome function can be turned into an affine maximizer by a nontrivial perturbation of one player’s type space (Lin et al., 2017).

Number theory provides functional-equation variants. Arithmetic functions commutable with sums of $f:(a,b)\to\mathbb{R}$17 squares satisfy

$f:(a,b)\to\mathbb{R}$18

for $f:(a,b)\to\mathbb{R}$19, and the classification reduces to the zero function, the identity on representable integers, or sign-perturbed identity on representable integers, with arbitrary values on non-representable integers (Lee, 2017). For multiplicative functions commutable with $f:(a,b)\to\mathbb{R}$20, the plus form forces the identity, while the minus form admits exactly the identity, the constant function, and the indicator of $f:(a,b)\to\mathbb{R}$21 for primes $f:(a,b)\to\mathbb{R}$22 (Park, 2022).

Statistics and systems security contribute still other meanings. In combined p-value function inference, compatibility means that the same combined p-value function determines hypothesis tests, point estimates, and confidence intervals, so that exclusion of a null value by the confidence interval matches rejection by the p-value and the median estimate lies inside the interval (Pawel et al., 13 Mar 2025). In secure email, Pretzel argues that provider-supplied functions such as spam filtering and topic extraction are compatible with default end-to-end encryption by expressing the underlying classifiers through secure two-party computation rather than plaintext access (Gupta et al., 2016).

These usages are not definitions of a single concept. This suggests instead that “compatible functions” is an umbrella label for formally verified coexistence: coexistence with composition theorems, semigroup laws, optimization grammars, differentiation rules, shared feedback constraints, counting formulas, inferential pipelines, or encrypted computation.