Remez Algorithm: Minimax Polynomial Approximation

• Remez algorithm is an iterative method for computing minimax polynomial approximations by minimizing the maximum uniform error over a compact interval.
• It leverages the Chebyshev alternation theorem to guarantee optimality by ensuring alternating extreme errors at critical nodes.
• Extensions of the algorithm address floating-point stability and extend its application to complex domains and certified function approximations.

The Remez algorithm is an iterative numerical scheme for computing the minimax (Chebyshev) polynomial approximation of a continuous function on a compact interval or, more generally, on compact subsets of the complex plane. The central goal is to find, for a given function $f$ and degree $n$, a polynomial $p_n^*$ of degree at most $n$ that minimizes the uniform error $\|f-p_n^*\|_\infty$ over the domain. The Remez algorithm is a foundational method in approximation theory, numerical analysis, floating-point code generation, computer-assisted proofs of inequalities, and—in its complex extensions—computation of Chebyshev polynomials on complex sets 0701020.

1. Minimax Approximation and the Chebyshev Alternation Theorem

Given a continuous function $f$ on $[a,b]$ and the space $\Pi_n$ of real polynomials of degree at most $n$, the minimax problem seeks a polynomial $$p_n^* = \operatorname{arg\,min}_{p \in \Pi_n} \|f - p\|_\infty$$ where $$\|f - p\|_\infty = \max_{x \in [a,b]} |f(x) - p(x)|$$. The optimal polynomial $p_n^*$ is called the degree-$n$ minimax (or Chebyshev) approximation, and the minimal error $E^*$ is known as the Chebyshev error 0701020.

Chebyshev’s alternation theorem provides a characterization of optimality: $p_n$ is a minimax polynomial if and only if there exist $n+2$ points $a \leq x_0 < x_1 < \cdots < x_{n+1} \leq b$ such that $f(x_i) - p_n(x_i) = (-1)^i E^*$ for all $i$. The error alternates in sign and attains its extremal value $E^*$ at each alternation point. This property is both necessary and sufficient for optimality and forms the theoretical underpinning of the Remez exchange process 0701020.

2. The Classical Remez Algorithm

The Remez algorithm is an iterative procedure that constructs a sequence of candidate polynomials $\{p_k\}$ and associated error estimates $\{E_k\}$ converging to $(p_n^*, E^*)$. The algorithm proceeds as follows 0701020:

1. Initialization: Select $n+2$ initial points $\{x_i^{(0)}\}$ in $[a,b]$ (often using Chebyshev nodes).
2. Linear System Solution: For iteration $k$, solve the $(n+2) \times (n+2)$ system:

$$\sum_{j=0}^{n} c_j^{(k)} (x_i^{(k)})^j + (-1)^i E_k = f(x_i^{(k)}),\quad i=0,\ldots,n+1$$

to obtain the polynomial coefficients $c_j^{(k)}$ and error $E_k$.

1. Polynomial Construction: Form $p_k(x) = \sum_{j=0}^n c_j^{(k)} x^j$.
2. Alternation Points Update: Identify $n+2$ new points where the signed error $f(x)-p_k(x)$ attains local extrema with alternating sign; these serve as the next interpolation nodes.
3. Stopping Criterion: Check if $|E_k - E_{k-1}| < \text{tol}$ or if alternation points have stabilized. If not, continue to the next iteration.

Provided mild regularity and nondegeneracy of $f$, the process yields at least linear convergence to the true minimax solution; rapid convergence is typical in practical settings [0701020].

3. Extensions: Incomplete Bases and Floating-Point Robustness

In floating-point implementations, small polynomial coefficients may trigger catastrophic cancellation during Horner evaluation (e.g., $a_i + x \cdot q_{i+1}$ when $a_i$ and $x\cdot q_{i+1}$ nearly negate each other). To eliminate this, the Remez algorithm can be adapted to target an incomplete monomial basis—i.e., constraining certain coefficients to be zero (0803.0439). The process involves:

• Identifying monomials likely to cause numerical instability (via static simulation of Horner evaluation and sign/magnitude interval analysis).
• Solving a constrained minimax problem on the reduced basis. The Chebyshev alternation theorem continues to apply under a generalized Haar condition.
• Automatically balancing approximation accuracy against numerical safety: coefficients that introduce cancellation are omitted, sometimes resulting in significant savings in operation counts and improved stability.

Empirical results confirm that this cancellation-safe Remez procedure reliably reduces hardware operation counts and prevents deleterious round-off propagation in high-accuracy function evaluations (0803.0439).

4. Applications in Certified Function Approximation and Inequality Proof

The Remez algorithm is central to computer-assisted proofs of function inequalities and the construction of certified elementary function approximations. For an inequality $f(x)\geq 0$ over $[a,b]$, Remez is employed to construct $p_n(x)$ with $|f(x)-p_n(x)|\leq\epsilon$, followed by verifying $p_n(x)-\epsilon\geq 0$ on the domain. Theoretical limits (Richardson’s undecidability) mean this approach is necessarily heuristic in the general case, but with rigorous methods—such as interval arithmetic for residual bounding, adaptive node refinement, and analytic-numeric certification (Markov/Bernstein derivative bounds to exclude hidden sign changes)—full mathematical certification can be achieved for many nontrivial functions [0701020].

Automated tools such as Dandelion further systematize this certification process by verifying, in a theorem-prover (HOL4), that a candidate polynomial satisfies the claimed $\|f-p\|_\infty\leq\epsilon$ for all $x \in [a,b]$. The approach divides verification into Taylor error bounding, polynomial error analysis (via Sturm sequences and extremal value isolation), and interval arithmetic for subexpression ranges. Dandelion demonstrates this method is competitive with, and often more thorough than, alternative formal-verification approaches (Becker et al., 2022).

5. Generalizations: The Complex Remez Algorithm

The Remez paradigm extends to compact subsets of the complex plane $E\subset\mathbb{C}$, where the task is to compute the unique monic polynomial $T_n^{(E)}(z)=z^n+\sum_{k=0}^{n-1} a_k^* z^k$ minimizing $\|T_n^{(E)}\|_E=\max_{z\in E}|T_n^{(E)}(z)|$ (Rubin, 8 May 2024). Tang’s complex Remez algorithm realizes this by:

• Working in the real-linear space $C_{\mathbb{R}}(E)$ with appropriate duality constraints,
• Introducing generalized alternation conditions via discrete sums of point masses with phase factors,
• Iteratively constructing candidate vectors $(t, \alpha, r)$ corresponding to support points, phases, and weights, updating the trial polynomial at each step,
• Ensuring convergence by monitoring the relative error gap.

This approach enables high-precision calculation of Chebyshev polynomials on various planar sets, analysis of their norms and zeros, and the study of geometric and asymptotic properties (e.g., Widom factor decay, symmetry reduction) for degrees well beyond those accessible to previous methods. Furthermore, numerical experiments support a correspondence between Chebyshev and Faber polynomials on level curves, with precise asymptotic behavior as the domain is dilated (Rubin, 8 May 2024).

6. Notable Implementational Strategies and Examples

Typical implementation initializes alternation points using Chebyshev nodes, employs Vandermonde-type systems for coefficient computation, and uses derivative root-finding for extremal point updates. In modern polynomial implementation tools, black-box evaluation and interval-based heuristics efficiently handle all function evaluations and error computations (Becker et al., 2022, 0803.0439).

Illustrative examples include:

• Construction of Remez polynomials for trigonometric, exponential, and inverse-trigonometric functions with cancellation-safe bases.
• Chebyshev polynomials on regular polygons, hypocycloids, and circular lunes in the complex plane—revealing universal patterns in norm decay and zero distribution (Rubin, 8 May 2024).
• Formal certification scenarios (e.g., Dandelion verifying supremum norm error bounds for polynomial approximations up to $\epsilon \sim 10^{-8}$ for degree 3 on nontrivial function domains) (Becker et al., 2022).

7. Limitations, Undecidability, and Future Research

Algorithmic approaches based on Remez cannot, due to undecidability, decide sign properties for arbitrary elementary functions; practical schemes thus blend heuristic, numeric, and analytic components [0701020]. Convergence may stagnate for "pathological" functions or mischosen initial nodes. Advances in verified computation, interval methods, and hybrid analytic–numeric certification continue to expand the scope and reliability of Remez-based procedures in computational mathematics and verified scientific computing 0701020.

Recent work on complex-domain Remez algorithms further motivates research into extremal polynomials on nonstandard domains, Chebyshev–Faber connections, and efficient routines for high-degree approximation with quantifiable, certified error bounds (Rubin, 8 May 2024). The flexibility of the Remez framework underpins its utility across traditional approximation theory, computer algebra, floating-point system design, and formal mathematics.