CRAN release v0.6.0

Fixed

• Fixed the Remez exchange silently converging to a reference-local fixed point that is not the global minimax, returning a materially suboptimal approximation with no warning (forexample, atan on $[0, 3]$ at degree 3 reported an expected error 1.8 times smaller than the returned approximation's true maximum error). Two structural defects were responsible: the root search only ever examined the intervals between consecutive current reference points, so a sign change between an endpoint and the outermost reference point was invisible; and the maximize/minimize schedule was seeded at the lower endpoint and flipped mechanically, so an error curve with one more oscillation than the schedule assumed desynced the exchange into returning roots instead of extrema. Both are fixed by the exchange redesign described under “Changed”.
• Fixed a crash (“segfault from C stack overflow”) evaluating a Chebyshev-basis approximation at several million or more points at once.
• Added a clear error, instead of undefined behavior, when an internal Chebyshev evaluation would require more than $2^{31}$ matrix cells.
• Fixed lower > upper (or lower == upper) being silently accepted and producing a badly suboptimal approximation with no warning; this now raises a clear error suggesting the arguments be swapped.
• Fixed the Remez-iteration stagnation check (isUnchanging) reporting convergence as “unchanging” when errors were still improving rapidly between iterations. The ratio test was checked in only one direction. Now, it requires the ratio to be close to 1 in either direction before treating the solution as stalled.
• Fixed opts$ztol using the monomial-basis coefficient scale even when the Chebyshev basis was requested, which could incorrectly zero out genuine small-but-relevant Chebyshev coefficients on ranges intersecting $[-1, 1]$.
• Fixed the degree $n+1$ restart's “uppermost coefficient is effectively 0” check silently accepting any negative top coefficient regardless of magnitude (a missing abs()), which could return a polynomial one degree lower than warranted.
• Fixed severe numerical conditioning loss when fitting the Chebyshev basis on ranges far from $[-1, 1]$ (e.g. degree 10 on $[5, 6]$ had a condition number around $10^{24}$), which could cap the achievable accuracy or force convergence warnings well short of the true minimax error (that $[5, 6]$ case: observed error improved from $\sim 1.8 \times 10^{-10}$ to $\sim 3 \times 10^{-12}$, matching the true value). See the Breaking change below.

Changed

• The Remez exchange was redesigned to be driven by the error curve itself rather than by the current reference's structure. Roots of the error equation are now located over the whole approximation interval from an oversampled grid. Each sign-region's search direction is taken from the error's own sign there instead of a mechanically alternated schedule, and the interval endpoints are always retained as exchange candidates. When the curve offers more alternating extrema than the reference holds, the new reference is the alternating window that contains the global extremum and maximizes its smallest error—the classical conditions for the leveled error to increase strictly toward the true minimax. Converged results for healthy inputs agree with version 0.5.0 at tolerance level but are not bitwise identical. Inputs that previously exited through a singular-matrix error or a stall may now exit through a different path as the iteration trajectory differs.
• Breaking: Chebyshev-basis approximation now internally maps the requested range $[l, u]$ to $[-1, 1]$ before evaluating $T_k$, fixing the conditioning issue noted above. a (and b) are therefore coefficients of $T_k$ of the internally mapped variable, not $T_k(x)$, whenever the fitted range is not $[-1, 1]$ Results on $[-1, 1]$ are unchanged. aMono and bMono remain coefficients in the natural, raw $x$ in every case, and are the stable way to consume a Chebyshev-basis result without needing to know about the internal map. See minimaxApprox for a documented accuracy caveat on this conversion at high degree on wide, far-off $[-1,1]$ ranges.
• Added upfront validation of fn, lower, upper, degree, and opts in minimaxApprox(), replacing cryptic low-level failures (e.g. “missing value where TRUE/FALSE needed”, “invalid argument type”) with clear, argument-specific error messages. fn must be a function whose first argument is x, although primitives such as exp and sin are still accepted; lower/upper must be finite non-missing numeric scalars; degree must be finite and non-missing; opts$maxiter/miniter/conviter must be single positive integers and opts$tol/convrat/tailtol/ ztol must be finite numeric scalars when supplied. The tailtol and ztol options may still be NULL by design.
• Corrected wording in the degree $n+1$ restart messages, which described the returned polynomial's length as its degree (a degree $n$ polynomial has $n+1$ coefficients).
• isOscil, used internally to test for the expected alternating-sign error pattern, no longer propagates NA when passed a NaN/NA error value. Previously this could have errored deep inside the convergence check instead of failing cleanly.
• The internal zero-basis perturbation, used in relative-error mode when the approximated function is exactly 0 at a candidate extremum, now escalates to a magnitude-scaled step when the fixed 1e-12 nudge would be a no-op (points with $|x| \gtrsim 4.5 \times 10^3$). It remains identical to the previous behavior at ordinary magnitudes.
• Functions that the requested degree resolves to, or near, machine floating-point precision—including functions that are exactly polynomials, such as minimaxApprox(function(x) x^2, 0, 1, 2), and well-behaved cases such as minimaxApprox(exp, -1, 1, 14)—no longer hard-error or emit misleading non-convergence warnings. This case can surface two ways, both now handled: (1) both the degree $n$ and degree $n+1$ Remez solves fail singular, previously “neither converged”; or (2) on some BLAS/LAPACK platforms the same case does not report singular but instead runs to maxiter (or stalls “unchanging”) while genuinely wandering at the machine-precision floor. In either case, when there is no minimax problem left to solve, the degree $n$ interpolant through the Chebyshev reference is returned instead, together with a warning that it is not technically a Remez result but that its error is below the level at which a distinct minimax solution can be discerned. Genuinely non-representable cases, where no polynomial comes near the function, still raise the original error or convergence warning.
• Updated the package Title and Description to reflect the expanded scope: minimax approximation by polynomials and rational functions via both the classical and barycentric algorithms.

Added 

• Added a reference-local certificate check to all approximation paths. At a converged exit, the returned approximation's maximum error on a dense grid is compared against its leveled error, and a warning is raised when the former materially exceeds the latter. In that case the reported expected error is a valid lower bound on the true minimax error (de la Vallée Poussin) rather than a certificate or gurantee of it. This can occur for degrees that are non-normal for the function (e.g. even or odd functions at generic degrees) or when the exchange stalls at a reference-local fixed point. The check is skipped where its ratio would be noise: results at the machine-precision floor and relative-error results with an exact zero of fn on the grid.
• Added a "Barycentric" option to the basis argument, implementing the barycentric-Remez algorithm for the best polynomial approximation of Pachón and Trefethen (2009). It represents the trial polynomial by its values at the reference points rather than by coefficients in a fixed basis, avoiding the ill-conditioned linear solve that limits the achievable degree in the classical bases. In consequence, it converges at degrees and on ranges where the Monomial and Chebyshev bases fail with a singular-matrix error—for example, minimaxApprox(exp, -1, 1, 14) (and degree 50), and exp on $[5, 6]$ at degree 10—and, using capacity scaling, remains stable on very wide or far-off-center intervals. Exactly-representable and machine-precision-resolved functions are handled natively (leveled error $0$, no rescue needed). Relative error is supported and a function that is exactly zero at a reference point (typically an interval endpoint), where the relative minimax does not exist, now raises a clear error. The returned object carries the barycentric representation and a coefficient-conversion residual diagnostic (convResid). See minimaxApprox.
• Added rational approximation support to the "Barycentric" basis, implementing a subset of the adaptive barycentric minimax algorithm of Filip, Nakatsukasa, Trefethen, and Beckermann (2018), with the initial reference taken from the error extrema of an adaptive Antoulas–Anderson (AAA) approximant (Nakatsukasa, Sète, and Trefethen, 2018). The leveled error is computed as an eigenvalue of a small symmetric matrix. Therefore, as for the polynomial case, there is no ill-conditioned linear solve. Rather, the classical smooth-function results are reproduced to convergence tolerance while cases such as $|x|$ converge to the certified minimax where reference-based classical iterations can struggle. This release only implements the algorithm using absolute error. Passing relErr = TRUE will return an error. Degenerate or defective requests (e.g. even degrees for an even function) are detected and reported with a clear error rather than repaired by degree reduction. A trial denominator with a zero in the interval is an error mirroring the classical rational path. The returned object carries a and b normalized as in the classical path, aMono and bMono, the barycentric representation, and a named two-component convResid. See minimaxApprox.
• Added the package's first vignette, “Using minimaxApprox”, covering the minimax criterion, the three bases and how to choose among them, rational approximation and its barycentric-basis limitations, relative error, the Warning/convResid diagnostics, and the handful of opts most users need.

Deprecated 

• The xi argument to minimaxApprox is deprecated and will be removed in the next release. The redesigned exchange algorithm derives its reference from the error curve at every iteration, making a user-supplied initial reference unnecessary. Passing xi now emits a deprecation warning.

Full Changelog: v0.5.0...v0.6.0