Hermite Rational Function Interpolation with Error Correction

Abstract

We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field $\mathsf {K}$ from function and function derivative values.

We present an interpolation algorithm that can locate and correct $\le E$ errors at distinct arguments $\xi \in \mathsf {K}$ where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such $\xi$ is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample $(f/g)^{(j)}(\xi _i)$ for $0 \le j \le \ell _i$, $1 \le i \le n$, $\xi _i$ distinct, where $(f/g)^{(j)}$ is the j-th derivative of the rational function f/g, $f,g\in \mathsf {K}[x]$, $\text {GCD}(f,g)=1$, $g\ne 0$, and where $N = \sum _{i=1}^n (\ell _i+1) \ge D_f + D_g +1+2E + 2\sum _{k=1}^E \ell _k$; $D_f$ is an upper bound for $\deg (f)$ and $D_g$ an upper bound for $\deg (g)$, which are input to our algorithm. The arguments $\xi _i$ can be poles, which is truly or falsely indicated by a function value $\infty$ with the corresponding $\ell _i=0$. Our results remain valid for fields $\mathsf {K}$ of characteristic $\ge 1 + \max _i \ell _i$. Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely $N (\log N)^{O(1)}$.

For polynomials, that is, $g=1$, and a uniform derivative profile $\ell _1 = \cdots = \ell _n$, our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 Welch-Berlekamp algorithm.

This research was supported by the National Science Foundation under Grant CCF-1717100 (Kaltofen and Yang).

A Appendix

Notation (in alphabetic order):
$\hat{a}_{i,j}$	the input value for the j-th derivative of f, or an error, at the i-th point
$\widehat{A}_{i,*}$	$=[\hat{a}_{i,0},\ldots ,\hat{a}_{i,\ell _i}]$, the row vector of values for the i-th point $\xi _i$
$\widehat{A}$	$=[\widehat{A}_{1,*},\ldots ,\widehat{A}_{n,*}]^T$, the collection of all input values
$\hat{b}_{i,j}$	$=\sum _{\tau =0}^j \left( {\begin{array}{c}j\\ \tau \end{array}}\right) \hat{a}_{i,\tau } P_{\infty }^{(j-\tau )}(\xi _i)$the value for the j-th derivative of H at the i-th point
$\beta$	the minimal integer such that there are $\ge D+1+2E+2\beta E$ values for derivatives of order $\le \beta$
$c_j$	the coefficient of $x^j$ in f
$D$	an upper bound of the degree of the polynomial interpolant
$D_f$	an upper bound of the degree of the numerator of the rational interpolant
$D_g$	an upper bound of the degree of the denominator interpolant
$\delta _\kappa$	$=\ell _{\lambda _\kappa } + 1 - \min \{\,j\mid \hat{a}_{\lambda _\kappa ,j} \text { is an error} \}$
E	an upper bound on the number of errors in the input values to the algorithm
$\xi _i$	the i-th interpolation point
$\xi _{\lambda _\kappa }$	$1 \le \kappa \le k$, are the points with erroneous values, namely, $\exists j$ s.t. $\hat{a}_{\lambda _\kappa ,j}$ is an error
$\epsilon _\kappa$	$= \min \{\,j\mid \hat{a}_{\lambda _\kappa ,j} \text { is an error} \} = \ell _{\lambda _\kappa } + 1 -\delta _\kappa$
$f$	polynomial interpolant or numerator of the rational interpolant for the correct values
$g$	the denominator of the rational interpolant for the correct values
$\bar{g}$	a factor of $g$ indicating true non-poles
$g_{\infty }$	a factor of $g$ indicating true poles
H	the polynomial Hermite interpolant for all input values (including $\le E$ errors)
$I_{\infty }$	$= \{i\mid \exists j \text { s.t. } \hat{a}_{i,j} = \infty \}$
k	the actual number of points with erroneous input values
$\mathsf {K}$	a field
$\ell _i$	the highest derivative order at the i-th point
$\varLambda$	the error locator polynomial
${\bar{\varLambda }}$	$= \prod _{\kappa \in \{1,\ldots ,k\}, \lambda _\kappa \notin I_{\infty }} (x-\xi _{\lambda _\kappa })^{\delta _\kappa }$
$\varLambda _{\infty }$	$= \prod _{\kappa \in \{1,\ldots ,k\}, \lambda _\kappa \in I_{\infty }}(x-\xi _{\lambda _\kappa })$
$m_j$	the number of input values for the j-th derivative of f
$M_j$	the number of input values for up to the j-th derivative of f
$n$	the number of distinct points
$n_\infty$	degree of $P_\infty$
$N$	the number of the input values
$P_{\infty }$	$=\prod _{\exists j \text { s.t. } \hat{a}_{i,j} = \infty }(x-\xi _i)$, the polynomial indicating all poles
$r_0$	$=(x-\xi _1)^{\ell _1+1}\cdots (x-\xi _n)^{\ell _{n}+1}$
$r_\gamma$	the $\gamma$-th remainder of the Euclidean polynomial remainder sequence $r_0, r_1, \ldots$
$s_\gamma$	the Bézout coefficient of $r_1$ in the $\gamma$-th extended Euclidean scheme: $s_\gamma r_1 + t_\gamma r_0 =r_\gamma$
$t_\gamma$	the Bézout coefficient of $r_0$ in the $\gamma$-th extended Euclidean scheme: $s_\gamma r_1 + t_\gamma r_0 =r_\gamma$