Equally-weighted formulas for numerical differentiation

Abstract

Equally-weighted formulas for numerical differentiation at a fixed pointx=a, which may be chosen to be 0 without loss in generality, are derived for (1) $$f^{(m)} (0) = k\left\{ {\sum\limits_{i = n + 1}^{2n} f (x_i ) - \sum\limits_{i = 1}^n f (x_i )} \right\} + R_{2n} $$ whereR 2n =0 whenf(x) is any (2n)th degree polynomial. Equation (1) is equivalent to (2) $$k\left\{ {\sum\limits_{i = n + 1}^{2n} {x_i^r } - \sum\limits_{i = 1}^n {x_i^r } } \right\} = \delta _m^r m!,r = 1,2, \ldots ,2n$$ ,r=1,2,..., 2n. By choosingf(x)=1/(zÂ?x),x i fori=1,..., n andx i fori=n+1,..., 2n are shown to be roots ofg n (z) andh n (z) respectively, satisfying (3) $$e^{ - (m - 1){! \mathord{\left/ {\vphantom {! k}} \right. \kern-\nulldelimiterspace} k}z^m } g_n (z) = h_n (z)\left( {1 + \frac{{c_1 }}{{z^{2n + 1} }} + \frac{{c_2 }}{{z^{2n + 2} }} + \cdots } \right)$$ . It is convenient to normalize withk=(mÂ?1)!. LetP s (z) denotez s · numerator of the (s+1)th diagonal member of the Pade table fore x , frx=1/z, that numerator being a constant factor times the general Laguerre polynomialL s Â?2sÂ?1 (x), and letP s (X i )=0, i=1, ...,s. Then for anym, solutions to (1) are had, for2n=2ms, forx i , i=1, ...,ms, andx i , i=ms+1,..., 2ms, equal to all them th rootsX i 1/m and (Â?X i )1/m respectively, and they give {(2s+1)mÂ?1}th degree accuracy. For2smÂ?2nÂ?(2s+1)mÂ?1, these (2sm)-point solutions are proven to be the only ones giving (2n)th degree accuracy. Thex i 's in (1) always include complex values, except whenm=1, 2n=2. For2sm 1, and either 2n<2m or(2s+1)mÂ?1<2n<(2s+2)m, it is proven that there are no non-trivial solutions to (1), real or complex. Form=1(1)6, tables ofx i are given to 15D, fori=1(1)2n, where 2n=2ms ands=1(1) [12/m], so that they are sufficient for attaining at least 24th degree accuracy in (1).