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/(z−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=(m−1)!. LetP _s (z) denotez ^s · numerator of the (s+1)^th diagonal member of the Padé table fore ^x, frx=1/z, that numerator being a constant factor times the general Laguerre polynomialL ^−2s−1 _s (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 ^1/m _i and (−X _i )^1/m respectively, and they give {(2s+1)m−1}^th degree accuracy. For2sm≦2n≦(2s+1)m−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<2n≦(2s+1)m−1,g _n (z) andh _n (z) are (n−sm)-parameter families of polynomials whose roots include those ofg _ms (z) andh _ms (z) respectively, and whose remainingn−ms roots are the same forg _n (z) andh _n (z). Form>1, and either 2n<2m or(2s+1)m−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 24^th degree accuracy in (1).