Equally-weighted formulas for numerical differentiation |
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Authors: | Herbert E Salzer |
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Institution: | (1) General Dynamics/Astronautics, P. O. Box 1128, San Diego 12, California |
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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)
whereR
2n
=0 whenf(x) is any (2n)th degree polynomial. Equation (1) is equivalent to (2)
,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)
. 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
s
–2s–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 (–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 24th degree accuracy in (1).Presented at the Twelfth International Congress of Mathematicians, Stockholm, Sweden, August 15–22, 1962.General Dynamics/Astronautics. A Division of General Dynamics Corporation. |
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Keywords: | |
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