Optimal quadratures for analytic functions |
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Authors: | M M Chawla B L Raina |
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Institution: | (1) Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-29, India |
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Abstract: | For integrals
–1
1
w(x)f(x)dx with
and with analytic integrands, we consider the determination of optimal abscissasx
i
o
and weightsA
i
o
, for a fixedn, which minimize the errorE
n
(f)=
–1
1
w(x)f(x)dx –
i
=1n
A
i
f(x
i
) over an appropriate Hilbert spaceH
2(E
; w(z)) of analytic functions. Simultaneously, we consider the simpler problem of determining intermediate-optimal weightsA
i
*, corresponding to (preassigned) Gaussian abscissasx
i
G
, which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA
i
* are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA
i
G
. In each case,A
i
G
=A
i
*+O(
–4n
), . For
, a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x
i
G
,A
i
*;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x
i
G
,A
i
*;i=1,...,n} to the optimal solution {x
i
o
,A
i
o
;i=1,...,n} in terms of
n
(), the maximum absolute remainder in the second set ofn normal equations. In each case,
n
() is, at least, of the order of
–4n
for large. |
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Keywords: | |
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