Re: -- Polynomial approximation for 1/(x+a) in large domain.

Srikanth <skt@xxxxxxxxxx> writes:

Hi

I would like to approximate the function 1/(x+a) (for a constant a) in
a large domain. I tried to minimize the L-infinity norm of the error
over my domain (I was working on a discrete version of the problem,
with irregular sampling over the domain, based on the regions that
were of more importance to me - I guess I could do a weighting as
well, but I couldn't figure out how...).

I found that so long as a is a decade less than the supremum of x, I
can get a fairly good approximation. But once x gets closer to a, the
approximation becomes lousy and if x>a, I get very very bad results.
Scaling x would obviously not work well here. I would like to know
what other approximation would be suitable - I'd rather not use a
rational approximation, since I start am approximating a rational
function. Would another norm be any better? Is there any other basis
that could approximate 1/(x+a) over a large domain?

For example, in Maple, to get the best approximation to 1/(x+10)
by a polynomial of degree 15 in L_infinity on the interval 0 <= x <= 50:

with(numapprox):
Digits:= 14;
f := minimax(1/(x+10), x = 0 .. 50, 15, 1, 'maxerror');

f := .99999906310500e-1+(-.99988589038603e-2+(.99768062607913e-3+
(-.98106144038814e-4+(.91640472830942e-5+(-.76635471189017e-6+
(.54029506705185e-7+(-.30712232518687e-8+(.13676678052275e-9+
(-.46777903958484e-11+(.12069071473839e-12+(-.22968484916164e-14+
(.31166257546383e-16+(-.28470001550345e-18+(.15669785061026e-20-
.39223991170004e-23*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x

maxerror;

.94667482e-7
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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