I think that Newton was pretty proud of himself for discovering that all
(interesting to him) functions could be decomposed into power series
(basis functions x^k).
High school & (most) college algebra classes never seem to get beyond
Newton's basis functions, even though they're just awful for most calculations.
I.e., we waste incredibly valuable student time teaching them stuff that
they won't use, and shouldn't use, because it's so bad. This is the equivalent
of teaching driver's ed by looking out of the rear window of a car through the
wrong end of a telescope.
Someone needs to come up with some high school courses based on Cheby
functions or splines or other functions with only short-range interactions.
At 10:19 AM 4/28/2013, Richard Fateman wrote:
>On 4/28/2013 9:59 AM, Henry Baker wrote:
>>Didn't Newton do this?
>
>If not Newton, then certainly Gauss. :)
>
>But the extensive references in the paper only go back to about 1970.
>The 1970 paper by Barton, Willers, Zahar can be redone in Maxima in a page
>or so; doing all this "right" takes much more work, if indeed it can be
>done completely satisfactorily at all.
>
>>At 09:45 AM 4/28/2013, Richard Fateman wrote:
>>>I came across this paper (dedicated to memory of Wm Schelter!)
>>>
>>>http://www.maia.ub.es/~angel/taylor/taylor.pdf
>>>
>>>on a software package for numerical solution of ODEs via Taylor series.
>>>
>>>Someone looking for a project might find this an interesting paper. There
>>>is a lot of history in this area, too.