Extra rules for empty lists will make the operations
work also on
finite-length inputs (polynomials) and allow coerced scalars to be finite series.
M. D. McIlroy, The music of streams (.ps.gz),
Information Processing Letters 77 (2001) 189-195.
series f = f : repeat 0 fromInteger c = series(fromInteger c)
negate (f:ft) = -f : -ft
(f:ft) + (g:gt) = f+g : ft+gt
(f:ft) * gs@(g:gt) = f*g : ft*gs + series(f)*gt
(f:ft) / (g:gt) = qs where qs = f/g : series(1/g)*(ft-qs*gt)
(f:ft) # gs@(0:gt) = f : gt*(ft#gs)
revert (0:ft) = rs where rs = 0 : 1/(ft#rs)
int fs = 0 : zipWith (/) fs [1..] -- integral from 0 to x
diff (_:ft) = zipWith (*) ft [1..] -- type (Num a,Enum a)=>[a]->[a]
tans = revert(int(1/(1:0:1)))From the usual differential relations between sine and cosine follows code
sins = int coss coss = 1 - int sinsWhen the operations are generalized to keep polynomials finite, the coefficients
pascal = 1/[1, −[1,1]]This formula expands to a list of lists:
[, [1,1], [1,2,1], [1,3,3,1], [1,4,6,4,1], ...]::[[Rational]]
(Sorry, package file extensions are .txt to placate some browsers.
And some browsers render minus signs in these programs as hyphens on the screen.)
Extensions to handle polynomials make a
practical package, doubled in size,
not as pretty, but much faster and capable of feats like pascal.
To see the dramatic speedup, try a bigger test like take 20 tans.
Why is finite more complicated than infinite? The end must be detected, if nothing else.
Aug 2007. Text, but not code, trivially modified.
Sep 2007. Misprint in definition of sins corrected; OK in the complete packages.
Explanation page and Pascal's triangle added. Minor text modifications.
Mar 2008. Tweak the introduction.
Apr 2008. Link code to explanations.
Change certain hyphens to minus signs. Explain and fix Enum nuisance.
Jul 2008. Trivial text edits.
Sep 2009. Mention operator (^^). Shortcut multiplication by 0 in practical package.
Oct 2009. Mention recip.
Apr 2012. Page translated to Romanian by Alexandra Seremina.
Mar 2013. Correct a typo in explanations.
Nov 2013. Simplified definition of pascal. Tweaked typesetting of formulas in explanations.
Dec 2013. Replace package file extensions .hs by .txt to mollify browsers.
Feb 2014. Show the expansion of pascal.
Sep 2016. One-word text edit.
Jan 2017. Ditto. Placate GHC default prelude by adding (Eq a) to contexts.
Cleverer int and diff in practical package; reflected in revised explanation.
May 2017. Rework explanations of coercion and the pascal example.
Trivial changes in punctuation characters.
Oct 2017. Rework coercion explanation futher; delete related -- comment
Mar 2018. Disambiguate explanation of multiplication: replace F'(g+xG) with (F')(g+xG).
Oct 2018. Correct spelling errors in explanations.