;ò ġÙR?c@sËdZdZdklZdfd„ƒYZeddƒZdefd„ƒYZeƒZ[deeZd efd „ƒYZ d efd „ƒYZ e ƒe ƒf\Z Z [ [ [[[[d Z dS(sSinusoids of multiples of angles, via polynomials. See S and C for details, or http://www.chaos.org.uk/~eddy/math/multiangle.html for theory. s: $Id: multiangle.py,v 1.10 2003/09/01 05:32:37 eddy Exp $ (s PolynomialsLazySeqcBsAtZedƒd„Zd„Zd„Zd„Zdd„ZRS(NicCs|g|_dS(N(sfirstsselfs _LazySeq__seq(sselfsfirst((s,/home/eddy/.sys/py/study/maths/multiangle.pys__init__scCsı|i|ƒ}toL|i}x@|djo.|i|i|ƒƒpt ‚|d}q"Wn|t |i ƒjo,|i t gd|t |i ƒ|_ n||i |<|SdS(Nii( sselfsgrowtoskeysvals __debug__sranksischecks coefficientsAssertionErrorslens _LazySeq__seqsNone(sselfskeysvalsi((s,/home/eddy/.sys/py/study/maths/multiangle.pys__growtos   , cCs|t|ƒjSdS(N(svalslong(sselfsval((s,/home/eddy/.sys/py/study/maths/multiangle.pyscheck!scCsZy|i|}Wn tj o|i|ƒSn X|tjo|i|ƒSn|SdS(N(sselfs _LazySeq__seqskeysanss IndexErrors_LazySeq__growtosNone(sselfskeysans((s,/home/eddy/.sys/py/study/maths/multiangle.pys __getitem__#s cCs­g}|djo4x||jo|i||ƒ||}qWn_|djo4xN||jo|i||ƒ||}qWWn||jotdƒ‚n|SdS(Nis,Sequence with zero step is a very bad idea !(sanssstepsloshisappendsselfs ValueError(sselfsloshisstepsans((s,/home/eddy/.sys/py/study/maths/multiangle.pys __getslice__,s     (s__name__s __module__s Polynomials__init__s_LazySeq__growtoschecks __getitem__s __getslice__(((s,/home/eddy/.sys/py/study/maths/multiangle.pysLazySeqs    iisMiddlecBsetZhded<dde@scCs.|||ddƒ|dd|ddS(Niiif0.5(spsx(spsx((s,/home/eddy/.sys/py/study/maths/multiangle.pysAscCs+d|d|| dƒd|ddS(Niiiif0.5(sxsp(spsx((s,/home/eddy/.sys/py/study/maths/multiangle.pysBsc CsÙt|jpt‚t|dƒ\} } | | d} |d| | djpt‚|| || f\}}d|| | d|| ƒ|| ƒ|||dƒ|} t| d„|ƒ pt‚| SdS(Niiif0.5cCs||ƒ|jS(N(spsa(spsa((s,/home/eddy/.sys/py/study/maths/multiangle.pysIs(sKsselfsAssertionErrorsdivmodskeysnsrsmsthissthatsdownsslabstwosupsanssfilterschecks( sselfskeysdownsupstwosslabscheckssthatsthissmsrsanssn((s,/home/eddy/.sys/py/study/maths/multiangle.pysgrowto>s B(s__name__s __module__szsgrowto(((s,/home/eddy/.sys/py/study/maths/multiangle.pysMiddle=ssseqCoscBs)tZdZddee ed„ZRS(s\Sequence of polynomials describing cos(n.t) in terms of cos(t) C[n](cos(t)) = cos(n.t) iicCsct|dƒ\}}|o|t||ƒ}n|||ƒ}|i |jpt ‚|SdS(Ni( sdivmodskeysnsrsxsKskatesanssselfstermsranksAssertionError(sselfskeyskatestermsxsnsrsans((s,/home/eddy/.sys/py/study/maths/multiangle.pysgrowtoUs (s__name__s __module__s__doc__stermszsgrowto(((s,/home/eddy/.sys/py/study/maths/multiangle.pysseqCosQs sseqSincBs(tZdZdedeed„ZRS(skSequence of polynomials describing sin(n.t)/sin(t) in terms of cos(t) sin(t)*S[n](cos(t)) == sin(n.t) iicCs†t|dƒ\}}|od|t|| ƒ}n/t||ƒhdd<dd<|d}|i |jpt ‚|SdS(Niiii˙˙˙˙( sdivmodskeysnsrsxsSstermsanssKsstemsranksAssertionError(sselfskeysstemstermsxsnsrsans((s,/home/eddy/.sys/py/study/maths/multiangle.pysgrowtobs .(s__name__s __module__s__doc__stermszsgrowto(((s,/home/eddy/.sys/py/study/maths/multiangle.pysseqSin]s sô $Log: multiangle.py,v $ Revision 1.10 2003/09/01 05:32:37 eddy Little tweaks. Defined term more clearly; deployed it more. Renamed z to x when tunnelling it. Delete LazySeq when used. Revision 1.9 2003/09/01 05:25:43 eddy Changed Middle to use addition-derived formulae. Refined a check. Revision 1.8 2003/08/17 20:25:35 eddy Added some assertions to Middle. Revision 1.7 2003/08/13 22:15:12 eddy Fixed d'oh; wrong sign for the term used by Cos. Revision 1.6 2003/08/12 22:57:43 eddy Eliminated A, B, Q, R in favour of K and halvings through S and C; i.e. I have now proved B-Q unification (and renamed T and D to K), along with eliminating A and R as mere aspects of C and S (respectively). Revision 1.5 2003/08/10 20:45:02 eddy Added implementation of the polynomial that unifies Q and B, albeit I haven't yet *proved* it does so. The assertions make me confident, though ;^> Revision 1.4 2003/08/10 12:45:42 eddy Renumbered S down by 1, so S[n].rank is n. Added assorted assertions. Revision 1.3 2003/08/10 11:47:33 eddy Fixed the S[1] tweak so it's lambda x: 1, not lambda x: x - d'oh ! Revision 1.2 2003/08/05 21:42:34 eddy Tweaked so S[1] is a polynomial, not 1.0; added comments. Initial Revision 1.1 2003/08/03 23:21:14 eddy N(s__doc__s_rcs_id_s polynomials PolynomialsLazySeqszsMiddlesKstermsseqCossseqSinsCsSs _rcs_log_( stermsLazySeqs_rcs_id_sKsCsMiddlesSsseqCoss Polynomialszs _rcs_log_sseqSin((s,/home/eddy/.sys/py/study/maths/multiangle.pys?s -