m vbFc@shdZdkZdklZdefdYZdefdYZdefdYZ[[dS( sPolynomials. Coefficients are assumed numeric. Only natural powers are considered. $Id: polynomial.py,v 1.19 2007/06/03 16:42:30 eddy Exp $ N(sLazytinvalidCoefficientcBstZdZRS(s"Invalid coefficient for polynomial(t__name__t __module__t__doc__(((t,/home/eddy/.sys/py/study/maths/polynomial.pyRs tunNaturalPowercBstZdZRS(s7Power of variable in polynomial is not a natural number(RRR(((RR s t PolynomialcBstZdZdZeieieieifdZ e dZ [ dZ eieifddZ dZ dZd Zd Zd ZeZd Zd ZedZ[dZdZedZ[dZedZ[dZedZ[dZdZeZdZ dZ!dZ"e"Z#dZ$e$Z%dZ&dZ'e'dZ(dZ)e'e)d Z*['[)d!Z+e+d"Z,e+eieifd#d$Z-[+d%Z.d&Z/d'Z0d(d(d)Z1d*Z2d+Z3d,Z4d-Z5d.Z6ei7d/Z8d0Z9d1Z:d2Z;d3Z<d4Z=d5Z>d6Z?d7Z@d8ZAd9kBlCZCeCd:ZD[Cd(d;d<ZEeFd=ZGd>ZHd?d(d@ZIdAZJRS(Bs:Model of the mathematical ring of polynomials (in one free variable). Supports arithmetic (+, -, *, /, %, divmod, power, negation), comparison (but cmp() will yield zero in some cases where == will say no), representation (as a python lambda expression), evaluation (i.e. calling as a function), repeated integration (as <<) and differentiation (as >>), use as boolean (only the zero polynomial is false) and (lazy) hashing. Lazy attributes: =============== rank -- highest power of the free variable with non-zero coefficient normalised -- same polynomial scaled so rank's coefficient is 1 derivative -- result of differentiating the polynomial assquares -- decompose as sum of scaled squares and a remainder sign -- like cmp(self, 0) but None when ill-defined or variable isreal -- are all coefficients real ? factors -- tuple of normalised irreducible factors and an optional scalar Note that a poly with true isreal considers positive-definite quadratic factors to be irreducible; assigning isreal = None will persuade a poly to believe quadratic factors are reducible. The value of p.assquares is of form (bok, rem) with rem either zero or of odd degree and p-rem equal to reduce(lambda y, (x, v): y + x*x*v, bok.items(), 0). The value of sign may presently sometimes be None when it needn't be; I've yet to find an example, but the computation in use only deals with `easy enough' cases. Note that cmp() yields zero when the difference's sign is None or zero; but == is true only when sign is zero (actual equality). Methods: ======== coefficient(n) -- yields coefficient of (: x**n ←x :) in polynomial hcf([poly, ...]) -- yields highest common factor of arbitrarily many integral([start=0, base=0]) -- integration unafter(poly) -- input to poly yielding self as output seek_root([guess=0, tol=1e-6]) -- find an input mapped to zero seek_factor([guess=None]) -- find a factor, if possible See individual methods' docs for details. cGsh|_y|\}||i|Wntj o|i|nmtttfj oWy|i |i ddfWn#tj o|i d|qX|i |nXdS(s Constructor. If exactly one argument is passed and it is a mapping, its keys are used as powers and values as coefficients of each power's term in the polynomial; if exactly one argument is passed and it is a sequence, it is interpreted as a mapping whose keys are indices into the list and values are entries in the list. Otherwise, each argument must be numeric and the arguments are interpreted as if they'd been passed as a sequence. Thus: Polynomial({ power: coeff ... }) == (: sum(: coeff * x**power :) <- x :) Polynomial([ co_0_, co_1_ ... ]) == Polynomial(( co_0_, co_1_ ... )) == Polynomial( co_0_, co_1_ ... ) == (: sum(: co_i_ * x**i <- i :) <- x :) As special cases of the last, Polynomial( value ) generates the constant polynomial value <- x and Polynomial() generates the zero polynomial - but note that this is the scalar zero polynomial; if a vector zero or dimensioned zero is desired, supply it as the value for a constant polynomial; but if your vector type supports slicing, you'd better pass it as Polynomial({0: zero}) instead. The polynomial 1 + 2.x + 3.x**2 <- x may thus be generated by any of: Polynomial(1,2,3), Polynomial((1,2,3)), Polynomial([1,2,3]) or Polynomial({0:1,1:2,2:3}). The last, dictionary, form is generally preferrable for sparse polynomials (ie, rank >> the number of non-zero coefficients): tuple form is better for short and sweet forms like that illustrated. Note that setting z = Polynomial(0, 1) provides a `free variable' that can then be used to generate polynomials the way many folk prefer; with this, Polynomial(1,2,3) can simply be written 3*z*z +2*z +1. If you want a polynomial's representation to use the name z (rather than x), simply set the polynomial's .variablename to 'z' (but note that this only applies to that polynomial, not to ones computed from it). iN(tselft_Polynomial__coefstargstargt_Polynomial__fromseqt ValueErrortAttributeErrort TypeErrortKeyErrortitemstgettNonet_Polynomial__storet_Polynomial__frombok(RR R ((Rt__init__6s%  cCsnt||jo)|t|jot|Sn|Sn/t|dot|do|SntdS(Nt__add__t__mul__(ttypetvaltoktypestlongthasattrR(RR((Rt _get_coeffks cCs1|o|||i|scCsVxO|iD]A\}}|| p |djo tq |i||q WdS(Ni(tbokRtkeyRtokRRR(RR)R+RR*((Rt __fromboks   cCsqx9|iiD](}|i|djo|i|=qqWyt|iiSWntj o dSnXdS(Nf0.0i(RRtkeysR*tmaxR (RtignoredR*((Rt_lazy_get_rank_scGs5x.|D]&}x|o|||}}qWqW|S(N(totherstotherR(RR1R2((Rthcfs cGs|i|idjS(Ni(RR3R1trank(RR1((RtcoprimescCsJ|idjo tdn|i|i}|djo|Sn||S(NisCan't normalise zeroi(RR4R Rtscale(RR/R6((Rt_lazy_get_normalised_s   cCs~y |iSWntj onX|ig}|i|d}ddi|}|p|d}n ||}||_|S(Nis lambda %s: s, t0( Rt_Polynomial__reprR t variablenametnamest_Polynomial__representttexttjointlambtans(RR?R=R;R@((Rt__repr__s    tzcCssy!|idjo |i}nWntj onXy|i|d|SWntj ot|SnXdS(Nii(tnumtimagtrealR R<R;tdepthtstr(RCR;RF((Rtformatsc Csxc|t|joO|dddjo|idq|itt|dddqWd||}}x|i D]}|i |}|o|djo d}nS|djo d}n<||||}d|jod |d }n |d 7}|djo||7}qK|d ||f7}n||||}|d djo|d|7}q|d |7}qW|d d jo |dSn|d djo |dSn|S(NiitatZitt-t t(s)*t*s%s**%ds +i(RFtlenR;tappendtchrtordtresulttnameRt_powersR*t coefficientRtfragtfmt( RR;RFRYRXRURR*RT((Rt __represents<*        cCs@h}x-|iiD]\}}||||x)|djo|i|d}}q W|S(sf << n -> nth integral of fliiN(R2Rtintegral(RR2((Rt __lshift__s  cCs5d|?x&|djo|i|d}}q W|S(sf >> n -> nth derivative of fliiN(R2Rt derivative(RR2((Rt __rshift__s  cCsMh}x:|iiD])\}}|o||||d V-> V and +: V-> V-> V, we can evaluate p(t) by substituting t in as the value of p's formal parameter. iiN( RRVR-R}t IndexErrorR!RWRTR*R (RR R-RTR*R}((Rt__call__s"    cCs|h}}x|o|i|iotd|d|n|i|i}t|||\}}|idjpt |i||ijpt |i d||