mò ^›ðEc@s_dZdZdklZlZlZd„Zd d„Zd d„Z ddd„Z d Z d S( sMCardan's solution of the cubic. See also HAKMEM note on cubics in search.py s5 $Id: cardan.py,v 1.8 2007/03/08 23:25:18 eddy Exp $ (scossacosspic Cs|p td‚|d|}|d|d|}||d|d|d|d|}ddd„} |pB|d jo | fSnd|d}| | |||fSn|p| d|ƒ|fSn|d|d}|d jo1|d}| ||ƒ| ||ƒ|fSn|d jpt‚|d}|d jowt ||dƒd}d|t |ƒ|d|t |dt dƒ|d|t |d t dƒ|fSn|d jo ||}||| |fSn| |}||||fS( s…Solves a cubic polynomial equation. Takes four arguments, the coefficients of the cube, square, linear and constant terms in the cubic, respectively. Finds all the inputs at which that cubic yields zero as output; returns these as a tuple. Repeated roots are appropriately repeated. Raises ValueError if there are no roots (or if *every* input yields zero output - i.e. all arguments are zero); this can only happen if the first argument is zero. To be specific: all entries in map(lambda x: ((a*x +b)*x +c)*x +d, Cardan(a, b, c, d)) will be tiny. See cardan(), which wraps Cardan and asserts this. s4Cardan solves cubics, use other tools for quadraticsf3.0if6.0if0.5f1.0cCs#|djo| | Sn||S(Ni(txtthird(RR((t(/home/eddy/.sys/py/study/maths/cardan.pytcuberoot*s iiN(tcubetAssertionErrortsquaretoffsettlineartEtconstanttFRtdisctacostatcostpi( RRRR R RR RR R((RtCardan s6  *   '  _   cCsÀ|dd||}y |iWnbtj oV|djo'|o td‚nd| d}q|djo|d}qn X|d}| d|d||}}||||fS(Nif4.0is-Positive definite quadratic has no real rootsx0.01.0f0.5( RRR R timagtAttributeErrortrealonlyt ValueErrortmidtgap(RRR RRRR ((Rt quadraticKs     cCs1|pp|p4y| d|SWqBtj otd‚qBXnyt||||ƒSWqwtj o fSqwXn|pAyt||||ƒ}Wntj o f}nXd|Snt ||||ƒ}|pt |ƒdjo|Sn|\}|p t d‚|t||||| |ƒS(Nf1.0s+Constant cubic has no roots (or everything)iis.zero root was meant to be dealt with earlier !(i( RRR RtZeroDivisionErrorRRRtansRtlentrootR(RRRR RRR((RtcubicWs,    if1e-14c Cs}t|||||ƒ}x^t||||d„|ƒD]>\}}|o+t |ƒ|jpt d||f‚q7q7W|S(NcCs"||||||||fS(N(Rtutstitc(RRRR R!((Rttss%s -> %s( RRRR R!RRtmapRtvtabsttolR( RRR R!RR&R$RR((Rtcardanps  /s™ $Log: cardan.py,v $ Revision 1.8 2007/03/08 23:25:18 eddy comment on HAKMEM Revision 1.7 2004/04/18 11:38:41 eddy added a comment assertion Revision 1.6 2003/07/26 22:34:05 eddy Separated out degenerate cases and added support for finding complex roots. Revision 1.5 2003/07/26 15:20:07 eddy missed out the factor of 1/2 in the quadratic special-case ! Revision 1.4 2003/07/26 13:19:47 eddy Halved F, introduced cuberoot() to deal with a gotcha. Revision 1.3 2003/07/26 12:49:03 eddy Made assertion's tolerance an optional parameter. Revision 1.2 2003/07/26 12:46:24 eddy Refined the assertion. Initial Revision 1.1 2003/07/26 12:37:25 eddy N( t__doc__t_rcs_id_tmathRR RRtNoneRRR't _rcs_log_( RRR)RRR,R RR'((Rt?s @