Y_Sc!@ sdZddlmZdaZddlTddlZddlmZddl m Z dZ dZ dZ dZd ZdbZdcZddZdRZdSZdTZdUZdVZdWZdXZdYZdZZd[Zd\Zd]Zedd^Z d_Z!d`Z"dS(es  Various bits of useful math not in the standard python library. Constants : - euler_gamma = 0.577215... - catalan = 0.915965... - golden_ratio = 1.618033... - bits_per_nat = log2(e) = 1/log(2) - sqrt_2pi = 2.50662... Special Functions : - gamma() -- Gamma function. - lngamma() -- Logarithm of the gamma function - factorial() -- Factorial function. - digamma() -- Digamma function (logarithmic derivative of gamma). - trigamma() -- Trigamma function (derivative of digamma). - cgamma() -- Complex math version of counterparts above. - clngamma() - cdigamma() - ctrigamma() - entropy() -- The entropy of a probability vector - incomplete_gamma() -- The 'upper' incomplete gamma function. - normalized_incomplete_gamma() -- - log2() -- Base 2 logarithms. - argmin() - argmax() i(tdivisiont euler_gammatcatalant golden_ratiot bits_per_nattsqrt_2pitgammatlngammat factorialtdigammattrigammatcgammatclngammatcdigammat ctrigammatentropytlog2tincomplete_gammatnormalized_incomplete_gammatargmaxtargmin(t*N(tcounti(tzipgox?g]O?gw?g+eG?g' @g@g`@g?gάL@g0Mg"=E,@g%{߿g:G?gŷKd?gtgmԞ$?g6*M+g&,?g@&ə%gU=w 6!?gѵvg >g?g@g@g8@g^@g@g@g@g&AgKAgAgAg2Ag(;L4BguwsBguwBg7Bgs6Cgh0{CgZACg Ƶ;(DglYaRwNDgΤ5Dg{zhRlDg!a?@)Eg~tEg凗Eggi=-FgQ[Fg -n'Fg3 VFg3 VIGcC s t|jS(sThe gamma function. Returns exact results for small integers. Will overflow for modest sized arguments. Use lngamma(z) instead. See: Eric W. Weisstein. "Gamma Function." From MathWorld, A Wolfram Web Resource. http://mathworld.wolfram.com/GammaFunction.html (R treal(tx((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRsc C sjt|}t|j}||krn|dkrAtddS|ttkrnttt|dSn|}|jdkrd|}nt}t}|d}|d}||}||d}d}x:tt|dddD]} ||| || 7}qWt |d||t j | |} |jdkr`t t j t || } nt| S(s2Gamma function with complex arguments and results.ig?gig?i(tcomplextfloorRtlent __factorialtintt__lanczos_gammat__lanczos_coefficientstrangeRtcmtexptpitsin( tztntzztgtctzhtzghtzptsstktf((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyR s.       #(cC s t|jS(s*The logarithm of the gamma function. (R R(R((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRsc C smt|}t|j}||krn|dkrAtddS|ttkrnttt|dSn|}|jdkrd|}nt}t}|d}|d}||}||d}d}x:tt|dddD]} ||| || 7}qWt |d||t j | |} |jdkr`t t j t || } nt j| S(s?The logarithm of the gamma function for a complex argument ig?gig?i(RRRRRRRR R!RR"R#R$R%tlog( R&R'R(R)R*R+R,R-R.R/R0((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyR s.       #(cC st|dS(s= The factorial function. factorial(z) == gamma(z+1) i(R(R&((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRscC s t|jS(s The digamma function, the logarithmic derivative of the gamma function. digamma(z) = d/dz ln( gamma(z) ) See: Eric W. Weisstein. "Digamma Function." From MathWorld-- A Wolfram Web Resource. http://mathworld.wolfram.com/DigammaFunction.html (R R(R((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyR sc C st|}t}t}|}|jdkr:d|}nd}d}x`tt|dddD]B}d||dd}|||}||}|||}qcW||d}||d} tj| |||| } |jdkr| ttj t|8} n| S(s'Digamma function with complex argumentsg?igiig?i( RRR RR!RR"R1R$ttan( R&R)R*R(R'tdR/tdztddtggR0((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyR s&  # cC s t|jS(sThe trigamma function, the derivative of the digamma function. trigamma(z) = d/dz digamma(z) = d/dz d/dz ln( gamma(z) ) See: Eric W. Weisstein. "Trigamma Function." From MathWorld-- A Wolfram Web Resource. http://mathworld.wolfram.com/TrigammaFunction.html (RR(R((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyR 1sc C st|}t}t}d}d}d}xltt|dddD]N}d||}|||}||7}||} || 7}|| |7}qGW||d7}|| ||d||}d||} ||d} | |d | | 7} | d| 7} | |7} | S( s>The trigamma function with complex arguments and return value.giiig?ig?g@(RRR R!R( R&R)R*tt1tt2tt3R/R4tdd1tdd2tresultR6((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyR:s* #    cC st||t|S(s0The 'upper' incomplete gamma function: oo - | -t a-1 incomplete_gamma(a,x) = | e t dt. | - x In Mathematica, Gamma[a,x]. Note that, very confusingly, the phrase 'incomplete gamma fucntion' can also refer to the same integral between 0 and x, (the 'lower' incomplete gamma function) or to the normalized versions, normalized_incomplete_gamma() ) See: Eric W. Weisstein. "Gamma Function." From MathWorld, A Wolfram Web Resource. http://mathworld.wolfram.com/IncompleteGammaFunction.html Bugs : This implementation is not very accurate for some arguments. (RR(taR((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRWsc C sd}d}d}|dks*|dkr9tdn|dkrIdS||dkrd|}|}xttd|D]c}||||9}||7}t|||kryd|t| |t|t|SqyWtd |nd}|d|}d|} d|} | } xtd|D]} | | |} |d }| | |} t| |kry|} n|| | } t| |kr|} nd| } | | }| |} t|d|kr1| t| |t|t|Sq1Wtd |d S( sThe upper, incomplete gamma function normalized so that the limiting values are zero and one. Q(a,x) = incomplete_gamma(a,x) / gamma(a) See: incomplete_gamma() Bugs : This implementation is not very accurate for some arguments. idg`sbO>gKH9isInvalid argumentsgg?is'Failed to converge after %d iterations.g@N(t ValueErrorR!tabsR#R1Rt RuntimeError(R=RtmaxitertepsilontsmallttermttotalR'tbR*R3thtitan((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRssL    /        +cC s t|dS(s" Return the base 2 logarithm of x i(R1(R((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRscC st|dkr!tdnd}d}x]|D]U}|dkrn||7}|tt| |7}q4|dkr4tdq4q4W||t|}|t|}|S(sh The entropy S = -Sum_i p_i ln p_i pvec is a frequency vector, not necessarily normalized. isZero length vectorgsNegative probability(RR>R1tfloat(tpvectbaseREtenttp((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRs    cC stt|tdS(sIReturn the index of the last occurrence of the maximum value in the list.i(tmaxRR(talist((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRscC stt|tdS(sJReturn the index of the first occurrence of the minimum value in the list.i(tminRR(RP((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyRs(s euler_gammascatalans golden_ratios bits_per_natssqrt_2pisgammaslngammas factorialsdigammastrigammascgammasclngammascdigammas ctrigammasentropyslog2sincomplete_gammasnormalized_incomplete_gammasargmaxsargming@(g?gάL@g0Mg"=E,@g%{߿g:G?gŷKd?gtgmԞ$?g6*M+g&,?g@&ə%gU=w 6!?gѵvg >(!g?g?g@g@g8@g^@g@g@g@g&AgKAgAgAg2Ag(;L4BguwsBguwBg7Bgs6Cgh0{CgZACg Ƶ;(DglYaRwNDgΤ5Dg{zhRlDg!a?@)Eg~tEg凗Eggi=-FgQ[Fg -n'Fg3 VFg3 VIG(#t__doc__t __future__Rt__all__tmathtcmathR"t itertoolsRt_py3kRRRRRRRR RRR RR RR R R RRRRR#RRR(((s:/home/psgendb/BIRCHDEV/pkg/weblogo-3.4/corebio/moremath.pyt=s   #  &     7