ó <Zc@s]ddlmZmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddlmZddlmZddlm Z de!fd„ƒYZ"de"eee e eeeeeeeeefd„ƒYZ#dS(iÿÿÿÿ(tgttlti(txrange(tSpecialFunctions(tRSCache(tQuadratureMethods(t LaplaceTransformInversionMethods(tCalculusMethods(tOptimizationMethods(t ODEMethods(t MatrixMethods(tMatrixCalculusMethods(tLinearAlgebraMethods(tEigen(tIdentificationMethods(tVisualizationMethods(tlibmptContextcBseZRS((t__name__t __module__(((s.lib/python2.7/site-packages/mpmath/ctx_base.pyRstStandardBaseContextcBsÐeZejZejZd„Zd„ZeZeZ d„Z d„Z d„Z d„Z d„Zd„Zd„Zd „Zd „Zd „Zeed „Zded „Zd„Zdd„Zdd„Zddd„Zd„Zd„Zd„Zd„Zd„Ze ej!ƒZ"e ej#ƒZ#e ej$ƒZ$e ej%ƒZ%e ej&ƒZ&e ej'ƒZ(e ej)ƒZ*e ej+ƒZ,e ej-ƒZ.dd„Z/dd„Z0d„Z1d„Z2d„Z3d„Z4RS( cCs[i|_tj|ƒtj|ƒtj|ƒtj|ƒtj|ƒtj|ƒdS(N(t_aliasesRt__init__RRRRR (tctx((s.lib/python2.7/site-packages/mpmath/ctx_base.pyR*s      cCsUxN|jjƒD]=\}}yt||t||ƒƒWqtk rLqXqWdS(N(RtitemstsetattrtgetattrtAttributeError(Rtaliastvalue((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt _init_aliases4s  cCsd|fGHdS(NsWarning:((Rtmsg((s.lib/python2.7/site-packages/mpmath/ctx_base.pytwarn@scCst|ƒ‚dS(N(t ValueError(RR((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt bad_domainCscCst|dƒr|jS|S(Ntreal(thasattrR#(Rtx((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt_reFscCst|dƒr|jS|jS(Ntimag(R$R'tzero(RR%((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt_imKscCs|S(N((RR%((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt _as_pointsPscKs|j|ƒ S(N(tconvert(RR%tkwargs((s.lib/python2.7/site-packages/mpmath/ctx_base.pytfnegSscKs|j|ƒ|j|ƒS(N(R+(RR%tyR,((s.lib/python2.7/site-packages/mpmath/ctx_base.pytfaddVscKs|j|ƒ|j|ƒS(N(R+(RR%R.R,((s.lib/python2.7/site-packages/mpmath/ctx_base.pytfsubYscKs|j|ƒ|j|ƒS(N(R+(RR%R.R,((s.lib/python2.7/site-packages/mpmath/ctx_base.pytfmul\scKs|j|ƒ|j|ƒS(N(R+(RR%R.R,((s.lib/python2.7/site-packages/mpmath/ctx_base.pytfdiv_scCsp|r@|r&td„|Dƒ|jƒStd„|Dƒ|jƒS|r`td„|Dƒ|jƒSt||jƒS(Ncss|]}t|ƒdVqdS(iN(tabs(t.0R%((s.lib/python2.7/site-packages/mpmath/ctx_base.pys escss|]}t|ƒVqdS(N(R3(R4R%((s.lib/python2.7/site-packages/mpmath/ctx_base.pys fscss|]}|dVqdS(iN((R4R%((s.lib/python2.7/site-packages/mpmath/ctx_base.pys hs(tsumR((Rtargstabsolutetsquared((s.lib/python2.7/site-packages/mpmath/ctx_base.pytfsumbscsk|dk rt||ƒ}n|rM|j‰t‡fd†|Dƒ|jƒStd„|Dƒ|jƒSdS(Nc3s%|]\}}|ˆ|ƒVqdS(N((R4R%R.(tcf(s.lib/python2.7/site-packages/mpmath/ctx_base.pys pscss|]\}}||VqdS(N((R4R%R.((s.lib/python2.7/site-packages/mpmath/ctx_base.pys rs(tNonetziptconjR5R((Rtxstyst conjugate((R:s.lib/python2.7/site-packages/mpmath/ctx_base.pytfdotks    cCs(|j}x|D]}||9}qW|S(N(tone(RR6tprodtarg((s.lib/python2.7/site-packages/mpmath/ctx_base.pytfprodts  icKs|j|||GHdS(s6 Equivalent to ``print(nstr(x, n))``. N(tnstr(RR%tnR,((s.lib/python2.7/site-packages/mpmath/ctx_base.pytnprintzscs2ˆdkrdˆj‰ny¡ˆj|ƒ}t|ƒ}t|ƒˆkrSˆjSˆj|ƒr¼tˆ|ˆƒ}t|jƒ|kr‘|jSt|jƒ|kr¼ˆj d|jƒSnWnnt k r-t |ˆj ƒrø|j ‡‡fd†ƒSt|dƒr.g|D]}ˆj|ˆƒ^qSnX|S(sÖ Chops off small real or imaginary parts, or converts numbers close to zero to exact zeros. The input can be a single number or an iterable:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = False >>> chop(5+1e-10j, tol=1e-9) mpf('5.0') >>> nprint(chop([1.0, 1e-20, 3+1e-18j, -4, 2])) [1.0, 0.0, 3.0, -4.0, 2.0] The tolerance defaults to ``100*eps``. idicsˆj|ˆƒS(N(tchop(ta(Rttol(s.lib/python2.7/site-packages/mpmath/ctx_base.pytŸst__iter__N(R;tepsR+R3R(t_is_complex_typetmaxR'R#tmpct TypeErrort isinstancetmatrixtapplyR$RI(RR%RKtabsxtpart_tolRJ((RRKs.lib/python2.7/site-packages/mpmath/ctx_base.pyRI€s&   'c Cs×|j|ƒ}|dkrH|dkrH|jd|j dƒ}}n|dkr]|}n|dkrr|}nt||ƒ}||kr’tSt|ƒ}t|ƒ}||krÃ||}n ||}||kS(sÖ Determine whether the difference between `s` and `t` is smaller than a given epsilon, either relatively or absolutely. Both a maximum relative difference and a maximum difference ('epsilons') may be specified. The absolute difference is defined as `|s-t|` and the relative difference is defined as `|s-t|/\max(|s|, |t|)`. If only one epsilon is given, both are set to the same value. If none is given, both epsilons are set to `2^{-p+m}` where `p` is the current working precision and `m` is a small integer. The default setting typically allows :func:`~mpmath.almosteq` to be used to check for mathematical equality in the presence of small rounding errors. **Examples** >>> from mpmath import * >>> mp.dps = 15 >>> almosteq(3.141592653589793, 3.141592653589790) True >>> almosteq(3.141592653589793, 3.141592653589700) False >>> almosteq(3.141592653589793, 3.141592653589700, 1e-10) True >>> almosteq(1e-20, 2e-20) True >>> almosteq(1e-20, 2e-20, rel_eps=0, abs_eps=0) False iiN(R+R;tldexptprecR3tTrue( Rtstttrel_epstabs_epstdifftabsstabstterr((s.lib/python2.7/site-packages/mpmath/ctx_base.pytalmosteq¤s !!          c Gs¤t|ƒdks+tdt|ƒƒ‚nt|ƒdksVtdt|ƒƒ‚nd}d}t|ƒdkr|d}n)t|ƒdkrª|d}|d}nt|ƒdkrÉ|d}n|j|ƒ|j|ƒ|j|ƒ}}}|||kstdƒ‚||kr9|dkr0gSt}n|dkrIgSt}g}d}|}x<|||}|d7}|||ƒr›|j|ƒqdPqdW|S(sa This is a generalized version of Python's :func:`~mpmath.range` function that accepts fractional endpoints and step sizes and returns a list of ``mpf`` instances. Like :func:`~mpmath.range`, :func:`~mpmath.arange` can be called with 1, 2 or 3 arguments: ``arange(b)`` `[0, 1, 2, \ldots, x]` ``arange(a, b)`` `[a, a+1, a+2, \ldots, x]` ``arange(a, b, h)`` `[a, a+h, a+h, \ldots, x]` where `b-1 \le x < b` (in the third case, `b-h \le x < b`). Like Python's :func:`~mpmath.range`, the endpoint is not included. To produce ranges where the endpoint is included, :func:`~mpmath.linspace` is more convenient. **Examples** >>> from mpmath import * >>> mp.dps = 15; mp.pretty = False >>> arange(4) [mpf('0.0'), mpf('1.0'), mpf('2.0'), mpf('3.0')] >>> arange(1, 2, 0.25) [mpf('1.0'), mpf('1.25'), mpf('1.5'), mpf('1.75')] >>> arange(1, -1, -0.75) [mpf('1.0'), mpf('0.25'), mpf('-0.5')] is+arange expected at most 3 arguments, got %iis+arange expected at least 1 argument, got %iiis0dt is too small and would cause an infinite loop(tlenRRtmpftAssertionErrorRRtappend( RR6RJtdttbtoptresulttiR\((s.lib/python2.7/site-packages/mpmath/ctx_base.pytarange×sD     /     c Os—t|ƒdkrK|j|dƒ}|j|dƒ}t|dƒ}nnt|ƒdkr£t|ddƒsvt‚|dj}|dj}t|dƒ}ntdt|ƒƒ‚|dkrÔtdƒ‚nd|ksê|drU|dkr|j|ƒgS|||j|dƒ}gt |ƒD]}|||^q.}||d |||j|ƒ}gt |ƒD]}|||^qy}|S( sÄ ``linspace(a, b, n)`` returns a list of `n` evenly spaced samples from `a` to `b`. The syntax ``linspace(mpi(a,b), n)`` is also valid. This function is often more convenient than :func:`~mpmath.arange` for partitioning an interval into subintervals, since the endpoint is included:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = False >>> linspace(1, 4, 4) [mpf('1.0'), mpf('2.0'), mpf('3.0'), mpf('4.0')] You may also provide the keyword argument ``endpoint=False``:: >>> linspace(1, 4, 4, endpoint=False) [mpf('1.0'), mpf('1.75'), mpf('2.5'), mpf('3.25')] iiiit_mpi_s*linspace expected 2 or 3 arguments, got %isn must be greater than 0tendpointiÿÿÿÿ( RdRetintR$RfRJRiRRR!R( RR6R,RJRiRGtstepRlR.((s.lib/python2.7/site-packages/mpmath/ctx_base.pytlinspace s,    ' 'cKs"|j|||j||fS(N(tcostsin(RtzR,((s.lib/python2.7/site-packages/mpmath/ctx_base.pytcos_sinNscKs"|j|||j||fS(N(tcospitsinpi(RRuR,((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt cospi_sinpiQscCstd|dd|ƒS(NiègÐ?i(Rp(Rtp((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt_default_hyper_maxprecTsic Cs!|j}zd}xö||d|_|j}|j}d}x}|ƒD]r}||7}|| r°|r°|j|ƒ} t|| ƒ}|j|ƒ} | | |jkr°Pq°n|d7}qHW|| } | | krØPn| |ksí|jrñPn|t|j| ƒ7}qW|SWd||_XdS(Ni iii(RYtninfR(tmagRPt_fixed_precisiontmin( Rttermst check_stepRYt extraprectmax_magR[tkttermtterm_magtsum_magt cancellation((s.lib/python2.7/site-packages/mpmath/ctx_base.pytsum_accuratelyas2      cCs+|j}zd}x||d|_|j}|j}|}d}x|ƒD]v} || 9}| |} ||sº|j| ƒ} t|| ƒ}|j||ƒ} | |jkrºPqºn|d7}qNW|| } | | krâPn| |ks÷|jrûPn|t|j| ƒ7}qW|SWd||_XdS(Ni iii(RYR|RBR}RPR~R(RtfactorsRRYR‚RƒRBR[R„tfactorR…R†R‡Rˆ((s.lib/python2.7/site-packages/mpmath/ctx_base.pytmul_accurately}s6        cCs|j|ƒ|j|ƒS(sConverts `x` and `y` to mpmath numbers and evaluates `x^y = \exp(y \log(x))`:: >>> from mpmath import * >>> mp.dps = 30; mp.pretty = True >>> power(2, 0.5) 1.41421356237309504880168872421 This shows the leading few digits of a large Mersenne prime (performing the exact calculation ``2**43112609-1`` and displaying the result in Python would be very slow):: >>> power(2, 43112609)-1 3.16470269330255923143453723949e+12978188 (R+(RR%R.((s.lib/python2.7/site-packages/mpmath/ctx_base.pytpowerscCs |j|ƒS(N(tzeta(RRG((s.lib/python2.7/site-packages/mpmath/ctx_base.pyt _zeta_int¯scs%dg‰‡‡‡‡fd†}|S(sù Return a wrapped copy of *f* that raises ``NoConvergence`` when *f* has been called more than *N* times:: >>> from mpmath import * >>> mp.dps = 15 >>> f = maxcalls(sin, 10) >>> print(sum(f(n) for n in range(10))) 1.95520948210738 >>> f(10) # doctest: +IGNORE_EXCEPTION_DETAIL Traceback (most recent call last): ... NoConvergence: maxcalls: function evaluated 10 times icsCˆdcd7<ˆdˆkr6ˆjdˆƒ‚nˆ||ŽS(Niis%maxcalls: function evaluated %i times(t NoConvergence(R6R,(tNtcounterRtf(s.lib/python2.7/site-packages/mpmath/ctx_base.pytf_maxcalls_wrappedÃs((RR“R‘R”((R‘R’RR“s.lib/python2.7/site-packages/mpmath/ctx_base.pytmaxcalls²s cs7i‰‡‡‡fd†}ˆj|_ˆj|_|S(s™ Return a wrapped copy of *f* that caches computed values, i.e. a memoized copy of *f*. Values are only reused if the cached precision is equal to or higher than the working precision:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> f = memoize(maxcalls(sin, 1)) >>> f(2) 0.909297426825682 >>> f(2) 0.909297426825682 >>> mp.dps = 25 >>> f(2) # doctest: +IGNORE_EXCEPTION_DETAIL Traceback (most recent call last): ... NoConvergence: maxcalls: function evaluated 1 times csƒ|r!|t|jƒƒf}n|}ˆj}|ˆkr`ˆ|\}}||kr`| Snˆ||Ž}||fˆ|<|S(N(ttupleRRY(R6R,tkeyRYtcprectcvalueR(RR“tf_cache(s.lib/python2.7/site-packages/mpmath/ctx_base.pytf_cachedßs   (Rt__doc__(RR“R›((RR“Ršs.lib/python2.7/site-packages/mpmath/ctx_base.pytmemoizeÊs   N(5RRRRt ComplexResultRRtFalseR~tverboseR R"R&R)R*R-R/R0R1R2R9R;RARERHRIRcRmRrRvRyR{t staticmethodtgcdt_gcdt list_primestisprimetbernfractmoebiustifact_ifacteulernumt _eulernumt stirling1t _stirling1t stirling2t _stirling2R‰RŒRRR•R(((s.lib/python2.7/site-packages/mpmath/ctx_base.pyRsT                 $3 I .       N($toperatorRRt libmp.backendRtfunctions.functionsRtfunctions.rszetaRtcalculus.quadratureRtcalculus.inverselaplaceRtcalculus.calculusRtcalculus.optimizationRt calculus.odesR tmatrices.matricesR tmatrices.calculusR tmatrices.linalgR tmatrices.eigenR tidentificationRt visualizationRtRtobjectRR(((s.lib/python2.7/site-packages/mpmath/ctx_base.pyts<