;ς ρFc@sdZdfd„ƒYZdS(sOIntegration made easy. $Id: integrate.py,v 1.4 2007/03/24 15:19:18 eddy Exp $ s IntegratorcBstZdZed„Zed„Zeed„Zeed„Zeed„Zdk l Z e d„Z [ d„d „d „Z d „d „Z e e e d „Z[ e e d„Z[ [ d„ZdkZdeideiƒeidƒdeideigd„Z[RS(sB Base class for integrators. Provides crude trapezium rule integrator, which derived classes might wish to over-ride. More importantly, defines an API for integrators: measure(func) -- integrator scaling self.integrand by func between(start, stop) -- integrates over an interval before(stop) -- as between, but with minus infinity as start beyond(start) -- as between, but with plus infinity as stop Both before() and beyond() presume that they're integrating a `tail' of a distribution: to compute an integral between one side of a distribution's mode and infinity on the other side, it is probably wisest to call between() on the given bound and a more-or-less arbitrary position on the same side of the mode as the infinity in question, then to use before() or beyond() to compute the remainder, using between()'s output as offset (see below). All three integrator methods take an optional argument, test, following the given required arguments; and an optional keyword argument, offset, after that. These are used in deciding when the integral has been determined accurately enough: after each iteration, the integrator calls test with the change in its estimate of the integral as first argument and, as second argument, the new estimate, optionally with offset added to it. If the change is small enough that further refinement is a waste of time, test should return true. The integrator will then return the given estimate added to an error bar whose width is the last change. By default, the code is geared up to deal with values of class Quantity (see basEddy.quantity) and, indeed, the integrators return Quantity()s. If the first estimate at the integral (plus optional offset) has a positive .width attribute, it is presumed to be a Quantity() and the default test simply compares the change in estimate with the .width of the new estimate; if the change is within this existing error bar, it is taken to be small enough. In the absence of .width, or if the .width is not positive, the default test simply looks to see whether the change in estimate is smaller than 1e-6 times the estimate (plus optional offset); in the absence of an offset, this last test will work poorly if the integral should yield zero. cCs-||_|tj ot|ƒ|_ndS(sInitialises an integrator. Required first argument is the function to be integrated. This will be stored as self.integrand. Optional second argument, width - only needed if .beyond() or .before() is liable to be called with bound zero - is a `unit' change in input. It should ideally be approximately the difference between highest and lowest inputs for which the integrand differs significantly from 0; e.g. if func describes a Gaussian distribution with variance 1, a width of about 5 would be prudent. N(sfuncsselfs integrandswidthsNonesabss_Integrator__unit(sselfsfuncswidth((s+/home/eddy/.sys/py/study/maths/integrate.pys__init__+s   cCsQ|tjo'y |i}Wq4tj oq4Xnt||id„|ƒSdS(s°New integrator scaling self.integrand pointwise by a given function. Required argument, func, is a function accepting the same inputs as self.integrand(). Optional argument, width, is as for the Integrator constructor; if omitted, the value used when self was constructed (if any) is used. Returns an Integrator whose integrand is (: func(x) * self.integrand(x) ←x :) which can be construed as integrating func using self as measure; or as integrating self using func as measure. If self is a probability distribution, its moments can be computed using func()s of form (: x**i ←x :) for i = 1, 2, 3 ...cCs||ƒ||ƒS(N(sfsxsi(sxsfsi((s+/home/eddy/.sys/py/study/maths/integrate.pysLsN(swidthsNonesselfs_Integrator__unitsAttributeErrors Integratorsfuncs integrand(sselfsfuncswidth((s+/home/eddy/.sys/py/study/maths/integrate.pysmeasure;s    cCs?|i|||d|i|ƒ|i|ƒd||ƒSdS(sΡIntegral over a range. Two required arguments, start and stop, give the bounds of the range. Also accepts the usual optional tolerance specifiers, test and offset: see class doc for details. f1.0f0.5N(sselfs_Integrator__betweensstartsstops integrandstestsoffset(sselfsstartsstopstestsoffset((s+/home/eddy/.sys/py/study/maths/integrate.pysbetweenNs   cCs%|i||i|ƒ ||ƒ SdS(sVIntegration from minus infinity. Equivalent to between() with start set to a value less than any at which self's distribution is distinguishable from zero. Required argument is the upper bound of the integral; also accepts the usual optional tolerance specifiers, test and offset: see class doc for details. N(sselfs_Integrator__outwardssstops_Integrator__stepstestsoffset(sselfsstopstestsoffset((s+/home/eddy/.sys/py/study/maths/integrate.pysbeforeZscCs#|i||i|ƒ||ƒSdS(sδIntegration to plus infinity. As before() but required argument is the lower bound of integration, the upper bound being greater than any value at which self's distribution is distinguishable from zero. N(sselfs_Integrator__outwardssstarts_Integrator__stepstestsoffset(sselfsstartstestsoffset((s+/home/eddy/.sys/py/study/maths/integrate.pysbeyondds(stophatcCs|||SdS(N(smidsHsspread(smidsspreadsH((s+/home/eddy/.sys/py/study/maths/integrate.pys__bluroscCst|ƒdt|ƒjS(Nf9.9999999999999995e-07(sabssdsn(sdsn((s+/home/eddy/.sys/py/study/maths/integrate.pyssscCst|iƒ|ijS(N(sabssdsbestsnswidth(sdsn((s+/home/eddy/.sys/py/study/maths/integrate.pystscCsΊyœ|i}y`d}x+|djo|i|df\}}qWx%|id|ijo |i}qFWWntj onX|d|jo|SnWntj onX|SdS(sReturns a sensible tolerance test. Where between, below and beyond are not given a tolerance test function, this determines one given an example value kindred to the final output (plus optional offset) relative to which the tolerance is computed. i iiN(segswidthswscountsbestsAttributeErrorsbywidths microclose(segs microclosesbywidthscountsw((s+/home/eddy/.sys/py/study/maths/integrate.pys __gettestrs$   cCs||S(N(sasb(sasb((s+/home/eddy/.sys/py/study/maths/integrate.pysŠscCst||ƒSdS(N(sreducesplussrow(srowsplus((s+/home/eddy/.sys/py/study/maths/integrate.pys__sumŠsc Cs ||} |tjo5y| | i}WqLtj o| d}qLXn|tjo||| ƒ}nd} xnoˆd| | f\} } || } |t || |id„td| ƒƒƒ|| } | | } || | |ƒo|| | ƒSqvq}WdS(NiiicCs||||ƒS(N(sfsbsiss(sisbsssf((s+/home/eddy/.sys/py/study/maths/integrate.pyss(sedgesgapsnowsoffsetsNonesbestsAttributeErrorstestsgettestsnswasshssumsmapsstartsselfs integrandsrangesdifsblur(sselfsstartsgapsedgestestsoffsetsblursgettestssumshsnowswassdifsn((s+/home/eddy/.sys/py/study/maths/integrate.pys __betweenŒs$     c Cs„t|i|ƒƒd}x(|i||ƒ|jo|d}qWx(|i||ƒ|jo|d}qGW|tjo|||ƒ}n||} |i || |d|ƒ| f\}}|tjo5y||i}Wqtj od|}qXnxznor|d}||} |i || |d||ƒ} || | f\}}|| ||ƒo||| ƒSqq WdS(Nf1000.0f7.0isoffsetiii(sabssselfs integrandsboundssmalls_Integrator__probesstepstestsNonesgettestsnextsbetweensoffsetstotalsbestsAttributeErrorsmoresblur( sselfsboundsstepstestsoffsetsblursgetteststotalssmallsnextsmore((s+/home/eddy/.sys/py/study/maths/integrate.pys __outwards₯s0  '   cCsŠy |iSWntj onXyt|id„ƒƒ}Wn%ttfj ot|ƒ}nX|o ||_n td‚|SdS(NcCsdS(Ni((sx((s+/home/eddy/.sys/py/study/maths/integrate.pysΏss?Integrator needs a width parameter for .before(0) or .beyond(0)( sselfs_Integrator__unitsAttributeErrorsabssboundscopysanss TypeErrors ValueError(sselfsboundsans((s+/home/eddy/.sys/py/study/maths/integrate.pys__stepΌs   Nif2.0iicCs&tt|i||d„|ƒƒSdS(s>Scale of integrand's values for inputs base + of order scale. cCst||||ƒƒS(N(sabssfsbsxss(sxsfsssb((s+/home/eddy/.sys/py/study/maths/integrate.pysΚsN(smaxsmapsselfs integrandsscalesbasessamples(sselfsbasesscalessamples((s+/home/eddy/.sys/py/study/maths/integrate.pys__probeΗs(s__name__s __module__s__doc__sNones__init__smeasuresbetweensbeforesbeyondsstudy.value.quantitystophats_Integrator__blurs_Integrator__gettests_Integrator__sums_Integrator__betweens_Integrator__outwardss_Integrator__stepsmathsexpspissqrtses_Integrator__probe(((s+/home/eddy/.sys/py/study/maths/integrate.pys Integrators& $       @N(s__doc__s Integrator(s Integrator((s+/home/eddy/.sys/py/study/maths/integrate.pys?s