Q(h|'\ddlZddlZddlmZmZddlmZm Z GddZ GddZ y)N) assert_equalassert_allclose) _iv_ratio _iv_ratio_cceZdZejj dgddZejj ddejdfejddfgdZ ejj ddej ejejgejj d eje j eje j ej ejejgd Zejj dd deje j ejgd Zejj d deje jfdeje jfdeje jdzfdeje j dfeje j ej$eje j fgdZejj d ddej$eje j eje j fgdZejj d eje j eje j feje j dz eje j feje j eje j dz fgdZy) TestIvRatiov,x,r))?UUUUUU?g.a0R#?)r UUUUUU?g<)?)r r gVS?)r UUUUUU?gP]k(?)r 笪?gjD?)6Z5Z?g&R͒U?)r窪?gZ?)r竪?gZr!?)r?g4e~u?)r}| @gG)ȿ?)Q@}P?g1a?)rj6i?gִN`?)r:m@g9Ƭ7?)r5T@g4+?)r翎H%@gJ]?)EdL@9L;w3@g'~V?)r^s!iFE@g/X?)rSR@g_8?)rPT`@g)X?)r>=s@g\h*?c6tt|||ddy)aThe reference values are computed using mpmath as follows. from mpmath import mp mp.dps = 100 def iv_ratio_mp(v, x): return mp.besseli(v, x) / mp.besseli(v - 1, x) def _sample(n, *, v): '''Return n positive real numbers x such that iv_ratio(v, x) are roughly evenly spaced over (0, 1). The formula is taken from [1]. [1] Banerjee A., Dhillon, I. S., Ghosh, J., Sra, S. (2005). "Clustering on the Unit Hypersphere using von Mises-Fisher Distributions." Journal of Machine Learning Research, 6(46):1345-1382. ''' r = np.arange(1, n+1) / (n+1) return r * (2*v-r*r) / (1-r*r) for v in (0.5, 1, 2.34, 56.789): xs = _sample(5, v=v) for x in xs: print(f"({v}, {x}, {float(iv_ratio_mp(v,x))}),") 缉ؗҼrrc0tt|||yziIf exactly one of v or x is inf and the other is within domain, should return 0 or 1 accordingly.Nrr&r's r,test_infzTestIvRatio.test_inf@s Xa^Q'r.r)\(\?r*cLtt||tjyzeIf at least one argument is out of domain, or if v = x = inf, the function should return nan.N)rr&npnanr(r)r*s r,test_nanzTestIvRatio.test_nanIs Xa^RVV,r.r c\tt|ddtt|ddy)z?If x is +/-0.0, return x to ensure iv_ratio is an odd function.Nr1r(r)s r, test_zero_xzTestIvRatio.test_zero_xRs& Xa%s+Xa&-r.v,x@xD{c<tt||d|z|z y)a9If x is much less than v, the bounds x x --------------------------- <= R <= ----------------------- v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) collapses to R ~= x/2v. Test against this asymptotic expression. r Nr1r8s r, test_tiny_xzTestIvRatio.test_tiny_xXs" Xa^c!eQY/r.rg7yACrBg\)c=Hc0tt||dy)aAIf x is much greater than v, the bounds x x --------------------------- <= R <= --------------------------- v-0.5+sqrt(x**2+(v+0.5)**2) v-0.5+sqrt(x**2+(v-0.5)**2) collapses to R ~= 1. Test against this asymptotic expression. ?Nr1r8s r, test_huge_xzTestIvRatio.test_huge_xks Xa^S)r.cx||z }|dtjd|zz }tt|||ddy)aIf both x and v are very large, the bounds x x --------------------------- <= R <= ----------------------- v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) collapses to R ~= x/(v+sqrt(x**2+v**2). Test against this asymptotic expression, and in particular that no numerical overflow occurs during intermediate calculations. rr"rr#N)r6hypotrr&r(r)r*texpecteds r, test_huge_v_xzTestIvRatio.test_huge_v_x{s: EBHHQN*+Au1Er.N__name__ __module__ __qualname__pytestmark parametrizer-r6infr2r7finfofloatsmallest_normalsmallest_subnormalr9maxr>sqrtrErJrQr.r,rr s" [[W',?-,?8 [[W BFFA A'( (  [[S4"&&"&&"&&"AB [[SHBHHUO$C$C#C$,BHHUO$F$F#F$&FF7BFFBFF#<=-=C-  [[S38288E?+>+>"GH.I.  [[U HBHHUO + +, HBHHUO . ./ HBHHUO . .q 01 %  a %  gbgghbhhuo&9&9:; % 0 0 [[U %$$ %xrxx':':;% *  * [[U %  hbhhuo112 %  q ("((5/"5"56 %  hbhhuo11A56% F  Fr.rceZdZejj dgddZejj ddejdfejddfgdZ ejj ddej ejejgejj d eje j eje j ej ejejgd Zejj dd deje j ejgd Zejj d deje jfdeje jfdeje jdzfdeje j dfeje j ej$eje j fgdZejj d ddej$eje j eje j fgdZejj d eje j eje j feje j dz eje j feje j eje j dz fgdZy) TestIvRatioCr ))r r g{s+?)r r ga*?)r r gTV6?)r r g`D)?)r rg,wU?)rrgvL[?)rrg>7R?)rrgL?)rrg5?)rrgZ?)rrg3zʈ?)rrgؤO??)rrgsF?)rrg1?)rrgL9Ԋ?)rrgCv`?)rrg -S?)rrg@ɣ?)rrgO?)rr g^VO?c6tt|||ddy)z8The reference values are one minus those of TestIvRatio.V瞯zTestIvRatioC.test_zero_xs& Z3'-Z4(#.r.r?r@rAcBtt||dd|z|z z y)a=If x is much less than v, the bounds x x --------------------------- <= R <= ----------------------- v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) collapses to 1-R ~= 1-x/2v. Test against this asymptotic expression. rIr Nrhr8s r,rEzTestIvRatioC.test_tiny_xs!" Z1%sCE19}5r.rFrGcBtt|||dz |z ddy)aKIf x is much greater than v, the bounds x x --------------------------- <= R <= --------------------------- v-0.5+sqrt(x**2+(v+0.5)**2) v-0.5+sqrt(x**2+(v-0.5)**2) collapses to 1-R ~= (v-0.5)/x. Test against this asymptotic expression. r rdrr#Nrer8s r,rJzTestIvRatioC.test_huge_xs!  1a(1S5!)%aHr.rKc~||z }d|dtjd|zz z }tt|||ddy)aIf both x and v are very large, the bounds x x --------------------------- <= R <= ----------------------- v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) collapses to 1 - R ~= 1 - x/(v+sqrt(x**2+v**2). Test against this asymptotic expression, and in particular that no numerical overflow occurs during intermediate calculations. rr"rr#N)r6rMrrfrNs r,rQzTestIvRatioC.test_huge_v_xs@ EqAA.// 1a((QGr.NrRr`r.r,rbrbs& [[W',A-,A [[W BFFA A'* *  [[S4"&&"&&"&&"AB [[SHBHHUO$C$C#C$,BHHUO$F$F#F$&FF7BFFBFF#<=/=C/  [[S38288E?+>+>"GH/I/  [[U HBHHUO + +, HBHHUO . ./ HBHHUO . .q 01 %  a %  gbgghbhhuo&9&9:; % 6 6 [[U %$$ %xrxx':':;% I  I [[U %  hbhhuo112 %  q ("((5/"5"56 %  hbhhuo11A56% H  Hr.rb) rVnumpyr6 numpy.testingrrscipy.special._ufuncsrr&rrfrrbr`r.r,rqs27 AFAFHiHiHr.