""" A fast implementation of the Voigt function.
"""
from __future__ import division
import numpy as np
from math import sqrt, pi
sqrtpi = sqrt(pi)
# u values corresponding to H table values below
U = np.arange(0, 20, 0.01)
# Table of h1 and h3 for use in Voigt function
H1 = np.array(
[-1.128379167096,-1.128153506307,-1.127476704444,-1.126349302806,
-1.124772202831,-1.122746665023,-1.120274307436,-1.117357103734,
-1.113997380833,-1.110197816115,-1.105961434223,-1.101291603457,
-1.096192031753,-1.090666762269,-1.084720168573,-1.078356949458,
-1.071582123367,-1.064401022458,-1.056819286313,-1.048842855295,
-1.040477963566,-1.031731131783,-1.022609159473,-1.013119117112,
-1.003268337903,-0.993064409287,-0.982515164176,-0.971628671947,
-0.960413229188,-0.948877350223,-0.937029757433,-0.924879371370,
-0.912435300708,-0.899706832010,-0.886703419362,-0.873434673861,
-0.859910352995,-0.846140349907,-0.832134682585,-0.817903482971,
-0.803456986018,-0.788805518705,-0.773959489031,-0.758929374994,
-0.743725713581,-0.728359089780,-0.712840125622,-0.697179469285,
-0.681387784257,-0.665475738580,-0.649453994194,-0.633333196387,
-0.617123963365,-0.600836875965,-0.584482467511,-0.568071213835,
-0.551613523468,-0.535119728024,-0.518600072770,-0.502064707417,
-0.485523677115,-0.468986913693,-0.452464227121,-0.435965297228,
-0.419499665677,-0.403076728196,-0.386705727090,-0.370395744018,
-0.354155693073,-0.337994314132,-0.321920166521,-0.305941622964,
-0.290066863852,-0.274303871809,-0.258660426580,-0.243144100224,
-0.227762252632,-0.212522027360,-0.197430347784,-0.182493913569,
-0.167719197466,-0.153112442425,-0.138679659026,-0.124426623230,
-0.110358874442,-0.096481713891,-0.082800203315,-0.069319163960,
-0.056043175874,-0.042976577504,-0.030123465589,-0.017487695339,
-0.005072880900, 0.007117603905, 0.019080624559, 0.030813284570,
0.042312923934, 0.053577117337, 0.064603672120, 0.075390626005,
0.085936244587, 0.096239018606, 0.106297661007, 0.116111103790,
0.125678494668, 0.134999193535, 0.144072768747, 0.152898993242,
0.161477840493, 0.169809480309, 0.177894274488, 0.185732772342,
0.193325706088, 0.200673986130, 0.207778696230, 0.214641088574,
0.221262578761, 0.227644740700, 0.233789301442, 0.239698135944,
0.245373261784, 0.250816833826, 0.256031138848, 0.261018590143,
0.265781722099, 0.270323184764, 0.274645738407, 0.278752248083,
0.282645678207, 0.286329087142, 0.289805621818, 0.293078512380,
0.296151066866, 0.299026665943, 0.301708757680, 0.304200852389,
0.306506517515, 0.308629372606, 0.310573084348, 0.312341361679,
0.313937950984, 0.315366631379, 0.316631210086, 0.317735517896,
0.318683404738, 0.319478735348, 0.320125385037, 0.320627235572,
0.320988171168, 0.321212074587, 0.321302823355, 0.321264286099,
0.321100318992, 0.320814762332, 0.320411437226, 0.319894142361,
0.319266651125, 0.318532708393, 0.317696027886, 0.316760289435,
0.315729136408, 0.314606173263, 0.313394963225, 0.312099026077,
0.310721836083, 0.309266820022, 0.307737355347, 0.306136768458,
0.304468333094, 0.302735268836, 0.300940739729, 0.299087853007,
0.297179657938, 0.295219144761, 0.293209243745, 0.291152824341,
0.289052694434, 0.286911599696, 0.284732223035, 0.282517184136,
0.280269039094, 0.277990280130, 0.275683335398, 0.273350568872,
0.270994280310, 0.268616705302, 0.266220015385, 0.263806318229,
0.261377657901, 0.258936015180, 0.256483307947, 0.254021391629,
0.251552059701, 0.249077044244, 0.246598016547, 0.244116587773,
0.241634309652, 0.239152675232, 0.236673119663, 0.234197021025,
0.231725701178, 0.229260426666, 0.226802409630, 0.224352808763,
0.221912730283, 0.219483228934, 0.217065309004, 0.214659925362,
0.212267984517, 0.209890345679, 0.207527821848, 0.205181180899,
0.202851146686, 0.200538400145, 0.198243580407, 0.195967285910,
0.193710075514, 0.191472469618, 0.189254951272, 0.187057967287,
0.184881929338, 0.182727215069, 0.180594169179, 0.178483104495,
0.176394303104, 0.174328017332, 0.172284470848, 0.170263859698,
0.168266353322, 0.166292095577, 0.164341205726, 0.162413779421,
0.160509889676, 0.158629587805, 0.156772904362, 0.154939850049,
0.153130416616, 0.151344577731, 0.149582289845, 0.147843493024,
0.146128111769, 0.144436055819, 0.142767220922, 0.141121489599,
0.139498731881, 0.137898806029, 0.136321559226, 0.134766828262,
0.133234440187, 0.131724212947, 0.130235956007, 0.128769470940,
0.127324552012, 0.125900986736, 0.124498556410, 0.123117036639,
0.121756197832, 0.120415805687, 0.119095621649, 0.117795403363,
0.116514905091, 0.115253878132, 0.114012071207, 0.112789230837,
0.111585101702, 0.110399426986, 0.109231948698, 0.108082407992,
0.106950545455, 0.105836101395, 0.104738816107, 0.103658430123,
0.102594684457, 0.101547320830, 0.100516081880, 0.099500711373,
0.098500954380, 0.097516557468, 0.096547268856, 0.095592838578,
0.094653018623, 0.093727563075, 0.092816228235, 0.091918772736,
0.091034957656, 0.090164546608, 0.089307305837, 0.088463004296,
0.087631413725, 0.086812308713, 0.086005466764, 0.085210668341,
0.084427696923, 0.083656339036, 0.082896384293, 0.082147625423,
0.081409858291, 0.080682881921, 0.079966498505, 0.079260513420,
0.078564735226, 0.077878975668, 0.077203049679, 0.076536775369,
0.075879974015, 0.075232470052, 0.074594091054, 0.073964667719,
0.073344033845, 0.072732026308, 0.072128485038, 0.071533252989,
0.070946176112, 0.070367103323, 0.069795886469, 0.069232380298,
0.068676442412, 0.068127933258, 0.067586716054, 0.067052656776,
0.066525624108, 0.066005489409, 0.065492126662, 0.064985412445,
0.064485225879, 0.063991448591, 0.063503964673, 0.063022660634,
0.062547425362, 0.062078150081, 0.061614728306, 0.061157055799,
0.060705030534, 0.060258552644, 0.059817524389, 0.059381850104,
0.058951436167, 0.058526190951, 0.058106024786, 0.057690849916,
0.057280580461, 0.056875132379, 0.056474423420, 0.056078373094,
0.055686902597, 0.055299934908, 0.054917394549, 0.054539207682,
0.054165302043, 0.053795606904, 0.053430053038, 0.053068572685,
0.052711099521, 0.052357568621, 0.052007916428, 0.051662080722,
0.051320000589, 0.050981616389, 0.050646869729, 0.050315703428,
0.049988061495, 0.049663889096, 0.049343132530, 0.049025739199,
0.048711657583, 0.048400837216, 0.048093228658, 0.047788783472,
0.047487454201, 0.047189194340, 0.046893958321, 0.046601701483,
0.046312380055, 0.046025951131, 0.045742372654, 0.045461603391,
0.045183602918, 0.044908331595, 0.044635750552, 0.044365821669,
0.044098507559, 0.043833771549, 0.043571577663, 0.043311890609,
0.043054675757, 0.042799899130, 0.042547527383, 0.042297527792,
0.042049868238, 0.041804517192, 0.041561443703, 0.041320617383,
0.041082008396, 0.040845587444, 0.040611325754, 0.040379195067,
0.040149167627, 0.039921216168, 0.039695313903, 0.039471434514,
0.039249552141, 0.039029641375, 0.038811677240, 0.038595635191,
0.038381491103, 0.038169221257, 0.037958802339, 0.037750211423,
0.037543425969, 0.037338423811, 0.037135183150, 0.036933682546,
0.036733900911, 0.036535817501, 0.036339411909, 0.036144664057,
0.035951554189, 0.035760062868, 0.035570170964, 0.035381859650,
0.035195110398, 0.035009904968, 0.034826225408, 0.034644054043,
0.034463373472, 0.034284166563, 0.034106416445, 0.033930106506,
0.033755220385, 0.033581741970, 0.033409655381, 0.033238945006,
0.033069595437, 0.032901591502, 0.032734918257, 0.032569560976,
0.032405505149, 0.032242736480, 0.032081240877, 0.031921004457,
0.031762013533, 0.031604254614, 0.031447714408, 0.031292379807,
0.031138237889, 0.030985275916, 0.030833481330, 0.030682841747,
0.030533344956, 0.030384978917, 0.030237731757, 0.030091591765,
0.029946547392, 0.029802587246, 0.029659700092, 0.029517874848,
0.029377100578, 0.029237366499, 0.029098661968, 0.028960976487,
0.028824299698, 0.028688621380, 0.028553931447, 0.028420219946,
0.028287477146, 0.028155693172, 0.028024858549, 0.027894963834,
0.027765999707, 0.027637956969, 0.027510826538, 0.027384599450,
0.027259266855, 0.027134820015, 0.027011250304, 0.026888549204,
0.026766708306, 0.026645719305, 0.026525574002, 0.026406264299,
0.026287782199, 0.026170119804, 0.026053269315, 0.025937223027,
0.025821973333, 0.025707512715, 0.025593833750, 0.025480929104,
0.025368791533, 0.025257413880, 0.025146789074, 0.025036910129,
0.024927770145, 0.024819362301, 0.024711679860, 0.024604716163,
0.024498464633, 0.024392918767, 0.024288072140, 0.024183918405,
0.024080451285, 0.023977664578, 0.023875552157, 0.023774107962,
0.023673326005, 0.023573200367, 0.023473725197, 0.023374894712,
0.023276703194, 0.023179144990, 0.023082214514, 0.022985906240,
0.022890214705, 0.022795134511, 0.022700660318, 0.022606786845,
0.022513508873, 0.022420821238, 0.022328718838, 0.022237196622,
0.022146249601, 0.022055872835, 0.021966061444, 0.021876810597,
0.021788115520, 0.021699971488, 0.021612373829, 0.021525317922,
0.021438799195, 0.021352813126, 0.021267355243, 0.021182421120,
0.021098006380, 0.021014106692, 0.020930717773, 0.020847835383,
0.020765455329, 0.020683573461, 0.020602185675, 0.020521287909,
0.020440876143, 0.020360946400, 0.020281494745, 0.020202517283,
0.020124010162, 0.020045969566, 0.019968391723, 0.019891272897,
0.019814609391, 0.019738397547, 0.019662633744, 0.019587314399,
0.019512435964, 0.019437994927, 0.019363987815, 0.019290411187,
0.019217261637, 0.019144535794, 0.019072230323, 0.019000341920,
0.018928867313, 0.018857803268, 0.018787146573, 0.018716894064,
0.018647042595, 0.018577589056, 0.018508530367, 0.018439863479,
0.018371585372, 0.018303693057, 0.018236183573, 0.018169053990,
0.018102301405, 0.018035922942, 0.017969915757, 0.017904277029,
0.017839003967, 0.017774093806, 0.017709543809, 0.017645351263,
0.017581513483, 0.017518027809, 0.017454891607, 0.017392102266,
0.017329657201, 0.017267553854, 0.017205789687, 0.017144362190,
0.017083268872, 0.017022507270, 0.016962074942, 0.016901969469,
0.016842188454, 0.016782729524, 0.016723590326, 0.016664768531,
0.016606261831, 0.016548067937, 0.016490184585, 0.016432609529,
0.016375340544, 0.016318375428, 0.016261711995, 0.016205348082,
0.016149281544, 0.016093510257, 0.016038032116, 0.015982845035,
0.015927946945, 0.015873335798, 0.015819009564, 0.015764966230,
0.015711203802, 0.015657720305, 0.015604513780, 0.015551582286,
0.015498923899, 0.015446536712, 0.015394418835, 0.015342568396,
0.015290983537, 0.015239662419, 0.015188603218, 0.015137804125,
0.015087263349, 0.015036979113, 0.014986949655, 0.014937173231,
0.014887648110, 0.014838372576, 0.014789344929, 0.014740563483,
0.014692026566, 0.014643732522, 0.014595679707, 0.014547866494,
0.014500291267, 0.014452952426, 0.014405848383, 0.014358977564,
0.014312338410, 0.014265929373, 0.014219748919, 0.014173795527,
0.014128067689, 0.014082563909, 0.014037282704, 0.013992222604,
0.013947382150, 0.013902759897, 0.013858354410, 0.013814164268,
0.013770188060, 0.013726424388, 0.013682871864, 0.013639529114,
0.013596394772, 0.013553467487, 0.013510745917, 0.013468228730,
0.013425914607, 0.013383802238, 0.013341890327, 0.013300177583,
0.013258662731, 0.013217344504, 0.013176221644, 0.013135292905,
0.013094557051, 0.013054012855, 0.013013659100, 0.012973494580,
0.012933518098, 0.012893728465, 0.012854124505, 0.012814705048,
0.012775468934, 0.012736415014, 0.012697542147, 0.012658849200,
0.012620335051, 0.012581998585, 0.012543838698, 0.012505854291,
0.012468044278, 0.012430407578, 0.012392943121, 0.012355649844,
0.012318526692, 0.012281572619, 0.012244786587, 0.012208167567,
0.012171714535, 0.012135426478, 0.012099302390, 0.012063341272,
0.012027542133, 0.011991903989, 0.011956425866, 0.011921106795,
0.011885945815, 0.011850941973, 0.011816094322, 0.011781401924,
0.011746863846, 0.011712479164, 0.011678246960, 0.011644166323,
0.011610236415, 0.011576456207, 0.011542824873, 0.011509341531,
0.011476005304, 0.011442815321, 0.011409770717, 0.011376870636,
0.011344114226, 0.011311500642, 0.011279029046, 0.011246698605,
0.011214508494, 0.011182457891, 0.011150545983, 0.011118771961,
0.011087135025, 0.011055634376, 0.011024269226, 0.010993038789,
0.010961942287, 0.010930978946, 0.010900147999, 0.010869448684,
0.010838880244, 0.010808441929, 0.010778132993, 0.010747952695,
0.010717900302, 0.010687975084, 0.010658176315, 0.010628503279,
0.010598955260, 0.010569531549, 0.010540231444, 0.010511054245,
0.010481999259, 0.010453065798, 0.010424253175, 0.010395560715,
0.010366987742, 0.010338533587, 0.010310197585, 0.010281979076,
0.010253877405, 0.010225891921, 0.010198021978, 0.010170266935,
0.010142626154, 0.010115099002, 0.010087684852, 0.010060383080,
0.010033193065, 0.010006114194, 0.009979145854, 0.009952287440,
0.009925538348, 0.009898897981, 0.009872365744, 0.009845941047,
0.009819623305, 0.009793411934, 0.009767306357, 0.009741306000,
0.009715410293, 0.009689618670, 0.009663930567, 0.009638345428,
0.009612862696, 0.009587481821, 0.009562202255, 0.009537023456,
0.009511944883, 0.009486966000, 0.009462086274, 0.009437305177,
0.009412622182, 0.009388036769, 0.009363548418, 0.009339156614,
0.009314860847, 0.009290660607, 0.009266555391, 0.009242544697,
0.009218628026, 0.009194804885, 0.009171074781, 0.009147437227,
0.009123891737, 0.009100437830, 0.009077075027, 0.009053802853,
0.009030620835, 0.009007528505, 0.008984525396, 0.008961611045,
0.008938784992, 0.008916046779, 0.008893395954, 0.008870832064,
0.008848354662, 0.008825963303, 0.008803657543, 0.008781436944,
0.008759301068, 0.008737249482, 0.008715281756, 0.008693397459,
0.008671596168, 0.008649877459, 0.008628240913, 0.008606686111,
0.008585212639, 0.008563820085, 0.008542508040, 0.008521276096,
0.008500123850, 0.008479050900, 0.008458056847, 0.008437141293,
0.008416303846, 0.008395544113, 0.008374861705, 0.008354256236,
0.008333727321, 0.008313274579, 0.008292897630, 0.008272596097,
0.008252369605, 0.008232217783, 0.008212140259, 0.008192136667,
0.008172206641, 0.008152349817, 0.008132565835, 0.008112854337,
0.008093214965, 0.008073647366, 0.008054151187, 0.008034726079,
0.008015371695, 0.007996087687, 0.007976873714, 0.007957729433,
0.007938654505, 0.007919648594, 0.007900711364, 0.007881842482,
0.007863041617, 0.007844308441, 0.007825642625, 0.007807043845,
0.007788511779, 0.007770046104, 0.007751646503, 0.007733312657,
0.007715044251, 0.007696840973, 0.007678702509, 0.007660628552,
0.007642618793, 0.007624672926, 0.007606790647, 0.007588971653,
0.007571215645, 0.007553522324, 0.007535891393, 0.007518322556,
0.007500815521, 0.007483369996, 0.007465985690, 0.007448662316,
0.007431399588, 0.007414197219, 0.007397054929, 0.007379972434,
0.007362949455, 0.007345985715, 0.007329080936, 0.007312234844,
0.007295447166, 0.007278717629, 0.007262045965, 0.007245431904,
0.007228875179, 0.007212375525, 0.007195932679, 0.007179546378,
0.007163216361, 0.007146942369, 0.007130724144, 0.007114561431,
0.007098453973, 0.007082401519, 0.007066403816, 0.007050460614,
0.007034571663, 0.007018736717, 0.007002955529, 0.006987227854,
0.006971553450, 0.006955932074, 0.006940363485, 0.006924847446,
0.006909383717, 0.006893972062, 0.006878612247, 0.006863304038,
0.006848047201, 0.006832841507, 0.006817686725, 0.006802582626,
0.006787528984, 0.006772525572, 0.006757572167, 0.006742668544,
0.006727814481, 0.006713009757, 0.006698254153, 0.006683547451,
0.006668889432, 0.006654279882, 0.006639718585, 0.006625205327,
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[docs]def voigt_wofz(a, u):
""" Compute the Voigt function using Scipy's wofz().
Parameters
----------
a: float
Ratio of Lorentzian to Gaussian linewidths.
u: array of floats
The frequency or velocity offsets from the line centre, in units
of the Gaussian broadening linewidth.
See the notes for `voigt` for more details.
"""
try:
from scipy.special import wofz
except ImportError:
s = ("Can't find scipy.special.wofz(), can only calculate Voigt "
" function for 0 < a < 0.1 (a=%g)" % a)
print s
else:
return wofz(u + 1j * a).real
[docs]def voigt_slow(a, u):
""" Calculate the voigt function to very high accuracy.
Uses numerical integration, so is slow. Answer is correct to 20
significant figures.
Note this needs `mpmath` or `sympy` to be installed.
"""
try:
import mpmath as mp
except ImportError:
from sympy import mpmath as mp
with mp.workdps(20):
z = mp.mpc(u, a)
result = mp.exp(-z*z) * mp.erfc(-1j*z)
return result.real
[docs]def voigt(a, u):
""" Compute the Voigt function using a fast approximation.
Parameters
----------
a : float
Ratio of Lorentzian to Gaussian linewidths (see below).
u : array of floats, shape (N,)
The frequency or velocity offsets from the line centre, in units
of the FWHM of the Gaussian broadening (see below).
Returns
-------
H : array of floats, shape (N,)
The Voigt function.
Notes
-----
The Voigt function is useful for calculating the optical depth as
function of frequency for an absorption transition associated with
an atom or ion.
The Voigt function H(a, u) is related to the Voigt profile
V(x, sigma, gamma)::
V(x, sigma, gamma) = H(a, u) / (sqrt(2*pi) * sigma)
where::
a = gamma / (sqrt(2) * sigma)
u = x / (sqrt(2) * sigma)
The Voigt profile is convolution of a Gaussian profile::
G(x, sigma) = exp(-0.5 * (x / sigma)^2) / (sigma * sqrt(2*pi))
and a Lorentzian profile::
L(x, gamma) = gamma / ((x^2 + gamma^2) * pi)
It is normalised; the integral of V over all x is 1.
This function uses a Taylor approximation to the Voigt function
for 0 < a < 0.1. (Harris 1948, ApJ, 108, 112). Relative error
with respect to `voigt_wofz` is < 10^-4.9 for a < 0.1. For larger
`a` the exact calculation is done in `voigt_wofz`.
"""
a = float(a)
if a > 0.1:
return voigt_wofz(a, u)
elif a < 0:
raise ValueError('a must be > 0 (%f)' % a)
u = np.abs(u)
out = np.empty_like(u)
u2 = u*u
cond = u > 19.99
if cond.any():
# Use asymptotic approximation.
iu2c = 1. / u2[cond]
iu2c2 = iu2c * iu2c
iu2c3 = iu2c2 * iu2c
iu2c4 = iu2c3 * iu2c
a2 = a**2
k2 = 1.5 + a2
k3 = 3.75 + 5 * a2
k4 = 26.25 * a2
out[cond] = a / sqrtpi * (iu2c + k2 * iu2c2 + k3 * iu2c3 + k4 * iu2c4)
# for u values with abs(u) <= 19.99 use lookup tables
notcond = ~cond
u = u[notcond]
u2 = u2[notcond]
expmu2 = np.exp(-u2)
out[notcond] = expmu2 + a*(np.interp(u, U, H1) + a*(
(1. - 2.*u2)*expmu2 + a*(np.interp(u, U, H3) + a*(
0.5 - 2.*u2 + 2./3.*u2*u2)*expmu2)))
return out