Algorithm 916: computing the Faddeyeva and Voigt functions

Transcription

1 Algorithm 96: computing th Faddyva and Voigt functions MOFREH R. ZAGHLOUL, Unitd Arab Emirats Univrsity AHMED N. ALI, Unitd Arab Emirats Univrsity W prsnt a MATLAB function for th numrical valuation of th Faddyva function w(z). Th function is basd on a nwly dvlopd accurat algorithm. In addition to its highr accuracy, th softwar provids a flxibl accuracy vs fficincy trad-off through a controlling paramtr that may b usd to rduc accuracy and computational tim and vic vrsa. Vrification of th flxibility, rliability and suprior accuracy of th algorithm is providd through comparison with standard algorithms availabl in othr libraris and softwar packags. Catgoris and Subjct Dscriptors: G..0 [Numrical Analysis]: Gnral-Computr Arithmtic; Numrical Algorithms; Multipl prcision arithmtic; G.. [Numrical Analysis]: Approximation-Spcial Functions Approximations; G.4 [Mathmatical Softwar]: Algorithm dsign and analysis; fficincy Gnral Trms: Algorithms Additional Ky Words and Phrass: Function valuation, accuracy, Matlab, Faddyva function Author s addrsss: M. Zaghloul and A. Ali, Dpartmnt of Physics, Collg of Scincs, Unitd Arab Emirats Univrsity, Al-Ain, 755, UAE.

2 . INTRODUCTION Th valu and significanc of th scald complmntary rror function for complx variabls, also known as th Faddyva function or th plasma disprsion function, is wll rcognizd in th litratur for its applications in svral filds of physics [Armstrong 967; Gautschi 967; 970; Hui t al. 978; Humlíčk 98; Dominguz t al. 987; Popp t al. 990; Lthr t al. 99; Schrir 99; Shippony t al. 993; Widman 994; Wlls 999; Luqu t al. 005; Ltchworth t al. 007; Abrarov t al. 00]. Plasma spctroscopy, nuclar physics, radiativ hat transfr, and nuclar magntic rsonanc ar a fw xampls of th filds for which fficint and accurat valuation of this function is rquird. Som of ths applications rquir a small numbr of valuations of th function whr accuracy is mor important than th computational tim whil othr applications rquir normous numbrs of function valuations, which imposs tight rstrictions on th computational tim. Accordingly, computational accuracy and computational tim ar issus of intrst that should b critically addrssd and invstigatd in dvloping any succssful algorithm for th computation of this function. Motivatd by its practical importanc and a lack of closd form xprssions for th calculation of th Faddyva function, numrical valuation of th function has bn th focus of rsarch ovr many dcads [Armstrong 967; Gautschi 967; 970; Hui t al. 978; Humlíčk 98; Dominguz t al. 987; Popp t al. 990; Lthr t al. 99; Schrir 99; Shippony t al. 993; Widman 994; Wlls 999; Luqu t al. 005; Ltchworth t al. 007; Abrarov t al. 00; Zaghloul 007]. As a rsult, a wid varity of algorithms for th calculation of this function hav bn dvlopd and prsntd in th litratur. Howvr, as it is shown in this study, most of ths algorithms los accuracy in som rgions of th computational domain. W introduc a nw algorithm for th calculation of th Faddyva function which provids flxibility, rliability and suprior accuracy. In sction w prsnt th dfinition of th function and brifly summariz som rlvant fundamntal mathmatical rlations. Thn, in sction 3, w stablish th analytical basis of th algorithm whil numrical analysis and computational dtails ar discussd in sction 4. A short dscription of th Matlab function is givn in sction 5. Vrification of th algorithm and comparisons with othr comptitiv cods in th litratur ar providd in sction 6.. DEFINITION AND FUNDAMENTAL MATHEMATICAL RELATIONS For a complx variabl z=x+iy, th Faddyva (plasma disprsion) function, w(z), th ral Voigt function, V(x,y), th imaginary Voigt function, L(x,y), th complx rror function, rf(z), th imaginary rror function, rfi(z) and th Dawson s intgral F(z) ar all closly rlatd to ach othr. On can summariz ths rlations as w( z) = = = z z z ( rf( iz)) = (+ rf( iz)) (+ i rfi( z)) z i = + F( z) = V ( x, y) + i L( x, y) z rfc( iz) for y > 0 ()

3 In th abov rlations, i =, rfc(z) is th complmntary rror function, rfi(z) is th imaginary rror function which is rlatd to th rror function by rfi(z)= - i rf(iz). As can b sn from th last lin in (), th ral and imaginary Voigt functions ar just th ral and imaginary parts of th Faddyva function for y>0, rspctivly. Th valuation of all of ths functions can, thrfor, b prformd through th rror function. Th rror function of a complx variabl z can b rgardd as a lin intgral in th complx plan givn by z = t rf ( z) dt () 0 Diffrnt paths can b takn to prform this lin intgral in th complx plan. For xampl, on may choos a linar path btwn th initial point (origin) and th final point z which givs an xprssion for th complx rror function of th form z = z t rf ( z) dt (3) 0 An altrnativ path can b followd through th lin sgmnts [0, 0 iy] and [0 x, iy] which givs, for th complx rror function th xprssion, x y y y x = t t + t rf ( z) cos(yt) dt i dt sin(yt) dt (4) In addition to th abov paths, anothr simpl and usful path from th initial to th final points can b takn through th lin sgmnts [0 x, iy=0] and [x, 0 iy] which rsults in x y y t t t rf ( z) = dt + sin(xt) dt + i cos(xt) dt (5) Th first trm on th right hand sid of (5) is th dfinition of th rror function of th ral variabl x. Equation (5) is th basis of th prsnt algorithm for th calculation of th Faddyva function as shown blow. For som limiting valus of th paramtrs x and y, analytical formula do xist for th ral and imaginary parts of th Faddyva function V ( x 0, y) rfcx( y) L( x 0, y) 0 V L ( x + y ) ) ( )[ y ( x + y )] ( x + y ) ) ( )[ x ( x + y )] V ( ± x, y 0) (6) y whr rfcx( y) = rfc( y) is th scald complmntary rror function of th ral argumnt y. Whn y 0, th imaginary part of th Faddyva function cannot b xprssd as simply, howvr, it can b xprssd in trms of Dawson s intgral of x (th ral part of z) whr L( ± x, y 0) F( x) (7) 3

4 and F(x) has wll rportd asymptotic xprssions for limiting valus of x [Armstrong 967]. Following Salzr 95 [Salzr 95], w can writ t a ( + a n t cosh (ant) ) ± E a n= whr th rlativ rror n / a E = cos(n t / a) is of th ordr of / a E ~. n= Tabl prsnts som rprsntativ valus of th rlativ rror E corrsponding to som valus of th paramtr a. Tabl : Rlativ rror, E, of th rprsntation of paramtr a. t givn by Eq. (8) as a function of th a E (8) 3- ANALYTICAL BASIS OF THE ALGORITHM t Rplacing in (5) by its rprsntation givn in (8), w gt, for th ral and imaginary parts of rf(z), th following xprssions R[rf( z)] = rf( x) + n= a x a n 4a n + 4 x 8a ( cos(xy)) + [ x x cosh(a n y) cos( x y) + a n sinh(a n y) sin( x y) ] (9) 4a Im[rf( z)] = n= (4x) a n 4a n + 4 x 8a sin( x y) + [ x cosh(a n y) sin( x y) + a n sinh(a n y) cos( x y) ] (0) Th rror in th xprssions in (9) and (0) for th ral and imaginary parts of th complx rror function is controllabl through th paramtr a as indicatd in Tabl. Both xprssions, howvr, rduc to th formula givn by Salzr [Salzr 95] and Abramowitz and Stgun [Abramowitz t al. 97] with a rlativ rror lss than th floating point rlativ accuracy, ε, on a 6 digit computational platform, by stting a qual to /. An xprssion for th Faddyva z function, w ( z) = R[ w( z)] + i Im[ w( z)] = rfc( iz), can b obtaind by substituting (9) and (0) into (). Th rsulting xprssions for th ral and imaginary parts of th Faddyva function ar thn givn by 4

5 R[ w( z)] = 8a y + Im[ w( z)] = 8a + a xsin( xy) rfcx ( y) cos(xy) + n= 4 a n 4 a n a n + 4 y a x rfcx ( y) sin (xy) + n= a n + 4 y [sin( xy) ( xy)] [cosh(a n x) cos( x y)] [sin(xy) (xy)] [ y sin(xy) + a n sinh(a n x)] () () Som commnts on th abov xprssions may b usful at this stag - Th sris in () and () hav an infinit numbr of trms and nd to b truncatd for practical us. Th ffct of such a truncation on th accuracy of th computations is prdictabl and can b controlld for a convrging sris, - Whil th xponntial factors in th sris ar dcaying with th indx, n, thy ar multiplid by growing hyprbolic functions and on cannot dirctly dtrmin whr to truncat th infinit sums for practical us, 3- Thr is a limitation on th valuation of th hyprbolic functions (rlatd to th largst positiv floating point numbr, R max, availabl on th computational platform). Sinc th argumnt of ths hyprbolic functions is (anx), this will impos a strict rstriction on th numbr of trms to b includd in th sums for a givn valu of x with possibl catastrophic consquncs on both accuracy and rliability, 4- In writing th abov xprssions, th scald complmntary rror function of a ral variabl is usd to rduc rounding rror associatd with th trm (-rf(y)), howvr, spcial car is ndd for th valuation of th rfcx(y) function to avoid ovrflow problms illustratd in [Zaghloul 007]. At prsnt, many softwar packags hav wll-bhavd algorithms for computing th scald complmntary rror function of a ral variabl, rfcx, and algorithms for accurat and fficint computation of this function ar availabl in th litratur, 5- Th valuation of th quantity xp(-x ) common to all of th abov trms, can suffr undrflow problms for larg valus of x. Ths problms can b avoidd by combining this quantity with othr larg quantitis whrvr possibl. To ovrcom th abov-statd concrns and rstrictions, th xprssions in () and () ar rwrittn in th forms sin( ) x a x xy R[ w( z)] = rfcx( y) cos(xy) + [sin( xy) ( xy)] (3) a y y + y cos(xy) Σ + Σ + Σ 3 and 5

6 x a x Im[ w( z)] = rfcx( y)sin(xy) + [sin(xy) /(xy)] (4) a + y sin(xy) Σ Σ 4 + Σ 5 whr ( a n + x ) n= a n y Σ = (5) + ( an+ x) n= a n y Σ = (6) + ( a n) 3 n= a n y Σ = (7) + an + ( a n x) Σ 4 = (8) n= a n + y a n ( an) Σ 5 = (9) n= a n + y Th convrgnc of th sris (5)-(9) can b vrifid by applying th simpl ratio-tst [Boas 006]. In addition, in all of th abov xprssions (5)-(9), th pr-xponntial factors (fractions in brackts) of th argumnts of th summations assum valus lss than for n>/a. For a=/ (which is sufficint to xprss by th xprssion givn in (8) to machin accuracy on a 6-digit computational platform, s Tabl ) th prxponntial factors will always assum valus lss than or qual to on for n. For valus of n, th trms in th sris Σ, Σ and Σ 4 dcras monotonically with n whil for rlativly larg valus of x th trms Σ 3 and Σ 5 incras with n from on to a crtain limit whn thy start dcaying monotonically with n. Th fact that all xponntial factors in th abov summations vntually dcay with incrasing n provids us with th possibility of obtaining a truncatd sris of practical us and computational fficincy as shown in th nxt sction. It is undrstood, howvr, that all of th abov summations ar to b prformd using a singl loop for computational fficincy. Th loop spcifications can b dtrmind onc a cutoff schm for th sris in (5)-(9) is stablishd. Morovr th computation of th xponntials in ths sris can b rducd to th computation of a singl xponntial and products using a singl computational loop. This rducs th dpndnc on using th intrinsic function to calculat ths xponntials within th loop and can sav a significant amount of computational tim. As th ral part of th Faddyva function is vn in x and its imaginary part is odd in x, w nd only considr th right half of th complx plan (x 0) sinc th vn/odd proprtis of th ral and imaginary parts of th function can b usd to find th corrsponding valus in th lft half of th plan. In addition, th valus of w (z) in th lowr half of th complx plan can b obtaind from valus in th uppr half using th rlationship z w( z) = w( z) t (0) 6

7 which is quivalnt to th symmtry rlations V ( x, y) = L( x, y) = x + y + y cos(xy) V (, y) sin(xy) L(, y) (0 ) Thus w do not los gnrality by considring th valuation of th function in just th first quadrant, howvr, w not that (0) rquirs th subtraction of two valus which lads to a loss of accuracy in th lowr half. Th practical usfulnss of th calculation of th partial drivativs of th ral and imaginary parts of th Faddyva function has bn pointd out by many authors [0,3,5]. Onc th Faddyva function has bn calculatd accuratly on can calculat ths partial drivativs with rlativ simplicity using th xprssions V ( x, y) = R[ zw ( z)] = [ y L( x, y) xv ( x, y)] () x V ( x, y) = Im[ zw ( z)] = [ x L( x, y) + yv ( x, y)] () y in conjunction with th rlations L( x, y) V ( x, y) L( x, y) V ( x, y = & = ) (3) y x x y Schrir [Schrir 99] rports numrical problms that can aris whn subtracting two numbrs of approximatly qual magnituds (whn V x ~ 0 ). Ltchworth and Bnnr [Ltchworth t al. 007] us a spcial algorithm to calculat ths partial drivativ (to accuracy <0.5%) but this incrass th computational tim by about 70%. 4- NUMERICAL ANALYSIS AND MACHINE LIMITATIONS 4.. High Accuracy Computations Du to th finit numbr of dcimal digits availabl to stor a ral numbr in floating-point arithmtic, thr ar machin limitations on th valuation of th abov summations. Th floating-point rlativ accuracy, ε, and th smallst positiv floatingpoint numbr, R min, in th usd computational platform impos rstrictions on th accuracy and caus practical machin-truncation of th sums. At any stag during th computation of th sums, th nw accumulatd sum aftr adding th trm α n+ of th sris can b writtn as Σ = Σ + α ( ( n) ) n n n+ = Σ +, (4) + n whr ( n) = α n+ Σ. n (4) implis that th sum of n+ trms will not diffr from th sum of n trms (i.. th sris will b ffctivly machin-truncatd) if th trm ( n) = α n+ Σ bcoms n lss than th floating-point rlativ accuracy, ε, or if α n+ <R min. For computational fficincy (shortr computational tim) w nd to spcify ths intrnally truncatd trms and xclud thm from th computational loop. Starting with th sum Σ and considring th possibility of machin-truncation of th sum du to th undrflow of th trms α n+, a simpl saf stimation for th last valu 7

8 Σ of th indx n to b includd in th valuation of th sum, n cut, can b drivd basd on th undrflow of th xponntial factors only (sinc th pr-xponntial factor is alrady lss than unity) Σ n ln( R ) x cut, R min (5) min a It is implicitly undrstood that (5) implis rounding to th narst intgr towards infinity. Similarly, th trms of th sums, Σ and Σ 4, hav th sam xponntial dpndnc, whil both of th pr-xponntial factors will hav valus lss than or qual to unity for n / a. In such a cas, th trm α n+ in both sums will rach valus <R min if th xponntial factor bcoms R min which givs Σ, Σ 4 ncut, R [ ln( Rmin) x] (6) min a Both (5) and (6) indicat that, for x ln( R min ), no trms from th argumnts of th sums Σ, Σ and Σ 4 will b ffctiv in th computations and that th valus of ths sums will b ffctivly truncatd. W not, howvr, that machintruncation of ths sums du to th R min limitation would also imply machin-truncation du to machin accuracy (i., (n) ε ) as sn in (4). For computational fficincy, this latr condition may b usd to brak th computational loop as additional cycls of th computations or additional trms of th sris will not chang th valus of th sums. Furthrmor, w can also mak th computations of th sums Σ 3 and Σ 5 vry fficint. As pointd out abov, th valus of th trms of ths two sris grow initially with n up to a crtain valu (pak) and thn dcay continuously as n incrass. Simpl invstigation of ths two sums shows that this pak is in th vicinity of n=x/a. Accordingly, if on starts calculating ths sums from around n=x/a and procds in both dirctions, th valus of th trms will dcras until thy gt machin-truncatd. For ach valu of th indx n usd in th calculation of th trms of th sums Σ, Σ and Σ 4 w can add to ach of th sums Σ 3 and Σ 5 two trms by marching on stp in ach dirction. Th sums ar truncatd whn th sum of th nwly addd two trms rlativ to th valu of th prviously accumulatd sum bcoms lss than th machin accuracy. Handling th computation of th sums, Σ 3 and Σ 5 this way lads to a significant saving in xcution tim by dramatically rducing th numbr of trms rquiring valuation which, in turn, lads to a smallr numbr of loop cycls. Th asymptotic xprssion in th first lin of (6) is usd for valus of x<r min. 4.. Accuracy Vs Efficincy Trad-off With 6-digit floating-point arithmtic and for valus of th paramtr a > ½, th xpansion in (8) bcoms lss accurat and its accuracy will b govrnd by th corrsponding valu of th rlativ rror, E, as shown in Tabl. It is not ncssary, thrfor, to kp th strict condition for truncating th sums, i.., (n)<ε sinc th accuracy of th computations will b govrnd principally by th rlativ rror E. Rcalling that th rlation btwn a and E can b writtn as E ~ /a it sms mor appropriat to choos E (namd tiny in th cod) instad of ε to tst for th convrgnc of th sums. That way w can rduc th numbr of trms includd in th sums by xcluding trms that will not ffctivly nhanc th accuracy and thus achiv 8

9 rasonabl acclration of th computations and rduc th computational tim. Noting that, changing th valu of a changs E (tiny) and vic vrsa, w could choos ithr a or tiny as th fr paramtr for controlling th accuracy and fficincy of th computations. This fr paramtr will b usd as an argumnt of th function. W hav chosn to us tiny and w calculat a intrnally from th abov rlation as tiny givs a bttr indication of th accuracy of th computations. This allows flxibility for accuracy vs fficincy trad-offs whil maintaining th ability to run th cod for high accuracy Calculation of th Exponntials and Othr Numrical Considrations Th cntral part of th prsnt algorithm dpnds on th valuation of th sums (5)-(9) which all hav xponntial trms. Th valuation of intrinsic functions lik th xponntial function is known to b slowr than othr simpl mathmatical oprations such as multiplication and/or division. A naïv computation of ths sums would rquir thr xponntial valuations pr computational loop. This would b computationally xpnsiv. Howvr, w may rduc this to just on xponntial valuation in ach cycl of th loop. For th cas of x < R ) w hav som flxibility in valuating th x ln( min trm ithr sparatly or by combining it with othr trms. In such a cas on can a writ all of th xponntials in (5)-(9) in trms of th xponntial n whr ( a n + x ) a n x = (7) + n ( a n x) a n x = ax n ( an) a n ax = x ax In th abov xprssions ax, and ar calculatd onc outsid th loop, for ach valu of x, and th products ar simply prformd using multiplis insid th computational loop. For x R ) only Σ 3 and Σ 5 (which hav th sam ln( min xponntial factor) contribut to th calculation of th ral and imaginary parts of th Faddyva function. Howvr, du to th natur of th computation of ths trms, th xponntial factor nds to b computd twic; onc for th stp to th right of n 0 =cil(x/a) and onc on th lft wing whr cil indicats rounding to th narst intgr towards infinity. Th indics for ths two factors ar rlatd to th loop indx, n, by n 3- plus = n 0 +(n-) and n 3-minus = n 0 - n if n 3-minus, rspctivly. Whn n 3-minus <, only th trm on th right wing is includd. Th xponntial factors for th trms on th lft and right wings ar rlatd by n a n x ( an3 plus x) ( 3 minus ) ( a + ax a n0 ) (4a n0 4ax a ) = (30) Clarly th scond xponntial factor on th right hand sid of (30) and th argumnt of th product may b calculatd onc outsid th loop, for ach valu of x, thus rducing th numbr of xponntial function valuations to only on pr cycl. Th product can b valuatd using just multiplis insid th loop, thus rducing th computational tim. (8) (9) 9

10 A fw mor important computational points ar rlatd to th calculation of th imaginary part of th Faddyva function using (4). Firstly, for valus of x<<, th two sums Σ 4 and Σ 5 bcom vry clos to ach othr and th subtraction of ths two sums could significantly affct th accuracy in rgions of th computational domain whr ths two trms ar th dominant trms in calculating th imaginary part of th Faddyva function. Howvr, this problm can b simply ovrcom by xprssing th sum of ths two trms (for x<<) in its original form as in (), i.. in trms of sinh(anx). Th first thr trms in th sris xpansion of sinh(x) will b sufficint to xprss sinh(x) to th machin accuracy for x 0 - and sinc an is usually <0 in th prsnt computations thn this is satisfactorily for x Th scond important point in th calculation of th imaginary part of th Faddyva function using (4) is rlatd to th calculation of th sum of th first thr trms on th right hand sid of th quation, for x < ln( R min ), which can b writtn safly in th form + + a n a sin ( xy) rfcx ( y) (3) y n= + a n y Th trms in th curly brackts ar only dpndnt on y and for y 5 w hav found that this sum is zro to machin accuracy. Using this prvnts rounding rrors affcting th accuracy of computations. Not that for vry small valus of x th rsult of th whol xprssion (3) is O(yx) whil th total of -Σ 4 + Σ 5 is O(x), th significanc of rounding rrors is thus clar for small valus of x and rlativly larg valus of y. 5. THE MATLAB FUNCTION FADDEYEVA.M Th function Faddyva(z,tiny) rturns, in gnral, an array of complx valus for th Faddyva function of th sam siz as th input array for th complx variabl z. Th input z is usually an array (with on or two dimnsions) but can b a singl scalar as wll. Whn z contains only imaginary valus z=iy, th function rturns th ral valus calculatd from th MATLAB built-in function rfcx(y). Th function is st for th calculation for th whol complx domain. Howvr, for ngativ valus of y and xp(y - x ) gratr than th largst floating point numbr in th computational platform, Faddyva cannot calculat th Faddyva function du to inscapabl ovrflow problms. Th function chcks for accptabl valus and issus an rror mssag for any points outsid this domain. Th valu of th scalar fr paramtr tiny can b chosn by th usr within th rang tiny min tiny 0-4 to control th accuracy and computational tim. Th valu of tiny min is a valu clos to but lss than th floating-point rlativ accuracy, ε. For xampl, for a 6 digit computational platform, tiny min can b takn roughly to b ~ (th valu of E corrsponding to a=/ in Tabl ) whil for a 3-digit computational platform tiny min can b takn to b roughly Th maximum valu of tiny=0-4 corrsponds to a= (th maximum valu for a for which th xpansion in (8) can b usd). Incrasing th valu of tiny within its abov mntiond rang will dcras th computational tim at th xpns of th computational accuracy and vic vrsa. Choosing a valu of tiny<tiny min will just incras th run tim without any improvmnt in th accuracy of computations which will thn b govrnd solly by th 0

11 machin charactristics. Valus of tiny<tiny min or tiny>0-4 rsult in tiny bing rst to tiny min or 0-4 rspctivly; a warning mssag is rturnd in both cass. It is to b notd that tiny<ε is usd only for th calculation of th corrsponding valu of th paramtr a and not for th truncation of th sums sinc th calculations cannot b claimd to b prformd for rlativ accuracy lss than th machin accuracy psilon, ε, in any cas. Accordingly, for th truncation of th sums, th maximum of tiny and ε is usd. 6. ALGORITHM VERIFICATION AND EFFICIENCY 6.. High Accuracy Computations Thr diffrnt indpndnt computational tchniqus ar usd to invstigat th accuracy of th prsnt algorithm - Mathmatica [Wolfram Rsarch, Inc. 008] provids th imaginary rror function rfi(z) as a spcial function which can b valuatd and thn usd in conjunction z with th rlation givn in (); that is w ( z) = ( + i rfi( z) ), to calculat th Faddyva function [Zaghloul 008]. Th arbitrary-prcision arithmtic usd in Mathmatica allows us to obtain highly accurat valus for th function rfi(z) although ths calculations ar vry xpnsiv computationally. This is only gnrally suitabl for applications whr th spd of arithmtic is not a rstrictiv factor, or whr prcis rsults for a small numbr of valuations ar rquird. W can, howvr, gnrat highly accurat valus of th function rfi(z) using Mathmatica by using larg numbrs of digits of prcision, - th simpl propr intgral givn in rfrnc [Zaghloul 007] can b usd to calculat th ral Voigt function (ral part of th Faddyva function), and 3- th Algorithm 680 [Popp t al. 990], which is widly usd in th litratur and is implmntd in many softwar packags and libraris, calculats th Faddyva function to a claimd accuracy of 4 significant digits. Th rlativly long computational tim associatd with th first two mthods maks thm infficint for us in applications rquiring a larg numbr of function valuations. For this rason, Algorithm 680 is rgardd as th comptitiv highly accurat algorithm du to its computational spd and claimd accuracy. Tabl prsnts sampl rsults calculatd using ths thr mthods with th rsults from th prsnt algorithm. Th valus of th complx variabl z usd in th computations in this tabl hav bn slctd to allow som conclusions to b drawn in addition to stablishing confidnc and rliability in th prsnt cod. Th valu of th rlativ rror in th calculation of th Faddyva function proposd hr is givn in Tabl 3 and compard with th othr approachs, taking th valus of w calculatd using th function rfi(z) from Mathmatica with high numbr of digits of accuracy as rfrnc valus. Looking closr into th valus givn in ths two tabls w conclud that; a) compard to calculating th Faddyva function using rfi(z) from Mathmatica, th prsnt algorithm is mor rliabl sinc w faild to calculat rfi(z) using Mathmatica for valus of x> and y> whil th

12 prsnt algorithm dos not suffr such a limitation. Not that for this domain Mathmatica cannot b usd as rfrnc, and thrfor, no comparison was prformd for this rang in Tabl 3. b) for th whol domain of computations, th prsnt algorithm shows vry high accuracy as shown in Tabl 3, whil th Algorithm 680 suffrs a catastrophic loss of accuracy in th vicinity of x=6.3 as wll as for small valus of y signifid by bold-fac numbrs in Tabls and 3. Th rlativ rror for th ral part of th Faddyva function from Algorithm 680 in this rgion of th first quadrant gos up to 00%. It has to b notd that x=6.3 is on of th built-in valus in Algorithm 680. To invstigat th ffctivnss of th prsnt algorithm compard to Algorithm 680, w calculatd th Faddyva function using both algorithms for,840,70 points of th complx variabl z distributd ovr th uppr half of th complx plan using th grid y=logspac(-0, 4, 7) and x=linspac(-00, 00, 4000) whr y=logspac(-0, 4, 7) gnrats a row vctor of 7 logarithmically qually spacd points btwn 0-0 and 0 4 and x=linspac(-00, 00, 4000) gnrats a row vctor of 4000 linarly qually spacd points btwn -00 and 00. Using Matlab (R009b), th computational tim takn by th prsnt algorithm was found to b <8.0% of that takn by Algorithm 680, which rprsnts a significant tim saving. Figurs (-a) and (-b) show surfac plots of th absolut rlativ rror δ V = V Vrf / Vrf and δ L = L Lrf / Lrf in th rsults obtaind from th prsnt algorithm using rsults from Algorithm 680, which is availabl in Matlab, as rfrnc valus. As can b clarly sn from ths figurs, th rsults from th prsnt algorithm show high agrmnt (around 3 significant digits) ovr th chosn computational domain xcpt for th rgion in th vicinity of x=6.3 and small valus of y whr Algorithm 680 badly loss its accuracy as indicatd abov. Th abov findings provid th ncssary vrification and confirm th high accuracy as wll as th rliability of th prsnt algorithm.

13 Figur -a Absolut rlativ rror δ V = V Vrf / Vrf in th calculations of th ral part of th Faddyva function from th prsnt algorithm using th rsults from Algorithm 680 as rfrnc valus. Figur -b Absolut rlativ rror δ = L L / L in th calculations of th imaginary part of th L rf Faddyva function from th prsnt algorithm using th rsults from Algorithm 680 as rfrnc valus. rf 3

14 6.. Efficint Computations with Lowr Accuracis Tabl 4 blow shows valus of th fr variabl tiny usd in th calling argumnt of our Matlab function and th corrsponding rlativ accuracy δ = ( V V )/ V and δ L = ( L Lrf )/ Lrf in th calculations, using valus calculatd with th highst accuracy obtainabl from th prsnt algorithm as rfrnc valus. Th nd to quantify th fficincy improvmnts obtainabl whn using th accuracy vs fficincy trad-off capability of th prsnt algorithm is th rason of using th highst accuracy computations from th prsnt algorithm as rfrnc valus. Th run tims rquird to calculat th function for,840,70 points gnratd using th grid y=logspac(-0, 4, 7) and x=linspac(-00, 00, 4000) rlativ to th run tim rquird to prform th sam computations using th highst accuracy computations from th prsnt algorithm ar also includd in th tabl. As can b sn from th tabl, running th prsnt algorithm at lowr accuracy improvs th fficincy of th computations and dcrass th computational tim by up to 45%. Compard to othr fficint and low-accuracy algorithms in th litratur [Hui t al. 978; Humlíčk 98; Popp t al. 990; Lthr t al. 99; Shippony t al. 993; Widman 994], th prsnt algorithm sms to b mor rliabl vn at low-accuracy. In addition, othr algorithms fail in som rgions of th computational domain, particularly nar th ral axis (vry small valus of y); for xampl, th Popp and Wijrs algorithm [Popp t al. 990], known for its accuracy, fails in this rgion, rturning rsults for th ral part of th Faddyva function that ar svral ordrs of magnitud away from th corrct valus. Figur shows a comparison btwn th calculations of th partial drivativ V ( x, y) x using th prsnt algorithm (run at th lowst accuracy) and calculations from Algorithm 680, for y=0-0, in th rgion x=[6.-6.5]. Hr w s that th rsults from th prsnt algorithm (vn whn run at th lowst accuracy) sm to b mor accurat and mor rliabl than computations from Popp and Wijrs algorithm in this rgion, whr th lattr loss its accuracy and fails to produc th corrct bhavior of V ( x, y) x. V rf rf 4

15 Figur V(x,y)/ x as calculatd, from th prsnt algorithm (tiny=0-4 ) and from Popp and Wijrs algorithm, using (), for y=0-0. Figur 3 shows a surfac plot of th rlativ rror δ V = ( V Vrf )/ Vrf for th rsults obtaind from Hui s algorithm [Hui t al. 978] using rsults from th prsnt algorithm as rfrnc valus. W not that th Matlab vrsion of Hui s algorithm (crf.m [Hui t al. 978]) mployd in this comparison, uss th p=5 rational approximation whr p is th dgr of nominator polynomial. As is clar from th plot, th rlativ rrors in th rsults of Hui s algorithm ar vry larg for small valus of y and rach 4 ordr of magnitud for mdium valus of x whn y=0-0. In addition to th larg rrors for mdium valus of x and small valus of y, Hui s algorithm producs ngativ valus for th Voigt function (ral part of th Faddyva function) for xampl, for y=0-5 and x=4. Th Voigt function is positiv ovr th whol first quadrant. 5

16 Figur 3 Absolut rlativ rror, δ V = ( V Vrf )/ V, in th calculations of th ral part of th rf Faddyva function from Hui s algorithm using rsults from th prsnt algorithm (tiny=tiny min ) as rfrnc valus. Figur 4 shows a comparison btwn th calculations of th partial drivativ V ( x, y) x using th prsnt algorithm (run at th lowst accuracy) and calculations from Hui s algorithm, using (), for y=0-0, in th rgion x=[7,5]. As w s from th figur, calculations from th prsnt algorithm sm to b mor accurat and mor rliabl than Hui s algorithm which fails to produc th corrct bhavior (ngativ V ( x, y) x ) or th corrct ordr of magnitud of V ( x, y) x in this rgion of th computational domain. W not that th rror contours givn in [Hui t al. 978] wr prsntd ithr for th modulus of th complx rror function or for th absolut valu of th Voigt function V(x,y). That was, probably, th rason why such failurs wr not clar from thir papr. Although Hui s algorithm taks about 0% of th computational tim takn by th prsnt algorithm (for tiny=0-4 ) and about 5% of th computational tim takn by th prsnt algorithm for th highst-accuracy computations, th fact that it fails to produc th corrct valus or corrct signs of th function or vn its corrct bhavior, in this rgion of th computational domain, poss important qustions about its rliability. 6

17 Figur 4 V(x,y)/ x as calculatd, from th prsnt algorithm (tiny=0-4 ) and from Hui s algorithm, using (), for y=0-0. Humlíčk [Humlíčk 98] rportd that any rational approximation suffrs invitabl failur nar th ral axis and h attmptd to ovrcom this failur in his algorithm, w4. Howvr, invstigating th rsults of Humlíčk s algorithm w found that it also suffrs complt failur nar th ral axis in th vicinity of x=5.5. Th rsults for y=0-0 show that Humlíčk s algorithm undrstimats th ral part of th Faddyva function by 8 ordrs of magnitud and by 3 ordrs of magnitud for y=0-5. Figur 5 shows a surfac plot of th rlativ rror in th calculation of V(x,y) using Humlíčk s algorithm taking th rsults from th prsnt algorithm as rfrnc. Tabl 5 shows that th computational tim using th Humlíčk s original cod is almost thr tims that usd by th prsnt algorithm (for th highst-accuracy computations). Evn using a mor fficint vrsion of Humlíčk s cod, modifid by th prsnt authors, w not that th computational tim takn by Humlíčk s algorithm is still longr than that takn by th prsnt algorithm (for th highst-accuracy computations). Figur 6, on th othr hand, shows a comparison btwn th calculations of th partial drivativ V ( x, y) x using () from th prsnt algorithm (run at th lowst accuracy) and thos calculations from Humlíčk s algorithm, for y=0-0, in th rgion x=[5.4,6.4]. This shows that Humlíčk s algorithm dos not produc th corrct bhavior of V ( x, y) x in this domain. 7

18 Figur 5 Absolut rlativ rror, δ V = ( V Vrf )/ Vrf, in th calculations of th ral part of th Faddyva function from Humlíčk s algorithm using rsults from th prsnt algorithm (tiny=tiny min ) as rfrnc valus. Figur 6 V(x,y)/ x as calculatd, from th prsnt algorithm (tiny=0-4 ) and from Humlíčk s algorithm, using (), for y=0-0. 8

19 As with Hui s algorithm, Widman s algorithm [Widman 994] also producs ngativ valus for th ral part of th Faddyva function nar th ral axis. Th ngativ valus for th Voigt function calculatd from Widman s algorithm appar for all valus of th paramtr N (numbr of trms in th rational sris) for y=0-0. A surfac plot of th rlativ rror in th calculations of th ral part of th Faddyva function using Widman s algorithm with N=56 taking th rsults of th prsnt algorithm as a rfrnc is shown in Figur 7. Th figur shows that th rrors rsulting from Widman s algorithm ar catastrophic for small valus of y and that th computd magnitud of V(x,y) is ovrstimatd by up to 6 ordrs of magnitud for y=0-0. Th situation bcoms vn wors for smallr valus of N. Tabl 5 shows that th run tim of th prsnt algorithm at highst accuracy is shortr than th run tim of Widman s algorithm with N=56. Figur 7 Absolut rlativ rror, δ V = ( V Vrf )/ Vrf, in th calculations of th ral part of th Faddyva function from Widman s algorithm, N=56, using rsults from th prsnt algorithm (tiny=tiny min ) as rfrnc valus. Figurs (8-a) and (8-b) show comparisons btwn th calculations of th partial drivativ V ( x, y) x using th prsnt algorithm (run at th lowst accuracy) and Widman s algorithm with N=8 and N=3, rspctivly. Th calculations shown in th figur ar for th rgion x=[6,5] and y=0-0 in Figur (8-a) and y=0-0 in Figur (8- b). From ths figurs, w s that Widman s algorithm dos not rproduc th corrct bhavior for V ( x, y) x in th rgions shown. 9

20 Figur 8-a V(x,y)/ x as calculatd, from th prsnt algorithm (tiny=0-4 ) and from Widman s algorithm with N=8, using (), for y=0-0. Figur 8-b V(x,y)/ x as calculatd, from th prsnt algorithm (tiny=0-4 ) and from Widman s algorithm with N=3, using (), for y=0-0. 0

21 Othr comptitiv algorithms in th litratur also show loss of accuracy in som rgions of th computational domain. For xampl, th algorithm by Shippony and Rad [Shippony t al. 993] xhibits th sam failur in calculating th ral part of th Faddyva function nar th ral axis. In particular w dtctd th sam loss of accuracy suffrd by Popp and Wijrs algorithm for vry small valus of y nar x=6.3. In addition, th Shippony and Rad algorithm producs ngativ valus for V(x,y) and/or L(x,y) in svral rgions of th computational domain. Just for xampl w rfr to th points at x=.5 and y=.5,.0,.5, 3.0, 3.5, 4.0, 5.0 tc. It has to b notd that ths failurs hav bn obtaind vn with th us of th corrction providd by Shippony and Rad in [Shippony t al. 003]. Th situation is no bttr with th Ltchworth and Bnnr s algorithm [Ltchworth t al. 007] whr similar failurs and loss of accuracy ar obtaind for vry small valus of y and valus of x gratr than but clos to x=5.76. For x=5.76 and y=0-0 Ltchwoth and Bnnr s algorithm rturns, for th ral part of Faddyva function th valu V= whil th valu rturnd from th prsnt algorithm is V= and that rturnd from using th function rfi(z) from Mathmatica TM is Th algorithm in [Zaghloul 007] rturns Figur 9 shows a similar comparison btwn th calculations of th partial drivativ V ( x, y) x using th prsnt algorithm (with tiny=0-4 ) and calculations from Ltchworth and Bnnr algorithm, in (), for y=0-0. Th failur of Ltchworth and Bnnr s algorithm for valus of x gratr than but clos to x=5.76 can b asily rcognizd from th figur. It has to b mphasizd hr that ths "comptitiv cods" wr (probably) not dsignd for such xtrm valus of y and th tsts prsntd hr ar mor a dmonstration of th "suprior accuracy" of our softwar vn for xtrm valus. Figur 9 V(x,y)/ x as calculatd, from th prsnt algorithm (tiny=0-4 ) and from Ltchworth & Bnnr s algorithm, using (), for y=0-0.

22 7. CONCLUSIONS An algorithm accompanid by a computr cod, in th form of a MATLAB TM function, for th numrical valuation of th Faddyva function w(z) is prsntd. Th algorithm is mor accurat and avoids failurs discovrd in othr comptitiv publishd algorithms. In addition to its suprior accuracy, th prsnt algorithm and computr cod allow a flxibl accuracy vs fficincy trad-off through controlling a fr paramtr tiny. By adjusting th valu of this paramtr th function can b run with a lowr accuracy and shortr computational tim or high accuracy and longr computational tim. Evn whn run at its lowst accuracy, th prsnt algorithm avoids major problms suffrd by othr comptitiv cods. For all lvls of accuracy th prsnt cod is safr and mor rliabl sinc it dos not rturn ngativ valus for ral and/or imaginary parts of th Faddyva function nor dos it suffr from th loss of accuracy xhibitd by som of th othr comptitiv cods. Th prsnt algorithm can, thrfor, b safly usd and implmntd in prsonal and commrcial libraris.

23 Tabl : Rsults from algorithms in th litratur in comparison with rsults from th prsnt algorithm for som slctd valus of z z Mathmatica TM Algorithm 680 [Zaghloul 007] Prsnt algorithm x y V L V L V V L Fails to valuat rfi $ Fails to valuat rfi Fails to valuat rfi Fails to valuat rfi Fails to valuat rfi * This is th corrct valu as calculatd using (4) in [Zaghloul 007]. Th valu givn in Tabl 4 in th sam rfrnc is calculatd using th asymptotic xprssion for y 0 $ Calculatd using th asymptotic xprssion for y 0 3

24 Tabl 3: Valus of th rlativ rrorsδ V = ( V Vrf ) / Vrf & δ L = ( L Lrf )/ Lrf in calculating th Faddyva function by diffrnt cods using valus of th function calculatd using rfi(z) from Mathmatica as rfrnc valus. z Algorithm 680 Zaghloul [007] Prsnt algorithm x y V L V V L Maximum no. of sris trms

25 Tabl 4: Accuracy vs fficincy trad-off of th prsnt algorithm (computations ar prformd on an array of,840,70 points gnratd using th grid y=logspac(-0, 4, 7) and x=linspac(- 00, 00, 4000) using Matlab (R009b) ). tiny δ V, max = Vtiny Vtiny min / Vtiny δ L, max = Ltiny L L Run tim (s) ε min tiny min / tiny min Tabl 5: Running tims of th prsnt algorithm (for thr valus of th paramtr tiny) compard with othr comptitiv algorithms (computations ar prformd on an array of,840,70 points gnratd using th grid y=logspac(-0, 4, 7) and x=linspac(-00, 00, 4000) using Matlab (R009b)) * Algorithm Run tim (s) Commnts Faddyva, tiny= ε Faddyva, tiny=-8 Faddyva, tiny= Popp & Wijrs [990] Larg rror in th vicinity of x=6.3 & vry small valus of y Humlíčk [98] (original) 3. Larg rror and loss of accuracy in th vicinity Humlíčk [98] (modifid) Widmann [994], N=6 Widmann [994], N=3 Widmann [994], N=64 Widmann [994], N=8 Widmann [994], N= of x=5.6 and vry small valus of y Ngativ valus for V(x,y) nar x-axis Incorrct bhavior and ordr of magnitud of V(x,y)/ x for vry small valus of y. Hui t al [978] 0.4 Larg rror for small valus of y Ngativ valus for V(x,y) (.g. at y=0-5 & x=4) Incorrct bhavior and ordr of magnitud of V(x,y)/ x for vry small valus of y. * Timing rsults dpnd on both hardwar and th vrsion of th softwar usd and can chang significantly. 5

26 ACKNOWLEDGMENTS Th authors would lik to acknowldg valuabl commnts and suggstions rcivd from th rviwrs. In particular th commnts and suggstions rcivd from th associat ditor, algorithm ditor and th fourth anonymous rfr wr xtrmly hlpful and insightful. W would lik also to thank Prof. C. Bnnr from Collg of William and Mary, Williamsburg, VA, USA, for snding us a copy of Ltchworth & Bnnr s computr cod. REFERENCES ABRAMOWITZ M. AND STEGUN, I. A. 97. Handbook of mathmatical functions, Dovr Publications, Inc., Nw York. ABRAROV S.M., QUINE B.M. AND JAGPAL R.K. 00. Rapidly convrgnt sris for highaccuracy calculation of th Voigt function. J. Quant. Spctrosc. & Radiat. Transfr, Vol., ABRAROV S.M., QUINE B.M. AND JAGPAL R.K. 00. High-accuracy approximation of th complx probability functions by Fourir xpansion of xponntial multiplir. Computr Physics Communications, Vol. 8, ARMSTRONG, B.H Spctrum Lin Profils: Th Voigt Function. J. Quant. Spctrosc. & Radiat. Transfr, Vol. 7, 6-88 BOAS M.L Mathmatical Mthods in th Physical Scincs, Third Edition. John Wily & Sons, Inc. NJ, USA. DOMINGUEZ, H.J., LLAMAS, H.F. PRIETO, A.C. AND ORTEGA, A.B A Simpl Rlationship btwn th Voigt Intgral and th Plasma Disprsion Function. Additional Mthods to Estimat th Voigt Intgral. Nuclar Instrumnts and Mthods in Physics Rsarch A. Vol. 78, GAUTSCHI, W Algorithm 363-Complx rror function, Commun. ACM, 635. GAUTSCHI, W Efficint Computation of th Complx Error Function. SIAM J. Numr. Anal., Vol. 7, HUI, A.K., ARMSTRONG, B.H. AND WRAY, A.A Rapid Computation of th Voigt and Complx Error Functions. J. Quant. Spctrosc. & Radiat. Transfr, Vol. 9, HUMLÍČEK J. 98. Optimizd Computation of th Voigt and Complx Probability Functions. J. Quant. Spctrosc. & Radiat. Transfr, Vol. 7, No. 4, LETCHWORTH K.L. AND BENNER D.C 007. Rapid and accurat calculation of th Voigt function. J. Quant. Spctrosc. & Radiat. Transfr, Vol. 07, LETHER F.G. AND WENSTON P.R. 99. Th numrical computation of th Voigt function by a corrctd midpoint quadratur rul for (-, ). Journal of Computational & Applid Mathmatics Vol. 34, 75-9 LUQUE, J. M. CALZADA, M. D. AND SAEZ, M A nw procdur for obtaining th Voigt function dpndnt upon th complx rror function. J. Quant. Spctrosc. & Radiat. Transfr, Vol. 94, 5-6. POPPE, G.P.M. AND C. WIJERS, M. J Mor Efficint Computation of th Complx Error Function. ACM Transactions on Mathmatical Softwar, Vol. 6, No., POPPE G.P.M. AND WIJERS, C.M.J Algorithm 680, Evaluation of th Complx Error Function. ACM Transactions on Mathmatical Softwar, Vol. 6, No., 47. 6

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