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Curve Fitting Functions
Contents
1. ORIGIN BASIC FUNCTIONS ..........................................................................................................................

2 2. CHROMATOGRAPHY FUNCTIONS ............................................................................................................... 23 3. EXPONENTIAL FUNCTIONS ........................................................................................................................ 30 4. GROWTH/SIGMOIDAL ................................................................................................................................ 69 5. HYPERBOLA FUNCTIONS ........................................................................................................................... 81 6. LOGARITHM FUNCTIONS ........................................................................................................................... 87 7. PEAK FUNCTIONS ...................................................................................................................................... 93 8. PHARMACOLOGY FUNCTIONS.................................................................................................................. 113 9. POWER FUNCTIONS ................................................................................................................................. 120 10. RATIONAL FUNCTIONS .......................................................................................................................... 140 11. SPECTROSCOPY FUNCTIONS .................................................................................................................. 155 12. WAVEFORM FUNCTIONS........................................................................................................................ 163

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1. Origin Basic Functions
Allometric1 Beta Boltzmann Dhyperbl ExpAssoc ExpDecay1 ExpDecay2 ExpDecay3 ExpGrow1 ExpGrow2 Gauss GaussAmp Hyperbl Logistic LogNormal Lorentz Pulse Rational0 Sine Voigt 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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Allometric1
Function

y = ax b
Brief Description Classical Freundlich model. Has been used in the study of allometry. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = power Initial Values: a = 1.0 (vary), b = 0.5 (vary) Lower Bounds: none Upper Bounds: none Script Access allometric1(x,a,b) Function File FITFUNC\ALLOMET1.FDF

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Beta
Function

? ? w2 + w3 ? 2 ?? x ? xc y = y 0 + A?1 + ? ? w ?1 ? ?? ? 2 ?? w1 ? ?
Brief Description The beta function. Sample Curve

?? ? ?? ??

w2 ?1

? ? w2 + w3 ? 2 ?? x ? x c ?1 ? ? ? ? w ?1 ? ?? 3 ?? w1 ? ?

?? ? ?? ??

w3 ?1

Parameters Number: 6 Names: y0, xc, A, w1, w2, w3 Meanings: y0 = offset, xc = center, A = amplitude, w1 = width, w2 = width, w3 = width Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), A = 5.0 (vary), w1 = 5.0 (vary), w2 = 2.0 (vary), w3 = 2.0 (vary) Lower Bounds: w1 > 0.0, w2 > 1.0, w3 > 1.0 Upper Bounds: none Script Access beta(x,y0,xc,A,w1,w2,w3) Function File FITFUNC\BETA.FDF

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Boltzmann
Function

y=

A1 ? A2 + A2 1 + e ( x ? x0 )/ dx

Brief Description Boltzmann function - produces a sigmoidal curve. Sample Curve

Parameters Number: 4 Names: A1, A2, x0, dx Meanings: A1 = initial value, A2 = final value, x0 = center, dx = time constant Initial Values: A1 = 0.0 (vary), A2 = 1.0 (vary), x0 = 0.0 (vary), dx = 1.0 (vary) Lower Bounds: none Upper Bounds: none Constraints dx ! = 0 Script Access boltzman(x,A1,A2,x0,dx) Function File FITFUNC\BOLTZMAN.FDF

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Dhyperbl
Function

y=

Px P 1x + 3 + P5 x P2 + x P4 + x

Brief Description Double rectangular hyperbola function. Sample Curve

Parameters Number: 5 Names: P1, P2, P3, P4, P5 Meanings: Unknowns 1-5 Initial Values: P1 = 1.0 (vary), P2 = 1.0 (vary), P3 = 1.0 (vary), P4 = 1.0 (vary), P5 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access dhyperbl(x,P1,P2,P3,P4,P5) Function File FITFUNC\DHYPERBL.FDF

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ExpAssoc
Function

y = y0 + A1 1 ? e ? x / t1 + A2 1 ? e ? x / t2
Brief Description Exponential associate. Sample Curve

(

)

(

)

Parameters Number: 5 Names: y0, A1, t1, A2, t2 Meanings: y0 = offset, A1 = amplitude, t1 = width, A2 = amplitude, t2 = width Initial Values: y0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary) Lower Bounds: t1 > 0, t2 > 0 Upper Bounds: none Script Access expassoc(x,y0,A1,t1,A2,t2) Function File FITFUNC\EXPASSOC.FDF

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ExpDecay1
Function

y = y0 + A1e ? (x ? x0 )/ t1
Brief Description Exponential decay 1 with offset. Sample Curve

Parameters Number: 4 Names: y0, x0, A1, t1 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = decay constant Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 10 (vary), t1 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdecay1(x,y0,x0,A1,t1) Function File FITFUNC\EXPDECY1.FDF

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ExpDecay2
Function

y = y0 + A1e ? ( x? x0 )/ t1 + A2 e ? (x ? x0 )/ t2
Brief Description Exponential decay 2 with offset. Sample Curve

Parameters Number: 6 Names: y0, x0, A1, t1, A2, t2 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = decay constant, A2 = amplitude, t2 = decay constant Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 10 (vary), t1 = 1.0 (vary), A2 = 10 (vary), t2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdecay2(x,y0,x0,A1,t1,A2,t2) Function File FITFUNC\EXPDECY2.FDF

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ExpDecay3
Function

y = y0 + A1e ? ( x? x0 )/ t1 + A2 e ? (x ? x0 )/ t2 + A3e ? (x ? x0 )/ t3
Brief Description Exponential decay 3 with offset. Sample Curve

Parameters Number: 8 Names: y0, x0, A1, t1, A2, t2, A3, t3 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = decay constant, A2 = amplitude, t2 = decay constant, A3 = amplitude, t3 = decay constant Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 10 (vary), t1 = 1.0 (vary), A2 = 10 (vary), t2 = 1.0 (vary), A3 = 10 (vary), t3 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdecay3(x,y0,x0,A1,t1,A2,t2,A3,t3) Function File FITFUNC\EXPDECY3.FDF

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ExpGrow1
Function

y = y 0 + A1e ( x ? x0 ) / t1
Brief Description Exponential growth 1 with offset. Sample Curve

Parameters Number: 4 Names: y0, x0, A1, t1 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = width Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary) Lower Bounds: t1 > 0.0 Upper Bounds: none Script Access expgrow1(x,y0,x0,A1,t1) Function File FITFUNC\EXPGROW1.FDF

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ExpGrow2
Function

y = y0 + A1e ( x? x0 )/ t1 + A2 e (x ? x0 )/ t2
Brief Description Exponential growth 2 with offset. Sample Curve

Parameters Number: 6 Names: y0, x0, A1, t1, A2, t2 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = width, A2 = amplitude, t2 = width Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary) Lower Bounds: t1 > 0.0, t2 > 0.0 Upper Bounds: none Script Access expgrow2(x,y0,x0,A1,t1,A2,t2) Function File FITFUNC\EXPGROW2.FDF

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Gauss
Function
?2 A y = y0 + e w π /2

( x ? xc )2
w2

Brief Description Area version of Gaussian function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 10 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gauss(x,y0,xc,w,A) Function File FITFUNC\GAUSS.FDF

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GaussAmp
Function
?

y = y0 + Ae

( x ? xc )2
2 w2

Brief Description Amplitude version of Gaussian peak function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 10 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gaussamp(x,y0,xc,w,A) Function File FITFUNC\GAUSSAMP.FDF

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Hyperbl
Function

y=

P 1x P2 + x

Brief Description Hyperbola function. Also the Michaelis-Menten model in enzyme kinetics. Sample Curve

Parameters Number: 2 Names: P1, P2 Meanings: P1 = amplitude, P2 = unknown Initial Values: P1 = 1.0 (vary), P2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access hyperbl(x,P1,P2) Function File FITFUNC\HYPERBL.FDF

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Logistic
Function

y=

A1 ? A2 + A2 p 1 + (x / x0 )

Brief Description Logistic dose response in pharmacology/chemistry. Sample Curve

Parameters Number: 4 Names: A1, A2, x0, p Meanings: A1 = initial value, A2 = final value, x0 = center, p = power Initial Values: A1 = 0.0 (vary), A2 = 1.0 (vary), x0 = 1.0 (vary), p = 1.5 (vary) Lower Bounds: p > 0.0 Upper Bounds: none Script Access logistic(x,A1,A2,x0,p) Function File FITFUNC\LOGISTIC.FDF

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LogNormal
Function

y = y0 +

A 2π wx

?[ln x / xc ]2

e

2 w2

Brief Description Log-Normal function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = amplitude Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: xc > 0, w > 0 Upper Bounds: none Script Access lognormal(x,y0,xc,w,A) Function File FITFUNC\LOGNORM.FDF

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Lorentz
Function

y = y0 +

2A w π 4(x ? xc )2 + w 2

Brief Description Lorentzian peak function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary),w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access lorentz(x,y0,xc,w,A) Function File FITFUNC\LORENTZ.FDF

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Pulse
Function
x ? x0 x0 ? ? ? ? x? t1 ? t2 ? y = y0 + A 1 ? e e ? ? ? ? p

Brief Description Pulse function. Sample Curve

Parameters Number: 6 Names: y0, x0, A, t1, P, t2 Meanings: y0 = offset, x0 = center, A = amplitude, t1 = width, P = power, t2 = width Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A = 1.0 (vary), t1 = 1.0 (vary), P = 1.0 (vary), t2 = 1.0 (vary) Lower Bounds: A > 0.0, t1 > 0.0, P > 0.0, t2 > 0.0 Upper Bounds: none Script Access pulse(x,y0,x0,A,t1,P,t2) Function File FITFUNC/PULSE.FDF

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Rational0
Function

y=

b + cx 1 + ax

Brief Description Rational function, type 0. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.24 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access rational0(x,a,b,c) Function File FITFUNC\RATION0.FDF

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Sine
Function

? x ? xc ? y = A sin ? π ? w ? ?
Brief Description Sine function. Sample Curve

Parameters Number: 3 Names: xc, w, A Meanings: xc = center, w = width, A = amplitude Initial Values: xc = 0.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access sine(x,xc,w,A) Function File FITFUNC\SINE.FDF

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Voigt
Function

y = y0 + A ?

2 ln 2 wL ∞ e ?t ? dt 2 2 2 ∫ ?∞ π 3 / 2 wG ? ? wL ? ? x ? xc ? ? ln 2 w ? ? +? ? 4 ln 2 w ? t ? ? G ? G ? ? ?

2

Brief Description Voigt peak function. Sample Curve

Parameters Number: 5 Names: y0, xc, A, wG, wL Meanings: y0 = offset, xc = center, A = amplitude, wG = Gaussian width, wL = Lorentzian width Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), wG = 1.0 (vary), wL = 1.0 (vary) Lower Bounds: wG > 0.0, wL > 0.0 Upper Bounds: none Script Access voigt5(x,y0,xc,A,wG,wL) Function File FITFUNC\VOIGT5.FDF

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2. Chromatography Functions
CCE ECS Gauss GaussMod GCAS Giddings 24 25 26 27 28 29

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CCE
Function

? ? ( x ? xc 1 ) ? ?0.5 k ( x ? x + ( x ? xc 3 )) y = y0 + A?e 2 w + B(1 ? 0.5(1 ? tanh (k 2 (x ? xc ))))e 3 c 3 ? ? ? ? ?
2

Brief Description Chesler-Cram peak function for use in chromatography. Sample Curve

Parameters Number: 9 Names: y0, xc1, A, w, k2, xc2, B, k3, xc3 Meanings: y0 = offset, xc1 = unknown, A = unknown, w = unknown, k2 = unknown, xc2 = unknown, B = unknown, k3 = unknown, xc3 = unknown Initial Values: y0 = 0.0 (vary), xc1 = 1.0 (vary), A = 1.0 (vary), w = 1.0 (vary), k2 = 1.0 (vary), xc2 = 1.0 (vary), B = 1.0 (vary), k3 = 1.0 (vary), xc3 = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access cce(x,y0,xc1,A,w,k2,xc2,B,k3,xc3) Function File FITFUNC\CHESLECR.FDF

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ECS
Function

? a 4 ? a3 ?? 2 3 ?1 + z z ? 3 + 4 z ? 6 z + 3 ? ? ? A ? ?0.5 z 2 ? 3! 4! ?? y = y0 + ?e 2 ? 10a3 6 ?? w 2π ? 4 2 z ? 15 z + 45 z ? 15 ?+ ?? ? 6 ! ? ?? ? ?

( (

)

(

)

)

where

z=

x ? xc w

Brief Description Edgeworth-Cramer peak function for use in chromatography. Sample Curve

Parameters Number: 6 Names: y0, xc, A, w, a3, a4 Meanings: y0 = offset, xc = center, A = amplitude, w = width, a3 = unknown, a4 = unknown Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), a3 = 1.0 (vary), a4 = 1.0 (vary) Lower Bounds: A > 0.0, w > 0.0 Upper Bounds: none Script Access ecs(x,y0,xc,A,w,a3,a4) Function File FITFUNC\EDGWTHCR.FDF

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Gauss
Function
?2 A y = y0 + e w π /2

( x ? xc )2
w2

Brief Description Area version of Gaussian function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 10 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gauss(x,y0,xc,w,A) Function File FITFUNC\GAUSS.FDF

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GaussMod
Function
?t ? ? A 2? f ( x) = y0 + e ? 0 ? t0 1? w ?
2

?

x ? xc t0



z

?∞

1 ?2 e dy 2π

y2

where

z=

x ? xc w ? w t0

Brief Description Exponentially modified Gaussian peak function for use in chromatography. Sample Curve

Parameters Number: 5 Names: y0, A, xc, w, t0 Meanings: y0 = offset, A = amplitude, xc = center, w = width, t0 = unknown Initial Values: y0 = 0.0 (vary), A = 1.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), t0 = 0.05 (vary) Lower Bounds: w > 0.0, t0 > 0.0 Upper Bounds: none Script Access gaussmod(x,y0,A,xc,w,t0) Function File FITFUNC\GAUSSMOD.FDF

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GCAS
Function
4 2 a A ? ? e ? z / 2 ?1 + ∑ i H i (z )? w 2π i =3 i! ? ?

f ( z ) = y0 + z=

x ? xc w H 3 = z 3 ? 3z H 4 = z 4 ? 6z 3 + 3
Brief Description Gram-Charlier peak function for use in chromatography. Sample Curve

Parameters Number: 6 Names: y0, xc, A, w, a3, a4 Meanings: y0 = offset, xc = center, A = amplitude, w = width, a3 = unknown, a4 = unknown Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), a3 = 0.01 (vary), a4 = 0.001 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gcas(x,y0,xc,A,w,a3,a4) Function File FITFUNC\GRMCHARL.FDF

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Giddings
Function

y = y0 +

A w

? x ? xc 2 xc x ? xc ? ?e w I1 ? ? x ? ? w ?

Brief Description Giddings peak function for use in chromatography. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access giddings(x,y0,xc,w,A) Function File FITFUNC\GIDDINGS.FDF

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3. Exponential Functions
Asymtotic1 BoxLucas1 BoxLucas1Mod BoxLucas2 Chapman Exp1P1 Exp1P2 Exp1P2md Exp1P3 Exp1P3Md Exp1P4 Exp1P4Md Exp2P Exp2PMod1 Exp2PMod2 Exp3P1 Exp3P1Md Exp3P2 ExpAssoc ExpDec1 ExpDec2 ExpDec3 ExpDecay1 ExpDecay2 ExpDecay3 ExpGro1 ExpGro2 ExpGro3 ExpGrow1 ExpGrow2 ExpLinear Exponential MnMolecular MnMolecular1 Shah Stirling YldFert YldFert1 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

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Asymptotic1
Function

y = a ? bc x
Brief Description Asymptotic regression model - 1st parameterization. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.1 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = asymptote, b = response range, c = rate Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access Asymptotic1(x,a,b,c) Function File FITFUNC\ASYMPT1.FDF

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BoxLucas1
Function

y = a 1 ? e ? bx

(

)

Brief Description A parameterization of Box Lucas model. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access boxlucas1(x,a,b) Function File FITFUNC\BOXLUC1.FDF

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BoxLucas1Mod
Function

y = a 1? bx

(

)

Brief Description A parameterization of Box Lucas model. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.5 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access boxlucas1mod(x,a,b) Function File FITFUNC\BOXLC1MD.FDF

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BoxLucas2
Function

y=

a1 e ? a2 x ? e ? a1x a1 ? a2

(

)

Brief Description A parameterization of Box Lucas model. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. p. 254 Sample Curve

Parameters Number: 2 Names: a1, a2 Meanings: a1 = unknown, a2 = unknown Initial Values: a1 = 2.0 (vary), a2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access boxlucas2(x,a1,a2) Function File FITFUNC\BOXLUC2.FDF

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Chapman
Function

y = a 1 ? e ? bx

(

)

c

Brief Description Chapman model. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.35 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access chapman(x,a,b,c) Function File FITFUNC\CHAPMAN.FDF

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Exp1P1
Function

y = e x? A
Brief Description One-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.5 Sample Curve

position:A=1 (A,1)

y(1)=1

y=0
Parameters Number: 1 Names: A Meanings: A = position Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access exp1p1(x,A) Function File FITFUNC\EXP1P1.FDF

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Exp1p2
Function

y = e ? Ax
Brief Description One-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.15 Sample Curve

Parameters Number: 1 Names: A Meanings: A = coefficient Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access exp1p2(x,A) Function File FITFUNC\EXP1P2.FDF

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Exp1p2md
Function

y = Bx
Brief Description One-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.16 Sample Curve

Parameters Number: 1 Names: B Meanings: B = position Initial Values: B = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access exp1p2md(x,B) Function File FITFUNC\EXP1P2MD.FDF

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Exp1p3
Function

y = Ae ? Ax
Brief Description One-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.13 Sample Curve

Parameters Number: 1 Names: A Meanings: A = coefficient Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access exp1p3(x,A) Function File FITFUNC\EXP1P3.FDF

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Exp1P3Md
Function

y = ? ln (B )B x
Brief Description One-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.14 Sample Curve

Parameters Number: 1 Names: B Meanings: B = coefficient Initial Values: B = 5.0 (vary) Lower Bounds: none Upper Bounds: none Script Access exp1p3md(x,B) Function File FITFUNC\EXP1P3MD.DFD

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Exp1P4
Function

y = 1 ? e ? Ax
Brief Description One-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.18 Sample Curve

Parameters Number: 1 Names: A Meanings: A = coefficient Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access exp1p4(x,A) Function File FITFUNC\EXP1P4.FDF

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Exp1P4Md
Function

y = 1? Bx
Brief Description One-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.19 Sample Curve

Parameters Number: 1 Names: B Meanings: B = coefficient Initial Values: B = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access exp1p4md(x,B) Function File FITFUNC\EXP1P4.FDF

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Exp2P
Function

y = ab x
Brief Description Two-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.2.9 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = position, b = position Initial Values: a = 1.0 (vary), b = 1.5 (vary) Lower Bounds: none Upper Bounds: none Script Access exp2p(x,a,b) Function File FITFUNC\EXP2P.FDF

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Exp2PMod1
Function

y = ae bx
Brief Description Two-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.2.10 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = rate Initial Values: a = 1.0 (vary), b = 1.5 (vary) Lower Bounds: none Upper Bounds: none Script Access exp2pmod1(x,a,b) Function File FITFUNC\EXP2PMD1.FDF

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Exp2PMod2
Function

y = e a+bx
Brief Description Two-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.2.11 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = rate Initial Values: a = 1.0 (vary), b =1.5 (vary) Lower Bounds: none Upper Bounds: none Script Access exp2pmod2(x,a,b) Function File FITFUNC\EXP2PMD2.FDF

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Page 45 of 166

Exp3P1
Function
b x+c

y = ae

Brief Description Three-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.33 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access exp3p1(x,a,b,c) Function File FITFUNC\EXP3P1.FDF

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Exp3P1Md
Function
a+ b x+c

y=e

Brief Description Three-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.34 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access exp3p1md(x,a,b,c) Function File FITFUNC\EXP3P1MD.FDF

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Exp3P2
Function

y = e a +bx +cx

2

Brief Description Three-parameter exponential function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.39 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access exp3p2(x,a,b,c) Function File FITFUNC\EXP3P2.FDF

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ExpAssoc
Function

y = y0 + A1 1 ? e ? x / t1 + A2 1 ? e ? x / t2
Brief Description Exponential associate. Sample Curve

(

)

(

)

Parameters Number: 5 Names: y0, A1, t1, A2, t2 Meanings: y0 = offset, A1 = amplitude, t1 = width, A2 = amplitude, t2 = width Initial Values: y0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary) Lower Bounds: t1 > 0, t2 > 0 Upper Bounds: none Script Access expassoc(x,y0,A1,t1,A2,t2) Function File FITFUNC\EXPASSOC.FDF

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Page 49 of 166

ExpDec1
Function

y = y0 + Ae ? x / t
Brief Description Exponential decay 1. Sample Curve

Parameters Number: 3 Names: y0, A, t Meanings: y0 = offset, A = amplitude, t = decay constant Initial Values: y0 = 0.0 (vary), A = 1.0 (vary), t = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdec1(x,y0,A,t) Function File FITFUNC\EXPDEC1.FDF

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ExpDec2
Function

y = y0 + A1e ? x / t1 + A2 e ? x / t2
Brief Description Exponential decay 2. Sample Curve

Parameters Number: 5 Names: y0, A1, t1, A2, t2 Meanings: y0 = offset, A1 = amplitude, t1 = decay constant, A2 = amplitude, t2 = decay constant Initial Values: y0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdec2(x,y0,A1,t1,A2,t2) Function File FITFUNC\EXPDEC2.FDF

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ExpDec3
Function

y = y0 + A1e ? x / t1 + A2 e ? x / t2 + A3 e ? x / t3
Brief Description Exponential decay 3. Sample Curve

Parameters Number: 7 Names: y0, A1, t1, A2, t2, A3, t3 Meanings: y0 = offset, A1 = amplitude, t1 = decay constant, A2 = amplitude, t2 = decay constant, A3 = amplitude, t3 = decay constant Initial Values: y0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary), A3 = 1.0 (vary), t3 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdec3(x,y0,A1,t1,A2,t2,A3,t3) Function File FITFUNC\EXPDEC3.FDF

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ExpDecay1
Function

y = y0 + A1e ? (x ? x0 )/ t1
Brief Description Exponential decay 1 with offset. Sample Curve

Parameters Number: 4 Names: y0, x0, A1, t1 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = decay constant Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 10 (vary), t1 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdecay1(x,y0,x0,A1,t1) Function File FITFUNC\EXPDECY1.FDF

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ExpDecay2
Function

y = y0 + A1e ? ( x? x0 )/ t1 + A2 e ? (x ? x0 )/ t2
Brief Description Exponential decay 2 with offset. Sample Curve

Parameters Number: 6 Names: y0, x0, A1, t1, A2, t2 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = decay constant, A2 = amplitude, t2 = decay constant Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 10 (vary), t1 = 1.0 (vary), A2 = 10 (vary), t2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdecay2(x,y0,x0,A1,t1,A2,t2) Function File FITFUNC\EXPDECY2.FDF

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ExpDecay3
Function

y = y0 + A1e ? ( x? x0 )/ t1 + A2 e ? (x ? x0 )/ t2 + A3e ? (x ? x0 )/ t3
Brief Description Exponential decay 3 with offset. Sample Curve

Parameters Number: 8 Names: y0, x0, A1, t1, A2, t2, A3, t3 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = decay constant, A2 = amplitude, t2 = decay constant, A3 = amplitude, t3 = decay constant Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 10 (vary), t1 = 1.0 (vary), A2 = 10 (vary), t2 = 1.0 (vary), A3 = 10 (vary), t3 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expdecay3(x,y0,x0,A1,t1,A2,t2,A3,t3) Function File FITFUNC\EXPDECY3.FDF

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Page 55 of 166

ExpGro1
Function

y = y 0 + A1e x / t1
Brief Description Exponential growth 1. Sample Curve

Parameters Number: 3 Names: y0, A1, t1 Meanings: y0 = offset, A1 = amplitude, t1 = growth constant Initial Values: y0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expgro1(x,y0,A1,t1) Function File FITFUNC\EXPGRO1.FDF

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Page 56 of 166

ExpGro2
Function

y = y0 + A1e x / t1 + A2 e x / t2
Brief Description Exponential growth 2. Sample Curve

Parameters Number: 5 Names: y0, A1, t1, A2, t2 Meanings: y0 = offset, A1 = amplitude, t1 = growth constant, A2 = amplitude, t2 = growth constant Initial Values: y0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expgro2(x,y0,A1,t1,A2,t2) Function File FITFUNC\EXPGRO2.FDF

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ExpGro3
Function

y = y0 + A1e x / t1 + A2 e x / t2 + A3e x / t3
Brief Description Exponential growth 3. Sample Curve

Parameters Number: 7 Names: y0, A1, t1, A2, t2, A3, t3 Meanings: y0 = offset, A1 = amplitude, t1 = growth constant, A2 = amplitude, t2 = growth constant, A3 = amplitude, t3 = growth constant Initial Values: y0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary), A3 = 1.0 (vary), t3 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access expgro3(x,y0,A1,t1,A2,t2,A3,t3) Function File FITFUNC\EXPGRO3.FDF

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Page 58 of 166

ExpGrow1
Function

y = y 0 + A1e ( x ? x0 ) / t1
Brief Description Exponential growth 1 with offset. Sample Curve

Parameters Number: 4 Names: y0, x0, A1, t1 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = width Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary),A1 = 1.0 (vary), t1 = 1.0 (vary) Lower Bounds: t1 > 0.0 Upper Bounds: none Script Access expgrow1(x,y0,x0,A1,t1) Function File FITFUNC\EXPGROW1.FDF

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Page 59 of 166

ExpGrow2
Function

y = y0 + A1e ( x? x0 )/ t1 + A2 e (x ? x0 )/ t2
Brief Description Exponential growth 2 with offset. Sample Curve

Parameters Number: 6 Names: y0, x0, A1, t1, A2, t2 Meanings: y0 = offset, x0 = center, A1 = amplitude, t1 = width, A2 = amplitude, t2 = width Initial Values: y0 = 0.0 (vary), x0 = 0.0 (vary), A1 = 1.0 (vary), t1 = 1.0 (vary), A2 = 1.0 (vary), t2 = 1.0 (vary) Lower Bounds: t1 > 0.0, t2 > 0.0 Upper Bounds: none Script Access expgrow2(x,y0,x0,A1,t1,A2,t2) Function File FITFUNC\EXPGROW2.FDF

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Page 60 of 166

ExpLinear
Function

y = p1e ? x / p2 + p3 + p 4 x
Brief Description Exponential linear combination. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. p. 298 Sample Curve

Parameters Number: 4 Names: p1, p2, p3, p4 Meanings: p1 = coefficient, p2 = unknown, p3 = offset, p4 = coefficient Initial Values: p1 = 1.0 (vary), p2 = 1.0 (vary), p3 = 1.0 (vary), p4 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access explinear(x,p1,p2,p3,p4) Function File FITFUNC\EXPLINEA.FDF

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Page 61 of 166

Exponential
Function

y = y0 + Ae R0 x
Brief Description Exponential. Sample Curve

Parameters Number: 3 Names: y0, A, R0 Meanings: y0 = offset, A = initial value, R0 = rate Initial Values: y0 = 0.0 (vary), A = 1.0 (vary), R0 = 1.0 (vary) Lower Bounds: A > 0.0 Upper Bounds: none Script Access exponential(x,y0,A,R0) Function File FITFUNC\EXPONENT.FDF

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Page 62 of 166

MnMolecular
Function

y = A 1 ? e ? k ( x? xc )
Brief Description

(

)

Monomolecular growth model. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. p. 328 Sample Curve

Parameters Number: 3 Names: A, xc, k Meanings: A = amplitude, xc = center, k = rate Initial Values: A = 2.0 (vary), xc = 1.0 (vary), k = 1.0 (vary) Lower Bounds: A > 0.0 Upper Bounds: none Script Access mnmolecular(x,A,xc,k) Function File FITFUNC\MMOLECU.FDF

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MnMolecular1
Function

y = A1 ? A2 e ? kx
Brief Description Monomolecular growth model. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. p. 328 Sample Curve

Parameters Number: 3 Names: A1, A2, k Meanings: A1 = offset, A2 = coefficient, k = coefficient Initial Values: A1 = 1.0 (vary), A2 = 1.0 (vary), k = 1.0 (vary) Lower Bounds: A1 > 0.0, A2 > 0.0 Upper Bounds: none Script Access mnmolecular1(x,A1,A2,k) Function File FITFUNC\MMOLECU1.FDF

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Page 64 of 166

Shah
Function

y = a + bx + cr x
Brief Description Shah model. Sample Curve

Parameters Number: 4 Names: a, b, c, r Meanings: a = offset, b = coefficient, c = coefficient, r = unknown Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 1.0 (vary), r = 0.5 (vary) Lower Bounds: r > 0.0 Upper Bounds: r < 1.0 Script Access shah(x,a,b,c,r) Function File FITFUNC\SHAH.FDF

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Page 65 of 166

Stirling
Function

? e kx ? 1 ? y = a + b? ? k ? ? ? ?
Brief Description Stirling model. Sample Curve

Parameters Number: 3 Names: a, b, k Meanings: a = offset, b = coefficient, k = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), k = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access stirling(x,a,b,k) Function File FITFUNC\STIRLING.FDF

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Page 66 of 166

YldFert
Function

y = a + br x
Brief Description Yield-fertilizer model in agriculture and learning curve in psychology. Sample Curve

Parameters Number: 3 Names: a, b, r Meanings: a = offset, b = coefficient, r = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), r = 0.5 (vary) Lower Bounds: r > 0.0 Upper Bounds: r < 1.0 Script Access yldfert(x,a,b,r) Function File FITFUNC\YLDFERT.FDF

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Page 67 of 166

YldFert1
Function

y = a + be ? kx
Brief Description Yield-fertilizer model in agriculture and learning curve in psychology. Sample Curve

Parameters Number: 3 Names: a, b, k Meanings: a = offset, b = coefficient, k = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), k = 0.5 (vary) Lower Bounds: k > 0.0 Upper Bounds: none Script Access yldfert1(x,a,b,k) Function File FITFUNC\YLDFERT1.FDF

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4. Growth/Sigmoidal
Boltzmann Hill Logistic SGompertz SLogistic1 SLogistic2 SLogistic3 SRichards1 SRichards2 SWeibull1 SWeibull2 70 71 72 73 74 75 76 77 78 79 80

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Boltzmann
Function

y=

A1 ? A2 + A2 1 + e ( x ? x0 )/ dx

Brief Description Boltzmann function - produces a sigmoidal curve. Sample Curve

Parameters Number: 4 Names: A1, A2, x0, dx Meanings: A1 = initial value, A2 = final value, x0 = center, dx = time constant Initial Values: A1 = 0.0 (vary), A2 = 1.0 (vary), x0 = 0.0 (vary), dx = 1.0 (vary) Lower Bounds: none Upper Bounds: none Constraints dx ! = 0 Script Access boltzman(x,A1,A2,x0,dx) Function File FITFUNC\BOLTZMAN.FDF

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Page 70 of 166

Hill
Function

y = Vmax

xn k n + xn

Brief Description Hill function. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. p. 120 Sample Curve

Parameters Number: 3 Names: Vmax, k, n Meanings: Vmax = unknown, k = unknown, n = unknown Initial Values: Vmax = 1.0 (vary), k = 1.0 (vary), n = 1.5 (vary) Lower Bounds: Vmax > 0 Upper Bounds: none Script Access hill(x,Vmax,k,n) Function File FITFUNC\HILL.FDF

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Page 71 of 166

Logistic
Function

y=

A1 ? A2 + A2 p 1 + (x / x0 )

Brief Description Logistic dose response in pharmacology/chemistry. Sample Curve

Parameters Number: 4 Names: A1, A2, x0, p Meanings: A1 = initial value, A2 = final value, x0 = center, p =power Initial Values: A1 = 0.0 (vary), A2 = 1.0 (vary), x0 = 1.0 (vary), p = 1.5 (vary) Lower Bounds: p > 0.0 Upper Bounds: none Script Access logistic(x,A1,A2,x0,p) Function File FITFUNC\LOGISTIC.FDF

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Page 72 of 166

SGompertz
Function

y = ae ? exp (? k (x ? xc ))
Brief Description Gompertz growth model for population studies, animal growth. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 330 331 Sample Curve

Parameters Number: 3 Names: a, xc, k Meanings: a = amplitude, xc = center, k = coefficient Initial Values: a = 1.0 (vary), xc = 1.0 (vary), k = 1.0 (vary) Lower Bounds: a > 0.0, k > 0.0 Upper Bounds: none Script Access sgompertz(x,a,xc,k) Function File FITFUNC\GOMPERTZ.FDF

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Page 73 of 166

SLogistic1
Function

y=

1+ e

? k ( x ? xc )

a

Brief Description Sigmoidal logistic function, type 1. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 328 330 Sample Curve

Parameters Number: 3 Names: a, xc, k Meanings: a = amplitude, xc = center, k = coefficient Initial Values: a = 1.0 (vary), xc = 1.0 (vary), k = 1.0 (vary) Lower Bounds: xc > 0 Upper Bounds: none Script Access slogistic1(x,a,xc,k) Function File FITFUNC\SLOGIST1.FDF

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Page 74 of 166

SLogistic2
Function

y=

a 1+ a ? y0 ?4Wmax x / a e y0

Brief Description Sigmoidal logistic function, type 2. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 328 330 Sample Curve

Parameters Number: 3 Names: y0, a, Wmax Meanings: y0 = initial value, a = amplitude, Wmax = maximum growth rate Initial Values: y0 = 0.5 (vary), a = 1.0 (vary), Wmax = 1.0 (vary) Lower Bounds: y0 > 0.0, a > 0.0, Wmax > 0.0 Upper Bounds: none Script Access slogistic2(x,y0,a,Wmax) Function File FITFUNC\SLOGIST2.FDF

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SLogistic3
Function

y=

a 1 + be ?kx

Brief Description Sigmoidal logistic function, type 3. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 328 330 Sample Curve

Parameters Number: 3 Names: a, b, k Meanings: a = amplitude, b = coefficient, k = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), k = 1.0 (vary) Lower Bounds: a > 0.0, b > 0.0, k >0.0 Upper Bounds: none Script Access slogistic3(x,a,b,k) Function File FITFUNC\SLOGIST3.FDF

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Page 76 of 166

SRichards1
Function

[ y = [a

y = a1?d ? e ?k (x ? xc )
1? d

+ e ? k ( x ? xc

]( ) ( ]

1 / 1? d ) 1 / 1? d )

,d <1 ,d >1

Brief Description Sigmoidal Richards function, type 1. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 332 337 Sample Curve

Parameters Number: 4 Names: a, xc, d, k Meanings: a = unknown, xc = center, d = unknown, k = coefficient Initial Values: a = 1.0 (vary), xc = 1.0 (vary), d = 5 (vary), k = 0.5 (vary) Lower Bounds: a > 0.0, k > 0.0 Upper Bounds: none Script Access srichards1(x,a,xc,d,k) Function File FITFUNC\SRICHAR1.FDF

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Page 77 of 166

SRichards2
Function

y = a 1 + (d ? 1)e ?k ( x? xc )
Brief Description

[

](

1 / 1? d )

,d ≠1

Sigmoidal Richards function, type 2. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 332 337 Sample Curve

Parameters Number: 4 Names: a, xc, d, k Meanings: a = unknown, xc = center, d = unknown, k = coefficient Initial Values: a = 1.0 (vary), xc = 1.0 (vary), d = 5.0 (vary), k = 1.0 (vary) Lower Bounds: a > 0.0, k > 0.0 Upper Bounds: none Script Access srichards2(x,a,xc,d,k) Function File FITFUNC\SRICHAR2.FDF

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Page 78 of 166

SWeibull1
Function

y = A 1 ? e ?(k (x ? xc ))
Brief Description

(

d

)

Sigmoidal Weibull function, type 1. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 338 339 Sample Curve

Parameters Number: 4 Names: A, xc, d, k Meanings: A = amplitude, xc = center, d = power, k = coefficient Initial Values: A = 1.0 (vary), xc = 1.0 (vary), d = 5.0 (vary), k = 1.0 (vary) Lower Bounds: A > 0.0, k > 0.0 Upper Bounds: none Script Access sweibull1(x,A,xc,d,k) Function File FITFUNC\WEIBULL1.FDF

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Page 79 of 166

SWeibull2
Function

y = A ? (A ? B )e ? (kx )
Brief Description

d

Sigmoidal Weibull function, type 2. Reference: Seber, G. A. F., Wild, C. J. 1989. Nonlinear Regression. John Wiley & Sons, Inc. pp. 338 339 Sample Curve

Parameters Number: 4 Names: a, b, d, k Meanings: a = unknown, b = unknown, d = power, k = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), d = 5.0 (vary), k = 1.0 (vary) Lower Bounds: a > 0.0, b > 0.0, k > 0.0 Upper Bounds: none Script Access sweibull2(x,a,b,d,k) Function File FITFUNC\WEIBULL2.FDF

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5. Hyperbola Functions
Dhyperbl Hyperbl HyperbolaGen HyperbolaMod RectHyperbola 82 83 84 85 86

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Dhyperbl
Function

y=

Px P 1x + 3 + P5 x P2 + x P4 + x

Brief Description Double rectangular hyperbola function. Sample Curve

Parameters Number: 5 Names: P1, P2, P3, P4, P5 Meanings: Unknowns 1-5 Initial Values: P1 = 1.0 (vary), P2 = 1.0 (vary), P3 = 1.0 (vary), P4 = 1.0 (vary), P5 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access dhyperbl(x,P1,P2,P3,P4,P5) Function File FITFUNC\DHYPERBL.FDF

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Hyperbl
Function

y=

P 1x P2 + x

Brief Description Hyperbola function. Also the Michaelis-Menten model in enzyme kinetics. Sample Curve

Parameters Number: 2 Names: P1, P2 Meanings: P1 = amplitude, P2 = unknown Initial Values: P1 = 1.0 (vary), P2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access hyperbl(x,P1,P2) Function File FITFUNC\HYPERBL.FDF

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HyperbolaGen
Function

y=a?

b (1 + cx )1 / d

Brief Description Generalized hyperbola function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.4.7 Sample Curve

Parameters Number: 4 Names: a, b, c, d Meanings: a = coefficient, b = coefficient, c = coefficient, d = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5, d = 0.5 Lower Bounds: none Upper Bounds: none Script Access hyperbolagen(x,a,b,c,d) Function File FITFUNC\HYPERGEN.FDF

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HyperbolaMod
Function

y=

x θ1 x + θ 2

Brief Description Modified hyperbola function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.2.18 Sample Curve

Parameters Number: 2 Names: T1, T2 Meanings: T1 = amplitude, T2 = unknown Initial Values: T1 = 1.0 (vary), T2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access hyperbolamod(x,T1,T2) Function File FITFUNC\HYPERBMD.FDF

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Page 85 of 166

RectHyperbola
Function

y=a

bx 1 + bx

Brief Description Rectangular hyperbola function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.2.16 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access recthyperbola(x,a,b) Function File FITFUNC\RECTHYPB.FDF

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6. Logarithm Functions
Bradley Log2P1 Log2P2 Log3P1 Logarithm 88 89 90 91 92

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Bradley
Function

y = a ln (? b ln( x) )
Brief Description Bradley model. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 3.3.7 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = unknown, b = unknown Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access bradley(x,a,b) Function File FITFUNC\BRADLEY.FDF

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Log2P1
Function

y = b ln (x ? a )
Brief Description Two-parameter logarithm function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.2.1 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = offset, b = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access log2p1(x,a,b) Function File FITFUNC\LOG2P1.FDF

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Page 89 of 166

Log2P2
Function

y = ln(a + bx )
Brief Description Two-parameter logarithm. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.2.3 Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = offset, b = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access log2p2(x,a,b) Function File FITFUNC\LOG2P2.FDF

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Page 90 of 166

Log3P1
Function

y = a ? b ln (x + c )
Brief Description Three-parameter logarithm function. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.32 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access log3p1(x,a,b,c) Function File FITFUNC\LOG3P1.FDF

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Logarithm
Function

y = ln (x ? A)
Brief Description One-parameter logarithm. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.1.1 Sample Curve

Parameters Number: 1 Names: A Meanings: A = center Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access logarithm(x,A) Function File FITFUNC\LOGARITH.FDF

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7. Peak Functions
Asym2Sig Beta CCE ECS Extreme Gauss GaussAmp GaussMod GCAS Giddings InvsPoly LogNormal Logistpk Lorentz PearsonVII PsdVoigt1 PsdVoigt2 Voigt Weibull3 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112

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Page 93 of 166

Asym2Sig
Function

y = y0 + A 1+ e
Brief Description

1
? x ? xc + w1 / 2 w2

? ? 1 ?1 ? x ? xc ? w1 / 2 ? ? w3 ? 1+ e

? ? ? ? ?

Asymmetric double sigmoidal. Sample Curve

Parameters Number: 6 Names: y0, xc, A, w1, w2, w3 Meanings: y0 = offset, xc = center, A = amplitude, w1 = width, w2 = width, w3 = width Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w1 = 1.0 (vary), w2 = 1.0 (vary), w3 = 1.0 (vary) Lower Bounds: w1 > 0.0, w2 > 0.0, w3 > 0.0 Upper Bounds: none Script Access asym2sig(x,y0,xc,A,w1,w2,w3) Function File FITFUNC\ASYMDBLS.FDF

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Page 94 of 166

Beta
Function

? ? w2 + w3 ? 2 ?? x ? xc y = y 0 + A?1 + ? ? w ?1 ? ?? ? 2 ?? w1 ? ?
Brief Description The beta function. Sample Curve

?? ? ?? ??

w2 ?1

? ? w2 + w3 ? 2 ?? x ? x c ?1 ? ? ? ? w ?1 ? ?? 3 ?? w1 ? ?

?? ? ?? ??

w3 ?1

Parameters Number: 6 Names: y0, xc, A, w1, w2, w3 Meanings: y0 = offset, xc = center, A = amplitude, w1 = width, w2 = width, w3 = width Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), A = 5.0 (vary), w1 = 5.0 (vary), w2 = 2.0 (vary), w3 = 2.0 (vary) Lower Bounds: w1 > 0.0, w2 > 1.0, w3 > 1.0 Upper Bounds: none Script Access beta(x,y0,xc,A,w1,w2,w3) Function File FITFUNC\BETA.FDF

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Page 95 of 166

CCE
Function

? ? ( x ? xc 1 ) ? 0.5 k ( x ? x + ( x ? xc 3 )) y = y 0 + A?e 2 w + B(1 ? 0.5(1 ? tanh (k 2 (x ? xC 2 ))))e 3 c 3 ? ? ? ? ?
2

Brief Description Chesler-Cram peak function for use in chromatography. Sample Curve

Parameters Number: 9 Names: y0, xc1, A, w, k2, xc2, B, k3, xc3 Meanings: y0 = offset, xc1 = unknown, A = unknown, w = unknown, k2 = unknown, xc2 = unknown, B = unknown, k3 = unknown, xc3 = unknown Initial Values: y0 = 0.0 (vary), xc1 = 1.0 (vary), A = 1.0 (vary), w = 1.0 (vary), k2 = 1.0 (vary), xc2 = 1.0 (vary), B = 1.0 (vary), k3 = 1.0 (vary), xc3 = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access cce(x,y0,xc1,A,w,k2,xc2,B,k3,xc3) Function File FITFUNC\CHESLECR.FDF

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Page 96 of 166

ECS
Function

? a 4 ? a3 ?? 2 3 ?1 + z z ? 3 + 4 z ? 6 z + 3 ? ? ? A ? ?0.5 z 2 ? 3! 4! ?? y = y0 + ?e 2 ? 10a3 6 ?? w 2π ? 4 2 z ? 15 z + 45 z ? 15 ?+ ?? ? 6 ! ? ?? ? ?

( (

)

(

)

)

where

z=

x ? xc w

Brief Description Edgeworth-Cramer peak function for use in chromatography. Sample Curve

Parameters Number: 6 Names: y0, xc, A, w, a3, a4 Meanings: y0 = offset, xc = center, A = amplitude, w = width, a3 = unknown, a4 = unknown Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), a3 = 1.0 (vary), a4 = 1.0 (vary) Lower Bounds: A > 0.0, w > 0.0 Upper Bounds: none Script Access ecs(x,y0,xc,A,w,a3,a4) Function File FITFUNC\EDGWTHCR.FDF

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Extreme
Function

? ? ? x ? xc ? ? ? x ? xc ? ? y = y0 + Ae ?? exp ?? ? ?? ? ? ? + 1? ? ? w ?? ? w ? ? ?
Brief Description Extreme function in statistics. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = amplitude Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access extreme(x,y0,xc,w,A) Function File FITFUNC\EXTREME.FDF

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Page 98 of 166

Gauss
Function
?2 A y = y0 + e w π /2

( x ? xc )2
w2

Brief Description Area version of Gaussian function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 10 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gauss(x,y0,xc,w,A) Function File FITFUNC\GAUSS.FDF

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Page 99 of 166

GaussAmp
Function
?

y = y0 + Ae

( x ? xc )2
2 w2

Brief Description Amplitude version of Gaussian peak function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 10 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gaussamp(x,y0,xc,w,A) Function File FITFUNC\GAUSSAMP.FDF

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GaussMod
Function
?t ? ? A 2? f ( x) = y0 + e ? 0 ? t0 1? w ?
2

?

x ? xc t0



z

?∞

1 ?2 e dy 2π

y2

where

z=

x ? xc w ? w t0

Brief Description Exponentially modified Gaussian peak function for use in chromatography. Sample Curve

Parameters Number: 5 Names: y0, A, xc, w, t0 Meanings: y0 = offset, A = amplitude, xc = center, w = width, t0 = unknown Initial Values: y0 = 0.0 (vary), A = 1.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), t0 = 0.05 (vary) Lower Bounds: w > 0.0, t0 > 0.0 Upper Bounds: none Script Access gaussmod(x,y0,A,xc,w,t0) Function File FITFUNC\GAUSSMOD.FDF

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Page 101 of 166

GCAS
Function
4 2 a A ? ? e ? z / 2 ?1 + ∑ i H i (z )? w 2π i =3 i! ? ?

f ( z ) = y0 + z=

x ? xc w H 3 = z 3 ? 3z H 4 = z 4 ? 6z 3 + 3
Brief Description Gram-Charlier peak function for use in chromatography. Sample Curve

Parameters Number: 6 Names: y0, xc, A, w, a3, a4 Meanings: y0 = offset, xc = center, A = amplitude, w = width, a3 = unknown, a4 = unknown Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), a3 = 0.01 (vary), a4 = 0.001 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gcas(x,y0,xc,A,w,a3,a4) Function File FITFUNC\GRMCHARL.FDF

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Page 102 of 166

Giddings
Function

y = y0 +

A w

? x ? xc 2 xc x ? xc ? ?e w I1 ? ? x ? ? w ?

Brief Description Giddings peak function for use in chromatography. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access giddings(x,y0,xc,w,A) Function File FITFUNC\GIDDINGS.FDF

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Page 103 of 166

InvsPoly
Function

y = y0 +

A ? x ? xc ? ? x ? xc ? ? x ? xc ? 1 + A1 ? 2 ? + A2 ? 2 ? + A3 ? 2 ? w ? w ? w ? ? ? ?
2 4 6

Brief Description Inverse polynomial peak function with center. Sample Curve

Parameters Number: 7 Names: y0, xc, w, A, A1, A2, A3 Meanings: y0 = offset, xc = center, w = width, A = amplitude, A1 = coefficient, A2 = coefficient, A3 = coefficient Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 1.0 (vary), A1 = 0.0 (vary), A2 = 0.0 (vary), A3 = 0.0 (vary) Lower Bounds: w > 0.0, A1 ≥ 0.0, A2 ≥ 0.0, A3 ≥ 0.0 Upper Bounds: none Script Access invspoly(x,y0,xc,w,A,A1,A2,A3) Function File FITFUNC\INVSPOLY.FDF

Last Updated 11/14/00

Page 104 of 166

LogNormal
Function

y = y0 +

A 2π wx

?[ln x / xc ]2

e

2 w2

Brief Description Log-Normal function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = amplitude Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: xc > 0, w > 0 Upper Bounds: none Script Access lognormal(x,y0,xc,w,A) Function File FITFUNC\LOGNORM.FDF

Last Updated 11/14/00

Page 105 of 166

Logistpk
Function
? x ? xc w 2

y = y0 +

4 Ae

x ? xc ? ? ? ?1 + e w ? ? ? ? ?

Brief Description Logistic peak function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = amplitude Initial Values: y0 = 0.0 (vary), xc = 1.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access logistpk(x,y0,xc,w,A) Function File FITFUNC\LOGISTPK

Last Updated 11/14/00

Page 106 of 166

Lorentz
Function

y = y0 +

2A w π 4(x ? xc )2 + w 2

Brief Description Lorentzian peak function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access lorentz(x,y0,xc,w,A) Function File FITFUNC\LORENTZ.FDF

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Page 107 of 166

PearsonVII
Function
1 / mu ? mu 2 mu e (Γ ( 2 ?1) ) ? 21 / mu ? 1 2? (x ? xc ) ? y=A ?1 + 4 π e (Γ ( mu ?1 / 2) ) ? w2 ?

Brief Description Pearson VII peak function. Sample Curve

Parameters Number: 4 Names: xc, A, w, mu Meanings: xc = center, A = amplitude, w = width, mu = profile shape factor Initial Values: xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), mu = 1.0 (vary) Lower Bounds: A > 0.0, w > 0.0, mu > 0.0 Upper Bounds: none Script Access pearson7(x,xc,A,w,mu) Function File FITFUNC\PEARSON7.FDF

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Page 108 of 166

PsdVoigt1
Function
4 ln 2 ? 2 w 4 ln 2 ? w2 ( x ? xc )2 ? y = y0 + A?mu e + (1 ? mu ) ? 2 2 πw ? ? ? π 4(x ? xc ) + w ?

Brief Description Pseudo-Voigt peak function type 1. Sample Curve

Parameters Number: 5 Names: y0, xc, A, w, mu Meanings: y0 = offset, xc = center, A = amplitude, w = width, mu = profile shape factor Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), mu = 0.5 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access psdvoigt1(x,y0,xc,A,w,mu) Function File FITFUNC\PSDVGT1.FDF

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Page 109 of 166

PsdVoigt2
Function
4 ln 2 2 ? wL 2 4 ln 2 ? wG 2 ( x ? xc ) ? ? ( ) y = y 0 + A?m u m e 1 + ? u π 4(x ? x c )2 + wL 2 w π ? ? G ? ?

Brief Description Pseudo-Voigt peak function type 2. Sample Curve

Parameters Number: 6 Names: y0, xc, A, wG, wL, mu Meanings: y0 = offset, xc = center, A = amplitude, wG = width, wL = width, mu = profile shape factor Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), wG = 1.0 (vary), wL = 1.0 (vary), mu = 0.5 (vary) Lower Bounds: wG > 0.0, wL > 0.0 Upper Bounds: none Script Access psdvoigt2(x,y0,xc,A,wG,wL,mu) Function File FITFUNC\PSDVGT2.FDF

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Page 110 of 166

Voigt
Function

y = y0 + A ?

2 ln 2 wL ∞ e ?t ? dt 2 2 2 ∫ ?∞ π 3 / 2 wG ? ? wL ? ? x ? xc ? ? ln 2 w ? ? +? ? 4 ln 2 w ? t ? ? G ? G ? ? ?

2

Brief Description Voigt peak function. Sample Curve

Parameters Number: 5 Names: y0, xc, A, wG, wL Meanings: y0 = offset, xc = center, A = amplitude, wG = Gaussian width, wL = Lorentzian width Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), wG = 1.0 (vary), wL = 1.0 (vary) Lower Bounds: wG > 0.0, wL > 0.0 Upper Bounds: none Script Access voigt5(x,y0,xc,A,wG,wL) Function File FITFUNC\VOIGT5.FDF

Last Updated 11/14/00

Page 111 of 166

Weibull3
Function
1

x ? xc ? w2 ? 1 ? w2 S= +? ? w ? ? w1 2 ? ? ? w2 ? 1 ? y = y 0 + A? ? w ? ? ? ? 2
Brief Description Weibull peak function. Sample Curve
1? w2 w2

[S ]

w2 ?1

e

? w2 ?1 ? ?[S ]w2 + ? ? w ? ? ? 2 ?

Parameters Number: 5 Names: y0, xc, A, w1, w2 Meanings: y0 = offset, xc = center, A = amplitude, w1 = width, w2 = width Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w1 = 1.0 (vary), w2 = 1.0 (vary) Lower Bounds: w1 > 0.0, w2 > 0.0 Upper Bounds: none Script Access weibull3(x,y0,xc,A,w1,w2) Function File FITFUNC\WEIBULL3.FDF

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Page 112 of 166

8. Pharmacology Functions
Biphasic DoseResp OneSiteBind OneSiteComp TwoSiteBind TwoSiteComp 114 115 116 117 118 119

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Biphasic
Function

y = Amin +

(Amax 1 ? Amin )
1 + 10

(( x ? x 0 _ 1)*h1)

+

(1 + 10 (

(Amax 2 ? Amin )

( x 0 _ 2 ? x )*h 2 )

)

Brief Description Biphasic sigmoidal dose response (7 parameters logistic equation). Sample Curve

Parameters Number: 7 Names: Amin, Amax1, Amax2, x0_1, x0_2, h1, h2 Meanings: Amin = bottom asymptote, Amax1 = first top asymptote, Amax2 = second top asymptote, x0_1 = first median, x0_2 = second median, h1 = slope, h2 = slope Initial Values: Amin = 0.0 (vary), Amax1 = 1.0 (vary), Amax2 = 1.0 (vary), x0_1 = 1.0 (vary), x0_2 = 10.0 (vary), h1 = 1.0 (vary), h2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access response2(x,Amin,Amax1,Amax2,x0_1,x0_2,h1,h2) Function File FITFUNC\BIPHASIC.FDF

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Page 114 of 166

DoseResp
Function

y = A1 +

A2 ? A1 1 + 10 (log x0 ? x ) p

Brief Description Dose-response curve with variable Hill slope given by parameter 'p'. Sample Curve

Parameters Number: 4 Names: A1, A2, LOGx0, p Meanings: A1 = bottom asymptote, A2 = top asymptote, LOGx0 = center, p = hill slope Initial Values: A1 = 1.0 (vary), A2 = 100.0 (vary), LOGx0 = -5.0 (vary), p = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access response1(x,A1,A2,LOGx0,p) Function File FITFUNC\DRESP.FDF

Last Updated 11/14/00

Page 115 of 166

OneSiteBind
Function

y=

Bmax x K1 + x

Brief Description One site direct binding. Rectangular hyperbola, connects to isotherm or saturation curve. Sample Curve

Parameters Number: 2 Names: Bmax, K1 Meanings: Bmax = top asymptote, K1 = median Initial Values: Bmax = 1.0 (vary), K1 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access binding1(x,Bmax,K1) Function File FITFUNC\BIND1.FDF

Last Updated 11/14/00

Page 116 of 166

OneSiteComp
Function

y = A2 +

A1 ? A2 1 + 10 ( x ? log x0 )

Brief Description One site competition curve. Dose-response curve with Hill slope equal to -1. Sample Curve

Parameters Number: 3 Names: A1, A2, log(x0) Meanings: A1 = top asymptote, A2 = bottom asymptote, log(x0) = center Initial Values: A1 = 10.0 (vary), A2 = 1.0 (vary), log(x0) = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access competition1(x,A1,A2,LOGx0) Function File FITFUNC\COMP1.FDF

Last Updated 11/14/00

Page 117 of 166

TwoSiteBind
Function

y=

Bmax 1 x Bmax 2 x + K1 + x K 2 + x

Brief Description Two site binding curve. Sample Curve

Parameters Number: 4 Names: Bmax1, Bmax2, k1, k2 Meanings: Bmax1 = first top asymptote, Bmax2 = second top asymptote, k1 = first median, k2 = second median Initial Values: Bmax1 = 1.0 (vary), Bmax2 = 1.0 (vary), k1 = 1.0 (vary), k2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access binding2(x,Bmax1,Bmax2,k1,k2) Function File FITFUNC\BIND2.FDF

Last Updated 11/14/00

Page 118 of 166

TwoSiteComp
Function

y = A2 +

(A1 ? A2 ) f
1 + 10

( x ? log x01 )

+

(A1 ? A2 )(1 ? f )
1 + 10 (x ? log x02 )

Brief Description Two site competition. Sample Curve

Parameters Number: 5 Names: A1, A2, log(x0_1), log(x0_2), f Meanings: A1 = top asymptote, A2 = bottom asymptote, log(x0_1) = first center, log(x0_2) = second center, f = fraction Initial Values: A1 = 10.0 (vary), A2 = 1.0 (vary), log(x0_1) = 1.0 (vary), log(x0_2) = 2.0 (vary), f = 0.5 (vary) Lower Bounds: none Upper Bounds: none Script Access competition2(x,A1,A2,LOGx0_1,LOGx0_2,f) Function File FITFUNC\COMP2.FDF

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9. Power Functions
Allometric1 Allometric2 Asym2Sig Belehradek BlNeld BlNeldSmp FreundlichEXT Gunary Harris LangmuirEXT1 LangmuirEXT2 Pareto Pow2P1 Pow2P2 Pow2P3 Power Power0 Power1 Power2 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139

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Allometric1
Function

y = ax b
Brief Description Classical Freundlich model. Has been used in the study of allometry. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = power Initial Values: a = 1.0 (vary), b = 0.5 (vary) Lower Bounds: none Upper Bounds: none Script Access allometric1(x,a,b) Function File FITFUNC\ALLOMET1.FDF

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Page 121 of 166

Allometric2
Function

y = a + bx c
Brief Description An extension of classical Freundlich model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = offset, b = coefficient, c = power Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 (vary) Lower Bounds: none Upper Bounds: none Script Access allometric2(x,a,b,c) Function File FITFUNC\ALLOMET2.FDF

Last Updated 11/14/00

Page 122 of 166

Asym2Sig
Function

y = y0 + A 1+ e
Brief Description

1
? x ? xc + w1 / 2 w2

? ? 1 ?1 ? x ? xc ? w1 / 2 ? ? w3 ? 1+ e

? ? ? ? ?

Asymmetric double sigmoidal. Sample Curve

Parameters Number: 6 Names: y0, xc, A, w1, w2, w3 Meanings: y0 = offset, xc = center, A = amplitude, w1 = width, w2 = width, w3 = width Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w1 = 1.0 (vary), w2 = 1.0 (vary), w3 = 1.0 (vary) Lower Bounds: w1 > 0.0, w2 > 0.0, w3 > 0.0 Upper Bounds: none Script Access asym2sig(x,y0,xc,A,w1,w2,w3) Function File FITFUNC\ASYMDBLS.FDF

Last Updated 11/14/00

Page 123 of 166

Belehradek
Function

y = a(x ? b )

c

Brief Description Belehradek model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = position, c = power Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access belehradek(x,a,b,c) Function File FITFUNC\BELEHRAD.FDF

Last Updated 11/14/00

Page 124 of 166

BlNeld
Function

y = a + bx f

(

)

?1 / c

Brief Description Bleasdale-Nelder model. Sample Curve

Parameters Number: 4 Names: a, b, c, f Meanings: a = coefficient, b = coefficient, c = coefficient, f = power Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5, f = 1.0 Lower Bounds: none Upper Bounds: none Script Access blneld(x,a,b,c,f) Function File FITFUNC\BLNELD.FDF

Last Updated 11/14/00

Page 125 of 166

BlNeldSmp
Function

y = (a + bx )

?1 / c

Brief Description Simplified Bleasdale-Nelder model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access blneldsmp(x,a,b,c) Function File FITFUNC\BLNELDSP.FDF

Last Updated 11/14/00

Page 126 of 166

FreundlichEXT
Function

y = ax bx

?c

Brief Description Extended Freundlich model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = power Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access freundlichext(x,a,b,c) Function File FITFUNC\FRENDEXT.FDF

Last Updated 11/14/00

Page 127 of 166

Gunary
Function

y=

x a + bx + c x

Brief Description Gunary model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access gunary(x,a,b,c) Function File FITFUNC\GUNARY.FDF

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Harris
Function

y = a + bx c

(

)

?1

Brief Description Farazdaghi-Harris model for use in yield-density study. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = power Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access harris(x,a,b,c) Function File FITFUNC\HARRIS.FDF

Last Updated 11/14/00

Page 129 of 166

LangmuirEXT1
Function

y=

abx1?c 1 + bx1?c

Brief Description Extended Langmuir model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access langmuirext1(x,a,b,c) Function File FITFUNC\LANGEXT1.FDF

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LangmuirEXT2
Function

y=

1 a + bx c ?1

Brief Description Extended Langmuir model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access langmuirext2(x,a,b,c) Function File FITFUNC\LANGEXT2.FDF

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Pareto
Function

y =1=

1 xA

Brief Description Pareto function. Sample Curve

Parameters Number: 1 Names: A Meanings: A = coefficient Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access pareto(x,A) Function File FITFUNC\PARETO.FDF

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Page 132 of 166

Pow2P1
Function

y = a 1 ? x ?b

(

)

Brief Description Two-parameter power function. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = power Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access pow2p1(x,a,b) Function File FITFUNC\POW2P1.FDF

Last Updated 11/14/00

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Pow2P2
Function

y = a(1 + x )

b

Brief Description Two-parameter power function. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = power Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access pow2p2(x,a,b) Function File FITFUNC/POW2P2.FDF

Last Updated 11/14/00

Page 134 of 166

Pow2P3
Function

y =1?

1 (1 + ax )b

Brief Description Two-parameter power function. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = power Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access pow2p3(x,a,b) Function File FITFUNC\POW2P3.FDF

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Page 135 of 166

Power
Function

y = xA
Brief Description One-parameter power function. Sample Curve

Parameters Number: 1 Names: A Meanings: A = power Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access power(x,A) Function File FITFUNC\POWER.FDF

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Power0
Function

y = y 0 + A x ? xc
Brief Description

p

Symmetric power function with offset. Sample Curve

Parameters Number: 4 Names: y0, xc, A, P Meanings: y0 = offset, xc = center, A = amplitude, P = power Initial Values: y0 = 0.0 (vary), xc = 5.0 (vary), A = 1.0 (vary), P = 0.5 (vary) Lower Bounds: A > 0.0 Upper Bounds: none Script Access power0(x,y0,xc,A,P) Function File FITFUNC\POWER0.FDF

Last Updated 11/14/00

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Power1
Function

y = A x ? xc

p

Brief Description Symmetric power function. Sample Curve

Parameters Number: 3 Names: xc, A, P Meanings: xc = center, A = amplitude, P = power Initial Values: xc = 0.0 (vary), A = 1.0 (vary), P = 2.0 (vary) Lower Bounds: A > 0.0, P > 0.0 Upper Bounds: none Script Access power1(x,xc,A,P) Function File FITFUNC\POWER1.FDF

Last Updated 11/14/00

Page 138 of 166

Power2
Function

y = A x ? xc y = A x ? xc

Pl Pu

, x < xc , x > xc

Brief Description Asymmetric power function. Sample Curve

Parameters Number: 4 Names: xc, A, pl, pu Meanings: xc = center, A = amplitude, p1 = power, pu = power Initial Values: xc = 0.0 (vary), A = 1.0 (vary), p1 = 2.0 (vary), pu = 2.0 (vary) Lower Bounds: A > 0.0, p1 > 0.0, pu > 0.0 Upper Bounds: none Script Access power2(x,xc,A,pl,pu) Function File FITFUNC\POWER2.FDF

Last Updated 11/14/00

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10. Rational Functions
BET BETMod Holliday Holliday1 Nelder Rational0 Rational1 Rational2 Rational3 Rational4 Reciprocal Reciprocal0 Reciprocal1 ReciprocalMod 141 142 143 144 145 146 147 148 149 150 151 152 153 154

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BET
Function

y=

abx 1 + (b ? 2)x ? (b ? 1)x 2

Brief Description BET model. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = coefficient Initial Values: a = 1.0 (vary), b = 5.0 (vary) Lower Bounds: none Upper Bounds: none Script Access bet(x,a,b) Function File FITFUNC\BET.FDF

Last Updated 11/14/00

Page 141 of 166

BETMod
Function

y=

x a + bx ? (a + b )x 2

Brief Description Modified BET model. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = coefficient Initial Values: a = 1.0 (vary), b = 5.0 (vary) Lower Bounds: none Upper Bounds: none Script Access betmod(x,a,b) Function File FITFUNC\BETMOD.FDF

Last Updated 11/14/00

Page 142 of 166

Holliday
Function

y = a + bx + cx 2
Brief Description

(

)

?1

Holliday model - a Yield-density model for use in agriculture. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access holliday(x,a,b,c) Function File FITFUNC\HOLLIDAY.FDF

Last Updated 11/14/00

Page 143 of 166

Holliday1
Function

y=

a a + bx + cx 2

Brief Description Extended Holliday model. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access holliday1(x,a,b,c) Function File FITFUNC\HOLLIDY1.FDF

Last Updated 11/14/00

Page 144 of 166

Nelder
Function

y=

x+a 2 b0 + b1 (x + a ) + b2 (x + a )

Brief Description Nelder model - a Yield-fertilizer model in agriculture. Sample Curve

Parameters Number: 4 Names: a, b0, b1, b2 Meanings: a = unknown, b0 = unknown, b1 = unknown, b2 = unknown Initial Values: a = 1.0 (vary), b0 = 1.0 (vary), b1 = 1.0 (vary), b2 = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access nelder(x,a,b0,b1,b2) Function File FITFUNC\NELDER.FDF

Last Updated 11/14/00

Page 145 of 166

Rational0
Function

y=

b + cx 1 + ax

Brief Description Rational function, type 0. Reference: Ratkowksy, David A. 1990. Handbook of Nonlinear Regression Models. Marcel Dekker, Inc. 4.3.24 Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access rational0(x,a,b,c) Function File FITFUNC\RATION0.FDF

Last Updated 11/14/00

Page 146 of 166

Rational1
Function

y=

1 + cx a + bx

Brief Description Rational function, type 1. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b =coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access rational1(x,a,b,c) Function File FITFUNC\RATION1.FDF

Last Updated 11/14/00

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Rational2
Function

y=

b + cx a+x

Brief Description Rational function, type 2. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access rational2(x,a,b,c) Function File FITFUNC\RATION2.FDF

Last Updated 11/14/00

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Rational3
Function

y=

b+x a + cx

Brief Description Rational function, type 3. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access rational3(x,a,b,c) Function File FITFUNC\RATION3.FDF

Last Updated 11/14/00

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Rational4
Function

y =c+

b x+a

Brief Description Rational function, type 4. Sample Curve

Parameters Number: 3 Names: a, b, c Meanings: a = coefficient, b = coefficient, c = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary), c = 0.5 Lower Bounds: none Upper Bounds: none Script Access rational4(x,a,b,c) Function File FITFUNC\RATION4.FDF

Last Updated 11/14/00

Page 150 of 166

Reciprocal
Function

y=

1 a + bx

Brief Description Two-parameter linear reciprocal function. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access reciprocal(x,a,b) Function File FITFUNC\RECIPROC.FDF

Last Updated 11/14/00

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Reciprocal0
Function

y=

1 1 + Ax

Brief Description One-parameter linear reciprocal function. Sample Curve

Parameters Number: 1 Names: A Meanings: A = coefficient Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access reciprocal0(x,A) Function File FITFUNC\RECIPR0.FDF

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Reciprocal1
Function

y=

1 x+ A

Brief Description One-parameter linear reciprocal function. Sample Curve

Parameters Number: 1 Names: A Meanings: A = position Initial Values: A = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access reciprocal1(x,A) Function File FITFUNC\RECIPR1.FDF

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Page 153 of 166

ReciprocalMod
Function

y=

a 1 + bx

Brief Description Two parameter linear reciprocal function. Sample Curve

Parameters Number: 2 Names: a, b Meanings: a = coefficient, b = coefficient Initial Values: a = 1.0 (vary), b = 1.0 (vary) Lower Bounds: none Upper Bounds: none Script Access reciprocalmod(x,a,b) Function File FITFUNC\RECIPMOD.FDF

Last Updated 11/14/00

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11. Spectroscopy Functions
GaussAmp InvsPoly Lorentz PearsonVII PsdVoigt1 PsdVoigt2 Voigt 156 157 158 159 160 161 162

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GaussAmp
Function
?

y = y0 + Ae

( x ? xc )2
2 w2

Brief Description Amplitude version of Gaussian peak function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 10 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access gaussamp(x,y0,xc,w,A) Function File FITFUNC\GAUSSAMP.FDF

Last Updated 11/14/00

Page 156 of 166

InvsPoly
Function

y = y0 +

A ? x ? xc ? ? x ? xc ? ? x ? xc ? 1 + A1 ? 2 ? + A2 ? 2 ? + A3 ? 2 ? w ? w ? w ? ? ? ?
2 4 6

Brief Description Inverse polynomial peak function with center. Sample Curve

Parameters Number: 7 Names: y0, xc, w, A, A1, A2, A3 Meanings: y0 = offset, xc = center, w = width, A = amplitude, A1 = coefficient, A2 = coefficient, A3 = coefficient Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 1.0 (vary), A1 = 0.0 (vary), A2 = 0.0 (vary), A3 = 0.0 (vary) Lower Bounds: w > 0.0, A1 ≥ 0.0, A2 ≥ 0.0, A3 ≥ 0.0 Upper Bounds: none Script Access invspoly(x,y0,xc,w,A,A1,A2,A3) Function File FITFUNC\INVSPOLY.FDF

Last Updated 11/14/00

Page 157 of 166

Lorentz
Function

y = y0 +

2A w π 4(x ? xc )2 + w 2

Brief Description Lorentzian peak function. Sample Curve

Parameters Number: 4 Names: y0, xc, w, A Meanings: y0 = offset, xc = center, w = width, A = area Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access lorentz(x,y0,xc,w,A) Function File FITFUNC\LORENTZ.FDF

Last Updated 11/14/00

Page 158 of 166

PearsonVII
Function
1 / mu ? mu 2 mu e (Γ ( 2 ?1) ) ? 21 / mu ? 1 2? (x ? xc ) ? y=A ?1 + 4 π e (Γ ( mu ?1 / 2) ) ? w2 ?

Brief Description Pearson VII peak function. Sample Curve

Parameters Number: 4 Names: xc, A, w, mu Meanings: xc = center, A = amplitude, w = width, mu = profile shape factor Initial Values: xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), mu = 1.0 (vary) Lower Bounds: A > 0.0, w > 0.0, mu > 0.0 Upper Bounds: none Script Access pearsonvii(x,xc,A,w,mu) Function File FITFUNC\PEARSON7.FDF

Last Updated 11/14/00

Page 159 of 166

PsdVoigt1
Function
4 ln 2 ? 2 w 4 ln 2 ? w2 ( x ? xc )2 ? y = y0 + A?mu e + (1 ? mu ) ? 2 2 πw ? ? ? π 4(x ? xc ) + w ?

Brief Description Pseudo-Voigt peak function type 1. Sample Curve

Parameters Number: 5 Names: y0, xc, A, w, mu Meanings: y0 = offset, xc = center, A = amplitude, w = width, mu = profile shape factor Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), w = 1.0 (vary), mu = 0.5 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access psdvoigt1(x,y0,xc,A,w,mu) Function File FITFUNC\PSDVGT1.FDF

Last Updated 11/14/00

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PsdVoigt2
Function
4 ln 2 2 ? wL 2 4 ln 2 ? wG 2 ( x ? xc ) ? ? ( ) y = y 0 + A?m u m e 1 + ? u π 4(x ? x c )2 + wL 2 w π ? ? G ? ?

Brief Description Pseudo-Voigt peak function type 2. Sample Curve

Parameters Number: 6 Names: y0, xc, A, wG, wL, mu Meanings: y0 = offset, xc = center, A = amplitude, wG = width, wL = width, mu = profile shape factor Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), wG = 1.0 (vary), wL = 1.0 (vary), mu = 0.5 (vary) Lower Bounds: wG > 0.0, wL > 0.0 Upper Bounds: none Script Access psdvoigt2(x,y0,xc,A,wG,wL,mu) Function File FITFUNC\PSDVGT2.FDF

Last Updated 11/14/00

Page 161 of 166

Voigt
Function

y = y0 + A ?

2 ln 2 wL ∞ e ?t ? dt 2 2 2 ∫ ?∞ π 3 / 2 wG ? ? wL ? ? x ? xc ? ? ln 2 w ? ? +? ? 4 ln 2 w ? t ? ? G ? G ? ? ?

2

Brief Description Voigt peak function. Sample Curve

Parameters Number: 5 Names: y0, xc, A, wG, wL Meanings: y0 = offset, xc = center, A = amplitude, wG = Gaussian width, wL = Lorentzian width Initial Values: y0 = 0.0 (vary), xc = 0.0 (vary), A = 1.0 (vary), wG = 1.0 (vary), wL = 1.0 (vary) Lower Bounds: wG > 0.0, wL > 0.0 Upper Bounds: none Script Access voigt5(x,y0,xc,A,wG,wL) Function File FITFUNC\VOIGT5.FDF

Last Updated 11/14/00

Page 162 of 166

12. Waveform Functions
Sine SineDamp SineSqr 164 165 166

Last Updated 11/14/00

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Sine
Function

? x ? xc ? y = A sin ? π ? w ? ?
Brief Description Sine function. Sample Curve

Parameters Number: 3 Names: xc, w, A Meanings: xc = center, w = width, A = amplitude Initial Values: xc = 0.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0 Upper Bounds: none Script Access sine(x,xc,w,A) Function File FITFUNC\SINE.FDF

Last Updated 11/14/00

Page 164 of 166

SineDamp
Function
? x t0

y = Ae

? x ? xc ? sin ? π ? w ? ?

Brief Description Sine damp function. Sample Curve

Parameters Number: 4 Names: xc, w, t0, A Meanings: xc = center, w = width, t0 = decay constant, A = amplitude Initial Values: xc = 0.0 (vary), w = 1.0 (vary), t0 = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 , t0 > 0.0 Upper Bounds: none Script Access sinedamp(x,xc,w,t0,A) Function File FITFUNC\SINEDAMP.FDF

Last Updated 11/14/00

Page 165 of 166

SineSqr
Function

? x ? xc ? y = A sin 2 ? π ? w ? ?
Brief Description Sine square function. Sample Curve

Parameters Number: 3 Names: xc, w, A Meanings: xc = center, w = width, A = amplitude Initial Values: xc = 0.0 (vary), w = 1.0 (vary), A = 1.0 (vary) Lower Bounds: w > 0.0 Upper Bounds: none Script Access sinesqr(x,xc,w,A) Function File FITFUNC\SINESQR.FDF

Last Updated 11/14/00

Page 166 of 166


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