1. Rapid and Accurate Calculation of the Voigt Function D. Chris Benner Kendra L. Letchworth 9th International HITRAN Conference June 26-28, 2006

2. Why is this routine needed? Spectra with higher signal to noise ratios require more accurate analysis routines.Spectra with higher signal to noise ratios require more accurate analysis routines. Most fitting programs and simulations perform millions of calculations, so routines must also be fast.Most fitting programs and simulations perform millions of calculations, so routines must also be fast. Our routine calculates the Voigt profile to a relative accuracy of 10 -6.Our routine calculates the Voigt profile to a relative accuracy of 10 -6. 100 times more accurate than routines such as Drayson & Humlíček but requires 2 to 4 times less calculation time.100 times more accurate than routines such as Drayson & Humlíček but requires 2 to 4 times less calculation time. Our routine includes the imaginary part of the function for applications to line-mixing.Our routine includes the imaginary part of the function for applications to line-mixing. Option of returning the derivatives of the real Voigt profile with respect to x and y. They are calculated with relative accuracy of less than 10 -3, without significant additional calculation time required.Option of returning the derivatives of the real Voigt profile with respect to x and y. They are calculated with relative accuracy of less than 10 -3, without significant additional calculation time required. For more information about relative speed and accuracy of various Voigt routines (not including ours) see:For more information about relative speed and accuracy of various Voigt routines (not including ours) see: F. Schreier, J.Q.S.R.T., Vol. 48, 1992. F. Schreier, J.Q.S.R.T., Vol. 48, 1992.

3. The Voigt Profile v - v 0 : distance from line centerv - v 0 : distance from line center α L : Lorentz half-widthα L : Lorentz half-width α D : Doppler half-widthα D : Doppler half-width B.H. Armstrong, J.Q.S.R.T., Vol. 7, 1966.

4. Mathematical Approximations: Gauss-Hermite Quadrature Quadrature Points - v i Quadrature Weights - w i Expression simplified using: Symmetry: reduces the number of points calculated from n to (n+1)/2 by taking w i (f(v i ) + f(-v i )) and simplifying expression.Symmetry: reduces the number of points calculated from n to (n+1)/2 by taking w i (f(v i ) + f(-v i )) and simplifying expression. Odd-order quadrature: a quadrature point always falls on v=0, giving the simple term at beginning of the equations.Odd-order quadrature: a quadrature point always falls on v=0, giving the simple term at beginning of the equations. Repeating expressions within equations: (y 2 +x 2 +v i 2 ) andRepeating expressions within equations: (y 2 +x 2 +v i 2 ) and (y 2 +x 2 +v i 2 ) 2 -4x 2 v i 2 can be calculated only once. (y 2 +x 2 +v i 2 ) 2 -4x 2 v i 2 can be calculated only once. Real Part: Real Part: Imaginary Part:

5. Mathematical Approximations: Taylor Series Expansion 3 rd order Taylor expansion using a table of pre-computed values of the complex Voigt function (K+iL) and its derivatives.3 rd order Taylor expansion using a table of pre-computed values of the complex Voigt function (K+iL) and its derivatives. Real Part: ∂ 2 K/∂x 2 =-∂ 2 K/∂y 2 decreases the number of stored derivatives of the real part (K) from 9 to 6.∂ 2 K/∂x 2 =-∂ 2 K/∂y 2 decreases the number of stored derivatives of the real part (K) from 9 to 6. Derivatives of the imaginary part (L) can be represented as functions of the derivatives of the real part since ∂L/∂x=-∂K/∂y and ∂L/∂y=∂K/∂x.Derivatives of the imaginary part (L) can be represented as functions of the derivatives of the real part since ∂L/∂x=-∂K/∂y and ∂L/∂y=∂K/∂x. Imaginary Part: Δx=x-x 0, Δy=y-y 0 where (x 0,y 0 ) is the closest gridpoint.Δx=x-x 0, Δy=y-y 0 where (x 0,y 0 ) is the closest gridpoint. A total of 8 values must be stored for each grid point (x 0,y 0 ) : K, L, ∂K/∂x, ∂K/∂y, ∂ 2 K/∂x∂y, ∂ 2 K/∂x 2, ∂ 3 K/∂x 3, ∂ 3 K/∂y 3.A total of 8 values must be stored for each grid point (x 0,y 0 ) : K, L, ∂K/∂x, ∂K/∂y, ∂ 2 K/∂x∂y, ∂ 2 K/∂x 2, ∂ 3 K/∂x 3, ∂ 3 K/∂y 3.

6. Mathematical Approximations: Lagrange Interpolating Polynomials Used only in small areas when other methods fail, so does not contribute significantly to calculation time.Used only in small areas when other methods fail, so does not contribute significantly to calculation time. Employ equal grid point spacing dx and dy.Employ equal grid point spacing dx and dy. v 1, v 2, v 3 are the three grid points and ∆v=(v- v 2 )/dvv 1, v 2, v 3 are the three grid points and ∆v=(v- v 2 )/dv Four polynomial interpolations of P(v) below are required for a spline interpolation of real or imaginary part.Four polynomial interpolations of P(v) below are required for a spline interpolation of real or imaginary part. A total of eight evaluations is required for one complete function evaluation.A total of eight evaluations is required for one complete function evaluation.

8. * Note that the Lagrange interpolation regions along the x- and y- axes are so small that they barely appear on the graph.

9. Programming Techniques Calculates the Voigt profile for an entire spectral line at one time, removing unnecessary subroutine calls.Calculates the Voigt profile for an entire spectral line at one time, removing unnecessary subroutine calls. Each parameter involving y is calculated only once per spectral line, saving calculation time.Each parameter involving y is calculated only once per spectral line, saving calculation time. To do this we require equal spacing in wavenumber; a version of the routine called for individual points is available, but not as time efficient.To do this we require equal spacing in wavenumber; a version of the routine called for individual points is available, but not as time efficient. Interpolation tables stored as binary files on the hard drive and read in only when needed.Interpolation tables stored as binary files on the hard drive and read in only when needed. All files take up a total of 1.5 MB of memory, a small price to pay for the increase in accuracy and speed.All files take up a total of 1.5 MB of memory, a small price to pay for the increase in accuracy and speed.

10. Accuracy Comparisons Less than 10 -6 Less than 10 -4 ↓ ↑ J. Humlíček, J.Q.S.R.T., Vol. 27, 1982.

11. Maximum Error 7x10 -4 Maximum Error 1.5x10 -2 How bad do some routines get at small y? Routines calculating only the Real Part R. J. Wells, J.Q.S.R.T., Vol. 62, 1999 J. H. Pierluissi, J.Q.S.R.T., Vol. 18, 1977. Twitty, Rarig, & Thompson, J.Q.S.R.T.,Vol. 24, 1980. S.R. Drayson, J.Q.S.R.T., Vol.16, 1976.

12. Speed Comparisons Humlíček * Drayson calculates only the real part. ** These times are close to the final values, but the routine with derivatives is still undergoing final testing. *