Efficient algorithm for generating spectra using line-by-lne methods

Abstract

A method is presented for efficient generation of spectra using line-by-line approaches. The only approximation is replacing the line shape function with an interpolation procedure, which makes the method independent of the line profile functional form. The resulting computational savings for large number of lines is proportional to the number of frequency points in the spectral range. Therefore, for large-scale problems the method can provide speedups of two orders of magnitude or more. A method was presented to generate line-by-line spectra efficiently. The first step was to replace the explicit calculation of the profile by the Newton divided-differences interpolating polynomial. The second step is to accumulate the lines effectively reducing their number to the number of frequency points. The final step is recognizing the resulting expression as a convolution and amenable to FFT methods. The reduction in computational effort for a configuration-to-configuration transition array with large number of lines is proportional to the number of frequency points. The method involves no approximations except for replacing the explicit profile evaluation by interpolation. Specifically, the line accumulation and convolution are exact given the interpolation procedure. Furthermore, the interpolation makes the method independent of the line profile functional form contrary to other schemes using FFT methods to generate line-by-line spectra but relying on the analytic form of the profile Fourier transform. Finally, the method relies on a uniform frequency mesh. For non-uniform frequency meshes, however, the method can be applied by using a suitable temporary uniform mesh and the results interpolated onto the final mesh with little additional cost.