Testing Aigau

Original sglitch

            The code with the AiGau showing the original glitch detailed below is in osglitch.zip.  #Changes

Data

02/28/2006  05:05a             385,077 AIGAUSS.OUT  -- data previously fitted – generated by starting with bracket and then directly integrating the Gaussian with Gauss Legendre quadrature.  – old data moved to osglitch.zip

03/06/2006  05:45a             159,968 FITMLAG.OUT  The fit – the Gauss Laguerre – generated by sglitch.wpj

03/06/2006  05:48a             146,720 LAGMLEG.OUT The Gauss Laguerre quadrature – the Gauss Legendre (AiGauss.out) – generated by sglitch3.wpj

Ides

03/06/2006  05:17a                 245 sglitch.wpj – compares aigau – fit to Gauss Laguerre integration

03/06/2006  05:30a                 251 sglitch2.wpj  -- compares aigau – fit to aigauss.out – original data

03/06/2006  05:35a                 251 sglitch3.wpj

Fortran

03/03/2006  04:18p               2,571 AIGAU.FOR  -- copied from ..\

03/06/2006  04:45a                 599 SGLITCH.FOR – generated for this work

03/06/2006  04:41a                 476 BRGLAGU.FOR  --copied from ..\aiglz – This folder contains Gauss Laguerre details.

03/05/2006  08:19p               2,277 GLAGU.FOR --copied from ..\aiglz

Figure 1  Difference between current approximation to Aigau and that from Gauss Laguerre integration.

Figure 2  The glitch clearly begins at -6.0

The value -6 is the last point for which the asymptotic expansion is used.  The result above implies that the fitting has been to a set of points containing this glitch.

The fitted data

Figure 3  The fitted data – aigau

sglitch2.wpj

Figure 4  Black is fit - Laguerre integration, blue is Laguerre - data fitted Gauss Quadrature on the Gaussian.

Conclusion

            Clearly out to about -2.5, Gauss Laguerre integration is more accurate than the direct Gauss Quadrature.

This is due to the fact that the exponential has been analytically removed from the Gauss Laguerre integral.  The glitch is an end point effect.  This will probably be best removed by extending the fit to -7 and using Gauss Laguerre for the z < -3.

Changes

            The codes in ../genint/Welcome.htm were changed to reflect this.  The Brack was refitted using codes in ../fitting/Welcome.htm and the results were displayed using sglitch.wpj

Figure 5  |AiGauss(z)-Aiglz(z)|/AiGauss(z)

            The relative error is less than 1e-15 and the error in the function AiGauss(z) is equal to the double precision limit.