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A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis

Received: 20 March 2022     Accepted: 6 April 2022     Published: 25 April 2022
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Abstract

Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].

Published in Applied and Computational Mathematics (Volume 11, Issue 2)
DOI 10.11648/j.acm.20221102.13
Page(s) 56-59
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Faddeyeva Function, Complex Probability Function, Voigt Function, W4 Algorithm

References
[1] Faddeyeva, V. N. And Terent’ev, N. M. Tables of Values of the Function W(z) for Complex Argument. Gosud. Izdat. The.-Teor. Lit., Moscow, 195; English Transl., Pergamon Press, New York 1961.
[2] Young C. Calculation of the Absorption Coefficient for Lines with Combined Doppler and Lorentz Broadening. J Quant Spectrosc Radiat Transfer 1965: 5: 549-552.
[3] Armstrong, B. H. Spectrum Line Profiles: The Voigt Function. J Quant Spectrosc Radiat Transfer 1967: 7: 61-88 doi: 10.1016/0022-4073(67)90057-X.
[4] Gautschi, W. Algorithm 363-Complex error function,” Commun. ACM 1969: 12: 635.
[5] Gautschi, W. Efficient Computation of the Complex Error Function. SIAM J. Numer. Anal 1970: 7: 187-198 doi: 10.1137/0707012.
[6] Hui, A. K., Armstrong, B. H. and Wray, A. A. Rapid Computation of the Voigt and Complex Error Functions. J Quant Spectrosc Radiat Transfer 1978: 19: 509-516. doi: 10.1016/0022-4073(78)90019-5.
[7] Humlíček J. Optimized Computation of the Voigt and Complex Probability Functions. J Quant Spectrosc Radiat Transfer 1982: 27: 4: 437-444 doi: 10.1016/0022-4073(82)90078-4.
[8] Dominguez, H. J., Llamas, H. F. Prieto, A. C. and Ortega, A. B. A Simple Relationship between the Voigt Integral and the Plasma Dispersion Function. Additional Methods to Estimate the Voigt Integral. Nuclear Instruments and Methods in Physics Research A 1987: 278: 625-626.
[9] Poppe, G. P. M. and Wijers, C. M. J. More Efficient Computation of the Complex Error Function. ACM Transactions on Mathematical Software 1990: 16: No. 1, 38-46.
[10] Poppe G. P. M. and Wijers, C. M. J. Algorithm 680, Evaluation of the Complex Error Function. ACM Transactions on Mathematical Software 1990: 16: No. 1: 47.
[11] Lether F. G. and Wenston P. R. The numerical computation of the Voigt function by a corrected midpoint quadrature rule for (-∞, ∞). Journal of Computational & Applied Mathematics 1991: 34: 75-92.
[12] Schreier, F. The Voigt and Complex Error Function: A Comparison of Computational Methods. J Quant Spectrosc Radiat Transfer 1992: 48: No. 5/6: 743-762 doi: 10.1016/0022-4073(92)90139-U.
[13] Shippony Z. and Read W. G. A highly accurate Voigt function algorithm. J Quant Spectrosc Radiat Transfer 1993: 50: 635-646.
[14] Weideman, J. A. C. Computation of the Complex Error Function. SIAM J. Numer. Anal 1994: 31: No. 5: 1497-1518.
[15] Kuntz M. A new implementation of the Humlicek algorithm for the calculation of the Voigt profile function. J Quant Spectrosc Radiat Transfer 1997: 57: No. 6: 819-824. doi 10.1016/S0022-4073(96)00162-8.
[16] Wells, R. J. Rapid Approximation to the Voigt/Faddeeva Function and its Derivatives. J Quant Spectrosc Radiat Transfer 1999: 62: 29-48. doi 10.1016/S0022-4073(97)00231-8.
[17] Shippony Z. and Read W. G. A correction to a highly accurate Voigt function algorithm. J Quant Spectrosc Radiat Transfer 2003: 78: 2: 255.
[18] Ruyten, W. Comment on “A new implementation of the Humlicek algorithm for the calculation of the Voigt profile function by M. Kuntz [JQSRT 57 (6) (1997) 819-824] J Quant Spectrosc Radiat Transfer 2004: 86: 231-233 doi 10.1016/j.jqsrt.2003.12.027.
[19] Luque, J. M. Calzada, M. D. And Saez, M. A new procedure for obtaining the Voigt function dependent upon the complex error function. J Quant Spectrosc Radiat Transfer 2005: 94: 151-161.
[20] Garcia T. T. Voigt profile fitting to quasar absorption lines: an analytic approximation to the Voigt-Hjerting function. Mon. Not. R. Astron. Soc. 2006: 369: 2025-2035. doi: 10.1111/j.1365-2966.2006.10450.x.
[21] Letchworth K. L. and Benner D. C. Rapid and accurate calculation of the Voigt function. J Quant Spectrosc Radiat Transfer 2007: 107: 173-192.
[22] Abrarov S. M., Quine B. M. and Jagpal R. K. Rapidly convergent series for high-accuracy calculation of the Voigt function. J Quant Spectrosc Radiat Transfer 2010: 111: 372-375. doi: 10.1016/j.jqsrt.2009.09.005.
[23] Imai K., Sizuki M. and Takahashi C. Evaluation of Voigt algorithms for the ISS/SMILES L2 data processing system. Advances in Space Research 2010: 45: 669-675. doi 10.1016/j.asr.2009.11.005.
[24] Schreier, F. Optimized implementations of rational approximations for the Voigt and complex error function. J Quant Spectrosc Radiat Transfer 2011: 112: 1010-1025 doi 10.1016/j.jqsrt.2010.12.010.
[25] Zaghloul, M. R. and Ali, A. N. Algorithm 916: Computing the Faddeyeva and Voigt Functions. ACM Trans. Math. Software (TOMS) 2011: 38: 2: article 15: 1-22.
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  • APA Style

    Mofreh Ramadan Zaghloul. (2022). A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Applied and Computational Mathematics, 11(2), 56-59. https://doi.org/10.11648/j.acm.20221102.13

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    ACS Style

    Mofreh Ramadan Zaghloul. A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Appl. Comput. Math. 2022, 11(2), 56-59. doi: 10.11648/j.acm.20221102.13

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    AMA Style

    Mofreh Ramadan Zaghloul. A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Appl Comput Math. 2022;11(2):56-59. doi: 10.11648/j.acm.20221102.13

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  • @article{10.11648/j.acm.20221102.13,
      author = {Mofreh Ramadan Zaghloul},
      title = {A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis},
      journal = {Applied and Computational Mathematics},
      volume = {11},
      number = {2},
      pages = {56-59},
      doi = {10.11648/j.acm.20221102.13},
      url = {https://doi.org/10.11648/j.acm.20221102.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221102.13},
      abstract = {Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].},
     year = {2022}
    }
    

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    T1  - A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis
    AU  - Mofreh Ramadan Zaghloul
    Y1  - 2022/04/25
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    N1  - https://doi.org/10.11648/j.acm.20221102.13
    DO  - 10.11648/j.acm.20221102.13
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20221102.13
    AB  - Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].
    VL  - 11
    IS  - 2
    ER  - 

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Author Information
  • Department of Physics, College of Sciences, United Arab Emirates University, AlAin, United Arab Emirates

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