Improved Steepest Descent Method using Modified Bessel Function K_(1/4 ) for Gamma Function Evaluation

Adam Badra Cahaya


Steepest descent method employs a Gaussian function when approximating an integral of a function. In this article we improve the approximation by using function in the form of  . As an example, we approximate the value of gamma function to provide improved approximation for Stirling formula that is often used for estimating factorial of a large number.


Steepest descent method; Stirling formula; gamma function

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W. A. Bowers, “More about approximations to n!,” Am. J. Phys., vol. 51, no. 9, p. 778, Sep. 1983, doi: 10.1119/1.13141.

W. Burnside, “A rapidly convergent series for log N,” Messenger Math, vol. 46, no. 1, pp. 157–159, 1917.

W. Schuster, “Improving Stirling’s formula,” Arch. der Math., vol. 77, pp. 170–176, 2001.

C. Leubner, “Generalised Stirling approximations to N!,” Eur. J. Phys., vol. 6, no. 4, pp. 299–301, Oct. 1985, doi: 10.1088/0143-0807/6/4/016.

S. S. Petrova and A. D. Solov’ev, “The Origin of the Method of Steepest Descent,” Hist. Math., vol. 24, no. 4, pp. 361–375, 1997, doi:

J. C. Meza, “Steepest descent,” WIREs Comput. Stat., vol. 2, no. 6, pp. 719–722, 2010, doi:

Y. Pu and J. Wang, “Fractional-order global optimal backpropagation machine trained by an improved fractional-order steepest descent method,” Front. Inf. Technol. Electron. Eng., vol. 21, no. 6, pp. 809–833, 2020, doi: 10.1631/FITEE.1900593.

J.-C. Son, M.-K. Baek, S.-H. Park, and D.-K. Lim, “Improved Immune Algorithm Combined with Steepest Descent Method for Optimal Design of IPMSM for FCEV Traction Motor,” Energies, vol. 14, no. 13, 2021, doi: 10.3390/en14133904.

E. J. Weniger and J. Cížek, “Rational approximations for the modified Bessel function of the second kind,” Comput. Phys. Commun., vol. 59, no. 3, pp. 471–493, 1990, doi:

Z.-H. Yang and Y.-M. Chu, “On approximating the modified Bessel function of the second kind,” J. Inequalities Appl., vol. 2017, no. 1, p. 41, 2017, doi: 10.1186/s13660-017-1317-z.

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