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Computation of matrix gamma function

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Abstract

Matrix functions have a major role in science and engineering. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other special matrix functions, is the matrix gamma function. This research article focus on the numerical computation of this function. Well-known techniques for the scalar gamma function, such as Lanczos, Spouge and Stirling approximations, are extended to the matrix case. This extension raises many challenging issues and several strategies used in the computation of matrix functions, like Schur decomposition and block Parlett recurrences, need to be incorporated to make the methods more effective. We also propose a fourth technique based on the reciprocal gamma function that is shown to be competitive with the other three methods in terms of accuracy, with the advantage of being rich in matrix multiplications. Strengths and weaknesses of the proposed methods are illustrated with a set of numerical examples. Bounds for truncation errors and other bounds related with the matrix gamma function will be discussed as well.

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References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1970)

    MATH  Google Scholar 

  2. Al-Mohy, A.H., Higham, N.J.: Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation. SIAM J. Matrix Anal. Appl. 30(4), 1639–1657 (2009)

    Article  MATH  Google Scholar 

  3. Al-Mohy, A.H., Higham, N.J.: Improved inverse scaling and squaring algorithms for the matrix logarithm. SIAM J. Sci. Comput. 34(4), C153–C169 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Al-Mohy, A.H., Higham, N.J., Relton, S.D.: Computing the Fréchet derivative of the matrix logarithm and estimating the condition number. SIAM J. Sci. Comput. 35(4), C394–C410 (2013)

    Article  MATH  Google Scholar 

  5. Askey, R., Roy, R.: Gamma function. In: Olver, F., Lozier, D., Boisvert, R., Clark, C. (eds.) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  6. Barradas, I., Cohen, J.E.: Iterated exponentiation, matrix–matrix exponentiation, and entropy. J. Math. Anal. Appl. 183, 76–88 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borwein, J.M., Corless, R.M.: Gamma and factorial in the Monthly. Am. Math. Mon. 125(5), 400–424 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourguet, L.: Sur les intégrales Eulériennes et quelques autres fonctions uniformes. Acta Mathematica 2, 261–295 (1883)

    Article  MathSciNet  MATH  Google Scholar 

  9. Braumann, C.A., Cortés, J.C., Jódar, L., Villafuerte, L.: On the random gamma function: theory and computing. J. Comput. Appl. Math. 335, 142–155 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cardoso, J.R., Sadeghi, A.: On the conditioning of the matrix–matrix exponentiation. Numer. Algorithms 79(2), 457–477 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Char, B.: On Stieltjes’ continued fraction for the gamma function. Math. Comput. 34(150), 547–551 (1980)

    MathSciNet  MATH  Google Scholar 

  12. Cortés, J.C., Jódar, L., Solís, F.J., Ku-Carrillo, R.: Infinite matrix products and the representation of the matrix gamma function. Abstract and Applied Analysis, vol. 2015, Article ID 564287. https://doi.org/10.1155/2015/564287 (2015)

  13. Davies, P.A., Higham, N.J.: A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2), 464–485 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Davis, P.J.: Leonhard Euler’s integral: a historical profile of the gamma function. Am. Math. Mon. 66, 849–869 (1959)

    MathSciNet  MATH  Google Scholar 

  15. Edwards, H.M.: Riemanns Zeta Function. Academic Press, Cambridge (1974)

    Google Scholar 

  16. Fekih-Ahmed, L.: On the power series expansion of the reciprocal gamma function, HAL archives. https://hal.archives-ouvertes.fr/hal-01029331v1 (2014). Accessed 16 Mar 2018

  17. Gautschi, W.: A computational procedure for incomplete gamma functions. ACM Trans. Math. Softw. 5(4), 466–481 (1979)

    Article  MATH  Google Scholar 

  18. Gautschi, W.: The incomplete gamma function since Tricomi. Atti Convegni Lincei 147, 203–237 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Gautschi, W.: A note on the recursive calculation of incomplete gamma functions. ACM Trans. Math. Softw. 25(1), 101–107 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Godfrey, P.: Lanczos implementation of the gamma function. http://www.numericana.com/answer/info/godfrey.htm (See also http://my.fit.edu/~gabdo/gamma.txt). Accessed 16 Mar 2018

  21. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)

    MATH  Google Scholar 

  22. Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Boston (1994)

    MATH  Google Scholar 

  23. Hale, N., Higham, N.J., Trefethen, L.: Computing \(A^{\alpha }\), \(\log (A)\), and related matrix functions by contour integrals. SIAM J. Numer. Anal. 46, 2505–2523 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2008)

    Book  MATH  Google Scholar 

  25. Higham, N.J.: The Matrix Function Toolbox. http://www.maths.manchester.ac.uk/~higham/mftoolbox/. Accessed 8 Feb 2018

  26. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, Paperback Edition. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  27. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  28. Jódar, L., Cortés, J.C.: On the hypergeometric matrix function. J. Comput. Appl. Math. 99, 205–217 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jódar, L., Cortés, J.C.: Some properties of gamma and beta functions. Appl. Math. Lett. 11(1), 89–93 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lanczos, C.: A precision approximation of the gamma function. J. Soc. Ind. Appl. Math. Ser. B Numer. Anal. 1, 86–96 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  31. Luke, Y.: The Special Functions and Their Approximations, vol. 1. Academic Press, New York (1969)

    MATH  Google Scholar 

  32. Mathias, R.: Approximation of matrix-valued functions. SIAM J. Matrix Anal. Appl. 14, 1061–1063 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  33. Parlett, B.N.: A recurrence among the elements of functions of triangular matrices. Linear Algebra Appl. 14, 117–121 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Paterson, M.S., Stockmeyer, L.J.: On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J. Comput. 2(1), 60–66 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pugh, G.R.: An analysis of the Lanczos gamma approximation. Ph.D. thesis, University of British Columbia (2004)

  36. Sastre, J., Jódar, L.: Asymptotics of the modified Bessel and incomplete gamma matrix functions. Appl. Math. Lett. 16(6), 815–820 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schmelzer, T., Trefethen, L.N.: Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45, 558–571 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Smith, D.: Algorithm 814: fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions. ACM Trans. Math. Softw. 27(4), 377–387 (2001)

    Article  MATH  Google Scholar 

  39. Spira, R.: Calculation of the gamma function by Stirling’s formula. Math. Comput. 25(114), 317–322 (1971)

    MathSciNet  MATH  Google Scholar 

  40. Spouge, J.: Computation of the gamma, digamma, and trigamma functions. SIAM J. Numer. Anal. 31(3), 931–944 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  41. Temme, N.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996)

    Book  MATH  Google Scholar 

  42. Trefethen, L.N., Weideman, J., Schmelzer, T.: Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Van Loan, C.: The sensitivity of the matrix exponential. SIAM J. Numer. Anal. 14(6), 971–981 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  44. Winitzki, S.: Computing the incomplete gamma function to arbitrary precision. In: Kumar, V., et al. (eds.) Lectures Notes on Computer Science, vol. 2667, pp. 790–798 (2003)

  45. Wrench, J.W.: Concerning two series for the gamma function. Math. Comput. 22, 617–626 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wrench, J.W.: Erratum: concerning two series for the gamma function. Math. Comput. 27, 681–682 (1973)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the anonymous reviewers for their helpful suggestions and comments. The work of the first author was supported by ISR-University of Coimbra (project UID/EEA/00048/2013) funded by “Fundação para a Ciência e a Tecnologia” (FCT). The work of the corresponding author is supported by Robat Karim branch, Islamic Azad University, Tehran, Iran.

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Authors and Affiliations

  1. Coimbra Polytechnic – ISEC, Coimbra, Portugal

    João R. Cardoso

  2. Institute of Systems and Robotics, University of Coimbra, Pólo II, Coimbra, Portugal

    João R. Cardoso

  3. Department of Mathematics, Robat Karim Branch, Islamic Azad University, Tehran, Iran

    Amir Sadeghi

Authors
  1. João R. Cardoso
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  2. Amir Sadeghi
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Correspondence to Amir Sadeghi.

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Communicated by Daniel Kressner.

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Cardoso, J.R., Sadeghi, A. Computation of matrix gamma function. Bit Numer Math 59, 343–370 (2019). https://doi.org/10.1007/s10543-018-00744-1

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