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Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation

  • Mike B. Giles
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 64)

Summary

This paper collects together a number of matrix derivative results which are very useful in forward and reverse mode algorithmic differentiation. It highlights in particular the remarkable contribution of a 1948 paper by Dwyer and Macphail which derives the linear and adjoint sensitivities of a matrix product, inverse and determinant, and a number of related results motivated by applications in multivariate analysis in statistics.

Keywords

Forward mode reverse mode numerical linear algebra 

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References

  1. 1.
    Bischof, C., Bücker, H., Lang, B., Rasch, A., Vehreschild, A.: Combining source transformation and operator overloading techniques to compute derivatives for MATLAB programs. In: Proceedings of the Second IEEE International Workshop on Source Code Analysis and Manipulation (SCAM 2002), pp. 65–72. IEEE Computer Society (2002)Google Scholar
  2. 2.
    Coleman, T., Verma, A.: ADMIT-1: Automatic differentiation and MATLAB interface toolbox. ACM Transactions on Mathematical Software 26(1), 150–175 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dwyer, P.: Some applications of matrix derivatives in multivariate analysis. Journal of the American Statistical Association 62(318), 607–625 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dwyer, P., Macphail, M.: Symbolic matrix derivatives. The Annals of Mathematical Statistics 19(4), 517–534 (1948)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Forth, S.: An efficient overloaded implementation of forward mode automatic differentiation in MATLAB. ACM Transactions on Mathematical Software 32(2), 195–222 (2006)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Giles, M.: An extended collection of matrix derivative results for forward and reverse mode algorithmic differentiation. Tech. Rep. NA08/01, Oxford University Computing Laboratory (2008)Google Scholar
  7. 7.
    Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York (2004)zbMATHGoogle Scholar
  8. 8.
    Griewank, A.: Evaluating derivatives: Principles and techniques of algorithmic differentiation. SIAM (2000)Google Scholar
  9. 9.
    Higham, N.: The scaling and squaring method for the matrix exponential revisited. SIAM Journal on Matrix Analysis and Applications 26(4), 1179–1193 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kubota, K.: Matrix inversion algorithms by means of automatic differentiation. Applied Mathematics Letters 7(4), 19–22 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Magnus, J., Neudecker, H.: Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons (1988)Google Scholar
  12. 12.
    Mathai, A.: Jacobians of matrix transformations and functions of matrix argument. World Scientific, New York (1997)zbMATHGoogle Scholar
  13. 13.
    Minka, T.: Old and new matrix algebra useful for statistics. http://research.microsoft.com/~minka/papers/matrix/ (2000)
  14. 14.
    Rao, C.: Linear statistical inference and its applications. Wiley, New York (1973)zbMATHCrossRefGoogle Scholar
  15. 15.
    Rogers, G.: Matrix derivatives. Marcel Dekker, New York (1980)zbMATHGoogle Scholar
  16. 16.
    Squire, W., Trapp, G.: Using complex variables to estimate derivatives of real functions. SIAM Review 10(1), 110–112 (1998)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Srivastava, M., Khatri, C.: An introduction to multivariate statistics. North Holland, New York (1979)zbMATHGoogle Scholar
  18. 18.
    Verma, A.: ADMAT: automatic differentiation in MATLAB using object oriented methods. In: SIAM Interdiscplinary Workshop on Object Oriented Methods for Interoperability, pp. 174–183. SIAM (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mike B. Giles
    • 1
  1. 1.Oxford University Computing LaboratoryOxfordUK

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