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Matrix function

From Simple English Wikipedia, the free encyclopedia

In mathematics, a function maps an input value to an output value. In the case of a matrix function, the input and the output values are matrices. One example of a matrix function occurs with the Algebraic Riccati equation, which is used to solve certain optimal control problems.

Matrix functions are special functions made by matrices.[1]

Definitions

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Most functions like exp ( x ) {\displaystyle \exp(x)} are defined as a solution of a differential equation.[2] But matrix functions will use a different way.

Suppose z {\displaystyle z} is a complex number and A {\displaystyle A} is a square matrix. If you have a polynomial:

f ( z ) := c 0 + c 1 z + + c m z m {\displaystyle f(z):=c_{0}+c_{1}z+\cdots +c_{m}z^{m}} ,

then it is reasonable to define

f ( A ) := c 0 I + c 1 A + + c m A m . {\displaystyle f(A):=c_{0}I+c_{1}A+\cdots +c_{m}A^{m}.}

Let's use this idea. When you have

f ( z ) := k = 0 c k z k {\displaystyle f(z):=\sum _{k=0}^{\infty }c_{k}z^{k}} ,

then you can introduce

f ( A ) := k = 0 c k A k . {\displaystyle f(A):=\sum _{k=0}^{\infty }c_{k}A^{k}.}

For example, the matrix version of the exponential function and the trigonometric functions are defined as follows:[1]

exp A := k = 0 1 k ! A k , {\displaystyle \exp A:=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k},}
sin A := k = 0 ( - 1 ) k ( 2 k + 1 ) ! A 2 k + 1 , cos A := k = 0 ( - 1 ) k ( 2 k ) ! A 2 k . {\displaystyle \sin A:=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}A^{2k+1},\quad \cos A:=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}A^{2k}.}

Importance

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Matrix functions are used at numerical methods for ordinary differential equations[3][4][5] and statistics.[1][6] This is why numerical analysts are studying how to compute them.[1] For example, the following functions are studied:

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References

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  1. 1 2 3 4 Higham, Nicholas J. (2008). Functions of matrices theory and computation. Philadelphia: Society for Industrial and Applied Mathematics.
  2. | Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions (Vol. 71). Cambridge University Press.
  3. | Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286.
  4. | Al-Mohy, A. H., & Higham, N. J. (2011). Computing the action of the matrix exponential, with an application to exponential integrators. SIAM journal on scientific computing, 33(2), 488-511.
  5. | Del Buono, N., & Lopez, L. (2003, June). A survey on methods for computing matrix exponentials in numerical schemes for ODEs. In International Conference on Computational Science (pp. 111-120). Springer, Berlin, Heidelberg.
  6. | James, A. T. (1975). Special functions of matrix and single argument in statistics. In Theory and Application of Special Functions (pp. 497-520). Academic Press.
  7. | Moler, C., & Van Loan, C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM review, 20(4), 801-836.
  8. | Moler, C., & Van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM review, 45(1), 3-49.
  9. | Higham, N. J. (2005). The scaling and squaring method for the matrix exponential revisited. SIAM Journal on Matrix Analysis and Applications, 26(4), 1179-1193.
  10. | Sidje, R. B. (1998). Expokit: A software package for computing matrix exponentials. ACM Transactions on Mathematical Software (TOMS), 24(1), 130-156.
  11. | Yuka Hashimoto,Takashi Nodera, Double-shift-invert Arnoldi method for computing the matrix exponential, Japan J. Indust. Appl. Math, pp727-738, 2018.
  12. | Bini, D. A., Higham, N. J., & Meini, B. (2005). Algorithms for the matrix pth root. Numerical Algorithms, 39(4), 349-378.
  13. | Hargreaves, G. I., & Higham, N. J. (2005). Efficient algorithms for the matrix cosine and sine. Numerical Algorithms, 40(4), 383-400.
  14. | Hale, N., Higham, N. J., & Trefethen, L. N. (2008). Computing A a , log ( A ) {\displaystyle A^{\alpha },\log(A)} , and related matrix functions by contour integrals. SIAM Journal on Numerical Analysis, 46(5), 2505-2523.
  15. | Miyajima, S. (2019). Verified computation of the matrix exponential. Advances in Computational Mathematics, 45(1), 137-152.
  16. | Miyajima, S. (2019). Verified computation for the matrix principal logarithm. Linear Algebra and its Applications, 569, 38-61.
  17. | Miyajima, S. (2018). Fast verified computation for the matrix principal pth root. Journal of Computational and Applied Mathematics, 330, 276-288.
  18. | Joao R. Cardoso, Amir Sadeghi, Computation of matrix gamma function, BIT Numerical Mathematics, (2019)

Further reading

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  • A Survey of the Matrix Exponential Formulae with Some Applications (2016), Baoying Zheng, Lin Zhang, Minhyung Cho, and Junde Wu. J. Math. Study Vol. 49, No. 4, pp. 393-428.
  • Higham, N. J. (2006). Functions of matrices. Manchester Institute for Mathematical Sciences, School of Mathematics, The University of Manchester.