Matrix function
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
Most functions like [math]\displaystyle{ \exp(x) }[/math] are defined as a solution of a differential equation.[2] But matrix functions will use a different way.
Suppose [math]\displaystyle{ z }[/math] is a complex number and [math]\displaystyle{ A }[/math] is a square matrix. If you have a polynomial:
- [math]\displaystyle{ f(z):=c_0+c_1z+\cdots+c_mz^m }[/math],
then it is reasonable to define
- [math]\displaystyle{ f(A):=c_0I+c_1A+\cdots+c_mA^m. }[/math]
Let's use this idea. When you have
- [math]\displaystyle{ f(z):=\sum_{k=0}^\infty c_kz^k }[/math],
then you can introduce
- [math]\displaystyle{ f(A):=\sum_{k=0}^\infty c_kA^k. }[/math]
For example, the matrix version of the exponential function and the trigonometric functions are defined as follows:[1]
- [math]\displaystyle{ \exp A:=\sum_{k=0}^\infty\frac{1}{k!}A^k, }[/math]
- [math]\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}. }[/math]
Importance
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:
Related pages
References
- ↑ 1.0 1.1 1.2 1.3 Higham, Nicholas J. (2008). Functions of matrices theory and computation. Philadelphia: Society for Industrial and Applied Mathematics.
- ↑ Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions (Vol. 71). Cambridge University Press.
- ↑ Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286.
- ↑ 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.
- ↑ 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.
- ↑ 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.
- ↑ Moler, C., & Van Loan, C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM review, 20(4), 801-836.
- ↑ 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.
- ↑ 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.
- ↑ Sidje, R. B. (1998). Expokit: A software package for computing matrix exponentials. ACM Transactions on Mathematical Software (TOMS), 24(1), 130-156.
- ↑ Yuka Hashimoto,Takashi Nodera, Double-shift-invert Arnoldi method for computing the matrix exponential, Japan J. Indust. Appl. Math, pp727-738, 2018.
- ↑ Bini, D. A., Higham, N. J., & Meini, B. (2005). Algorithms for the matrix pth root. Numerical Algorithms, 39(4), 349-378.
- ↑ Hargreaves, G. I., & Higham, N. J. (2005). Efficient algorithms for the matrix cosine and sine. Numerical Algorithms, 40(4), 383-400.
- ↑ Hale, N., Higham, N. J., & Trefethen, L. N. (2008). Computing [math]\displaystyle{ A^\alpha,\log(A) }[/math], and related matrix functions by contour integrals. SIAM Journal on Numerical Analysis, 46(5), 2505-2523.
- ↑ Miyajima, S. (2019). Verified computation of the matrix exponential. Advances in Computational Mathematics, 45(1), 137-152.
- ↑ Miyajima, S. (2019). Verified computation for the matrix principal logarithm. Linear Algebra and its Applications, 569, 38-61.
- ↑ Miyajima, S. (2018). Fast verified computation for the matrix principal pth root. Journal of Computational and Applied Mathematics, 330, 276-288.
- ↑ Joao R. Cardoso, Amir Sadeghi, Computation of matrix gamma function, BIT Numerical Mathematics, (2019)
Further reading
- 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.