行列値関数のソースを表示

'''行列値関数'''（ぎょうれつちかんすう、{{lang-en-short|matrix-valued functions}}）とは[[行列]]を変数に持つ[[特殊関数]]の総称である<ref name="higham">Higham, N. J. (2008). Functions of matrices: theory and computation. [[SIAM (学会)|SIAM]].</ref><ref name="chiba">千葉克裕; 行列の関数とジョルダン標準形, 2010. サイエンティスト社</ref>。
==定義==
[[ガンマ関数]]などを除けば、通常の特殊関数は多くの場合に常[[微分方程式]]の解 ([[可積分系]]の厳密解) として定義される。しかし行列値関数の場合は異なる。
===初等関数===
<math>\forall A\in\mathbb{C}^{n\times n},\quad f:\mathbb{C}\to\mathbb{C}</math>が与えられたとする。このとき<math>f(z)</math>がどのような関数であれば<math>f(A)</math>に行列としての意味を持たせられるかを考える。自然に思いつくのは多項式の場合：
:<math>f(z)=c_0+c_1z+\cdots+c_mz^m</math>
このときは当然ながら、
:<math>f(A)=c_0I+c_1A+\cdots+c_mA^m</math>
と定義するのが合理的である。この考えを発展させることで
:<math>f(z):=\sum_{k=0}^\infty c_kz^k</math>
と定義されているときには
:<math>f(A):=\sum_{k=0}^\infty c_kA^k</math>
と定義すればよいということが言える (もちろん行列からなる無限列の[[収束]]を適切に定義することも必要不可欠である)<ref name="higham"/><ref name="chiba"/>。例えば[[行列指数関数]]などの初等関数は次のように定められる<ref name="higham"/><ref name="chiba"/>:
:<math>\exp A:=\sum_{k=0}^\infty\frac{1}{k!}A^k,</math>
:<math>\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>
もしも<math>f(z)</math>がベキ級数表示を持たない場合はLagrange-Sylvester 多項式という道具を使って<math>f(A)</math>を定めることができる<ref name="chiba"/>。

===特殊関数===
代表的な[[特殊関数]]、具体的には
*[[ガンマ関数]]<ref>Jodar, L., & Cortés, J. C. (1998). Some properties of Gamma and Beta matrix functions. Applied Mathematics Letters, 11(1), 89-93.</ref>
*[[エアリー関数]]<ref>Kontsevich, M. (1992). Intersection theory on the moduli space of curves and the matrix Airy function. [[:en:Communications in Mathematical Physics]], 147(1), 1-23.</ref>
*[[ベッセル関数]]<ref>Herz, C. S. (1955). Bessel functions of matrix argument. [[Annals of Mathematics]](2), 61, 474-523.</ref>
*[[直交多項式]]<ref>Sinap, A., & Van Assche, W. (1996). Orthogonal matrix polynomials and applications. [[:en:Journal of Computational and Applied Mathematics]], 66(1-2), 27-52.</ref>
*[[超幾何級数]]<ref>Koev, P., & Edelman, A. (2008). Hypergeometric function of a matrix argument. Department of Mathematics, Massachusetts Institute of Technology (April 11 2008). URL https://math.mit.edu/~plamen/software/mhgref.html.</ref>
などの関数<ref>Gross, K. I., & Richards, D. S. P. (1987). Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Transactions of the American Mathematical Society, 301(2), 781-811.</ref><ref>Abdalla, M. (2018). Special matrix functions: Characteristics, achievements and future directions. Linear and Multilinear Algebra, 1-28.</ref>、もしくはその[[q類似]]についても行列バージョンを考えることができる<ref name="Gauss" group="Q">Salem, A. (2014). The basic Gauss hypergeometric matrix function and its matrix <math>q</math>-difference equation. Linear and Multilinear Algebra, 62(3), 347-361.</ref><ref name="gamma" group="Q">Salem, A. (2012). On a <math>q</math>-gamma and a <math>q</math>-beta matrix functions. Linear and Multilinear Algebra, 60(6), 683-696.</ref><ref group="Q">Salem, A. (2016). The <math>q</math>-Laguerre matrix polynomials. SpringerPlus, 5(1), 550.</ref><ref group="Q">Salem, A. (2017). On the Discrete <math>q</math>-Hermite Matrix Polynomials. International Journal of Applied and Computational Mathematics, 3(4), 3147-3158.</ref><ref group="Q">Dwivedi, R., & Sahai, V. (2019). On the matrix versions of <math>q</math>-zeta function, <math>q</math>-digamma function and <math>q</math>-polygamma function. Asian-European Journal of Mathematics.</ref><ref group="Q">Dwivedi, R., & Sahai, V. (2019). On the basic hypergeometric matrix functions of two variables. Linear and Multilinear Algebra, 67(1), 1-19.</ref>。例えば、行列からなる[[無限乗積]]の収束を適切に定義したうえで<ref>Trench, W. F. (1999). Invertibly convergent infinite products of matrices. [[:en:Journal of computational and applied mathematics]], 101(1-2), 255-263.</ref><ref>Trench, W. F. (1995). Invertibly convergent infinite products of matrices, with applications to difference equations. Computers & Mathematics with Applications, 30(11), 39-46.</ref>、[[qポッホハマー記号]]の行列バージョンは次のように定義できる<ref name="Gauss" group="Q"/><ref name="gamma" group="Q"/>。
:<math>(A;q)_n\!:=\!\prod_{k=0}^{n-1}(I-Aq^k), \quad (A;q)_{\infty}\!:=\!\lim_{n\to\infty}(A;q)_n,\quad|q|<1,\quad A\in\mathbb{C}^{n\times n} </math>
これを使って、[[:en:q-gamma function]]の行列バージョンも導入できる<ref name="gamma" group="Q"/>。
:<math>\Gamma_q(A):=(q;q)_\infty(q^A;q)_\infty^{-1}(1-q)^{I-A},\quad|q|<1 </math>

==工学的重要性==
[[行列指数関数]]は[[:en:exponential integrator]]などの[[常微分方程式の数値解法]]において必要である他<ref group="A">Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. [[:en:Acta Numerica]], 19, 209-286.</ref><ref group="A">Al-Mohy, A. H., & Higham, N. J. (2011). Computing the action of the matrix exponential, with an application to exponential integrators. [[:en:SIAM journal on scientific computing]], 33(2), 488-511.</ref><ref group="A">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.</ref><ref group="A">行列の指数関数に基づく連立線形常微分方程式の大粒度並列解法とその評価 ([[日本応用数理学会]]論文誌 Vol.19, No.3, 2009, pp.293--312) 則竹渚宇, 今倉暁, 山本有作, 張紹良
</ref><ref group="A">橋本悠香, & 野寺隆. (2016). 線形発展方程式のための Inexact Shift-invert Arnoldi 法. [[情報処理学会]]論文誌, 57(10), 2250-2259.</ref><ref group="A">A Note on Inexact Rational Krylov Method for Evolution Equations by Yuka Hashimoto and Takashi Nodera (2016), research report by the Department of Mathematics, Faculty of Science and Technology, Keio University.</ref><ref group="A">Hashimoto, Y., & Nodera, T. (2016). Inexact shift-invert Arnoldi method for evolution equations. ANZIAM Journal, 58, 1-27.</ref><ref group="A">Hashimoto, Y., & Nodera, T. (2016). Shift-invert rational Krylov method for evolution equations. ANZIAM Journal, 58, 149-161.</ref>、[[統計学]]などにおいて重要視されている<ref name="higham"/><ref>Leach, B. G. (1969). Bessel functions of matrix argument with statistical applications (Doctoral dissertation).</ref><ref>James, A. T. (1975). Special functions of matrix and single argument in statistics. In Theory and Application of Special Functions (pp. 497-520). [[:en:Academic Press]].</ref>。このような背景の下で、[[数値線形代数]]の研究者たちは行列値関数の高精度計算<ref name="higham"/><ref name="hpc"> 数値線形代数の数理とHPC, 櫻井鉄也, 松尾宇泰, 片桐孝洋編（シリーズ応用数理 / [[日本応用数理学会]]監修, 第6巻）[[共立出版]], 2018.8</ref><ref name="dh" group="B">Davies, P. I., & Higham, N. J. (2003). A Schur-Parlett algorithm for computing matrix functions. [[:en:SIAM Journal on Matrix Analysis and Applications]], 25(2), 464-485.</ref>・[[精度保証付き数値計算]]<ref>Miyajima, S. (2019). Verified computation of the matrix exponential. Advances in Computational Mathematics, 45(1), 137-152.</ref><ref>Miyajima, S. (2019). Verified computation for the matrix principal logarithm. Linear Algebra and its Applications, 569, 38-61.</ref><ref>Miyajima, S. (2018). Fast verified computation for the matrix principal pth root. [[:en:Journal of Computational and Applied Mathematics]], 330, 276-288.</ref><ref>Shinya Miyajima, Verified computation for the matrix Lambert W function, Applied Mathematics and Computation, Volume 362, Pages 1-15, December 2019.</ref>の研究に積極的に取り組んでいる。具体的には、以下の関数が取り組まれている。
*[[行列指数関数]]<ref name="19ways" group="B">Moler, C., & Van Loan, C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM review, 20(4), 801-836.</ref><ref name="25years" group="B">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.</ref><ref name="visit" group="B">Higham, N. J. (2005). The scaling and squaring method for the matrix exponential revisited. [[:en:SIAM Journal on Matrix Analysis and Applications]], 26(4), 1179-1193.</ref><ref group="B">Sidje, R. B. (1998). Expokit: A software package for computing matrix exponentials. ACM Transactions on Mathematical Software (TOMS), 24(1), 130-156.</ref><ref group="B">Yuka Hashimoto,Takashi Nodera, Double-shift-invert Arnoldi method for computing the matrix exponential, Japan J. Indust. Appl. Math, pp727-738, 2018.</ref>
*[[行列の平方根]]<ref name="bhm" group="B">Bini, D. A., Higham, N. J., & Meini, B. (2005). Algorithms for the matrix pth root. Numerical Algorithms, 39(4), 349-378.</ref><ref name="dhr" group="B">Deadman, E., Higham, N. J., & Ralha, R. (2012, June). Blocked Schur algorithms for computing the matrix square root. In International Workshop on Applied Parallel Computing (pp. 171-182). Springer, Berlin, Heidelberg.</ref><ref group="B">F. Tatsuoka, T. Sogabe, Y. Miyatake, S.-L. Zhang A cost-efficient variant of the incremental Newton iteration for the matrix pth root, J. Math. Res. Appl. 37 (2017), pp. 97-106.</ref><ref group="B">S. Mizuno, Y. Moriizumi, T. S. Usuda, T. Sogabe, An initial guess of Newton's method for the matrix square root based on a sphere constrained optimization problem, JSIAM Letters, 8 (2016), pp.17-20.</ref>
*行列の[[三角関数]]<ref name="hh" group="B">Hargreaves, G. I., & Higham, N. J. (2005). Efficient algorithms for the matrix cosine and sine. Numerical Algorithms, 40(4), 383-400.</ref>
*行列の実数乗<ref group="B">立岡文理，曽我部知広，宮武勇登，張紹良，[[二重指数関数型数値積分公式]]を用いた行列実数乗の計算，[[日本応用数理学会]]論文誌，Vol．28，No．3，2018，pp. 142-161</ref><ref name="tref" group="B">Hale, N., Higham, N. J., & Trefethen, L. N. (2008). Computing <math>A^\alpha,\log(A)</math>, and related matrix functions by contour integrals. [[:en:SIAM Journal on Numerical Analysis]], 46(5), 2505-2523.</ref>
*[[行列の対数]]<ref name="tref" group="B"/><ref group="B">Tatsuoka, F., Sogabe, T., Miyatake, Y., & Zhang, S. L. (2019). Algorithms for the computation of the matrix logarithm based on the double exponential formula. arXiv preprint arXiv:1901.07834.</ref>
*行列の[[ガンマ関数]]<ref group="B">Joao R. Cardoso, Amir Sadeghi, Computation of matrix gamma function, [[:en:BIT Numerical Mathematics]], (2019)</ref>
*行列の[[超幾何級数]]<ref group="B">Koev, P., & Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. [[:en:Mathematics of Computation]], 75(254), 833-846.</ref><ref group="B">Hashiguchi, H., Numata, Y., Takayama, N., & Takemura, A. (2013). The holonomic gradient method for the distribution function of the largest root of a Wishart matrix. Journal of Multivariate Analysis, 117, 296-312.</ref>

==関連項目==
===関連分野===
*[[行列解析]] ([[:en:matrix analysis|en]]) ([[線型代数学]]の中で行列値関数の性質を調べる分科)
*[[数値線形代数]]

===研究者===
*[[クリーブ・モラー]] ([[:en:Cleve Moler|en]])
*[[ニコラス・ハイアム]] ([[:en:Nicholas Higham|en]]) (行列値関数の研究で多くの業績がある<ref name="higham"/><ref name="dh" group="B"/><ref name="visit" group="B"/><ref name="bhm" group="B"/><ref name="dhr" group="B"/><ref name="hh" group="B"/>)

===主な行列値関数===
*[[行列指数関数]]
*[[行列の対数]]
*[[行列の平方根]]
==出典==
{{reflist|2}}
===<math>q-</math>類似===
{{reflist|group="Q"|2}}
===ODEの数値計算===
{{reflist|group="A"|2}}
===行列値関数の高精度計算===
{{reflist|group="B"|2}}

==参考文献==
* Higham, N. J. (2006). Functions of matrices. Manchester Institute for Mathematical Sciences, School of Mathematics, The University of Manchester.
* Higham, N. J. (2002). The matrix computation toolbox.
* 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.

==外部リンク==
*[https://mathtrain.jp/matrixexp 行列の指数関数とその性質]
*[http://www.sakuraacademia.jp/gyouretsu.html 行列関数の数値計算アルゴリズム開発]
*[https://researchmap.jp/jojsp7g9g-26434/ 行列引数超幾何関数の数値計算と holonomic Ｄ加群]、[[researchmap]]より
*{{PDFlink|[http://park.itc.u-tokyo.ac.jp/atstat/takemura-talks/120121-takemura-slide.pdf ウィシャート分布に現れる行列変数の超幾何関数に対するホロノミック勾配法]}}
*{{PDFlink|[https://nlagrouporg.files.wordpress.com/2019/06/anla19_cardoso.pdf Computation of matrix gamma function]}}
===Highamによる著作===
*{{PDFlink|[https://www.maths.manchester.ac.uk/~higham/talks/gautschi18.pdf Matrix Functions and their Sensitivity]}}
*{{PDFlink|[https://www.maths.manchester.ac.uk/~higham/talks/napier_log14.pdf The Matrix Logarithm: from Theory to Computation]}}
*{{PDFlink|[https://www.maths.manchester.ac.uk/~higham/talks/fun11.pdf Functions of a Matrix: Theory, Applications and Computation]}}
*{{PDFlink|[https://www.maths.manchester.ac.uk/~higham/talks/fun17.pdf Challenges in Multivalued Matrix Functions]}}
====行列指数関数====
*{{GitHub|higham/expmv}}
*{{PDFlink|[https://www.maths.manchester.ac.uk/~higham/talks/exp10.pdf How and How Not to Compute the Exponential of a Matrix]}}
*{{PDFlink|[https://www.maths.manchester.ac.uk/~higham/talks/an16_cvl.pdf Charlie Van Loan and the Matrix Exponential]}}
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{{デフォルトソート:きようれつちかんすう}}
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[[category:特殊関数]]
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