行列値関数とは - わかりやすく解説 Weblio辞書

ウィキペディア

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行列値関数

出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/01/19 15:06 UTC 版)

行列値関数（ぎょうれつちかんすう、英: matrix-valued functions）とは行列を変数に持つ特殊関数の総称である^[1]^[2]。

演算・操作	乗法転置逆行列サラスの方法基本変形対角化分解 LU QR コレスキーシューア特異値固有値
不変量	行列式跡階数固有値と固有ベクトル固有多項式最小多項式単因子階数標準形スミス標準形ジョルダン標準形有理標準形小行列式
クラス	正則基本対称エルミート正規直交回転ユニタリ冪零射影正定値ユニモジュラー置換区分

行列式

多重線型代数

数値線形代数

基本的な概念	浮動小数点数値的安定性疎行列 Z-行列 M行列条件数ゲルシュゴリンの定理クリロフ部分空間
ソフトウェア	Matrix Laboratory Basic Linear Algebra Subprograms (BLAS) LAPACK 数値解析ソフトの比較/一覧
ライブラリ	Armadillo INTLAB OpenBLAS 線型代数学ライブラリの比較
反復法・技法	SOR法共役勾配法 GMRES法固有値問題の数値解法ランチョス法 QR法べき乗法逆べき乗法ヤコビ法 (固有値問題) シュトラッセンのアルゴリズムハウスホルダー変換ギブンス回転
人物	ジェームズ・H・ウィルキンソンクリーブ・モラーニコラス・ハイアムジーン・ゴラブリチャード・バルガ

行列値関数

その他

出典

ODEの数値計算

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^ ^a ^b ^c ^d 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
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^ Miyajima, S. (2019). Verified computation of the matrix exponential. Advances in Computational Mathematics, 45(1), 137-152.
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行列値関数の高精度計算

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^ Salem, A. (2016). The $q$ -Laguerre matrix polynomials. SpringerPlus, 5(1), 550.
^ Salem, A. (2017). On the Discrete $q$ -Hermite Matrix Polynomials. International Journal of Applied and Computational Mathematics, 3(4), 3147-3158.
^ Dwivedi, R., & Sahai, V. (2019). On the matrix versions of $q$ -zeta function, $q$ -digamma function and $q$ -polygamma function. Asian-European Journal of Mathematics.
^ Dwivedi, R., & Sahai, V. (2019). On the basic hypergeometric matrix functions of two variables. Linear and Multilinear Algebra, 67(1), 1-19.

^ Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. en:Acta Numerica, 19, 209-286.
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^ 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.
^ 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.
^ Hashimoto, Y., & Nodera, T. (2016). Inexact shift-invert Arnoldi method for evolution equations. ANZIAM Journal, 58, 1-27.
^ Hashimoto, Y., & Nodera, T. (2016). Shift-invert rational Krylov method for evolution equations. ANZIAM Journal, 58, 149-161.

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^ ^a ^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.
^ 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.
^ ^a ^b Bini, D. A., Higham, N. J., & Meini, B. (2005). Algorithms for the matrix pth root. Numerical Algorithms, 39(4), 349-378.
^ ^a ^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.
^ 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.
^ 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.
^ ^a ^b Hargreaves, G. I., & Higham, N. J. (2005). Efficient algorithms for the matrix cosine and sine. Numerical Algorithms, 40(4), 383-400.
^ 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
^ ^a ^b Hale, N., Higham, N. J., & Trefethen, L. N. (2008). Computing $A^{\alpha },\log(A)$ , and related matrix functions by contour integrals. en:SIAM Journal on Numerical Analysis, 46(5), 2505-2523.
^ 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.
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