固有値計算と特異値計算(電子書籍版)

出版社： 丸善出版
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発行日： 2020-02-28
分野： その他  >  一般
ISBN： 9784621304730
電子書籍版： 2020-02-28 （電子書籍版）
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固有値計算とは正方行列の特徴量である固有値・固有ベクトルを求めること、特異値計算とは一般の行列の特徴量である特異値・特異ベクトルを求めることを指す。特に多変量解析の主成分分析、重回帰分析、情報圧縮などの一手法として、コンピュータ性能の向上によって一般的になりつつあり、自然科学、工学をはじめとしたあらゆる分野での応用が進んでいる。本書では固有値計算・特異値計算における基礎理論の解説に重点を置き、より正確に、かつより速く値を求める方法を解説し、アルゴリズムを通して理解を深められるつくりとした。

目次

  • 表紙
  • 第2期の刊行にあたって
  • 序文
  • プログラムファイルのダウンロード
  • 目次
  • 1 あらまし
  • 1.1 固有値問題
  • 1.2 固有値と固有ベクトルの性質
  • 1.3 どの固有値が必要か
  • 1.4 アルゴリズム概観
  • 1.4.1 密行列の場合
  • 1.4.2 帯行列の場合
  • 1.4.3 疎行列の場合
  • 1.5 固有値と固有ベクトルの品質
  • 1.6 固有値問題ソフトウェア
  • 1.7 プログラム
  • 2 密行列の固有値計算
  • 2.1 単一固有値の計算法
  • 2.1.1 最大固有値の計算法 ( べき乗法 )
  • 2.1.2 最小固有値の計算法 ( 逆反復法 )
  • 2.1.3 2番目, 3番目の固有値の計算法
  • 2.2 ヤコビ法
  • 2.2.1 ヤコビ法のアルゴリズム
  • 2.2.2 ヤコビ法の収束
  • 2.2.3 固有ベクトルの計算
  • 2.3 QR 法
  • 2.3.1 QR 分解
  • 2.3.2 QR 法のアルゴリズム
  • 2.3.3 QR 法の収束
  • 2.3.4 原点シフトによる加速
  • 2.3.5 陰的ダブルシフト QR 法
  • 2.3.6 固有ベクトルの計算
  • 2.4 ハウスホルダー3重対角化を用いる方法
  • 2.4.1 ハウスホルダー変換
  • 2.4.2 ハウスホルダー QR 分解
  • 2.4.3 エルミート行列の3重対角化
  • 2.4.4 ハウスホルダー逆変換
  • 2.4.5 ハウスホルダー順変換に対する性能改善
  • 2.4.6 3重対角行列の固有値計算
  • 2.5 非対称行列の固有値計算
  • 2.6 一般化固有値問題
  • 2.6.1 B が対称正定値の場合
  • 2.6.2 一般の場合
  • 2.6.3 QZ 法
  • 2.7 非線形固有値問題
  • 2.7.1 非線形固有値問題の解法
  • 2.8 最新のアルゴリズム
  • 2.8.1 MRRR 法
  • 2.8.2 スペクトラル分割統治法
  • 3 疎行列の固有値計算
  • 3.1 レイリー - リッツ法
  • 3.2 非対称行列の解法
  • 3.2.1 アーノルディ法
  • 3.2.2 ヤコビ - ダビッドソン法
  • 3.3 対称行列の解法
  • 3.3.1 ランチョス法
  • 3.3.2 LOBPCG 法
  • 3.4 量子力学に現れる固有値問題
  • 3.4.1 ハバードモデルのハミルトニアン
  • 3.4.2 ハミルトニアンの固有値計算
  • 3.5 プログラム
  • 4 櫻井 - 杉浦法
  • 4.1 行列のスペクトル分解
  • 4.2 周回積分による固有ベクトルの抽出
  • 4.3 数値積分による近似
  • 4.4 非線形固有値問題への適用
  • 4.5 櫻井 - 杉浦法のアルゴリズム
  • 5 反復改良法
  • 5.1 連立1次方程式に対する反復改良法
  • 5.1.1 アルゴリズム
  • 5.1.2 数値実験
  • 5.2 固有値問題に対する反復改良法
  • 5.2.1 アルゴリズム
  • 5.2.2 数値実験
  • 5.3 実対称行列の全固有ベクトルに対する反復改良法
  • 5.3.1 固有分解
  • 5.3.2 アルゴリズム
  • 5.3.3 収束定理
  • 5.3.4 数値実験
  • 6 特異値問題
  • 6.1 特異値の性質
  • 6.1.1 他の数値計算との関係
  • 6.2 特異値計算アルゴリズム
  • 6.2.1 2重対角化アルゴリズム
  • 6.2.2 2重対角化アルゴリズム ( ブロック版 )
  • 6.2.3 2重対角行列の特異値計算
  • 7 高精度特異値分解
  • 7.1 QD 法
  • 7.2 陽的シフトつき QD 法
  • 7.3 DQDS 法
  • 7.3.1 実装の概略
  • 7.3.2 シフト戦略
  • 7.3.3 収束判定条件の設計
  • 7.4 OQDS 法
  • 7.4.1 特異ベクトル計算
  • 7.4.2 OQDS 法の行列要素表示
  • 7.4.3 シフト戦略
  • 7.4.4 ギブンス回転と一般化ギブンス回転の実装
  • 7.4.5 収束判定条件の設計
  • 7.5 プログラム
  • 参考文献
  • 索引
  • 奥付

この書籍の参考文献

参考文献のリンクは、リンク先の都合等により正しく表示されない場合がありますので、あらかじめご了承下さい。

本参考文献は電子書籍掲載内容を元にしております。

参考文献

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