Fundamentals of matrix computations
Fundamentals of matrix computations
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DOI：
10.2307/2153000
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发表时间：
1991
期刊：
影响因子：
--
通讯作者：
D. S. Watkins
中科院分区：
文献类型：
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作者：
D. S. Watkins
前言。致谢。1高斯消去法及其变种。1.1矩阵乘法。1.2线性方程组。1.3三角系统。1.4正定系统的Cholesky分解。1.5带状正定系统。1.6稀疏正定系统。1.7高斯消去和LU分解。1.8高斯消去法和旋转法。1.9稀疏高斯消去法。2线性系统的灵敏度。2.1向量和矩阵范数。2.2条件编号。2.3摄动系数矩阵。2.4利用残差进行后验误差分析。2.5舍入误差向后稳定性。2.6舍入误差的传播。2.7高斯消去法的后向误差分析。2.8缩放。2.9成分敏感度分析。3最小二乘问题。3.1离散平方问题。3.2正交矩阵、旋转器和反射器。3.3最小二乘问题的解。3.4格拉姆-施密特进程。3.5几何逼近。3.6更新QR分解。4奇异值分解。4.1引言。4.2奇异值的一些基本应用。4.3奇异值分解与最小二乘问题。4.4最小二乘问题的灵敏度。5特征值和特征向量I.5.1微分方程组。5.2基本事实。5.3 Power方法和一些简单的扩展。5.4相似变换。5.5化为Hessenberg形式和三对角线形式。5.6弗朗西斯算法。5.7使用Francis算法计算特征向量。5.8 SVD恢复。6特征值和特征向量II.6.1特征空间和不变子空间。6.2子空间迭代和同步迭代。6.3 Krylov子空间和Francis算法。6.4大型稀疏特征值问题。6.5隐式重新启动。6.6雅可比-戴维森算法及相关算法。7特征值和特征向量III.7.1特征值和特征向量的灵敏度。7.2对称特征值问题的方法。7.3积本征值问题。7.4广义特征值问题。线性系统的8种迭代方法。8.1模型问题。8.2经典迭代法。8.3迭代法的收敛。8.4下降法最陡下降法。8.5关于停止标准。8.6预处理器。8.7共轭梯度法。8.8 CG算法的推导。8.9 CG算法的收敛速度。8.10不定和非对称问题。参考资料。索引。MatLab术语索引。
Preface. Acknowledgments. 1 Gaussian Elimination and Its Variants. 1.1 Matrix Multiplication. 1.2 Systems of Linear Equations. 1.3 Triangular Systems. 1.4 Positive Definite Systems Cholesky Decomposition. 1.5 Banded Positive Definite Systems. 1.6 Sparse Positive Definite Systems. 1.7 Gaussian Elimination and the LU Decomposition. 1.8 Gaussain Elimination and Pivoting. 1.9 Sparse Gaussian Elimination. 2 Sensitivity of Linear Systems. 2.1 Vector and Matrix Norms. 2.2 Condition Numbers. 2.3 Perturbing the Coefficient Matrix. 2.4 A Posteriori Error Analysis Using the Residual. 2.5 Roundoff Errors Backward Stability. 2.6 Propagation of Roundoff Errors. 2.7 Backward Error Analysis of Gaussian Elimination. 2.8 Scaling. 2.9 Componentwise Sensitivity Analysis. 3 The Least Squares Problem. 3.1 The Discrete Square Problem. 3.2 Orthogonal Matrices, Rotators and Reflectors. 3.3 Solution of the Least Squares Problem. 3.4 The Gram-Schmidt Process. 3.5 Geometric Approach. 3.6 Updating the QR Decomposition. 4 The Singular Value Decomposition. 4.1 Introduction. 4.2 Some Basic Applications of Singular Values. 4.3 The SVD and the Least Squares Problem. 4.4 Sensitivity of the Least Squares Problem. 5 Eigenvalues and Eigenvectors I. 5.1 Systems of Differential Equations. 5.2 Basic Facts. 5.3 The Power Method and Some Simple Extensions. 5.4 Similarity Transforms. 5.5 Reduction to Hessenberg and Tridiagonal Forms. 5.6 Francis's Algorithm. 5.7 Use of Francis's Algorithm to Calculate Eigenvectors. 5.8 The SVD Revisted. 6 Eigenvalues and Eigenvectors II. 6.1 Eigenspaces and Invariant Subspaces. 6.2 Subspace Iteration and Simultaneous Iteration. 6.3 Krylov Subspaces and Francis's Algorithm. 6.4 Large Sparse Eigenvalue Problems. 6.5 Implicit Restarts. 6.6 The Jacobi-Davidson and Related Algorithms. 7 Eigenvalues and Eigenvectors III. 7.1 Sensitivity of Eigenvalues and Eigenvectors. 7.2 Methods for the Symmetric Eigenvalue Problem. 7.3 Product Eigenvalue Problems. 7.4 The Generalized Eigenvalue Problem. 8 Iterative Methods for Linear Systems. 8.1 A Model Problem. 8.2 The Classical Iterative Methods. 8.3 Convergence of Iterative Methods. 8.4 Descent Methods Steepest Descent. 8.5 On Stopping Criteria. 8.6 Preconditioners. 8.7 The Conjugate-Gradient Method. 8.8 Derivation of the CG Algorithm. 8.9 Convergence of the CG Algorithm. 8.10 Indefinite and Nonsymmetric Problems. References. Index. Index of MATLAB Terms.