一类实矩阵对广义奇异值的表达公式

沈卫杰; 汤天宇; 徐玮玮

doi:10.6054/j.jscnun.2021100

一类实矩阵对广义奇异值的表达公式

南京信息工程大学数学与统计学院, 南京 210044

基金项目:

国家自然科学基金项目 11971243

江苏省自然科学基金项目 BK20181405

详细信息

通讯作者:
徐玮玮, Email: 002415@nuist.edu.cn

中图分类号: O241.6
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出版历程
- 收稿日期: 2021-04-01
- 网络出版日期: 2022-01-09
- 刊出日期: 2021-12-24

The Formula of Generalized Singular Values of a Class of Real Matrix Pairs

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

摘要

摘要: 应用双随机矩阵的性质, 首先得到了一类实对角矩阵迹函数优化问题的解析解, 再由该解析解得到了计算实矩阵对第i个广义奇异值的表达公式, 最后数值算例验证了结论的有效性.
- 实矩阵对 /
- 广义奇异值 /
- 广义奇异值分解 /
- 迹函数
Abstract: Using the properties of doubly stochastic matrices, the analytic solutions to a class of real diagonal matrix optimization problems are obtained, and then the formula for the i-th generalized singular value of a real matrix pair is obtained from the analytic solutions. Finally, numerical examples verify the validity of the conclusion.
- real matrix pair /
- generalized singular value /
- generalized singular value decomposition /
- trace function

HTML全文

特征值和奇异值在数值计算领域中有着重要的应用^[1]. 矩阵对的广义奇异值分解(Genera-lized Singular Value Decomposition, GSVD)已成为许多实际应用中重要的计算方法^[2-8]. 近年来, 学者们对大规模数据矩阵中的GSVD进行了大量研究, 如: ZHANG等^[5]应用神经动力学网络开发了一种用于求解GSVD的近似值的神经网络模型; WEI等^[6]使用随机投影捕获矩阵行为, 提出一种用于计算GSVD的低秩逼近的随机算法; 刘圆圆^[7]引入随机方法, 提出了一种改进的GSVD方法, 将其用于求解具有一般形式的大规模线性离散病态问题；XU等^[9-10]研究了一类酉约束矩阵优化问题的解析解.

受文献[9-10]启发, 本文将利用迹函数的矩阵优化问题来计算实矩阵对的广义奇异值, 提出计算一类实矩阵对的任意广义奇异值的新公式, 并用数值算例来验证本文结论的有效性.

1. 预备知识

本文中, ${\mathbb{R}}$ ^m×n表示m行n列实矩阵的集合, ${\mathbb{O}}$ _n表示n阶正交矩阵的集合; I_n为n阶单位矩阵, O_m×n为m×n零矩阵, 用diag(α₁, …, α_n)表示对角元素为α₁, …, α_n的对角矩阵；det(·)表示行列式函数, tr(·)表示迹函数；rank(A)表示矩阵A的秩, A^T和A^-1分别表示A的转置矩阵和A的逆矩阵, σ_i(A)表示矩阵A的第i个奇异值.

下面给出一些基本定义和定理.

定义1 ^[11] 若A ∈ ${\mathbb{R}}$ ^m×n, B ∈ ${\mathbb{R}}$ ^p×n, 有rank(A^T, B^T)=n, 则称矩阵对{A, B}是(m, p, n)实矩阵对.

定义2 ^[11] 若(α, β)≠(0, 0), det(β² A^T A-α² B^T B)=0, α, β≥0, 则称(α, β)是实矩阵对{A, B}的广义奇异值. {A, B}的广义奇异值(α, β)的集合记作σ{A, B}.

定理1 ^[12] 若{A, B}是一个(m, p, n)实矩阵对, 则存在正交矩阵U ∈ ${\mathbb{R}}$ ^m×m和V ∈ ${\mathbb{R}}$ ^p×p以及非奇异矩阵W ∈ ${\mathbb{R}}$ ^n×n, 使得

$\boldsymbol{U}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{W}^{-1}=\boldsymbol{\varSigma}_{A}, \boldsymbol{V}^{\mathrm{T}} \boldsymbol{B} \boldsymbol{W}^{-1}=\boldsymbol{\varSigma}_{B},$

(1)

$\boldsymbol{\varSigma}_{\boldsymbol{A}}=\left(\begin{array}{ll} \boldsymbol{\varLambda} & \\ & \boldsymbol{O}_{(m-r-s) \times(n-r-s)} \end{array}\right), \boldsymbol{\varSigma}_{{\boldsymbol{B}}}=\left(\begin{array}{ll} \boldsymbol{O}_{(p+r-n) \times r} & \\ & \boldsymbol{\varOmega} \end{array}\right),$

(2)

其中,

$\boldsymbol{\varLambda}=\operatorname{diag}\left(\alpha_{1}, \cdots, \alpha_{r+s}\right), \boldsymbol{\varOmega}=\operatorname{diag}\left(\beta_{r+1}, \cdots, \beta_{n}\right),$

$1=\alpha_{1}=\cdots=\alpha_{r}>\alpha_{r+1} \geqslant \cdots \geqslant \alpha_{r+s}>\alpha_{r+s+1}=\cdots=\alpha_{n}=0,$

$0=\beta_{1}=\cdots=\beta_{r}<\beta_{r+1} \leqslant \cdots \leqslant \beta_{r+s}<\beta_{r+s+1}=\cdots=\beta_{n}=1,$

$\alpha_{i}^{2}+\beta_{i}^{2}=1 \quad(1 \leqslant i \leqslant n) .$

定义3 ^[13] 设A=(a_ij)∈ ${\mathbb{R}}$ ^n×n, 如果对所有的i, j都有a_ij≥0, 则称A是非负矩阵.

定义4 ^[13] 设A=(a_ij)∈ ${\mathbb{R}}$ ^n×n是非负矩阵, 如果A满足

$\sum\limits_{j=1}^{n} a_{i j}=1 \quad(i=1,2, \cdots, n),$

$\sum\limits_{i=1}^{n} a_{i j}=1 \quad(j=1,2, \cdots, n),$

则称A为双随机矩阵.

2. 主要结论

本节将通过迹函数的优化问题给出一类实矩阵对广义奇异值的表达公式. 首先, 给出一些必要的引理.

引理1 ^[14] 如果D=(d_ij)是一个n阶双随机矩阵, 并且

$x_{1} \geqslant \cdots \geqslant x_{n} \geqslant 0, y_{1} \geqslant \cdots \geqslant y_{n} \geqslant 0,$

则

$\sum\limits_{i, j=1}^{n} d_{i j} x_{i} y_{j} \leqslant \sum\limits_{i=1}^{n} x_{i} y_{i} .$

引理2 令Γ=diag(γ₁, …, γ_n)和Δ=diag(δ₁, …, δ_n)是 ${\mathbb{R}}$ ^n×n上的2个对角矩阵, γ₁≥…≥γ_n≥0, δ₁≥…≥δ_n≥0, 则

$\max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{\varGamma}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varPhi} \boldsymbol{\varGamma}\right)=\sum\limits_{i=1}^{n} \gamma_{i}^{2} \delta_{i} .$

证明令Φ^*= I_n, 则

$\begin{aligned} &\max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{\varGamma}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varPhi} \boldsymbol{\varGamma}\right) \geqslant \operatorname{tr}\left(\boldsymbol{\varGamma}^{\mathrm{T}}\left(\boldsymbol{\varPhi}^{*}\right)^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varPhi}^{*} \boldsymbol{\varGamma}\right)= \\ &\ \ \ \ \ \ \ \ \operatorname{tr}\left(\boldsymbol{\varGamma}^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varGamma}\right)=\sum\limits_{i=1}^{n} \gamma_{i}^{2} \delta_{i} . \end{aligned}$

另一方面, 因为tr(AB)=tr(BA), 有

$\begin{gathered} \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{\varGamma}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varPhi} \boldsymbol{\varGamma}\right)=\max \limits_{\boldsymbol{\varPhi} \in \boldsymbol{O}_{n}} \operatorname{tr}\left(\boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varPhi} \boldsymbol{\varGamma} \boldsymbol{\varGamma}^{\mathrm{T}}\right)= \\ \ \ \ \ \ \ \ \ \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varPhi} \boldsymbol{\varGamma}^{2}\right)=\max \limits_{\boldsymbol{\varPhi}=\left(\varphi_{i j}\right) \in \mathbb{O}_{n}}\left(\sum\limits_{i, j=1}^{n} \varphi_{i j}^{2} \delta_{i} \gamma_{j}^{2}\right), \end{gathered}$

则可知矩阵(φ_ij²)是双随机矩阵. 由引理1, 有

$\max \limits_{{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{\varGamma}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{\varDelta} \boldsymbol{\varPhi} \boldsymbol{\varGamma}\right)=\max \limits_{\boldsymbol{\varPhi}=\left(\varphi_{i j}\right) \in \mathbb{O}_{n}}\left(\sum\limits_{i, j=1}^{n} \varphi_{i j}^{2} \delta_{i} \gamma_{j}^{2}\right) \leqslant \sum\limits_{i=1}^{n} \gamma_{i}^{2} \delta_{i} .$

证毕.

由引理2可得以下推论.

推论1 令Γ=diag(γ₁, …, γ_n)和Δ=diag(δ₁, …, δ_n)是 ${\mathbb{R}}$ ^n×n上的2个对角矩阵, 0≤γ₁≤…≤γ_n, 0≤δ₁≤…≤δ_n, 则

下面给出一类(n, n, n)实矩阵对{A, B}的广义奇异值{(α_i, β_i)}_i=1ⁿ的表达公式.

定理2 设{A, B}是(n, n, n)实矩阵对, 其广义奇异值为{(α_i, β_i)}_i=1ⁿ. 则对于1≤i≤n, 有

$\begin{aligned} \alpha_{i}^{2}=& \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i} \boldsymbol{\varPhi} \boldsymbol{A}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}\right)^{-1}\right)-\\ & \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i-1} \boldsymbol{\varPhi} \boldsymbol{A}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}\right)^{-1}\right), \end{aligned}$

(3)

其中,

$\boldsymbol{Q}_{i}=\operatorname{diag}\left(\boldsymbol{I}_{i}, \boldsymbol{O}_{(n-i) \times(n-i)}\right).$

(4)

证明令{A, B}是由式(1)和式(2)定义的(n, n, n)实矩阵对, 则有

$\boldsymbol{\varSigma}_{\boldsymbol{A}}^{\mathrm{T}} \boldsymbol{\varSigma}_{\boldsymbol{A}}+\boldsymbol{\varSigma}_{\boldsymbol{B}}^{\mathrm{T}} \boldsymbol{\varSigma}_{\boldsymbol{B}}=\boldsymbol{I}_{n}.$

进一步地, 有

$\left(\boldsymbol{U}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{W}^{-1}\right)^{\mathrm{T}}\left(\boldsymbol{U}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{W}^{-1}\right)+\left(\boldsymbol{V}^{\mathrm{T}} \boldsymbol{B} \boldsymbol{W}^{-1}\right)^{\mathrm{T}}\left(\boldsymbol{V}^{\mathrm{T}} \boldsymbol{B} \boldsymbol{W}^{-1}\right)=\boldsymbol{I}_{n},$

化简得

$\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}=\boldsymbol{W}^{\mathrm{T}} \boldsymbol{W}.$

对任意的1≤i≤n, 令Q_i由式(4)定义. 因为tr(AB)=tr(BA), 则由引理2, 可得

$\begin{aligned} \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} &\ \operatorname{tr}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i} \boldsymbol{\varPhi} \boldsymbol{A}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}\right)^{-1}\right)=\\ & \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i} \boldsymbol{\varPhi} \boldsymbol{A}\left(\boldsymbol{W}^{\mathrm{T}} \boldsymbol{W}\right)^{-1}\right)=\\ & \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\left(\boldsymbol{W}^{\mathrm{T}}\right)^{-1} \boldsymbol{A}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i} \boldsymbol{\varPhi} \boldsymbol{A} \boldsymbol{W}^{-1}\right)=\\ & \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\left(\boldsymbol{A} \boldsymbol{W}^{-1}\right)^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i} \boldsymbol{\varPhi} \boldsymbol{A} \boldsymbol{W}^{-1}\right)=\\ & \max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{\varSigma}_{A}^{\mathrm{T}} \boldsymbol{U}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i} \boldsymbol{\varPhi} \boldsymbol{U} \boldsymbol{\varSigma}_{A}\right)=\alpha_{1}^{2}+\cdots+\alpha_{i}^{2} . \end{aligned}$

类似地,

$\max \limits_{\boldsymbol{\varPhi} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{\varPhi}^{\mathrm{T}} \boldsymbol{Q}_{i-1} \boldsymbol{\varPhi} \boldsymbol{A}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}\right)^{-1}\right)=\alpha_{1}^{2}+\cdots+\alpha_{i-1}^{2} .$

因此, 式(3)得证. 证毕.

注1 (1)由定理1知广义奇异值{(α_i, β_i)}_i=1ⁿ满足

$\alpha_{i}^{2}+\beta_{i}^{2}=1 \quad(1 \leqslant i \leqslant n),$

因此, 由定理2易得β_i² (1≤i≤n).

(2) 定理2通过矩阵迹函数的优化问题给出了一类(n, n, n)实矩阵对{A, B}的广义奇异值{(α_i, β_i)}_i=1ⁿ的表达公式, 提供了一种用迹函数极大值优化问题来表示实矩阵对广义奇异值的新思路.

3. 数值算例

本节给出一些数值算例来验证定理2的有效性.

例1 设

$\boldsymbol{A}=\left(\begin{array}{llll} 2 & 3 & 4 & 2 \\ 1 & 5 & 9 & 4 \\ 5 & 4 & 1 & 6 \\ 8 & 2 & 4 & 6 \end{array}\right), \boldsymbol{B}=\left(\begin{array}{llll} 5 & 4 & 2 & 6 \\ 8 & 4 & 6 & 1 \\ 2 & 5 & 4 & 9 \\ 3 & 5 & 7 & 4 \end{array}\right),$

利用MATLAB R2020a的命令gsvd, 可以得到实矩阵对{A, B}的广义奇异值为

$\begin{gathered} \boldsymbol{\sigma}\{\boldsymbol{A}, \boldsymbol{B}\}=\left\{\left(\alpha_{i}, \boldsymbol{\beta}_{i}\right)\right\}=\{(0.993,0.115),(0.877,0.480), \\ (0.606,0.795),(0.149,0.989)\} . \end{gathered}$

应用引理2, 记H=A(A^TA+B^TB)^-1A^T, 有

其中, σ_j(Q_i)为Q_i的第j个奇异值, σ_j(H)为矩阵A(A^T A+B^T B)^-1 A^T的第j个奇异值, 并记 $f({\boldsymbol{Q}_i})= \max\limits_{{\mathit{\pmb{\Phi}}} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{A}^{\mathrm{T}} {\mathit{\pmb{\Phi}}}^{\mathrm{T}} \boldsymbol{Q}_{i} {\mathit{\pmb{\Phi}}} \boldsymbol{A}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}\right)^{-1}\right)$ . 利用MATLAB R2020a的命令svd, 可得矩阵H的奇异值: σ₁(H)=0.987, σ₂(H)=0.770, σ₃(H)=0.368, σ₄(H)=0.022.

由计算得到的α_i²、f(Q_i)和f(Q_i)-f(Q_i-1)(表 1)可知式(3)成立，从而验证了定理2的结论是正确的.

表 1 已知实矩阵对{A, B} 计算的α_i²和f(Q_i)- f(Q_i-1)

Table 1. The α_i² and f(Q_i)- f(Q_i-1) calculated by known real matrix pair {A, B}

i	α_i	α_i²	f(Q_i)	f(Q_i)-f(Q_i-1)
1	0.993	0.987	0.987	0.987
2	0.877	0.770	1.756	0.770
3	0.606	0.368	2.124	0.368
4	0.149	0.022	2.146	0.022

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| 显示表格

例2 设n=100. 使用MATLAB R2020a中的命令rand, 随机生成满足

$1=\alpha_{1}=\cdots=\alpha_{r}>\alpha_{r+1} \geqslant \cdots \geqslant \alpha_{r+s}>\alpha_{r+s+1}=\cdots=\alpha_{n}=0,$

$0=\beta_{1}=\cdots=\beta_{r}<\beta_{r+1} \leqslant \cdots \leqslant \beta_{r+s}<\beta_{r+s+1}=\cdots=\beta_{n}=1$

和

$\alpha_{i}^{2}+\beta_{i}^{2}=1 \quad(1 \leqslant i \leqslant n)$

的广义奇异值{(α_i, β_i)}_i=1ⁿ (表 2).

表 2 广义奇异值{(α_i, β_i)}_i=1ⁿ

Table 2. The generalized singular values {(α_i, β_i)}_i=1ⁿ

i	α_i	β_i	i	α_i	β_i
1	0.996	0.085	94	0.087	0.996
2	0.988	0.156	95	0.051	0.999
3	0.987	0.159	96	0.039	0.999
4	0.970	0.244	97	0.030	1.000
5	0.950	0.314	98	0.022	1.000
6	0.947	0.322	99	0.021	1.000
7	0.928	0.373	100	0.007	1.000
…	…	…

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于是, 实矩阵对{A, B}有广义奇异值分解

$\boldsymbol{A}=\boldsymbol{U} \operatorname{diag}\left(\alpha_{1}, \cdots, \alpha_{n}\right) \boldsymbol{W}, \boldsymbol{B}=\boldsymbol{V} \operatorname{diag}\left(\beta_{1}, \cdots, \beta_{n}\right) \boldsymbol{W},$

其中，正交矩阵U、V以及非奇异矩阵W可由MATLAB R2020a中命令orth和randn生成. 这里记 $f({\boldsymbol{Q}_i})= \max\limits_{{\mathit{\pmb{\Phi}}} \in \mathbb{O}_{n}} \operatorname{tr}\left(\boldsymbol{A}^{\mathrm{T}} {\mathit{\pmb{\Phi}}}^{\mathrm{T}} \boldsymbol{Q}_{i} {\mathit{\pmb{\Phi}}} \boldsymbol{A}\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}\right)^{-1}\right)$ . 于是, 计算α_i²、f(Q_i)和f(Q_i)-f(Q_i-1), 由结果(表 3)可知式(3)成立，从而验证了定理2的结论是正确的.

表 3 已知广义奇异值计算的α_i²和f(Q_i)-f(Q_i-1)

Table 3. The α_i² and f(Q_i)-f(Q_i-1) calculated with known generalized singular values

i	α_i²	f(Q_i)	f(Q_i)-f(Q_i-1)
1	0.993	0.993	0.993
2	0.976	1.968	0.976
3	0.975	2.943	0.975
4	0.941	3.884	0.941
5	0.902	4.785	0.902
6	0.897	5.682	0.897
7	0.861	6.543	0.861
…	…	…	…
94	0.008	34.596	0.008
95	0.003	34.599	0.003
96	0.002	34.601	0.002
97	0.001	34.602	0.001
98	0.000	34.602	0.000
99	0.000	34.602	0.000
100	0.000	34.602	0.000

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表 1 已知实矩阵对{A, B} 计算的α_i²和f(Q_i)- f(Q_i-1)

Table 1 The α_i² and f(Q_i)- f(Q_i-1) calculated by known real matrix pair {A, B}

i	α_i	α_i²	f(Q_i)	f(Q_i)-f(Q_i-1)
1	0.993	0.987	0.987	0.987
2	0.877	0.770	1.756	0.770
3	0.606	0.368	2.124	0.368
4	0.149	0.022	2.146	0.022

下载: 导出CSV

表 2 广义奇异值{(α_i, β_i)}_i=1ⁿ

Table 2 The generalized singular values {(α_i, β_i)}_i=1ⁿ

i	α_i	β_i	i	α_i	β_i
1	0.996	0.085	94	0.087	0.996
2	0.988	0.156	95	0.051	0.999
3	0.987	0.159	96	0.039	0.999
4	0.970	0.244	97	0.030	1.000
5	0.950	0.314	98	0.022	1.000
6	0.947	0.322	99	0.021	1.000
7	0.928	0.373	100	0.007	1.000
…	…	…

下载: 导出CSV

表 3 已知广义奇异值计算的α_i²和f(Q_i)-f(Q_i-1)

Table 3 The α_i² and f(Q_i)-f(Q_i-1) calculated with known generalized singular values

i	α_i²	f(Q_i)	f(Q_i)-f(Q_i-1)
1	0.993	0.993	0.993
2	0.976	1.968	0.976
3	0.975	2.943	0.975
4	0.941	3.884	0.941
5	0.902	4.785	0.902
6	0.897	5.682	0.897
7	0.861	6.543	0.861
…	…	…	…
94	0.008	34.596	0.008
95	0.003	34.599	0.003
96	0.002	34.601	0.002
97	0.001	34.602	0.001
98	0.000	34.602	0.000
99	0.000	34.602	0.000
100	0.000	34.602	0.000

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