# Partial Singular Value Assignment for Large-Scale Systems

^{*}

## Abstract

**:**

## 1. Introduction

**PSVA:**Given matrices A, B and the desired (non-zero) singular values $\{{\theta}_{1},\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}{\theta}_{p}\}$, find a matrix F such that$$A+B{F}^{\top}$$

- The core innovation of our devised approach is rooted in the construction of a sparse basis within the null space of the orthogonal projection of matrix A. This strategy offers a notable advantage by obviating the necessity for the full SVD of a large-scale matrix.
- Our algorithm brings to the forefront the possibility of computational savings, especially when dealing with matrices of significant scale and sparse structure (i.e., the number of non-zeros in each column of the matrix is bounded by a constant much smaller than n). Then, the proposed algorithm strives to assign the desired singular values with exceptional computational efficiency, hopefully scaling with the computational complexity of $O\left(n\right)$ floating-point operations.
- To validate the practical utility of our proposed algorithm, we conduct a series of numerical experiments. These experiments not only confirm the feasibility of our approach but also demonstrate its effectiveness in real-world scenarios.

**Definition 1**

**.**Two pairs of matrices $(A,B)$ and $(C,D)$ are equivalent if there are orthogonal matrices U and V, a matrix E and a nonsingular matrix R with compatible sizes such that

## 2. PSVA for Single Singular Value

**Single PSVA:**Given a matrix $A\in {\mathbb{R}}^{n\times n}$, a vector $b\in {\mathbb{R}}^{n\times 1}$ and the desired (non-zero) singular values $\theta $, find a vector $f\in {\mathbb{R}}^{n\times 1}$ such that$$A+b{f}^{\top}$$

**Theorem 1.**

**Proof.**

**Remark 1.**

Algorithm 1 PSVA for the Single Singular Value |

Input: A large-scale sparse matrix $A\in {\mathbb{R}}^{n\times n}$, $b\in {\mathbb{R}}^{n\times 1}$ and the desired singular value $\theta $. Output: A vector $f\in {\mathbb{R}}^{n\times 1}$ such that the minimal singular value of $A+b{f}^{\top}$ is $\theta $. 1. Compute ${f}_{1}=-{\left({\left({b}^{\top}b\right)}^{-1}{b}^{\top}A\right)}^{\top}$; 2. Compute the economic QR decomposition of b, i.e., $b={q}_{1}r$; 3. Compute the null space of the matrix $D=A+b{f}_{1}^{\top}$; 4. Form the vector $f={f}_{1}+\theta {v}_{2}/r$ such that the smallest singular value of $A+b{f}^{\top}$ is $\theta $. |

**Remark 2.**

## 3. PSVA for p Singular Values

**p****-PSVA:**Given a large-scale sparse matrix $A\in {\mathbb{R}}^{n\times n}$ with p smallest singular values $\{{\sigma}_{1},\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}{\sigma}_{p}\}$ being close to (or equal to) zero, and a matrix $B\in {\mathbb{R}}^{n\times p}$, find a matrix $F\in {\mathbb{R}}^{n\times p}$ such that$$A+B{F}^{\top}$$

**Theorem 2.**

**Proof.**

**Remark 3.**

Algorithm 2 PSVA for p Singular Values |

Input: A large-scale sparse matrix $A\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times p}$ and the desired singular values ${\theta}_{1},\cdots ,{\theta}_{p}$. Output: A matrix $F\in {\mathbb{R}}^{n\times p}$ such that p smallest singular values of $A+B{F}^{\top}$ are $\{{\theta}_{1},\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}{\theta}_{p}\}$. 1. Compute the economic QR decomposition of B, i.e., $B={Q}_{1}{R}_{1}$ with ${Q}_{1}=[{Q}_{11},\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}{Q}_{1p}]$ and
$$R=\left[\begin{array}{cccc}{r}_{11}& \cdots & \cdots & {r}_{1p}\\ & \ddots & & \vdots \\ & & {r}_{p-1,p-1}& \vdots \\ & & & {r}_{pp}\end{array}\right].$$
2. Compute the null space of $D{V}_{2}=(I-{Q}_{1}{Q}_{1}^{\top})A{V}_{2}=0$ with ${V}_{2}=[{V}_{21},\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}{V}_{2p}]$. 3. Compute ${f}_{p}$ by Algorithm 1 with the available A, ${Q}_{1p}$ and ${\theta}_{p}$. 4. For $i=p-1:1$
$${f}_{i}=\frac{1}{{r}_{ii}}\left[{\theta}_{i}{V}_{2i}-{A}^{\top}{Q}_{1i}-\sum _{j=i+1}^{p}{r}_{ij}{f}_{j}\right]$$
End. 5. Form the matrix $F=[{f}_{1},\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}{f}_{p}]$ such that the smallest p singular values of $A+B{F}^{\top}$ are $\{{\theta}_{1},\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}{\theta}_{p}\}$. |

**Theorem 3**

**.**For any matrix

**Remark 4.**

## 4. Numerical Examples

**Example 1.**

`rand(n,1)`”). We define the matrices

**Example 2.**

**Example 3.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Problem | CPU | DesSV | AsiSV | RelErr | CondS | CondPSVA |
---|---|---|---|---|---|---|

dubcova1 | 1.29 | 0.016; 0.015 | $1.59\times {10}^{-2}$; $1.49\times {10}^{-2}$ | $1.21\times {10}^{-13}$; $3.34\times {10}^{-13}$ | $2.74\times {10}^{2}$ | $3.19\times {10}^{2}$ |

gridgena | 9.77 | 0.16; 0.15 | $1.59\times {10}^{-1}$; $1.49\times {10}^{-1}$ | $1.08\times {10}^{-6}$; $2.52\times {10}^{-7}$ | $1.55\times {10}^{5}$ | $1.84\times {10}^{5}$ |

onetone2 | 2.48 | 0.001; 0.005 | $1.00\times {10}^{-3}$; $5.00\times {10}^{-3}$ | $9.39\times {10}^{-15}$; $8.76\times {10}^{-15}$ | $3.08\times {10}^{6}$ | $3.92\times {10}^{6}$ |

wathen100 | 4.29 | 0.10; 0.05 | $1.00\times {10}^{-1}$; $5.00\times {10}^{-2}$ | $9.99\times {10}^{-15}$; $3.99\times {10}^{-14}$ | $5.81\times {10}^{3}$ | $7.39\times {10}^{3}$ |

poli_large | 0.06 | 0.010; 0.005 | $1.00\times {10}^{-2}$; $5.00\times {10}^{-3}$ | $5.63\times {10}^{-15}$; $6.99\times {10}^{-15}$ | $2.51\times {10}^{3}$ | $3.75\times {10}^{3}$ |

msc10848 | 4.57 | 80.0; 81.0 | $7.99\times {10}^{1}$; $8.09\times {10}^{1}$ | $8.06\times {10}^{-9}$; $7.34\times {10}^{-9}$ | $7.60\times {10}^{9}$ | $7.86\times {10}^{9}$ |

bodyy4 | 0.95 | 1.0; 0.9 | $9.99\times {10}^{-1}$; $8.99\times {10}^{-1}$ | $3.99\times {10}^{-15}$; $3.99\times {10}^{-1}$ | $8.06\times {10}^{2}$ | $9.24\times {10}^{2}$ |

bodyy5 | 1.21 | 1.0; 0.9 | $1.00\times {10}^{0}$; $8.99\times {10}^{-1}$ | $2.66\times {10}^{-14}$; $8.99\times {10}^{-14}$ | $7.87\times {10}^{3}$ | $8.93\times {10}^{3}$ |

bodyy6 | 1.11 | 1.0; 0.9 | $9.99\times {10}^{-1}$; $8.99\times {10}^{-1}$ | $1.29\times {10}^{-14}$; $9.99\times {10}^{-15}$ | $7.69\times {10}^{4}$ | $8.68\times {10}^{4}$ |

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Huang, Y.; Tang, Q.; Yu, B.
Partial Singular Value Assignment for Large-Scale Systems. *Axioms* **2023**, *12*, 1012.
https://doi.org/10.3390/axioms12111012

**AMA Style**

Huang Y, Tang Q, Yu B.
Partial Singular Value Assignment for Large-Scale Systems. *Axioms*. 2023; 12(11):1012.
https://doi.org/10.3390/axioms12111012

**Chicago/Turabian Style**

Huang, Yiting, Qiong Tang, and Bo Yu.
2023. "Partial Singular Value Assignment for Large-Scale Systems" *Axioms* 12, no. 11: 1012.
https://doi.org/10.3390/axioms12111012