### 摘要

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors {x_{(i)}} with x_{(i)}∈R^{n1×⋯×nd} in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram-Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α_{i} in an arbitrary tensor v=∑_{i}α_{i}x_{(i)}. The orthonormal Q factor in the QR decomposition X≡[x_{(1)},⋯,x_{(p)}]=QR cannot be computed but expressed as XR^{-1} when required. The resulting algorithm has an O(p^{2}dn) computational complexity, with n=maxn_{i}. Some illustrative examples in the numerical solution of tensor linear equations are presented.

原文 | 英語 |
---|---|

頁（從 - 到） | 292-316 |

頁數 | 25 |

期刊 | Linear Algebra and Its Applications |

卷 | 491 |

DOIs | |

出版狀態 | 已發佈 - 2016 二月 15 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

## 指紋 深入研究「Q-less QR decomposition in inner product spaces」主題。共同形成了獨特的指紋。

## 引用此

*Linear Algebra and Its Applications*,

*491*, 292-316. https://doi.org/10.1016/j.laa.2015.08.035