The purpose of this work is to present an effective tool for computing different QR-decompositions of a complex nonsingular square matrix. The concept of the discrete signal-induced heap transform (DsiHT, Grigoryan 2006) is used. This transform is fast, has a unique algorithm for any length of the input vector/signal and can be used with different complex basic 2x2 transforms. The DsiHT zeroes all components of the input signal while moving or heaping the energy of the signal into one component, such as the first. We describe three different types of QR-decompositions that use the basic transforms with the T, G, and M-type complex matrices we introduce, and also without matrices, but using analytical formulas. We also present the mixed QR-decomposition, when different type DsiHTs are used at different stages of the algorithm. The number of such decompositions is greater than 3^((N-1)), for an NxN complex matrix. Examples of the QR-decomposition are described in detail for the 4x4 and 6x6 complex matrices and compared with the known method of Householder transforms. The precision of the QR-decompositions of NxN matrices, when N are 6, 13, 17, 19, 21, 40, 64, 100, 128, 201, 256, and 400 is also compared. The MATLAB-based scripts of the codes for QR-decompositions by the described DsiHTs are given.

翻译：本文旨在提出一种计算复非奇异方阵不同QR分解的有效工具。我们采用了离散信号诱导堆变换（DsiHT，Grigoryan 2006）的概念。该变换具有快速性，对任意长度的输入向量/信号均采用唯一算法，并可结合不同的复基本2×2变换使用。DsiHT在将信号能量"堆叠"至某一分量（如第一个分量）的同时，将输入信号的所有分量归零。我们描述了三种不同类型的QR分解，分别使用所引入的T型、G型和M型复矩阵的基本变换，以及无需矩阵而仅使用解析公式的分解方式。我们还提出了混合QR分解，即在算法的不同阶段使用不同类型的DsiHT。对于N×N复矩阵，这类分解的数量超过3^((N-1))。本文详细描述了4×4和6×6复矩阵的QR分解实例，并与已知的Householder变换方法进行了比较。此外，我们还比较了当N取6、13、17、19、21、40、64、100、128、201、256和400时，N×N矩阵QR分解的精度。文中给出了基于MATLAB的DsiHT QR分解代码脚本。