New Method of Givens Rotations for Triangularization of Square Matrices

This paper describes a new method of QR-decomposition of square nonsingular matrices (real or complex) by the Givens rotations through the unitary discrete heap transforms. This transforms can be defined by a different path, or the order of processing components of input data, which leads to different realizations of the QR-decomposition. The heap transforms are fast, because of a simple form of decomposition of their matrices. The direct calculation of the N-point discrete heap transform requires no more than 5(N-1) multiplications, 2(N-1) additions, plus 3(N-1) trigonometric operations. The QR-decomposition of the square matrix N × N uses about 4/3N3 multiplications and N(N-1)/2 square roots. This number varies depending on the path of the heap transform, and it is shown that “the optimal path” allows for significant reduction of number of operations in QR-decomposition. The heap transform and its matrix can be described analytically, and therefore, this transform can also be applied to the QR-decomposition of complex matrices.

Effective Methods of QR-Decompositions of Square Complex Matrices by Fast Discrete Signal-Induced Heap Transforms

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 2 × 2 transforms. The DsiHT is zeroing all components of the input signal while moving or heaping the energy of the signal to one component, for instance the first one. We describe three different types of QR-decompositions that use the basic transforms with the T, G, and M-type complex matrices we introduce, as well as without matrices but using analytical formulas. We also present the mixed QR-decomposition, when different type DsiHTs are used in different stages of the algorithm. The number of such decompositions is greater than 3(N-1), for an N × N complex matrix. Examples of the QR-decomposition are described in detail for the 4 × 4 and 6 × 6 complex matrices and compared with the known method of Householder transforms. The precision of the QR-decompositions of N × N 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.