New Method of Givens Rotations for Triangularization of Square Matrices

Abstract

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.