Fast qz decomposition and its applications

Abstract

The ability to introduce zeros in a selective fashion makes the Givens Rotations an important zeroing tool in certain structured matrix problems. Evans and Yalamov [2] combined two Givens Rotations in one step to annihilate two elements simultaneously in order to transform the original matrix to a “Z” form pattern. The composite scheme was called the QZ decomposition method and is suitable for parallel computation, which is confirmed by the numerical results [1].

In this paper, firstly the fast computation of the QZ decomposition is given, which eliminates the square roots and reduces the number of multiplications by 37.5%. Finally, the applications of the fast QZ decomposition method to the linear system of equations, least squares problem and the weighted least squares problem are considered.

Keywords:

• Givens rotations
• QZ decomposition
• linear system of equations
• least squares problem
• weighted least squares problem and parallel computation

AMS Subject clasification:

• 65F10
• 65F20

C.R. Categories:

• G.1.3

^∗This author's work was supported by Natural Science Foundation of China (Grant no. 19871011), and Natural Science Foundation of Liaoning Province of China (Grant no. 962172)

^†Correasponding author

^∗This author's work was supported by Natural Science Foundation of China (Grant no. 19871011), and Natural Science Foundation of Liaoning Province of China (Grant no. 962172)

^†Correasponding author

Notes

^∗This author's work was supported by Natural Science Foundation of China (Grant no. 19871011), and Natural Science Foundation of Liaoning Province of China (Grant no. 962172)

^†Correasponding author