New Method of Givens Rotations for Triangularization of Square Matrices

Affiliation(s)

Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, USA.

Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, USA.

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 4/3N^{3} 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.

Cite this paper

Grigoryan, A. (2014) New Method of Givens Rotations for Triangularization of Square Matrices.*Advances in Linear Algebra & Matrix Theory*, **4**, 65-78. doi: 10.4236/alamt.2014.42004.

Grigoryan, A. (2014) New Method of Givens Rotations for Triangularization of Square Matrices.

References

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http://dx.doi.org/10.1017/CBO9780511810817

[2] Moon, T.K. and Stirling, W.C. (2000) Mathematical Methods and Algorithms for Signal Processing. Prentice Hall, Upper Saddle River.

[3] Morrison, D.D. (1960) Remarks on the Unitary Triangularization of a Nonsymmetric Matrix. Journal ACM, 7, 185-186.

http://dx.doi.org/10.1145/321021.321030

[4] Golub, G.H. and Van Loan, C.F. (1996) Matrix Computations. 3rd Edition, Johns Hopkins, Balimore.

[5] Householder, A.S. (1958) Unitary Triangulation of a Nonsymmetric Matrix. Journal ACM, 5, 339-342.

http://dx.doi.org/10.1145/320941.320947

[6] Anderson, E., Bai, Z., Bischof, C.H., et al. (1999) LAPACK Users’ Guide. 3rd Edition, SIAM Philadelphia, Philadelphia.

http://dx.doi.org/10.1137/1.9780898719604

[7] Bindel, D., Demmel J., Kahan, W. and Marques, O. (2001) On Computing Givens Rotations Reliably and Efficiently. LAPACK Working Note 148, University of Tennessee, UT-CS-00-449.

[8] Demmel, J., Grigori, L., Hoemmen, M. and Langou, J. (2012) Communication-Optimal Parallel and Sequential QR and LU Factorizations. SIAM Journal on Scientific Computing, 34, 206-239.

http://dx.doi.org/10.1137/080731992

[9] Grigoryan, A.M. and Grigoryan, M.M. (2006) Nonlinear Approach of Construction of Fast Unitary Transforms. In: Proceedings of the 40th Annual Conference on Information Sciences and Systems (CISS 2006), Princeton University, Princeton, 1073-1078.

http://dx.doi.org/10.1109/CISS.2006.286625

[10] Grigoryan, A.M. and Grigoryan, M.M. (2007) Discrete Unitary Transforms Generated by Moving Waves. Proceedings of the International Conference: Wavelets XII, SPIE: Optics + Photonics 2007, San Diego.

[11] Grigoryan, A.M. and Grigoryan, M.M. (2007) New Discrete Unitary Haar-Type Heap Transforms. Proceedings of the International Conference: Wavelets XII, SPIE: Optics + Photonics, 6701.

[12] Grigoryan, A.M. and Grigoryan, M.M. (2009) Brief Notes in Advanced DSP: Fourier Analysis with MATLAB. CRC Press, Taylor and Francis Group, Philadelphia.

[1] Horn, R.A. and Charles, R.J. (1985) Matrix Analysis. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511810817

[2] Moon, T.K. and Stirling, W.C. (2000) Mathematical Methods and Algorithms for Signal Processing. Prentice Hall, Upper Saddle River.

[3] Morrison, D.D. (1960) Remarks on the Unitary Triangularization of a Nonsymmetric Matrix. Journal ACM, 7, 185-186.

http://dx.doi.org/10.1145/321021.321030

[4] Golub, G.H. and Van Loan, C.F. (1996) Matrix Computations. 3rd Edition, Johns Hopkins, Balimore.

[5] Householder, A.S. (1958) Unitary Triangulation of a Nonsymmetric Matrix. Journal ACM, 5, 339-342.

http://dx.doi.org/10.1145/320941.320947

[6] Anderson, E., Bai, Z., Bischof, C.H., et al. (1999) LAPACK Users’ Guide. 3rd Edition, SIAM Philadelphia, Philadelphia.

http://dx.doi.org/10.1137/1.9780898719604

[7] Bindel, D., Demmel J., Kahan, W. and Marques, O. (2001) On Computing Givens Rotations Reliably and Efficiently. LAPACK Working Note 148, University of Tennessee, UT-CS-00-449.

[8] Demmel, J., Grigori, L., Hoemmen, M. and Langou, J. (2012) Communication-Optimal Parallel and Sequential QR and LU Factorizations. SIAM Journal on Scientific Computing, 34, 206-239.

http://dx.doi.org/10.1137/080731992

[9] Grigoryan, A.M. and Grigoryan, M.M. (2006) Nonlinear Approach of Construction of Fast Unitary Transforms. In: Proceedings of the 40th Annual Conference on Information Sciences and Systems (CISS 2006), Princeton University, Princeton, 1073-1078.

http://dx.doi.org/10.1109/CISS.2006.286625

[10] Grigoryan, A.M. and Grigoryan, M.M. (2007) Discrete Unitary Transforms Generated by Moving Waves. Proceedings of the International Conference: Wavelets XII, SPIE: Optics + Photonics 2007, San Diego.

[11] Grigoryan, A.M. and Grigoryan, M.M. (2007) New Discrete Unitary Haar-Type Heap Transforms. Proceedings of the International Conference: Wavelets XII, SPIE: Optics + Photonics, 6701.

[12] Grigoryan, A.M. and Grigoryan, M.M. (2009) Brief Notes in Advanced DSP: Fourier Analysis with MATLAB. CRC Press, Taylor and Francis Group, Philadelphia.