Sanzheng Qiao
Personal Information
Name:  Sanzheng Qiao 
Position:  Professor Emeritus 
Office Location:  ITB246 
Telephone Number:  +1 (905)5259140 ext. 27234 
Facsimile Numbers:  +1 (905)5240340 
Electronic Address:  qiao@mcmaster.ca 
Biography
Sanzheng Qiao was born in Huangpu District, near People's Square, in
Shanghai,
China in 1945. He received the B.S. degree from Shanghai Teacher's
University, Shanghai, China in 1966, and the M.S. degree in computer
science and Ph.D. degree in applied mathematics from
Cornell University,
Ithaca, NY, in 1986 and 1987 respectively.
He was an assistant professor of computer science at
Ithaca College,
Ithaca, NY
from 1987 to 1988. From 1989 to 1993, he was an assistant professor
in the Department of Computer Science and Systems at
McMaster University,
Hamilton, Ontario, Canada, and an associate professor from 1993 to
1999. He is now a professor in Department of Computing and
Software at McMaster University.
He is a faculty member of Software Quality
Research Laboratory (SQRL)
at McMaster University.
His research interests include numerical
linear algebra, distributed/parallel scientific computing,
numerical methods for signal processing, and numerical software.
Course Information
Software

Erlang B Calculator.

Takagi factorization or symmetric SVD (
matlab
)

Fast Square Hankel/Toeplitz SVD (
matlab
)

Fast General Hankel/Toeplitz SVD (
matlab
)

Block Lanczos tridiagonalization of complex symmetric matrices (
matlab,
C
)
Publications

Ling Lu, Wei Xu and Sanzheng Qiao.
A fast SVD for multilevel block Hankel matrices with minimal
memory storage.
To appear in Numerical Algorithms. 2015.
DOI: 10.1007/s1107501499300
(pdf)

Wen Zhang, Sanzheng Qiao and Yimin Wei.
The diagonal reduction algorithm using fast Givens.
Computer Mathematics.
Editors: Ruyong Feng, Wenshin Lee and Yosuke Sato.
453465, 2014.
ISBN 9783662437988 (Print) 9783662437998 (Online)
(pdf)

Filip Jeremic and Sanzheng Qiao.
A Parallel Jacobitype Lattice Basis Reduction Algorithm.
International Journal of Numerical Analysis and Modeling,
Series B.
Vol. 5, No. 12, 2014, 112.
(pdf)

Zhaofei Tian and Sanzheng Qiao.
A Hybrid Method for Lattice Basis Reduction.
Technical Report CAS1401SQ,
Department of Computing and Software, McMaster University,
1280 Main Street, Hamilton, Ontario, Canada L8S 4K1.
(pdf)

Filip Jeremic and Sanzheng Qiao.
A GPU Implementation of a Jacobi Method for Lattice
Basis Reduction.
To appear in Proceedings of
International Workshop on DataIntensive Scientific Discovery (DISD),
2013, Shanghai University, Shanghai, China, August 14, 2013.
(pdf)

Zhaofei Tian and Sanzheng Qiao.
An Enhanced Jacobi Method for LatticeReductionAided MIMO
Detection.
Proceedings of 2013 IEEE China Summit and International Conference
on Signal and Information Processing (ChinaSIP),
July 610, 2013, Beijing, China, 3943.
DOI: 10.1109/ChinaSIP.2013.6625293
(pdf)

Huaian Diao, Weiguo Wang, Yimin Wei, and Sanzheng Qiao.
On Condition Numbers for MoorePenrose Inverse and Linear
Least Squares Problem Involving Kronecker Products.
Numerical Linear Algebra with Applications, 2013; 20:4459.
DOI: 10.1002/nla.1823.
(pdf)

Wen Zhang, Sanzheng Qiao, and Yimin Wei.
HKZ and Minkowski Reduction Algorithms for
LatticeReductionAided MIMO Detection.
IEEE Transactions on Signal Processing.
Vol. 60, No. 11, 2012, 59635976
DOI 10.1109/TSP.2012.2210708
(pdf)

Zhaofei Tian, Wen Zhang, Sanzheng Qiao.
A Polynomial Time Jacobi Method for Lattice Basis Reduction.
Technical Report CAS1204SQ, Department of Computing and
Software, McMaster University, Hamilton, Ontario, Canada,
L8S 4K1.
(pdf)

Zhaofei Tian and Sanzheng Qiao.
A Complexity Analysis of a Jacobi Method for Lattice Basis Reduction.
Proceedings of C3S2E'12 the Fifth International C* Conference on Computer
Science and Software Engineering. Editors: Bipin C. Desai, Sudhir Mudur,
Emil Vassey.
Montreal, Quebec, Canada, June 2628, 2012.
ACM International Conference Proceedings Series, ACM Press,
New York, NY, 6380.
(pdf)

Sanzheng Qiao.
A Jacobi Method for Lattice Basis Reduction.
Proceedings of 2012 Spring World Congress on Engineering
and Technology (SCET2012). Vol. 2. IEEE.
May 2730 2012, Xi'an, China.
649652.
(pdf)

Wen Zhang, Sanzheng Qiao, and Yimin Wei.
A Diagonal Lattice Reduction Algorithm for MIMO Detection.
IEEE Signal Processing Letters. Vol. 19, No. 5, 2012, 311314.
DOI 10.1109/LSP.2012.2191614
(pdf)

Wen Zhang, Sanzheng Qiao, and Yimin Wei.
Practical HKZ and Minkowski Lattice Reduction Algorithms.
Department of Computing and Software, Technical Report
CAS1104SQ, McMaster University, Hamilton Ontario, Canada L8S 4K1.
(pdf)

Yixian Luo and Sanzheng Qiao.
A Parallel LLL Algorithm.
Proceedings of the Forth International C* Conference on
Computer Science and Software Engineering, 2011, Montreal, QC,
Canada, May 1618, 2011. ACM International Conference Proceedings
Series, ACM Press, 93101.
(pdf)

Wen Zhang, Sanzheng Qiao, and Yimin Wei.
Practical Algorithms for Constructing HKZ and Minkowski
Reduced Bases.
Department of Computing and Software, Technical Report
CAS1103SQ, McMaster University, Hamilton Ontario, Canada L8S 4K1.
(pdf)

Wen Zhang, Yimin Wei, and Sanzheng Qiao.
A Delayed Sizereduction Technique for Speeding Up the
LLL Algorithm.
Department of Computing and Software, Technical Report
CAS1102SQ, McMaster University, Hamilton Ontario, Canada L8S 4K1.
(pdf)

Yixian Luo and Sanzheng Qiao.
A Parallel LLL Algorithm.
Department of Computing and Software, Technical Report
CAS1003SQ, McMaster University, Hamilton Ontario, Canada L8S 4K1.
(pdf)

F.T. Luk, S. Qiao, and W. Zhang.
A Lattice Basis Reduction Algorithm.
Institute for Computational Mathematics Technical Report 1004.
Hong Kong Baptist University, Kowloon, Hong Kong, China.
(pdf)

F.T. Luk and S. Qiao.
A Pivoted LLL Algorithm.
Linear Algebra Appl., 434(11), 2011, pp. 22962307.

Sanzheng Qiao.
A Deterministic Method for Choosing Search Radii in
Sphere Decoding.
Proceedings of The 5th International Conference on
Wireless Communications, Networking and Mobile Computing.
Beijing, China, September 2426, 2009.
http://ieeexplore.ieee.org/Xplore/guesthome.jsp
DOI: 10.1109/WICOM.2009.5304275

Weijia Yang and Sanzheng Qiao.
A Novel Anonymization Algorithm: Privacy Protection
and Knowledge Preservation.
Expert Systems With Applications.
37(1), 2010, pp. 756766.
(pdf)
DOI: http://dx.doi.org/10.1016/j.eswa.2009.05.097

Franklin T. Luk and Sanzheng Qiao.
Conditioning Properties of the LLL Algorithm,
in Mathematics for Signal and Information Processing
, edited by Schmalz, Ritter, Barrera, and Luk,
Proc. of SPIE Vol. 7444 (2009), article 744417.
(pdf)

Wei Xu and Sanzheng Qiao.
A Twisted Factorization Method for Symmetric SVD of a
Complex Symmetric Tridiagonal Matrix.
Numerical Linear Algebra with Applications.
16(2009), 801815.
(pdf)

Liangmin Zhou, Yiqin Lin, Yimin Wei, Sanzheng Qiao.
Perturbation Analysis and Condition Numbers of Symmetric
Algebraic Riccati Equations.
Automatica.
45(2009) 10051011.
(pdf)

Don Morgan and Sanzheng Qiao.
Analysis of Damped MassSpring Systems for Sound Synthesis.
EURASIP Journal on Audio, Speech, and Music Processing.
Vol. 2009, Article ID 947823, 19 pages, 2009.
DOI: 10.1155/2009/947823.
http://www.hindawi.com/journals/asmp/2009/947823.html
(pdf)

Liangmin Zhou, Lijing Lin, Yimin Wei, and Sanzheng Qiao.
Perturbation Analysis and Condition Numbers of Scaled
Total Least Squares Problems.
Numerical Algorithms,
51, 2009, 381399.
(pdf)
DOI 10.1007/s1107500992690

Fei Zhao and Sanzheng Qiao.
Radius Selection Algorithms for Sphere Decoding.
Proceedings of C3S2E09, ACM International
Conference Proceedings Series.
Edited by Bipin C. Desai and Carson K. Leung.
Concordia University,
Montreal, QC, May 1921, 2009, pp. 169174.
(pdf)

Kevin Browne, Sanzheng Qiao, and Yimin Wei.
A Lanczos Bidiagonalization Algorithm for Hankel Matrices.
Linear Algebra and Its Applications,
430(2009), 15311543.
(pdf)

Sanzheng Qiao, Wei Xu, and Yimin Wei.
An Algorithm for Solving Scaled Total Least Squares Problems.
The Proceedings of the 2008 International Conference
on Scientific Computing, Worldcomp'08.
Las Vegas, NE, July 1417, 2008, pp.211217.
(pdf)

Don Morgan and Sanzheng Qiao.
Accuracy and Stability in MassSpring Systems
for Sound Synthesis.
Proceedings of C3S2E08, ACM International
Conference Proceedings Series,
edited by Bipin C. Desai. Concordia University,
Montreal, QC, May 1213, 2008, pp. 6980.
(pdf)

Sanzheng Qiao.
Integer least squares: Sphere decoding and the LLL
algorithm.
Proceedings of C3S2E08, ACM International
Conference Proceedings Series,
edited by Bipin C. Desai. Concordia University,
Montreal, QC, May 1213, 2008, pp. 2328.
(pdf)

Wei Xu and Sanzheng Qiao.
A DivideandConquer Method for the Takagi Factorization.
SIAM J. Matrix Anal. Appl.,
30/1(2008), 142153.
(pdf)

Wei Xu, Sanzheng Qiao, and Yimin Wei.
A Note on the Scaled Total Least Squares Problem.
Linear Algebra and Its Applications,
428/2+3(2008), 469478.
(pdf)
DOI link: http://dx.doi.org/10.1016/j.laa.2007.03.015.

Wei Xu and Sanzheng Qiao.
A Fast SVD Algorithm for Square Hankel Matrices.
Linear Algebra and Its Applications,
428/2+3(2008), 550563.
(pdf)
DOI link: http://dx.doi.org/10.1016/j.laa.2007.05.027

Franklin T. Luk and Sanzheng Qiao.
Numerical Properties of the LLL Algorithm,
in Advanced Signal Processing Algorithms, Architectures,
and Implementations XVII, edited by Franklin T. Luk,
Proc. of SPIE Vol. 66973, (2007).
(pdf)

Yimin Wei, Huaian Diao, and Sanzheng Qiao.
Condition Number for Weighted Least Squares Problem.
Journal of Computational Mathematics,
Vol.25, No.5, 2007, 561572.
 Wei Xu, Yimin Wei, and Sanzheng Qiao,
Condition Numbers for Structured Least Squares,
BIT, Vol. 46, No. 1, pp. 203225, March 2006.
(pdf)

Yimin Wei, Wei Xu, Sanzheng Qiao, and Huaian Diao.
Componentwise Condition Numbers for Generalized Matrix
Inversion and Linear Least Squares.
Numerical Mathematics: A Journal of Chinese Universities,
(English Series).
14(3), Aug. 2005, pp.277286.
(pdf)

Guohong Liu and Sanzheng Qiao,
A High Performance C Package for Tridiagonalization of
Complex Symmetric Matrices,
Technical Report No. CAS 0504SQ,
Department of Computing and Software, McMaster University,
Hamilton, Ontario, Canada. L8S 4K1. August 2005.
(ps,
pdf)

Wei Xu and Sanzheng Qiao,
A DivideandConquer Method for the Takagi Factorization,
Technical Report No. CAS 0501SQ,
Department of Computing and Software, McMaster University,
Hamilton, Ontario, Canada. L8S 4K1. February 2005.
(ps,
pdf)

Sanzheng Qiao, Guohong Liu, and Wei Xu.
Block Lanczos Tridiagonalization of Complex Symmetric Matrices,
in Advanced Signal Processing Algorithms, Architectures,
and Implementations XV, edited by Franklin T. Luk,
Proceedings of the SPIE Vol. 5910, 2005, pp. 285295.
(ps,
pdf)

Wei Xu, Sanzheng Qiao, and Yimin Wei,
An Algorithm for Solving RankDeficient Scaled Total Least
Square Problems,
Technical Report No. CAS 0404SQ,
Department of Computing and Software, McMaster University,
Hamilton, Ontario, Canada. L8S 4K1. August 2004.
(ps,
pdf)

Sanzheng Qiao,
Orthogonalization Techniques for the Lanczos Tridiagonalization
of Complex Symmetric Matrices,
Advanced Signal Processing Algorithms, Architectures,
and Implementations XIV, edited by Franklin T. Luk,
Proc. of SPIE Vol. 5559, 2004, pp. 423434.
(ps,
pdf)

Yimin Wei, Huaian Diao, and Sanzheng Qiao,
Condition Number for Weighted Linear Least Squares Problem
and Its Condition Number,
Technical Report No. CAS 0402SQ,
Department of Computing and Software, McMaster University,
Hamilton, Ontario, Canada. L8S 4K1. May 2004.
(ps,
pdf)

Guorong Wang, Yimin Wei, and Sanzheng Qiao,
Generalized Inverses: Theory and Computations,
Graduate Series in Mathematics 5,
Science Press, Beijing/New York, April 2004.
ISBN 7030124375.
Preface and Table of Contents
(ps,
pdf)

Huaian Diao, Yimin Wei, and Sanzheng Qiao,
Displacement Rank of the Drazin Inverse,
J. Comput. Appl. Math.,
167(2004). 147161.
(pdf)

Guorong Wang and Sanzheng Qiao,
Solving Constrained Matrix Equations and Cramer Rule,
Applied Mathematics and Computation,
159/2 (2004) 333340.
(ps,
pdf)

F.T. Luk and S. Qiao,
A fast singular value algorithm for Hankel matrices,
Fast Algorithms for Structured Matrices:
Theory and Applications, Contemporary Mathematics 323,
Editor V. Olshevsky,
American Mathematical Society. 2003. 169177.
(ps,
pdf)

Wei Xu, Yimin Wei, and Sanzheng Qiao,
An Analysis of RankDeficient Scaled Total Least
Squares Problem,
Technical Report No. CAS 0310SQ,
Department of Computing and Software, McMaster University,
Hamilton, Ontario, Canada. L8S 4K1. September 2003.
(ps,
pdf)

W.E. Larimore, F.T. Luk, and Sanzheng Qiao,
Prediction Error Computation on a Grid,
Proceedings of
International Conference on Parallel Algorithms and
Computing Environments,
Editors Jun Zou and Xuebin.
Chinese University of Hong Kong, Hong Kong,
October 811, 2003, pp. 103106.
(ps,
pdf)

Chengshu Guo and Sanzheng Qiao,
A Stable Lanczos Tridiagonalization of Complex
Symmetric Matrices,
Technical Report No. CAS 0308SQ,
Department of Computing and Software, McMaster University,
Hamilton, Ontario, Canada. L8S 4K1. July 2003.
(Revised, September 2003.)
(ps,
pdf)

Yimin Wei and Sanzheng Qiao,
The representation and approximation of the Drazin inverse
of a linear operator in Hilbert space,
Applied Mathematics and Computation,
138 (2003) 7789.
(ps,
pdf)

F.T. Luk, S. Qiao, and D. Vandervoorde,
Exponential Decomposition and Hankel Matrix,
Mathematics in Signal Processing V,
Editors J.G. McWhirter and I.K. Proudler,
Oxford University Press. June, 2002, 275285.

Sanzheng Qiao and Xiaohong Wang,
Computing the Singular Vaues of 2by2 Complex Matrices,
Software Quality Research Laboratory Report No. 5,
Department of Computing and Software, McMaster University,
Hamilton, Ontario, Canada. L8S 4L7. June 2002.
(ps,
pdf).
Revised December 2002
 Xiaohong Wang and Sanzheng Qiao,
A Parallel Jacobi Method for the Takagi Factorization,
Proceedings of the International Conference on
Parallel and Distributed Processing Techniques and
Applications, Vol. I, Editor H.R. Arabnia,
pp. 206212,
Las Vegas, Nevada, USA, June, 2002.

S. Qiao and Jun Zhu,
Exponential Decomposition Using Displacement Structure,
Advanced Signal Processing Algorithms, Architectures,
and Implementations XI, Franklin T. Luk, Editor,
Proc. SPIE Vol. 4474, pp. 246253, 2001.
 F.T. Luk and S. Qiao,
A Fast Singular Value Algorithm for Hankel Matrices,
Advanced Signal Processing Algorithms, Architectures,
and Implementations XI, Franklin T. Luk, Editor,
Proc. SPIE Vol. 4474, pp. 254261, 2001.
 Yu Wu and Sanzheng Qiao,
Parallel implementation of a validated method for
initial value problems
Proceedings of the International Conference on
Parallel and Distributed Processing Techniques and
Applications, Vol. II, Editor: H.R. Arabnia,
pp. 699704,
Las Vegas, Nevada, USA, June 25  June 28, 2001.
 S. Qiao and D.L. Parnas,
Limitations of Backward Error Analysis
SERG Report 389,
Dept. of Computing and Software, McMaster University,
Hamilton, Ontario, Canada, L8S 4L7,
August, 2000.
 Dan Yan, Polina Galkin, and Sanzheng Qiao,
Global Meeting Plus (GMP)
Proceedings of the International Conference on
Parallel and Distributed Processing Techniques and
Applications, Vol. III, Editor: H.R. Arabnia,
pp. 16891695,
Las Vegas, Nevada, USA, June 26  June 29, 2000.
 S. Qiao and S. Wang,
Automatically Testing Numerical Stability
SERG Report 385,
Dept. of Computing and Software, McMaster University,
Hamilton, Ontario, Canada, L8S 4L7,
May 2000.
 W. Gao and S. Qiao,
Multithreaded Implementation of a Biomolecular Sequence
Alignment Algorithm,
Proceedings of Canadian Conference on Electrical
and Computer Engineering,
May 710, 2000, Halifax, Nova Scotia, Canada.
 Adam Bojanczyk, Sanzheng Qiao, and Allan O. Steinhardt,
Unifying Unitary and Hyperbolic Transformations,
Linear Algebra and Its Applications, 316(2000), pp.183197.
 F.T. Luk and S. Qiao,
A Fast Eigenvalue Algorithm for Hankel Matrices,
Linear Algebra and Its Applications, 316(2000), pp.171182.
(ps,
pdf)
 F.T. Luk and S. Qiao,
Analysis of a Fast Hankel Eigenvalue Algorithm,
in
Advanced Signal Processing Algorithms, Architectures,
and Implementations IX, Franklin T. Luk, Editor,
Proc. SPIE Vol. 3807, pp. 324333, 1999.

S. Qiao and H. Zhang,
An Automatic Logfile Analyzer for Parallel Programs,
Proceedings of the International Conference on
Parallel and Distributed Processing Techniques and
Applications, Vol. III, Editor: H.R. Arabnia,
pp. 13711376, Las Vegas, Nevada, USA, June 28  July 1, 1999.
(
compressed ps, revised)

W. Xiong, J. Li, R.M.M. Chen, and S. Qiao,
A Fast Decomposition of Banded Symmetric Toeplitz Matrices
for Parallel Processing,
Proceedings of IEEE International Symposium on Circuits and
Systems, Vol. III, pp. 259262,
Orlando, Florida, May 30  June 2, 1999.

S. Qiao,
Analysis of a Fast Eigenvalue Algorithm for Hankel Matrices,
Technical Report CAS9905,
Department of Computing and Software, McMaster University,
Ontario, Canada, L8S 4L7,
June 1999.

S. Qiao,
Analysis of the Unified Transformations,
Technical Report CAS9903,
Department of Computing and Software, McMaster University,
Ontario, Canada, L8S 4L7,
March 1999.
 K. Kreyman, D.L. Parnas, and S. Qiao,
Inspection Procedures for Critical Programs that Model
Physical Phenomena,
Technical Report No. 368,
Communications Research Laboratory, McMaster University,
Ontario, Canada, L8S 4K1,
February 1999.
(
pdf)
 S. Qiao and L. Qiao.
A Robust and Efficient Algorithm for Evaluating
Erlang B Formula.
Technical Report CAS9803,
Department of Computing and Software, McMaster University,
Ontario, Canada, L8S 4L7,
August 1998.
(ps,
pdf)
An
Erlang B Calculator, (new) is available.
 F.T. Luk and S. Qiao,
A Fast Eigenvalue Algorithm
for Hankel Matrices,
Advanced Signal Processing: Algorithms, Architectures,
and Implementations VIII, Franklin T. Luk, Editor,
Proc. SPIE Vol. 3461, pp.249256, July 1998.
 S. Qiao and W.SY. Wang,
Evaluating Phylogenetic Trees by Matrix Decomposition,
Anthropological Science,
106(1), 122, 1998.

S. Qiao,
Unifying Unitary and Hyperbolic Rotations and Reflectors,
Technical Report No. 9805, Department of Computer Science
and Systems, McMaster University, Ontario, Canada,
January 1998.
 S. Qiao and W. SY Wang,
A Matrix Representation of Phylogenetic Trees,
The Proceedings of
the Third Annual International Computing and
Combinatorics Conference (COCOON'97),
Editors: Tao Jiang and T.D. Lee,
Lecture Notes in Computer Science 1276, Springer, pp.274283,
Shanghai, China, August, 1997.
 F.T. Luk and S. Qiao,
Using ComplexOrthogonal Transformations to Diagonalize a
Complex Symmetric Matrix,
Advanced Signal Processing: Algorithms, Architectures,
and Implementations VII, Franklin T. Luk, Editor,
Proc. SPIE Vol. 3162, pp.418425, July 1997.
 S. Qiao and Sheldon Siutung Tam,
Parallel Implementation of a Matrix Decomposition
Method for Evaluatiing Phylogenetic Trees,
The Proceedings of the International Conference on
Parallel and Distributed Processing Techniques and
Applications (PDPTA'97), Volume II, Editor: H.R. Arabnia,
pp.774777,
Las Vegas, NV, June, 1997.
 S. Qiao, R.M.M. Chen, and X.D. Jia,
A QR Decomposition Based Algorithm for Computing
Transfer Functions From State Equation,
1997 IEEE International Symposium on
Circuits and Systems,
Hong Kong, June, 1997.
 M.K. Ng, R.J. Plemmons, and S. Qiao,
Regularized Blind Deconvolution using Recursive
Inverse Filtering,
Workshop on Scientific Computing,
Hong Kong, March, 1997. (invited lecture)
A revision appeared in
IEEE Trans. Image Processing, Vol. 9, No. 6,
2000. pp. 11301134.
Click
here for more on aerooptics research.
 F.T. Luk and S. Qiao,
Decaying Exponentials and Signal Decomposition,
Mathematics in Signal Processing,
Edited by J.G. McWhirter and I.K. Proudler,
Oxford University Press, 1998, pp. 175186.
ISBN 0198502028. (Based on the proceedings of a conference
on Mathematics in Signal Processing. Organized by the Institute
of Mathematics and its Applications and held at the
University of Warwick, UK, 1719 December 1996.)
 F.T. Luk and S. Qiao,
A Symmetric RankRevealing Toeplitz Matrix Decomposition,
Journal of VLSI Signal Processing,
Vol. 14, No. 1, pp.1928, Oct. 1996.
 F.T. Luk and S. Qiao,
An Adaptive Algorithm for Interference Cancelling in Array Processing,
Advanced Signal Processing Algorithms, Architectures, and Implementations
VI, Franklin T. Luk, Editor, SPIE Proceedings Vol. 2846,
151161(1996).
 F.T. Luk and S. Qiao,
RankRevealing Decomposition of Symmetric Toeplitz Matrices,
Advanced Signal Processing Algorithms
, Franklin T. Luk, Editor, SPIE Proceedings Vol. 2563,
293301(1995).
 S. Qiao,
Approximating the PSVD and QSVD,
SVD and Signal Processing, III:
Algorithms, Architectures and Applications,
edited by Marc Moonen and Bart De Moor,
Elsevier, pp. 149155, 1995.
 F.T. Luk and S. Qiao,
A New Matrix Decomposition for Signal Processing,
Automatica, a Journal of IFAC,
Vol. 30, No. 1, pp. 3943, Jan. 1994.
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Revised: August, 2014
Sanzheng Qiao