RenCang Li's Publication
Journal Publications

A converse to the BauerFike type theorem,
Linear Algebra and its Applications, 109 (1988), 167178.

On perturbation theorems for the generalized
eigenvalues of regular matrix pencils, Math. Numer. Sinica,
11:1 (1989), 1019 (Chinese).
English transl. Chinese J.
Numer. Math. Appl., 11:2 (1989), 2435.

Perturbation bounds for generalized eigenvalues. I,
Math. Numer. Sinica, 11:1 (1989), 196204 (Chinese).
English transl. Chinese J. Numer. Math. Appl.,
11 (1989), 19.

Perturbation bounds for generalized eigenvalues. II,
Math. Numer. Sinica, 11 (1989), 239247 (Chinese).
English transl. Chinese J. Numer. Math. Appl.,
11 (1989), 3443.

QR decomposition and nonlinear eigenvalue problems,
Math. Numer. Sinica, 11:4 (1989), 374385. (Chinese).

On the variations of the spectra of matrix pencils,
Linear Algebra and its Applications, 139 (1990), 147164.

Computing the Critical Points of a Stability Problem,
Math. Numer. Sinica, 12:3 (1990), 250258. (Chinese).

On eigenvalues of a Rayleigh quotient matrix,
Linear Algebra and its Applications, 169 (1992), 249255.
Note:
The argument that led to Theorem 3 is flawed, namely (13) in general incorrect
because \Gamma and \Sigma may not commute. But the attacked problem is
recently solved in

Compute multiple nonlinear eigenvalues,
J. Comp. Math., 10:1 (1992), 120.

Algorithms for inverse eigenvalue problems,
J. Comp. Math., 10:2 (1992), 97111.

Solution of Linear Matrix Equation AXDBXC=S and Perturbation of
Eigenspaces of a Matrix Pencil,
J. Comp. Math., 10 (1992), Supplementary Issue, pp. 93109.

A perturbation bound for definite pencils,
Linear Algebra and its Applications, 179 (1993), 191202.

A perturbation bound for the generalized polar decomposition,
BIT, 33 (1993), 304308.

Norms of certain matrices with applications to
variations of the spectra of matrices and matrix pencils,
Linear Algebra and its Applications, 182 (1993), 199234.

Bounds on perturbations of generalized singular values and
of associated subspaces,
SIAM Journal on Matrix Analysis and Applications,
14 (1993), 195234.

On eigenvalue variations of Rayleigh quotient matrix pencils of
a definite pencils, Linear Algebra and its Applications,
208/209 (1994), 471483.

On perturbations of matrix pencils with real
spectra, Math. Comp., 62 (1994), 231265.

New perturbation bounds for the unitary polar
factor, SIAM Journal on Matrix Analysis and Applications, 16 (1995).
 (with ZhiHao Cao and JinJun Xie)
A sharp version of Kahan's theorem on clustered eigenvalues,
Linear Algebra and its Applications, 245 (1996), 147156.
 (with Rajendra Bhatia)
On perturbations of matrix pencils with real
spectra, II. Math. Comp., 65 (1996), 637645.

Relative perturbation bounds for the unitary polar factor,
BIT, 37 (1997), 6775.
 (with Rajendra Bhatia and F. Kittaneh)
Some inequalities for commutators
and an application to spectral variation. II,
Linear and Multilinear Algebra, 43 (1997), 207219.
 (with William Kahan)
Unconventional schemes for a class of ordinary
differential equationswith applications to the Kortewegde
Vries (KdV) equation, J. Computational Physics,
134 (1997), 316331.
 (with William Kahan)
Composition constants for raising the orders of
unconventional schemes for ordinary differential
equations, Math. Comp., 66 (1997), 10891099.
 (with Rajendra Bhatia and F. Kittaneh)
Eigenvalues of
Symmetrizable Matrices, BIT, 38 (1998), 111.

Spectral Variations and Hadamard Products: Some Problems,
Linear Algebra and its Applications, 278 (1998),
317326.
Note: Paolo Tilli showed that
the conjecture in the paper was false in its generality, but remains
open in the most relevant cases from matrix perturbation theory.
 Paolo Tilli,
On some problems involving invariant norms and Hadamard
products, appeared in
Structured Matrices in Mathematics, Computer Science, and Engineering II
edited by Vadim Olshevsky, Comtemporary Mathematics, vol 2812001, 319324.
 M. Romeo and P. Tilli,
On some variational problems in the theory of unitarily
invariant norms and {Hadamard} products,
Linear Algebra and Its Applications,
326 (2001), pp.161172.

Relative perturbation theory: I eigenvalue and singular value
variations, SIAM Journal on Matrix Analysis and Applications,
19 (1998), 956982.
Note I: In establishing the theory, a socalled prelative
distance was proposed, among others, and proved to be a metric, indeed,
on the set of real numbers, but was conjectured to be a metric as well
on the set of complex numbers. That was finally resolved by
David Day and Anders Barrlund. Interestingly enough, this prelative
was later generalized. Interested readers may like to read the
following papers by them
 David Day,
Is A Metric on
, Technical Report Sand No. 981754,
Sandia National Laboratories, Albuquerque, NM, 1998.
 Anders Barrlund,
The pRelative Distance is a Metric,
SIAM Journal on Matrix Analysis and Applications,
Volume 21, Number 2, pp. 699702, 2000.
 Peter A. H鋝t?
A new weighted metric: the relative metric I,
Journal of Mathematical Analysis and Applications
Volume 274, Issue 1 , 1 October 2002, Pages 3858.
 Peter A. H鋝t?
A new weighted metric: the relative metric II,
http://arxiv.org/math.MG/0108026.
Note II: Much of results were later beautifully
extended and/or improved by C.K. Li and R. Mathias by means of
minmax principle; while ours were through a matrix decomposition
technique which holds an advantage over the minmax principle technique
in studying eigenspace variations in the 2nd part of this series.
Note III:
Another interesting reading is
SIAM News, Volume 37, Number 9, November 2004.

Relative perturbation theory: II eigenspace and singular space
variations, SIAM Journal on Matrix Analysis and Applications,
20 (1999), 471492.

Relative perturbation theory: (III) more bounds on eigenvalue
variation, Linear Algebra and its Applications, 266 (1997),
337345.

Relative Perturbation Theory: IV $\sin2\theta$ Theorems,
Linear Algebra and its Applications, 311 (2000), 4560.

A bound on the solution to a structured Sylvester equation
with an application to relative perturbation theory,
SIAM Journal on Matrix Analysis and Applications, 21 (1999), 440445.
 (with G. W. Stewart)
A new relative perturbation theorem
for singular value problem,
Linear Algebra and its Applications,
313 (2000), 4151.
 (with Rajendra Bhatia and William Kahan)
Pinchings and norms of scaled
triangular matrices, Linear and Multilinear Algebra,
50 (2002), 1521.

On perturbations of matrix pencils with real spectra, a Revisit,
Mathematics of Computation, 72 (2003), 715728.
 (with Ninoslav Truhar)
A Theorem for Graded Indefinite Hermitian Matrices,
Linear Algebra and its Applications, 359 (2003),
263276.
 (with Qiang Ye)
A Krylov Subspace Method for Quadratic Matrix Polynomials
with Application to Constrained Least Squares Problems,
SIAM Journal on Matrix Analysis and Applications, 25:2 (2003), 405428.
 (with Ernie Croot and Hui June Zhu)
The ABC conjecture and correctly rounded reciprocal square
roots, Theoretical Computer Science, 315 (2004), 405417.

Near Optimality of Chebyshev Interpolation For Elementary Function
Computations, IEEE Transactions on Computers, 53 (2004), 678687.

Accuracy of Computed Eigenvectors via Optimizing a Rayleigh Quotient,
BIT, 44:3 (2004), 585593.
Note: After its publication,
K. Veselic kindly brought to my attention that
the main result in this paper had already appeared in
And in fact, their Theorem 4 is stronger. It turns out that
a simple modification to our proof of Theorem 2
will lead to their Theorem 4.
 (with ChiKwong Li)
A Note on Eigenvalues of Perturbed Hermitian Matrices,
Linear Algebra and its Applications, 395 (2005), 183190.
Note There are several embarrassing typos in the published version. We
apologize for that. Please check out this
 (with Zhaojun Bai)
StructurePreserving Model Reduction using a Krylov Subspace
Projection Formulation,
Communications in Mathematical Sciences,
3:2 (2005), 179199.

Relative Perturbation Bounds For Positive Polar Factors Of Graded Matrices,
SIAM Journal on Matrix Analysis and Applications,
27 (2005), 424433.
 (with L. Hoffnung and Q. Ye)
Krylov Type Subspace Methods
for Matrix Polynomials,
Linear Algebra and its Applications,
415 (2006), 5281.

Lower bounds for the condition number of a real confluent
Vandermonde matrix,
Mathematics of Computation, 75 (2006), 19871995.

Asymptotically Optimal Lower Bounds For the Condition Number of a
Real Vandermonde Matrix, SIAM Journal on Matrix Analysis and Applications, 28:3 (2006), 829844.
 (with ChiKwong Li and Qiang Ye)
Eigenvalues of An Alignment Matrix in Nonlinear Manifold Learning,
Communications in Mathematical Sciences, 5:2 (2007), 313329.

Convergence of CG and GMRES on a Tridiagonal Toeplitz Linear System
, BIT, 47 (2007), 577599.
 (with Nicholas J. Higham and Fran鏾ise Tisseur)
Backward Error of Polynomial Eigenproblems
Solved by Linearization,
SIAM Journal on Matrix Analysis and Applications, 29:4 (2007), 12181241.
 (with Qiang Ye and Hongyuan Zha)
Analysis of An Alignment Algorithm for Nonlinear Dimensionality Reduction,
BIT, 47 (2007), 873885.

On Meinardus' Examples For the Conjugate Gradient Method,
Mathematics of Computation, 77 (2008), 335352.

Hard Cases For Conjugate Gradient Method
, International Journal for Information & Systems Sciences,
4:1 (2008), 1529.

Vandermonde Matrices with Chebyshev Nodes,
Linear Algebra and its Applications, 428 (2008), 18031832.
 (with Wei Zhang)
The Rate of Convergence of GMRES on a Tridiagonal Toeplitz Linear System,
Numerische Mathematik, 112 (2009), 167293.
 (with Wei Zhang)
The Rate of Convergence of GMRES on a Tridiagonal Toeplitz Linear System, II,
Linear Algebra and its Applications, 431 (2009), 24252436.
 (with Sylvie Boldo and Marc Daumas)
Formally Certified Argument
Reduction with a FusedMultiplyAdd,
IEEE Transactions on Computers, 58:8 (2009), 11391145.
 (with Peter Benner and Ninoslav Truhar)
On the ADI Method for Sylvester Equations,
Journal of Computational and Applied Mathematics,
233:4 (2009), 10351045.

Sharpness in Rates of Convergence for the Symmetric Lanczos Method,
Mathematics of Computation, 79:269 (2010), 419435.
 (with Ninoslav Truhar, Zoran Tomljanovic)
Analysis of the solution of the Sylvester equation using Low Rank ADI with exact shifts,
Systems & Control Letters, 59:34 (2010), 248257.
 (with WenWei Lin and ChernShuh Wang)
Structured Backward Error for Palindromic Polynomial Eigenvalue Problems,
Numerische Mathematik, 116:1 (2010), 95122.
 (with Yunkai Zhou)
Bounding the Spectrum of Large Hermitian Matrices,
Linear Algebra and its Applications, 435 (2011), 480493.
 (with Zhaojun Bai, Cherung Lee, and Shufang Xu)
Stable Solutions of Linear Systems Involving Long Chain of Matrix Multiplications,
Linear Algebra and its Applications, 435 (2011), 659673.
 (with XiaoWen Chang)
Multiplicative Perturbation Analysis for QR Factorizations,
Numerical Algebra, Control and Optimization, 1:2 (2011), 301316.
 (with Yuji Nakatsukasa,
Ninoslav Truhar, and Shufang Xu)
Perturbation of Partitioned Hermitian Definite Generalized Eigenvalue Problem,
SIAM Journal on Matrix Analysis and Applications, 32:2 (2011), 642663.
Errata
 (with William Kahan)
A Family of Anadromic Numerical Methods for Matrix Riccati Differential Equations,
Mathematics of Computation, 81:277 (2012), 233265.
 (with Dario Rocca, Zhaojun Bai, and Giulia Galli)
A block variational procedure for the iterative diagonalization of nonHermitian randomphase approximation matrices,,
The Journal of Chemical Physics, 136 (2012), 034111.
 (with Rajendra Bhatia)
An Interpolating Family of Means,
Comm. Stoch. Anal., 6:1 (2012), 1531.
 (with Jungong Xue and Shufang Xu)
Accurate Solutions of MMatrix Sylvester Equations,
Numerische Mathematik, 120:4 (2012), 639670.
 (with Jungong Xue and Shufang Xu)
Accurate Solutions of MMatrix Algebraic Riccati Equations,
Numerische Mathematik, 120:4 (2012), 671700.
 (with WeiGuo Wang and WeiChao Wang)
AlternatingDirectional Doubling Algorithm for MMatrix Algebraic Riccati Equations,
SIAM Journal on Matrix Analysis and Applications, 33:1 (2012), 170194.
 (with Yuji Nakatsukasa,
Ninoslav Truhar, and Weiguo Wang)
Perturbation of Multiple Eigenvalues of Hermitian Matrices,
Linear Algebra and its Applications, 437 (2012), 202213.
 (with Zhaojun Bai)
Minimization Principle for Linear Response Eigenvalue Problem, I: Theory,
SIAM Journal on Matrix Analysis and Applications, 33:4 (2012), 10751100.
 (with Zhaojun Bai)
Minimization Principle for Linear Response Eigenvalue Problem, II: Computation,
SIAM Journal on Matrix Analysis and Applications, 34:2 (2013), 392416.
 (with Xin Liang and Zhaojun Bai)
Trace Minimization Principles for Positive SemiDefinite Pencils,
Linear Algebra and its Applications, 438 (2013), 30853106.
 (with WeiGuo Wang and WeiChao Wang)
Deflating Irreducible Singular MMatrix Algebraic Riccati Equations,
Numerical Algebra, Control and Optimization,
3 (2013), 491518.
 (with Zhongming Teng)
Convergence Analysis of Lanczostype Methods for the Linear Response Eigenvalue Problem,
Journal of Computational and Applied Mathematics, 247 (2013), 1733.
 (with Xin Liang)
Extensions of Wielandt's Minmax Principles for Positive SemiDefinite Pencils,
Linear and Multilinear Algebra, 62:8 (2014), 10321048.
 (with Zhaojun Bai)
Minimization Principles and Computation for the Generalized Linear Response Eigenvalue Problem,
BIT Numerical Mathematics, 54 (2014), 3154.
 (with LeiHong Zhang
and Jungong Xue)
RayleighRitz Approximation for the Linear Response Eigenvalue Problem,
SIAM Journal on Matrix Analysis and Applications, 35 (2014), 765782.
 (with Kris Garrett)
GIP Integrators for Matrix Riccati Differential Equations,
Applied Mathematics and Computation, 241 (2014), 283297.
 (with LeiHong Zhang)
Maximization of the Sum of the Trace Ratio On the Stiefel Manifold, I: Theory,
SCIENCE CHINA Mathematics, 57:12 (2014), 24952508.
 (with Qiang Ye)
Simultaneous similarity reductions for a pair of matrices to condensed forms,
Communications in Mathematics and Statistics, 2(2014), 139153.
 (with LeiHong Zhang)
Maximization of the Sum of the Trace Ratio On the Stiefel Manifold, II: Computation,
SCIENCE CHINA Mathematics, 58:7 (2015), 15491566.
 (with Xuefeng Wang)
Monotonicity of Unitarily Invariant Norms,
Linear Algebra and its Applications, 466(2015), 254266.
 (with Linzhang Lu and Fei Yuan)
A New Look at the Doubling Algorithm for a Structured
Palindromic Quadratic Eigenvalue Problem,
Numerical Linear Algebra with Applications, 22 (2015), 393409.
There are a few typos in Theorem 3.2 and its proof. This is a
correction of it.
 (with LeiHong Zhang)
Convergence of the Block Lanczos Method For Eigenvalue Clusters,
Numerische Mathematik, 131:1 (2015), 83113.
 (with LeiHong Zhang , Wei Hong Yang, and Chungen Shen)
A Krylov Subspace Method For Large Scale Second Order Cone Linear Complementarity Problem,
SIAM Journal on Scientific Computing, 37:4 (2015), A2046A2075.
 (with Xin Liang) The Hyperbolic Quadratic Eigenvalue Problem,
Forum of Mathematics, Sigma, 3:e13 (2015), 93 pages, doi:10.1017/fms.2015.14.
 (with LeiHong Zhang
and WenWei Lin)
Backward Perturbation Analysis and Residualbased Error
Bounds For the Linear Response Eigenvalue Problem,
BIT Numerical Mathematics, 55:3 (2015), 869896.
 (with Zhongming Teng and Linzhang Lu)
Clusterrobust accuracy bounds for Ritz subspaces,
Linear Algebra and its Applications, 480 (2015), 1126.
 (with Zhongming Teng and Linzhang Lu)
Perturbation of Partitioned Linear Response Eigenvalue Problems,
Electronic Transactions on Numerical Analysis ,
44 (2015), 624638.
 (with TsungMing Huang, WeiQiang Huang, and WenWei Lin)
A New TwoPhase StructurePreserving Doubling Algorithm for Critically Singular
MMatrix Algebraic Riccati Equations,
Numerical Linear Algebra with Applications,
23 (2016), 291313.
 (with Jinrui Guan, Linzhang Lu, and Rongxia Shao)
SelfCorrective Iterations (SCI) for Generalized Diagonally Dominant Matrices,
Journal of Computational and Applied Mathematics, 302 (2016), 285300.
 (with Zhaojun Bai and WenWei Lin)
Linear Response Eigenvalue Problem Solved by Extended Locally Optimal
Preconditioned Conjugate Gradient Methods,
SCIENCE CHINA Mathematics, 59:8 (2016), 14431460.
 (with Akira Imakura and ShaoLiang Zhang)
Locally optimal and heavy ball GMRES methods,
Japan Journal on Industrial Applied Mathematics, 33:2 (2016), 471499.
 (with Zhongming Teng and Yunkai Zhou)
A Block ChebyshevDavidson Method for Linear Response Eigenvalue Problems,
Advances in Computational Mathematics, 42:5 (2016), 11031128.
 (with C. Kristopher Garrett and Zhaojun Bai)
A Nonlinear QR Algorithm for Banded Nonlinear Eigenvalue Problems,
Transactions on Mathematical Software, 43:1 (2016), 4:14:19.
 (with Linzhang Lu, Teng Wang, YuehCheng Kuo, and WenWei Lin)
A Fast Algorithm For Fast Train Palindromic Quadratic Eigenvalue Problems,
SIAM Journal on Scientific Computing, 38:6 (2016), A3410A3429.
 (with Tiexiang Li and WenWei Lin)
A Symmetric StructurePreserving GammaQR Algorithm for Linear Response Eigenvalue Problems,
Linear Algebra and its Applications, 520:1 (2017), 191214.
 (with Jungong Xue)
Highly Accurate Doubling Algorithms for Mmatrix Algebraic Riccati Equations,
Numerische Mathematik, 135 (2017), 733767.
 (with Weiguo Wang and Leihong Zhang)
Error Bounds For Approximate Deflating Subspaces For Linear
Response Eigenvalue Problems,
Linear Algebra and its Applications, 528 (2017), 273289.
 (with LeiHong Zhang and Chungen Shen)
On the Generalized Lanczos TrustRegion Method,
SIAM Journal on Optimization, 27:3 (2017), 21102142.
 (with TsungMing Huang, WenWei Lin, and Linzhang Lu)
Optimal Parameters for Doubling Algorithms,
Journal of Mathematical Study, 50:4 (2017), 339357.
 (with Yunfeng Cai, Leihong Zhang, and Zhaojun Bai)
On an EigenvectorDependent Nonlinear Eigenvalue Problem,
SIAM Journal on Matrix Analysis and Applications, 39:3 (2018), 13601382.
 (with Changli Liu) Structured Backward Error for Palindromic Polynomial Eigenvalue Problems, II:
Approximate Eigentriplets, Frontiers of Mathematics in China, 13:6 (2018), 13971426.
 (with Xiang Wang, Xing Li, and LeiHong Zhang) An Efficient Numerical Method for the Symmetric Positive Definite SecondOrder Cone Linear
Complementarity Problem, Journal of Scientific Computing, (2019).
Software
 Contribution to LAPACK. I provided subroutines for
solving secular equations which sit at the kernel
of the divideandconquer methods for the symmetric eigenvalue problem and singular value problem.
 Constructing high order palindromic composition schemes:
composition.txt and
composition.tar.gz, both at
NETLIB's ODE
 HPUX libm for Itanium
 Formal definitions and theorems, with proofs, about floatingpoint
numbers. Contributors: Sylvie Boldo*, Marc Daumas, Laurent Fousse*, RenCang Li and Guillaume Melquiond*, JeanMichel Muller, Laurence Rideau*, and Laurent Thery*. Contributors that were directly interacting with Coq are indicated by a star.
 MATLAB codes for CGtype methods for eigenvalue problems,
prepared for the 2013 SIAM Gene Golub SIAM Summer School, Fudan university, Shanghai.
 GIP Integrator Package for matrix Riccati differential equations
(GIPACK). This preliminary release contains MATLAB functions for using the integrators. Detailed discussion on the integrators
can be found at GIP Integrators for Matrix Riccati Differential Equations,
Applied Mathematics and Computation, 241 (2014), 283297.
 Unstructrually Banded Nonlinear Eigenvalue Software.
The package contains both Matlab and C++ implementations of solving unstructurally banded nonlinear eigenvalues with a Kublanovskaya type method.
 Expertly extended Locally Optimal Block Preconditioned 4d Conjugate Gradient
(xeLOBP4dCG). This preliminary release contains MATLAB functions for using the methods. Detailed discussion on the methods
can be found at
Linear Response Eigenvalue Problem Solved by Extended Locally Optimal
Preconditioned Conjugate Gradient Methods,
SCIENCE CHINA Mathematics, 59:8 (2016), 14431460.
 MATLAB codes for doubling algorithms for nonlinear matrix equations to go with
 (with TsungMing Huang and WenWei Lin)
StructurePreserving Doubling Algorithms For Nonlinear Matrix Equations.
SIAM, Philadelphia, September 2018.
PhD Thesis
Raising the Orders of Unconventional Schemes for Ordinary Differential
Equations,
University of California at Berkeley, 1995.
Conference Proceedings
 (with Anders H. Andersen, William S. Rayens, and Lee X. Blonder)
Mathematical problems in the application of multilinear models to facial emotion processing experiments,
Proc. SPIE 4121, Mathematical Modeling, Estimation, and Imaging, 77 (October 4, 2000); doi:10.1117/12.402429.
 (with Sylvie Boldo and
Marc Daumas)
Theorems on Efficient Argument Reductions,
Proceedings of the 16th IEEE Symposium on Computer Arithmetic, pp.129136,
2003.
 (with Zhaojun Bai)
StructurePreserving Model Reduction,
Proceedings of PARA'04,
J. Dongarra, K. Madsen, and J. Wa{\' s}niewski (eds.),
Lecture Notes in Computer Science, 3732 (2006), 323332, Springer.
Presented at
Minisymposium  Substructuring, Dimension Reduction and Applications,
organized by Zhaojun Bai and RenCang Li for
PARA'04 workshop on stateoftheart in Scientific Computing,
Technical University of Denmark, Copenhagen, Denmark, June 2023, 2004.

Rayleigh Quotient Based Optimization Methods For Eigenvalue Problems, Summary of lectures delivered
at Gene Golub SIAM Summer School 2013, Fudan University, Shanghai, China, July 22 to August 2, 2013.
Suggested reading
A Scorching Summer School in Shanghai by Jing Leng, Charles Puelz, and Leo Taslaman, SIAM News, November 1, 2013.
 (with Hongteng Xu, Hongyuan Zha, and Mark A. Davenport)
Active Manifold Learning via Gershgorin Circle Guided Sample Selection,
AAAI15 (2015, Association for the Advancement of Artificial Intelligence),
Austin, Texas, January 2529, 2015.
 (with Zhaojun Bai)
Recent Progress in Linear Response Eigenvalue Problems. In: Sakurai T., Zhang SL., Imamura T., Yamamoto Y., Kuramashi Y., Hoshi T. (eds) Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing. EPASA 2015. Lecture Notes in Computational Science and Engineering, vol 117. Springer, Cham
Book Chapters
 (with Z. Bai)
Stability and Accuracy Assessments. Sections, 4.8, 5.7,7.13, and 8.8 in
Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide,
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst (editors),
SIAM, Philadelphia, 2000.
 Matrix Perturbation Theory as Chapter 15 in
Handbook of Linear Algebra,
L. Hogben, R. Brualdi, A. Greenbaum and R. Mathias (editors),
Chapman & Hall/CRC, New York, 2007.
Revised Matrix Perturbation Theory as Chapter 21 in
Handbook of Linear Algebra, 2nd edition,
Chapman & Hall/CRC, New York, 2014.
 (with Z. Bai and Y. Su)
A Unified Krylov Projection Framework for
StructurePreserving Model Reduction, in
Model Order Reduction: Theory, Research Aspects and Applications,
Series: Mathematics in Industry,
Subseries: The European Consortium for Mathematics in Industry, Vol. 13,
Schilders, Wilhelmus H.A.; van der Vorst, Henk A.; Rommes, Joost (Eds.),
Springer, 2008.

Rayleigh Quotient Based Optimization Methods for Eigenvalue Problems, in
Matrix Functions and Matrix Equations, Series in Contemporary Applied Mathematics: Vol. 19,
Zhaojun Bai, Weiguo Gao, Yangfeng Su (Eds.), pp.76108, World Scientific, Singapore, 2015.
Book
 (with TsungMing Huang and WenWei Lin)
StructurePreserving Doubling Algorithms For Nonlinear Matrix Equations.
SIAM, Philadelphia, September 2018.
Selected Technical Reports (unpublished)
 (with Huan Ren)
An efficient tridiagonal eigenvalue solver on CM 5 with
Laguerre's iteration,
Technical ReportUCB//CSD94848,
Computer Science Division, University of California at Berkeley,
1994. (
cover)

Relations between the field of values of a matrix and those of
its Schur complements,
Technical ReportUCB//CSD94849,
Computer Science Division, University of California at Berkeley,
1994. (
cover)

Reciproot algorithmcorrectly rounded?
Technical ReportUCB//CSD94850,
Computer Science Division, University of California at Berkeley,
1994.

Solving secular equations stably and efficiently,
LAPACK working notes 89, (1993).
Also Technical ReportUCB//CSD94851,
Computer Science Division, University of California at Berkeley,
1994.

Linear systems with coefficient matrices having fields of values
not containing the origin,
Technical ReportUCB//CSD94853,
Computer Science Division, University of California at Berkeley,
1994.
 An Efficient Implementation of a Spectral Transform Method
for Solving the Shallow Water Equations, Manuscript, Oak
Ridge National Laboratory, 1996.

A MultiResolution Approach for Calculating Primary Eigenvectors
of a Large Set of Images, Technical Report, 9813,
Department of Mathematics, University of Kentucky, June 1998.

Unconventional Reflexive Numerical Methods for Matrix
Differential Riccati Equations, Technical Report, 200036,
Department of Mathematics, University of Kentucky, November 2000.

Test Positive Realness Of A General Transfer Function Matrix,
Technical Report, 200020,
Department of Mathematics, University of Kentucky, April 2000.

Structural Preserving Model Reductions, Technical Report, 200402,
Department of Mathematics, University of Kentucky, January 2004.

Asymptotically Optimal Lower Bounds For the Condition Number of a
Real Vandermonde Matrix, Technical Report, 200405,
Department of Mathematics, University of Kentucky, June 2004.

Sharpness in Rates of Convergence For CG and Symmetric Lanczos Methods,
Technical Report, 200501,
Department of Mathematics, University of Kentucky, January 2005.

Vandermonde Matrices with Chebyshev Nodes,
Technical Report, 200502,
Department of Mathematics, University of Kentucky, January 2005.
Problems
 (with David Day) A Bound on the Eigenvalue Gaps:
Problem 10499,
The American Mathematical Monthly, , 103:1 (1996), p.75;
Solution,
The American Mathematical Monthly, 105:1 (1998), p.71.
Libm for HPUX Itanium
 (with James W. Thomas, Jon P. Okada, and Peter Markstein)
The Libm Library and FloatingPoint Arithmetic in HPUX for Itanium?Based Systems
Research supported in part by
Any opinions, findings, and conclusions or recommendations expressed in these
material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Last modified Wednesday September 21, 2005