# Ren-Cang Li's Publication

## Journal Publications

1. A converse to the Bauer-Fike type theorem, Linear Algebra and its Applications, 109 (1988), 167-178.
2. On perturbation theorems for the generalized eigenvalues of regular matrix pencils, Math. Numer. Sinica, 11:1 (1989), 10-19 (Chinese). English transl. Chinese J. Numer. Math. Appl., 11:2 (1989), 24-35.
3. Perturbation bounds for generalized eigenvalues. I, Math. Numer. Sinica, 11:1 (1989), 196-204 (Chinese). English transl. Chinese J. Numer. Math. Appl., 11 (1989), 1-9.
4. Perturbation bounds for generalized eigenvalues. II, Math. Numer. Sinica, 11 (1989), 239-247 (Chinese). English transl. Chinese J. Numer. Math. Appl., 11 (1989), 34-43.
5. QR decomposition and nonlinear eigenvalue problems, Math. Numer. Sinica, 11:4 (1989), 374-385. (Chinese).
6. On the variations of the spectra of matrix pencils, Linear Algebra and its Applications, 139 (1990), 147-164.
7. Computing the Critical Points of a Stability Problem, Math. Numer. Sinica, 12:3 (1990), 250-258. (Chinese).
8. On eigenvalues of a Rayleigh quotient matrix, Linear Algebra and its Applications, 169 (1992), 249-255.
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
9. Compute multiple nonlinear eigenvalues, J. Comp. Math., 10:1 (1992), 1-20.
10. Algorithms for inverse eigenvalue problems, J. Comp. Math., 10:2 (1992), 97-111.
11. Solution of Linear Matrix Equation AXD-BXC=S and Perturbation of Eigenspaces of a Matrix Pencil, J. Comp. Math., 10 (1992), Supplementary Issue, pp. 93-109.
12. A perturbation bound for definite pencils, Linear Algebra and its Applications, 179 (1993), 191-202.
13. A perturbation bound for the generalized polar decomposition, BIT, 33 (1993), 304-308.
14. Norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils, Linear Algebra and its Applications, 182 (1993), 199-234.
15. Bounds on perturbations of generalized singular values and of associated subspaces, SIAM Journal on Matrix Analysis and Applications, 14 (1993), 195-234.
16. On eigenvalue variations of Rayleigh quotient matrix pencils of a definite pencils, Linear Algebra and its Applications, 208/209 (1994), 471-483.
17. On perturbations of matrix pencils with real spectra, Math. Comp., 62 (1994), 231-265.
18. New perturbation bounds for the unitary polar factor, SIAM Journal on Matrix Analysis and Applications, 16 (1995).
19. (with Zhi-Hao Cao and Jin-Jun Xie) A sharp version of Kahan's theorem on clustered eigenvalues, Linear Algebra and its Applications, 245 (1996), 147-156.
20. (with Rajendra Bhatia) On perturbations of matrix pencils with real spectra, II. Math. Comp., 65 (1996), 637-645.
21. Relative perturbation bounds for the unitary polar factor, BIT, 37 (1997), 67-75.
22. (with Rajendra Bhatia and F. Kittaneh) Some inequalities for commutators and an application to spectral variation. II, Linear and Multilinear Algebra, 43 (1997), 207-219.
23. (with William Kahan) Unconventional schemes for a class of ordinary differential equations--with applications to the Korteweg-de Vries (KdV) equation, J. Computational Physics, 134 (1997), 316-331.
24. (with William Kahan) Composition constants for raising the orders of unconventional schemes for ordinary differential equations, Math. Comp., 66 (1997), 1089-1099.
25. (with Rajendra Bhatia and F. Kittaneh) Eigenvalues of Symmetrizable Matrices, BIT, 38 (1998), 1-11.
26. Spectral Variations and Hadamard Products: Some Problems, Linear Algebra and its Applications, 278 (1998), 317-326.
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.
27. Relative perturbation theory: I eigenvalue and singular value variations, SIAM Journal on Matrix Analysis and Applications, 19 (1998), 956-982.
Note I: In establishing the theory, a so-called p-relative 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 p-relative was later generalized. Interested readers may like to read the following papers by them
Note II: Much of results were later beautifully extended and/or improved by C.-K. Li and R. Mathias by means of min-max principle; while ours were through a matrix decomposition technique which holds an advantage over the min-max 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.
28. Relative perturbation theory: II eigenspace and singular space variations, SIAM Journal on Matrix Analysis and Applications, 20 (1999), 471-492.
29. Relative perturbation theory: (III) more bounds on eigenvalue variation, Linear Algebra and its Applications, 266 (1997), 337-345.
30. Relative Perturbation Theory: IV $\sin2\theta$ Theorems, Linear Algebra and its Applications, 311 (2000), 45-60.
31. 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), 440-445.
32. (with G. W. Stewart) A new relative perturbation theorem for singular value problem, Linear Algebra and its Applications, 313 (2000), 41-51.
33. (with Rajendra Bhatia and William Kahan) Pinchings and norms of scaled triangular matrices, Linear and Multilinear Algebra, 50 (2002), 15-21.
34. On perturbations of matrix pencils with real spectra, a Revisit, Mathematics of Computation, 72 (2003), 715-728.
35. (with Ninoslav Truhar) A Theorem for Graded Indefinite Hermitian Matrices, Linear Algebra and its Applications, 359 (2003), 263-276.
36. (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), 405-428.
37. (with Ernie Croot and Hui June Zhu) The ABC conjecture and correctly rounded reciprocal square roots, Theoretical Computer Science, 315 (2004), 405-417.
38. Near Optimality of Chebyshev Interpolation For Elementary Function Computations, IEEE Transactions on Computers, 53 (2004), 678-687.
39. Accuracy of Computed Eigenvectors via Optimizing a Rayleigh Quotient, BIT, 44:3 (2004), 585-593.
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.
40. (with Chi-Kwong Li) A Note on Eigenvalues of Perturbed Hermitian Matrices, Linear Algebra and its Applications, 395 (2005), 183-190.
Note There are several embarrassing typos in the published version. We apologize for that. Please check out this
41. (with Zhaojun Bai) Structure-Preserving Model Reduction using a Krylov Subspace Projection Formulation, Communications in Mathematical Sciences, 3:2 (2005), 179-199.
42. Relative Perturbation Bounds For Positive Polar Factors Of Graded Matrices, SIAM Journal on Matrix Analysis and Applications, 27 (2005), 424-433.
43. (with L. Hoffnung and Q. Ye) Krylov Type Subspace Methods for Matrix Polynomials, Linear Algebra and its Applications, 415 (2006), 52-81.
44. Lower bounds for the condition number of a real confluent Vandermonde matrix, Mathematics of Computation, 75 (2006), 1987-1995.
45. Asymptotically Optimal Lower Bounds For the Condition Number of a Real Vandermonde Matrix, SIAM Journal on Matrix Analysis and Applications, 28:3 (2006), 829-844.
46. (with Chi-Kwong Li and Qiang Ye) Eigenvalues of An Alignment Matrix in Nonlinear Manifold Learning, Communications in Mathematical Sciences, 5:2 (2007), 313-329.
47. Convergence of CG and GMRES on a Tridiagonal Toeplitz Linear System , BIT, 47 (2007), 577-599.
48. (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), 1218-1241.
49. (with Qiang Ye and Hongyuan Zha) Analysis of An Alignment Algorithm for Nonlinear Dimensionality Reduction, BIT, 47 (2007), 873-885.
50. On Meinardus' Examples For the Conjugate Gradient Method, Mathematics of Computation, 77 (2008), 335-352.
51. Hard Cases For Conjugate Gradient Method , International Journal for Information & Systems Sciences, 4:1 (2008), 15-29.
52. Vandermonde Matrices with Chebyshev Nodes, Linear Algebra and its Applications, 428 (2008), 1803-1832.
53. (with Wei Zhang) The Rate of Convergence of GMRES on a Tridiagonal Toeplitz Linear System, Numerische Mathematik, 112 (2009), 167-293.
54. (with Wei Zhang) The Rate of Convergence of GMRES on a Tridiagonal Toeplitz Linear System, II, Linear Algebra and its Applications, 431 (2009), 2425-2436.
55. (with Sylvie Boldo and Marc Daumas) Formally Certified Argument Reduction with a Fused-Multiply-Add, IEEE Transactions on Computers, 58:8 (2009), 1139-1145.
56. (with Peter Benner and Ninoslav Truhar) On the ADI Method for Sylvester Equations, Journal of Computational and Applied Mathematics, 233:4 (2009), 1035-1045.
57. Sharpness in Rates of Convergence for the Symmetric Lanczos Method, Mathematics of Computation, 79:269 (2010), 419-435.
58. (with Ninoslav Truhar, Zoran Tomljanovic) Analysis of the solution of the Sylvester equation using Low Rank ADI with exact shifts, Systems & Control Letters, 59:3-4 (2010), 248-257.
59. (with Wen-Wei Lin and Chern-Shuh Wang) Structured Backward Error for Palindromic Polynomial Eigenvalue Problems, Numerische Mathematik, 116:1 (2010), 95-122.
60. (with Yunkai Zhou) Bounding the Spectrum of Large Hermitian Matrices, Linear Algebra and its Applications, 435 (2011), 480-493.
61. (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), 659-673.
62. (with Xiao-Wen Chang) Multiplicative Perturbation Analysis for QR Factorizations, Numerical Algebra, Control and Optimization, 1:2 (2011), 301-316.
63. (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), 642-663.
Errata
64. (with William Kahan) A Family of Anadromic Numerical Methods for Matrix Riccati Differential Equations, Mathematics of Computation, 81:277 (2012), 233-265.
65. (with Dario Rocca, Zhaojun Bai, and Giulia Galli) A block variational procedure for the iterative diagonalization of non-Hermitian random-phase approximation matrices,, The Journal of Chemical Physics, 136 (2012), 034111.
66. (with Rajendra Bhatia) An Interpolating Family of Means, Comm. Stoch. Anal., 6:1 (2012), 15-31.
67. (with Jungong Xue and Shufang Xu) Accurate Solutions of M-Matrix Sylvester Equations, Numerische Mathematik, 120:4 (2012), 639-670.
68. (with Jungong Xue and Shufang Xu) Accurate Solutions of M-Matrix Algebraic Riccati Equations, Numerische Mathematik, 120:4 (2012), 671-700.
69. (with Wei-Guo Wang and Wei-Chao Wang) Alternating-Directional Doubling Algorithm for M-Matrix Algebraic Riccati Equations, SIAM Journal on Matrix Analysis and Applications, 33:1 (2012), 170-194.
70. (with Yuji Nakatsukasa, Ninoslav Truhar, and Wei-guo Wang) Perturbation of Multiple Eigenvalues of Hermitian Matrices, Linear Algebra and its Applications, 437 (2012), 202-213.
71. (with Zhaojun Bai) Minimization Principle for Linear Response Eigenvalue Problem, I: Theory, SIAM Journal on Matrix Analysis and Applications, 33:4 (2012), 1075-1100.
72. (with Zhaojun Bai) Minimization Principle for Linear Response Eigenvalue Problem, II: Computation, SIAM Journal on Matrix Analysis and Applications, 34:2 (2013), 392-416.
73. (with Xin Liang and Zhaojun Bai) Trace Minimization Principles for Positive Semi-Definite Pencils, Linear Algebra and its Applications, 438 (2013), 3085-3106.
74. (with Wei-Guo Wang and Wei-Chao Wang) Deflating Irreducible Singular M-Matrix Algebraic Riccati Equations, Numerical Algebra, Control and Optimization, 3 (2013), 491-518.
75. (with Zhongming Teng) Convergence Analysis of Lanczos-type Methods for the Linear Response Eigenvalue Problem, Journal of Computational and Applied Mathematics, 247 (2013), 17-33.
76. (with Xin Liang) Extensions of Wielandt's Min-max Principles for Positive Semi-Definite Pencils, Linear and Multilinear Algebra, 62:8 (2014), 1032-1048.
77. (with Zhaojun Bai) Minimization Principles and Computation for the Generalized Linear Response Eigenvalue Problem, BIT Numerical Mathematics, 54 (2014), 31-54.
78. (with Lei-Hong Zhang and Jungong Xue) Rayleigh-Ritz Approximation for the Linear Response Eigenvalue Problem, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 765-782.
79. (with Kris Garrett) GIP Integrators for Matrix Riccati Differential Equations, Applied Mathematics and Computation, 241 (2014), 283-297.
80. (with Lei-Hong Zhang) Maximization of the Sum of the Trace Ratio On the Stiefel Manifold, I: Theory, SCIENCE CHINA Mathematics, 57:12 (2014), 2495-2508.
81. (with Qiang Ye) Simultaneous similarity reductions for a pair of matrices to condensed forms, Communications in Mathematics and Statistics, 2(2014), 139-153.
82. (with Lei-Hong Zhang) Maximization of the Sum of the Trace Ratio On the Stiefel Manifold, II: Computation, SCIENCE CHINA Mathematics, 58:7 (2015), 1549-1566.
83. (with Xuefeng Wang) Monotonicity of Unitarily Invariant Norms, Linear Algebra and its Applications, 466(2015), 254-266.
84. (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), 393-409.
There are a few typos in Theorem 3.2 and its proof. This is a correction of it.
85. (with Lei-Hong Zhang) Convergence of the Block Lanczos Method For Eigenvalue Clusters, Numerische Mathematik, 131:1 (2015), 83-113.
86. (with Lei-Hong 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), A2046-A2075.
87. (with Xin Liang) The Hyperbolic Quadratic Eigenvalue Problem, Forum of Mathematics, Sigma, 3:e13 (2015), 93 pages, doi:10.1017/fms.2015.14.
88. (with Lei-Hong Zhang and Wen-Wei Lin) Backward Perturbation Analysis and Residual-based Error Bounds For the Linear Response Eigenvalue Problem, BIT Numerical Mathematics, 55:3 (2015), 869-896.
89. (with Zhongming Teng and Linzhang Lu) Cluster-robust accuracy bounds for Ritz subspaces, Linear Algebra and its Applications, 480 (2015), 11-26.
90. (with Zhongming Teng and Linzhang Lu) Perturbation of Partitioned Linear Response Eigenvalue Problems, Electronic Transactions on Numerical Analysis , 44 (2015), 624-638.
91. (with Tsung-Ming Huang, Wei-Qiang Huang, and Wen-Wei Lin) A New Two-Phase Structure-Preserving Doubling Algorithm for Critically Singular M-Matrix Algebraic Riccati Equations, Numerical Linear Algebra with Applications, 23 (2016), 291-313.
92. (with Jinrui Guan, Linzhang Lu, and Rongxia Shao) Self-Corrective Iterations (SCI) for Generalized Diagonally Dominant Matrices, Journal of Computational and Applied Mathematics, 302 (2016), 285-300.
93. (with Zhaojun Bai and Wen-Wei Lin) Linear Response Eigenvalue Problem Solved by Extended Locally Optimal Preconditioned Conjugate Gradient Methods, SCIENCE CHINA Mathematics, 59:8 (2016), 1443-1460.
94. (with Akira Imakura and Shao-Liang Zhang) Locally optimal and heavy ball GMRES methods, Japan Journal on Industrial Applied Mathematics, 33:2 (2016), 471-499.
95. (with Zhongming Teng and Yunkai Zhou) A Block Chebyshev-Davidson Method for Linear Response Eigenvalue Problems, Advances in Computational Mathematics, 42:5 (2016), 1103-1128.
96. (with C. Kristopher Garrett and Zhaojun Bai) A Nonlinear QR Algorithm for Banded Nonlinear Eigenvalue Problems, Transactions on Mathematical Software, 43:1 (2016), 4:1-4:19.
97. (with Linzhang Lu, Teng Wang, Yueh-Cheng Kuo, and Wen-Wei Lin) A Fast Algorithm For Fast Train Palindromic Quadratic Eigenvalue Problems, SIAM Journal on Scientific Computing, 38:6 (2016), A3410-A3429.
98. (with Tiexiang Li and Wen-Wei Lin) A Symmetric Structure-Preserving GammaQR Algorithm for Linear Response Eigenvalue Problems, Linear Algebra and its Applications, 520:1 (2017), 191-214.
99. (with Jungong Xue) Highly Accurate Doubling Algorithms for M-matrix Algebraic Riccati Equations, Numerische Mathematik, 135 (2017), 733-767.
100. (with Wei-guo Wang and Lei-hong Zhang) Error Bounds For Approximate Deflating Subspaces For Linear Response Eigenvalue Problems, Linear Algebra and its Applications, 528 (2017), 273-289.
101. (with Lei-Hong Zhang and Chungen Shen) On the Generalized Lanczos Trust-Region Method, SIAM Journal on Optimization, 27:3 (2017), 2110-2142.
102. (with Tsung-Ming Huang, Wen-Wei Lin, and Linzhang Lu) Optimal Parameters for Doubling Algorithms, Journal of Mathematical Study, 50:4 (2017), 339-357.
103. (with Yunfeng Cai, Leihong Zhang, and Zhaojun Bai) On an Eigenvector-Dependent Nonlinear Eigenvalue Problem, SIAM Journal on Matrix Analysis and Applications, 39:3 (2018), 1360-1382.
104. (with Changli Liu) Structured Backward Error for Palindromic Polynomial Eigenvalue Problems, II: Approximate Eigentriplets, Frontiers of Mathematics in China, 13:6 (2018), 1397-1426.
105. (with Xiang Wang, Xing Li, and Lei-Hong Zhang) An Efficient Numerical Method for the Symmetric Positive Definite Second-Order Cone Linear Complementarity Problem, Journal of Scientific Computing, (2019).

## Software

1. Contribution to LAPACK. I provided subroutines for solving secular equations which sit at the kernel of the divide-and-conquer methods for the symmetric eigenvalue problem and singular value problem.
2. Constructing high order palindromic composition schemes: composition.txt and composition.tar.gz, both at NETLIB's ODE
3. HP-UX libm for Itanium
4. Formal definitions and theorems, with proofs, about floating-point numbers. Contributors: Sylvie Boldo*, Marc Daumas, Laurent Fousse*, Ren-Cang Li and Guillaume Melquiond*, Jean-Michel Muller, Laurence Rideau*, and Laurent Thery*. Contributors that were directly interacting with Coq are indicated by a star.
5. MATLAB codes for CG-type methods for eigenvalue problems, prepared for the 2013 SIAM Gene Golub SIAM Summer School, Fudan university, Shanghai.
6. 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), 283-297.
7. Unstructrually Banded Nonlinear Eigenvalue Software. The package contains both Matlab and C++ implementations of solving unstructurally banded nonlinear eigenvalues with a Kublanovskaya type method.
8. 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), 1443-1460.
9. MATLAB codes for doubling algorithms for nonlinear matrix equations to go with
1. (with Tsung-Ming Huang and Wen-Wei Lin) Structure-Preserving 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

1. (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.
2. (with Sylvie Boldo and Marc Daumas) Theorems on Efficient Argument Reductions, Proceedings of the 16th IEEE Symposium on Computer Arithmetic, pp.129-136, 2003.
3. (with Zhaojun Bai) Structure-Preserving Model Reduction, Proceedings of PARA'04, J. Dongarra, K. Madsen, and J. Wa{\' s}niewski (eds.), Lecture Notes in Computer Science, 3732 (2006), 323-332, Springer.
Presented at Mini-symposium - Substructuring, Dimension Reduction and Applications, organized by Zhaojun Bai and Ren-Cang Li for PARA'04 workshop on state-of-the-art in Scientific Computing, Technical University of Denmark, Copenhagen, Denmark, June 20-23, 2004.
4. 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.
5. (with Hongteng Xu, Hongyuan Zha, and Mark A. Davenport) Active Manifold Learning via Gershgorin Circle Guided Sample Selection, AAAI-15 (2015, Association for the Advancement of Artificial Intelligence), Austin, Texas, January 25-29, 2015.
6. (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

1. (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.
2. 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.
3. (with Z. Bai and Y. Su) A Unified Krylov Projection Framework for Structure-Preserving 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.
4. 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.76-108, World Scientific, Singapore, 2015.

## Book

1. (with Tsung-Ming Huang and Wen-Wei Lin) Structure-Preserving Doubling Algorithms For Nonlinear Matrix Equations. SIAM, Philadelphia, September 2018.

## Selected Technical Reports (unpublished)

1. (with Huan Ren) An efficient tridiagonal eigenvalue solver on CM 5 with Laguerre's iteration, Technical ReportUCB//CSD-94-848, Computer Science Division, University of California at Berkeley, 1994. ( cover)
2. Relations between the field of values of a matrix and those of its Schur complements, Technical ReportUCB//CSD-94-849, Computer Science Division, University of California at Berkeley, 1994. ( cover)
3. Reciproot algorithm-correctly rounded? Technical ReportUCB//CSD-94-850, Computer Science Division, University of California at Berkeley, 1994.
4. Solving secular equations stably and efficiently, LAPACK working notes 89, (1993). Also Technical ReportUCB//CSD-94-851, Computer Science Division, University of California at Berkeley, 1994.
5. Linear systems with coefficient matrices having fields of values not containing the origin, Technical ReportUCB//CSD-94-853, Computer Science Division, University of California at Berkeley, 1994.
6. An Efficient Implementation of a Spectral Transform Method for Solving the Shallow Water Equations, Manuscript, Oak Ridge National Laboratory, 1996.
7. A Multi-Resolution Approach for Calculating Primary Eigenvectors of a Large Set of Images, Technical Report, 98-13, Department of Mathematics, University of Kentucky, June 1998.
8. Unconventional Reflexive Numerical Methods for Matrix Differential Riccati Equations, Technical Report, 2000-36, Department of Mathematics, University of Kentucky, November 2000.
9. Test Positive Realness Of A General Transfer Function Matrix, Technical Report, 2000-20, Department of Mathematics, University of Kentucky, April 2000.
10. Structural Preserving Model Reductions, Technical Report, 2004-02, Department of Mathematics, University of Kentucky, January 2004.
11. Asymptotically Optimal Lower Bounds For the Condition Number of a Real Vandermonde Matrix, Technical Report, 2004-05, Department of Mathematics, University of Kentucky, June 2004.
12. Sharpness in Rates of Convergence For CG and Symmetric Lanczos Methods, Technical Report, 2005-01, Department of Mathematics, University of Kentucky, January 2005.
13. Vandermonde Matrices with Chebyshev Nodes, Technical Report, 2005-02, Department of Mathematics, University of Kentucky, January 2005.

## Problems

1. (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 HP-UX Itanium

1. (with James W. Thomas, Jon P. Okada, and Peter Markstein) The Libm Library and Floating-Point Arithmetic in HP-UX for Itanium?Based Systems

#### Research supported in part by

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