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. On eigenvalues of a Rayleigh quotient matrix, Linear Algebra and its Applications, 169 (1992), 249--255.
8. Compute multiple nonlinear eigenvalues, J. Comp. Math., 10 (1992), 1--20.
9. Algorithms for inverse eigenvalue problems, J. Comp. Math., 10 (1992), 97--111.
10. 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.
11. A perturbation bound for definite pencils, Linear Algebra and its Applications, 179 (1993), 191--202.
12. A perturbation bound for the generalized polar decomposition, BIT, 33 (1993), 304--308.
13. 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.
14. Bounds on perturbations of generalized singular values and of associated subspaces, SIAM Journal on Matrix Analysis and Applications, 14 (1993), 195--234.
15. On eigenvalue variations of Rayleigh quotient matrix pencils of a definite pencils, Linear Algebra and its Applications, 208/209 (1994), 471--483.
16. On perturbations of matrix pencils with real spectra, Math. Comp., 62 (1994), 231--265.
17. New perturbation bounds for the unitary polar factor, SIAM Journal on Matrix Analysis and Applications, 16 (1995).
18. (with Z.-H. Cao and J.-J. Xie) A sharp version of Kahan's theorem on clustered eigenvalues, Linear Algebra and its Applications, 245 (1996), 147--156.
19. (with R. Bhatia) On perturbations of matrix pencils with real spectra, II. Math. Comp., 65 (1996), 637--645.
20. Relative perturbation bounds for the unitary polar factor, BIT, 37 (1997), 67--75.
21. (with R. Bhatia and F. Kittaneh) Some inequalities for commutators and an application to spectral variation. II, Linear and Multilinear Algebra, 43 (1997), 207--219.
22. (with W. 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.
23. (with W. Kahan) Composition constants for raising the orders of unconventional schemes for ordinary differential equations, Math. Comp., 66 (1997), 1089--1099.
24. (with R. Bhatia and F. Kittaneh) Eigenvalues of Symmetrizable Matrices, BIT, 38 (1998), 1-11.
25. 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.
    • 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, 319-324.
    • 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.161--172.
26. 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
    • David Day, Is A Metric on , Technical Report Sand No. 98-1754, Sandia National Laboratories, Albuquerque, NM, 1998.
    • Anders Barrlund, The p-Relative Distance is a Metric, SIAM Journal on Matrix Analysis and Applications, Volume 21, Number 2, pp. 699-702, 2000.
    • Peter A. Hästö, A new weighted metric: the relative metric I, Journal of Mathematical Analysis and Applications Volume 274, Issue 1 , 1 October 2002, Pages 38-58.
    • Peter A. Hästö, 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 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.
    • Chi-Kwong Li and Roy Mathias, The Lidskii-Mirsky-Wielandt theorem - additive and multiplicative versions, Numerische Mathematik, Volume 81, Number 3, January 1999, Pages: 377 - 413.
27. Relative perturbation theory: II eigenspace and singular space variations, SIAM Journal on Matrix Analysis and Applications, 20 (1999), 471--492.
28. Relative perturbation theory: (III) more bounds on eigenvalue variation, Linear Algebra and its Applications, 266 (1997), 337--345.
29. Relative Perturbation Theory: IV $\sin2\theta$ Theorems, Linear Algebra and its Applications, 311 (2000), 45--60.
30. 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.
31. (with G. W. Stewart) A new relative perturbation theorem for singular value problem, Linear Algebra and its Applications, 313 (2000), 41--51.
32. (with R. Bhatia and W. Kahan) Pinchings and norms of scaled triangular matrices, Linear and Multilinear Algebra, 50 (2002), 15--21.
33. On perturbations of matrix pencils with real spectra, a Revisit, Mathematics of Computation, 72 (2003), 715--728.
34. (with Ninoslav Truhar) A Theorem for Graded Indefinite Hermitian Matrices, Linear Algebra and its Applications, 359 (2003), 263--276.
35. (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.
36. (with Ernie Croot and Hui June Zhu) The ABC conjecture and correctly rounded reciprocal square roots, Theoretical Computer Science, 315 (2004), 405--417.
37. Near Optimality of Chebyshev Interpolation For Elementary Function Computations, IEEE Transactions on Computers, 53 (2004), 678--687.
38. 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
    • J. Kovac-Striko and K. Veselic, Some Remarks on the Spectra of Hermitian Matrices, Linear Algebra and its Applications, 145 (1991), 221--229.
    And in fact, Their Theorem 4 is stronger.
39. (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
    • corrected version.
40. (with Zhaojun Bai) Structure-Preserving Model Reduction using a Krylov Subspace Projection Formulation, Communications in Mathematical Sciences, 3:2 (2005), 179--199.
41. Relative Perturbation Bounds For Positive Polar Factors Of Graded Matrices, SIAM Journal on Matrix Analysis and Applications, 27 (2005), 424--433.
42. (with L. Hoffnung and Q. Ye) Krylov Type Subspace Methods for Matrix Polynomials, Linear Algebra and its Applications, 415 (2006), 52--81.
43. Lower bounds for the condition number of a real confluent Vandermonde matrix, Mathematics of Computation, 75 (2006), 1987-1995.
44. 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.
45. (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.
46. Convergence of CG and GMRES on a Tridiagonal Toeplitz Linear System , BIT, 47 (2007), 577-599.
47. (with Nicholas J. Higham and Françoise Tisseur) Backward Error of Polynomial Eigenproblems Solved by Linearization, SIAM Journal on Matrix Analysis and Applications, 29:4 (2007), 1218-1241.
48. (with Qiang Ye and Hongyuan Zha) Analysis of An Alignment Algorithm for Nonlinear Dimensionality Reduction, BIT, 47 (2007), 873-885.
49. On Meinardus' Examples For the Conjugate Gradient Method, Mathematics of Computation, 77 (2008), 335-352.
50. Hard Cases For Conjugate Gradient Method , International Journal for Information & Systems Sciences, 4:1 (2008), 15-29.
51. Vandermonde Matrices with Chebyshev Nodes, Linear Algebra and its Applications, 428 (2008), 1803-1832.
52. (with W. Zhang) The Rate of Convergence of GMRES on a Tridiagonal Toeplitz Linear System, Numerische Mathematik, to appear.
53. (with S. Boldo and M. Daumas) Formally Certified Argument Reduction with a Fused-Multiply-Add, IEEE Transactions on Computers, to appear.

PhD Thesis

Conference Proceedings

1. (with S. Boldo and M. Daumas) Theorems on Efficient Argument Reductions, Proceedings of the 16th IEEE Symposium on Computer Arithmetic, pp.129--136, 2003.
2. (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.

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. Chapter 15 in Handbook of Linear Algebra, L. Hogben, R. Brualdi, A. Greenbaum and R. Mathias (editors), Chapman & Hall/CRC, New York, 2007.
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.

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.

Research supported in part by

• DOE's Applied Mathematical Sciences Research Program (while at ORNL, 1995-1996)
• National Science Foundation under Grant No. ACI-9721388
• National Science Foundation CAREER award under Grant No. CCR-9875201
• National Science Foundation under Grant No. DMS-0510664
• National Science Foundation under Grant No. DMS-0702335
• National Science Foundation under Grant No. DMS-0810506

Last modified Wednesday September 21, 2005