Ren-Cang Li's Publication

My research touches two areas of Numerical Analysis: Numerical Linear Algebra and Unconventional Methods for Solving Ordinary Differential Equations introduced to me by Prof. W. Kahan sometime in 1994 when I was a graduate student at University of California at Berkeley. The following is a list of my papers published or unpublished.

Unconventional Methods for Solving Ordinary Differential Equations

  1. Raising the Orders of Unconventional Schemes for Ordinary Differential Equations, PhD thesis , University of California at Berkeley, 1995.
  2. (with W. Kahan) Composition constants for raising the orders of unconventional schemes for ordinary differential equations, Math. Comp., 66, 1089--1099 (1997).
  3. (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, 316-331 (1997).
  4. Unconventional Reflexive Numerical Methods for Matrix Differential Riccati Equations, Technical Report 2000-36, Department of Mathematics University of Kentucky, 2000.

Numerical Linear Algebra

Perturbation Theory for Generalized Eigenvalue and Singular Value Problems

  1. On Perturbation Bounds for Eigenvalues of a Matrix, Preprint. Computing Center, Academia Sinica, Beijing, P. R. China, 1985.
  2. A converse to the Bauer-Fike type theorem, Linear Algebra Appl., 109, 167-178, 1988.
  3. On perturbation theorems for the generalized eigenvalues of regular matrix pencils, (in Chinese), Math. Numer. Sinica, 11:1, 10-19, 1989. English transl. Chinese J. Numer. Math. Appl., 11:2, 24-35, 1989.
  4. Perturbation bounds for generalized eigenvalues. I, (in Chinese), Math. Numer. Sinica, 11:1, 196-204, 1989. English transl. Chinese J. Numer. Math. Appl., 11:2, 1-9, 1989.
  5. Perturbation bounds for generalized eigenvalues. II, (in Chinese), Math. Numer. Sinica, 11, 239-247, 1989. English transl. Chinese J. Numer. Math. Appl., 11, 34-43, 1989.
  6. On the variations of the spectra of matrix pencils, Linear Algebra Appl., 139, 147-164, 1990.
  7. A perturbation bound for definite pencils, Linear Algebra Appl., 179, 191-202, 1993.
  8. Norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils, Linear Algebra Appl., 182, 199-234, 1993.
  9. Bounds on perturbations of generalized singular values and of associated subspaces, SIAM J. Matrix Anal. Appl., 14, 195-234, 1993.
  10. On eigenvalues of Rayleigh quotient matrix pencils of a definite pencils, Linear Algebra Appl., 208/209, 471-483, 1994.
  11. On perturbations of matrix pencils with real spectra, Mathematics of Computations, 62, 231-265, 1994.
  12. (with R. Bhatia) On perturbations of matrix pencils with real spectra. II, Mathematics of Computations, 65, 637-645, 1996.
  13. On perturbations of matrix pencils with real spectra, a Revisit, Technical Report 99-18, Department of Mathematics University of Kentucky, 1999.

Relative Perturbation Theories

  1. Relative perturbation theory: (I) eigenvalue and singular value variations, Technical Report UCB//CSD-94-855, Computer Science Division, Department of EECS, University of California at Berkeley, 1994. Revised January 1996. A shortened version in SIAM Journal on Matrix Analysis and Applications, 19 (1998), 956 -- 982.
    David Day's $\varrho_{\infty}(a,b) = \frac{\vert a - b\vert} {\max(\vert a\vert,\vert b \vert)}$ Is A Metric on ${\Bbb C}$ September, 1996.
    Anders Barrlund's The p-Relative Distance is a Metric, June 1998.
  2. Relative perturbation theory: (II) Eigenspace and Singular Subspace Variations, Technical Report UCB//CSD-94-856, Computer Science Division, Department of EECS, University of California at Berkeley, 1994. Revised January and April 1996. A shortened version in SIAM Journal on Matrix Analysis and Applications, Volume 20, Number 2, 1999, pp. 471-492.
  3. Relative perturbation theory: (III) more bounds on eigenvalue variation, Linear Algebra Appl., 266, 337-345, 1997.
  4. Relative Perturbation Bounds for the Unitary Polar Factor, BIT, 37, 67-75, 1997.
  5. Thoughts About Relative Perturbation Theory For Indefinite Hermitian Matrices, Work in Progress.
  6. A Bound On The Solution To A Structured Sylvester Equation With An Application To Relative Perturbation Theory Technical Report 98-21, Department of Mathematics University of Kentucky, 1998. SIAM Journal on Matrix Analysis and Applications, 21, 440-445, 2000.
  7. (with G. W. Stewart) A New Relative Perturbation Theorem for Singular Subspaces, Linear Algebra and its Application, 313, 41-51, 2000.
  8. Relative perturbation theory: IV $\sin2\theta$ Theorems, Technical Report 99-12, Department of Mathematics, University of Kentucky, 1999. Linear Algebra and its Application, 311, 45-60, 2000.
  9. (with Ninoslav Truhar) A $\sin2\theta$ Theorem for Graded Indefinite Hermitian Matrices, Technical Report 2000-33, Department of Mathematics, University of Kentucky, 2000. Linear Algebra and its Application, 359, 263-276, 2003.
  10. Positive Polar Factors of Graded Matrices, Technical Report 2003-11, Department of Mathematics, University of Kentucky, 2003.

Eigenvalue Computations and Applications

  1. (with Huan Ren) An efficient tridiagonal eigenvalue solver on CM 5 with Laguerre's iteration, Technical Report UCB//CSD-94-848, Computer Science Division, University of California at Berkeley, 1994. ( cover and the paper).
  2. Solving secular equations stably and efficiently, LAPACK working notes # 89, 1993. Also Technical Report UCB//CSD-94-851, Computer Science Division, University of California at Berkeley, 1994. ( cover and the paper).
  3. A Multi-Resolution Approach For Calculating Primary Eigenvectors Of a Large Set of Images, Technical Report 98-13, Department of Mathematics University of Kentucky, 1998.
  4. Fast Partial Eigenvalue Decomposition With Wavelet Transformation For Large Images, Department of Mathematics, University of Kentucky, 1999.
  5. Test Positive Realness Of A General Transfer Function Matrix, Technical Report 2000-20, Department of Mathematics University of Kentucky, 2000.
  6. (with Qiang Ye) A Krylov Subspace Method for Quadratic Matrix Polynomials with Application to Constrained Least Squares Problems, SIAM J. Matrix Anal. Appl., to appear.
  7. (with L. Hoffnung, Qiang Ye) Krylov Type Subspace Methods for Matrix Polynomials, 2003

Other Papers in Numerical Linear Algebra

  1. On eigenvalues of a Rayleigh quotient matrix, Linear Algebra Appl., 169, 249-255, 1992.
  2. A perturbation bound for the generalized polar decomposition, BIT, 33, 304-308, 1993.
  3. Relations between the field of values of a matrix and those of its Schur complements, Technical Report UCB//CSD-94-849, Computer Science Division, University of California at Berkeley, 1994. ( cover and the paper).
  4. Linear systems with coefficient matrices having fields of values not containing the origin, Technical Report UCB//CSD-94-853, Computer Science Division, University of California at Berkeley, 1994. ( cover and the paper).
  5. Reciproot algorithm--correctly rounded? Technical Report UCB//CSD-94-850, Computer Science Division, University of California at Berkeley, 1994. ( cover and the paper).
  6. New perturbation bounds for the unitary polar factor, SIAM Journal on Matrix Analysis and Applications, 16, 327-332, 1995
  7. (Z.-H. Cao and J.-J. Xie) A sharp version of Kahan's theorem on clustered eigenvalues, Linear Algebra Appl., 245, 147-156, 1996.
  8. (with R. Bhatia and F. Kittaneh) Some inequalities for commutators and an application to spectral variation. II, Linear and Multilinear Algebra, 43, 207-220, 1997.
  9. (with R. Bhatia and F. Kittaneh) Eigenvalue of symmetrizable matrices, BIT, 38, 1-11, 1998.
  10. Challenges in Matrix Theory --- Spectral Variations and Hadamard Products: Some Problems, Linear Algebra and its Applications, 278 (1998), 317--326.
  11. (with R. Bhatia and W. Kahan) Pinchings and Norms of Scaled Triangular Matrices, Linear and Multilinear Algebra, 50:1 (2002), 15--21.
  12. A Note on Eigenvalues of Perturbed Hermitian Matrices , Technical Report 2000-24, Department of Mathematics University of Kentucky, 2000.
  13. Accuracy of Computed Eigenvectors via Optimizing a Rayleigh Quotient , Technical Report 2000-28, Department of Mathematics University of Kentucky, 2000.

Non-linear Eigenvalue and Inverse Eigenvalue Problems

  1. QR decomposition and nonlinear eigenvalue problems, (in Chinese), Math. Numer. Sinica, 11:4, 374-385, 1989.
  2. Compute critical points of a stability problem, (in Chinese), Math. Numer. Sinica, 1990.
  3. Compute multiply nonlinear eigenvalues, J. Comp. Math., 10, 1-20, 1992.
  4. Algorithms for inverse eigenvalue problems, J. Comp. Math., 10, 97-111, 1992.
  5. Algorithms for inverse eigenvalue problems. II, J. Comp. Math., 1992.

Approximation Theory and Elementary Function Computations

  1. Near Optimality of Chebyshev Interpolation For Elementary Function Computations, IEEE Transcations on Computers, 53, 678-687, 2004.
  2. (with Ernie Croot and Hui June Zhu) The ABC Conjecture and Correctly Rounded Reciprocal Square Root, Theoretical Computer Science, 315 405-417, 2004.
  3. (with S. Boldo and M. Daumas) Theorems on Efficient Argument Reductions, Proceedings of the 16th IEEE Symposium on Computer Arithmetic, pp.129-136, 2003.
Last modified Sunday June 14, 2002