Linear Systems and Condition Estimation

Generating Extreme-Scale Matrices with Specified Singular Values or Condition Numbers (with M. Fasi), SIAM J. Sci. Comput., 43(1):A663–A684, 2021.
Matrices with Tunable Infinity-Norm Condition Number and No Need for Pivoting in LU Factorization (with M. Fasi). SIAM J. Matrix Anal. Appl. 42(1):417–435, 2021. Blog post.
Random Matrices Generating Large Growth in LU Factorization with Pivoting (with D. J. Higham and S. Pranesh), SIAM J. Matrix Anal. Appl., 42(1):185–201, 2021.
Solving Block Low-Rank Linear Systems by LU Factorization is Numerically Stable (with T. Mary). IMA J. Numer. Anal., 42(2):951–980, 2021.
Exploiting Lower Precision Arithmetic in Solving Symmetric Positive Definite Linear Systems and Least Squares Problems (with S. Pranesh). SIAM J. Sci. Comput., 43(1):A258–A277, 2021.
Three-Precision GMRES-based Iterative Refinement for Least Squares Problems (with E. Carson and S. Pranesh). SIAM J. Sci. Comput., 42(6):A4063-A4083, 2020.
Mixed-Precision Iterative Refinement Using Tensor Cores on GPUs to Accelerate Solution of Linear Systems (with A. Haidar, H. Bayraktar, S. Tomov and J. Dongarra), Proc. Roy. Soc. London A, 476 (2243):20200110, 2020.
Error Analysis For Standard and GMRES-Based Iterative Refinement in Two and Three-Precisions. MIMS EPrint 2019.19, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, November 2019. 8 pp.
Adaptive Precision in Block-Jacobi Preconditioning for Iterative Sparse Linear System Solvers (with Hartwig Anzt, Jack Dongarra, Goran Flegar and Enrique S. Quintana-Orti). Concurrency Computat.: Pract. Exper., 31(6), e4460, 2019.
Squeezing a Matrix Into Half Precision, with an Application to Solving Linear Systems (with S. Pranesh and M. Zounon). SIAM J. Sci. Comput., 41(4), A2536-A2551, 2019. Blog post.
A New Preconditioner that Exploits Low-Rank Approximations to Factorization Error (with T. Mary), SIAM J. Sci. Comput., 41(1):A59-A82, 2019.
Harnessing GPU Tensor Cores for Fast FP16 Arithmetic to Speed Up Mixed-Precision Iterative Refinement Solvers (with A. Haidar, S. Tomov, and J. Dongarra). In Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis, SC ’18 (Dallas, TX), IEEE Press, Piscataway, NJ, USA, 2018, pages 47:1-47:11. Blog post.
Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions (with Erin Carson). SIAM J. Sci. Comput., 40(2):A817-A847, 2018.
A New Analysis of Iterative Refinement and Its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems (with Erin Carson). SIAM J. Sci. Comput. 39(6):A2834-A2856, 2017. Page about the paper at University of Manchester’s In Abstract.
Estimating the Largest Elements of a Matrix (with Samuel Relton), SIAM J. Sci. Comput., 38(5):C584-C601, 2016. MATLAB codes on GitHub.
Numerical Conditioning MIMS EPrint 2013.76, November 2013. In Claude Brezinski and Ahmed Sameh, editors, Walter Gautschi. Selected Works with Commentaries, volume 1, pages 37-40. Birkhauser, New York, 2014. Published version.
Detecting the Causes of Ill-Conditioning in Structural Finite Element Models (with Ramaseshan Kannan, Stephen Hendry and Françoise Tisseur), Computers and Structures, 133:79-89, 2014. Published version.
Performance Analysis of Asynchronous Jacobi’s Method Implemented in MPI, SHMEM and OpenMP (with Iain Bethune, J. Mark Bull and Nicholas J. Dingle), Int. J. High Performance Computing Applications, 28(1):97-111, 2014.
Investigating the Performance of Asynchronous Jacobi’s Method for Solving Systems of Linear Equations (with Iain Bethune, J. Mark Bull and Nicholas J. Dingle), MIMS EPrint 2011.82, October 2011.
Gaussian Elimination, WIREs Comp. Stat., 3(3):230-238, 2011. Published version.
Cholesky Factorization. WIREs Comp. Stat., 1(2):251-254, 2009.
LAPACK-Style Codes for Pivoted Cholesky and QR Updating (with Sven Hammarling and Craig Lucas), in B. Kagstrom, E. Elmroth, J. Dongarra, and J. Wasniewski, editors, Applied Parallel Computing. State of the Art in Scientific Computing, 8th International Workshop, PARA 2006, number 4699 in Lecture Notes in Computer Science, pages 137-146. Springer-Verlag, Berlin, 2007.
Parallel Implementation of a Block Algorithm for Matrix 1-Norm Estimation (with Sheung Hun Cheng), In R. Sakellariou, J. Keane, J. Gurd, and L. Freeman, editors, Euro-Par 2001, Parallel Processing, volume 2150 of Lecture Notes in Computer Science, pages 568-577. Springer-Verlag, Berlin, 2001.
Implementation for LAPACK of a Block Algorithm for Matrix 1-Norm Estimation (with Sheung Hun Cheng), NA Report 393, August 2001.
A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra (with Françoise Tisseur), SIAM J. Matrix Anal. Appl., 21(4): 1185-1201, 2000.
Stability of Block $LDL^T$ Factorization of a Symmetric Tridiagonal Matrix, Linear Algebra Appl., 287: 181-189, 1999.
Factorizing Complex Symmetric Matrices with Positive Definite Real and Imaginary Parts, Math. Comp., 67(224):1591-1599, Oct. 1998.
A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization (with Sheung Hun Cheng), SIAM J. Matrix Anal. Appl., 19(4):1097-1110, 1998.
Iterative Refinement for Linear Systems and LAPACK, IMA J. Numer. Anal., 17(4): 495-509, 1997.
Stability of the Diagonal Pivoting Method with Partial Pivoting, SIAM J. Matrix Anal. Appl., 18(1): 52-65, Jan. 1997. More readable version (SIAM PDF has very thin fraction bars.)
Stability of Block $LU$ Factorization (with J. W. Demmel and R. S. Schreiber), Numerical Linear Algebra with Applications, 2(2):173-190, 1995
Stability of Parallel Triangular System Solvers, SIAM J. Sci. Comput., 16(2):400-413, Mar. 1995
Stability of the Partitioned Inverse Method for Parallel Solution of Sparse Triangular Systems (with Alex Pothen), SIAM J. Sci. Comput., 15(1):139-148, Jan. 1994.
Improved Error Bounds for Underdetermined System Solvers (with James W. Demmel), SIAM J. Matrix Anal. Appl., 14(1):1-14, January 1993.
Componentwise Error Analysis for Stationary Iterative Methods (with Philip A. Knight), In C. D. Meyer and R. J. Plemmons, editors, Linear Algebra, Markov Chains, and Queueing Models, volume 48 of IMA Volumes in Mathematics and its Applications, pages 29-46, Springer-Verlag, New York, 1993.
A Strassen-Type Matrix Inversion Algorithm for the Connection Machine (with S. M. Balle and P. C. Hansen), Technical Report CNC/1993/028, Centre for Novel Computing, University of Manchester, England, October 1993.
Finite Precision Behavior of Stationary Iteration for Solving Singular Systems (with Philip A. Knight), Linear Algebra and Appl., 192:165-186, 1993.
Perturbation Theory and Backward Error for $AX-XB=C$ , BIT, 33:124-136, 1993.
Estimating the Matrix $p$-norm, Numer. Math., 62:539-555, 1992.
Stability of Methods for Matrix Inversion (with Jeremy J. Du Croz), IMA J. Numer. Anal., 12:1-19, 1992.
Stability of Block Algorithms with Fast Level-3 BLAS (with James W. Demmel), ACM Trans. Math. Software, 18(3):274-291, Sept. 1992.
Backward Error and Condition of Structured Linear Systems (with Desmond J. Higham), SIAM J. Matrix Anal. Appl., 13(1):162-175, Jan. 1992.
Componentwise Perturbation Theory for Linear Systems with Multiple Right-Hand Sides (with Desmond J. Higham), Linear Algebra and Appl., 174:111-129, 1992.
Bounding the error in Gaussian elimination for tridiagonal systems, SIAM J. Matrix Anal. Appl., 11(4):521-530, Oct. 1990.
Iterative refinement enhances the stability of QR factorization methods for solving linear equations, BIT, 31:447-468, 1991.
Stability analysis of algorithms for solving confluent Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11(1):23-41, 1990.
Experience with a matrix norm estimator, SIAM J. Sci. Stat. Comput., 11(4):804-809, July 1990.
How Accurate is Gaussian Elimination?, In D. F. Griffiths and G. A. Watson, editors, Numerical Analysis 1989, Proceedings of the 13th Dundee Conference, volume 228 of Pitman Research Notes in Mathematics, pages 137-154. Longman Scientific and Technical, Essex, UK, 1990.
Analysis of the Cholesky Decomposition of a Semi-Definite Matrix, In M. G. Cox and S. J. Hammarling, editors, Reliable Numerical Computation , pages 161-185, Oxford University Press, 1990.
Exploiting Fast Matrix Multiplication within the Level 3 BLAS, ACM Trans. Math. Soft., 16(4):352-368, Dec. 1990.
The Accuracy of Solutions to Triangular Systems, SIAM J. Numer. Anal., 26(5):1252-1265, 1989.
Large Growth Factors in Gaussian Elimination with Pivoting (with Desmond J. Higham), SIAM J. Matrix Anal. Appl., 10(2):155-164, 1989.
FORTRAN Codes for Estimating the One-Norm of a Real or Complex Matrix, with Applications to Condition Estimation (Algorithm 674), ACM Trans. Math. Soft., 14(4):381-396, Dec. 1988.
Fast solution of Vandermonde-like systems involving orthogonal polynomials, IMA Journal of Numerical Analysis, 8:473-486, 1988.
Error Analysis of the Bjorck-Pereyra Algorithms for Solving Vandermonde Systems, Numer. Math., 50(5):613-632, 1987.
A survey of condition number estimation for triangular matrices, SIAM Review, 29(4):575-596, 1987.
Efficient algorithms for computing the condition number of a tridiagonal matrix, SIAM J. Sci. Stat. Comput., 7(1):150-165, 1986.

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