Eigenvalues and condition numbers of random matrices
SIAM Journal on Matrix Analysis and Applications
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Matrix computations (3rd ed.)
Stable and Efficient Algorithms for Structured Systems of Linear Equations
SIAM Journal on Matrix Analysis and Applications
Structured matrices and polynomials: unified superfast algorithms
Structured matrices and polynomials: unified superfast algorithms
A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems
SIAM Journal on Matrix Analysis and Applications
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
SIAM Journal on Matrix Analysis and Applications
Schur aggregation for linear systems and determinants
Theoretical Computer Science
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The known algorithms for linear systems of equations perform significantly slower where the input matrix is ill conditioned, that is lies near a matrix of a smaller rank. The known methods counter this problem only for some important but special input classes, but our novel randomized augmentation techniques serve as a remedy for a typical ill conditioned input and similarly facilitates computations with rank deficient input matrices. The resulting acceleration is dramatic, both in terms of the proved bit-operation cost bounds and the actual CPU time observed in our tests. Our methods can be effectively applied to various other fundamental matrix and polynomial computations as well.