Additive preconditioning and aggregation in matrix computations

Authors:
Victor Y. Pan;Dmitriy Ivolgin;Brian Murphy;Rhys Eric Rosholt;Islam Taj-Eddin;Yuqing Tang;Xiaodong Yan
Affiliations:
Department of Mathematics and Computer Science, Lehman College of the City University of New York, Bronx, NY 10468, USA and Ph.D. Programs in Mathematics and Computer Science, The City University ...;Ph.D. Program in Computer Science, The City University of New York, New York, NY 10036, USA;Department of Mathematics and Computer Science, Lehman College of the City University of New York, Bronx, NY 10468, USA;Department of Mathematics and Computer Science, Lehman College of the City University of New York, Bronx, NY 10468, USA;Ph.D. Program in Computer Science, The City University of New York, New York, NY 10036, USA;Ph.D. Program in Computer Science, The City University of New York, New York, NY 10036, USA;Ph.D. Program in Computer Science, The City University of New York, New York, NY 10036, USA
Venue:
Computers & Mathematics with Applications
Year:
2008

Citing 29
Cited 4

How can we speed up matrix multiplication?

SIAM Review
Direct methods for sparse matrices

Direct methods for sparse matrices
Matrix multiplication via arithmetic progressions

Journal of Symbolic Computation - Special issue on computational algebraic complexity
Fast and efficient parallel solution of sparse linear systems

SIAM Journal on Computing
Fast Gaussian elimination with partial pivoting for matrices with displacement structure

Mathematics of Computation
A rank-one reduction formula and its applications to matrix factorizations

SIAM Review
Matrix computations (3rd ed.)

Matrix computations (3rd ed.)
Iterative methods for solving linear systems

Iterative methods for solving linear systems
Solving a Polynomial Equation: Some History and Recent Progress

SIAM Review
Modular arithmetic for linear algebra computations in the real field

Journal of Symbolic Computation
Sign determination in residue number systems

Theoretical Computer Science - Special issue on real numbers and computers
Two Purposes for Matrix Factorization: A Historical Appraisal

SIAM Review
Multivariate polynomials, duality, and structured matrices

Journal of Complexity
On accurate floating-point summation

Communications of the ACM
Structured matrices and polynomials: unified superfast algorithms

Structured matrices and polynomials: unified superfast algorithms
Numerical Linear Algebra for High Performance Computers

Numerical Linear Algebra for High Performance Computers
Design, implementation and testing of extended and mixed precision BLAS

ACM Transactions on Mathematical Software (TOMS)
Computer Arithmetic in Theory and Practice

Computer Arithmetic in Theory and Practice
Accuracy and Stability of Numerical Algorithms

Accuracy and Stability of Numerical Algorithms
An iterated eigenvalue algorithm for approximating roots of univariate polynomials

Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Preconditioning techniques for large linear systems: a survey

Journal of Computational Physics
Accurate and Efficient Floating Point Summation

SIAM Journal on Scientific Computing
The aggregation and cancellation techniques as a practical tool for faster matrix multiplication

Theoretical Computer Science - Algebraic and numerical algorithm
Improved algorithms for computing determinants and resultants

Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations

Numerische Mathematik
Accurate Sum and Dot Product

SIAM Journal on Scientific Computing
The schur aggregation for solving linear systems of equations

Proceedings of the 2007 international workshop on Symbolic-numeric computation
Null space and eigenspace computations with additive preprocessing

Proceedings of the 2007 international workshop on Symbolic-numeric computation
Effect of small rank modification on the condition number of a matrix

Computers & Mathematics with Applications

Eigen-solving via reduction to DPR1 matrices

Computers & Mathematics with Applications
Schur aggregation for linear systems and determinants

Theoretical Computer Science
A new error-free floating-point summation algorithm

Computers & Mathematics with Applications
Additive preconditioning for matrix computations

CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications

Quantified Score

Hi-index	0.09

Visualization

Abstract

We combine our novel SVD-free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations. Our analysis and experiments show the power of our algorithms, guide us in selecting most effective policies of preconditioning and aggregation, and provide some new insights into these and related subjects. Compared to the popular SVD-based multiplicative preconditioners, our additive preconditioners are generated more readily and for a much larger class of matrices. Furthermore, they better preserve matrix structure and sparseness and have a wider range of applications (e.g., they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem).