An efficient implementation of a conformal mapping method based on the Szego¨ kernel
SIAM Journal on Numerical Analysis
Fast parallel algorithms for QR and triangular factorization
SIAM Journal on Scientific and Statistical Computing
Error analysis of the bjo¨rck-pereyra algorithms for solving vandermonde systems
Numerische Mathematik
Effectively well-conditioned linear systems
SIAM Journal on Scientific and Statistical Computing
The algebraic eigenvalue problem
The algebraic eigenvalue problem
Fast array algorithms for structured matrices
Fast array algorithms for structured matrices
A matrix problem with application to rapid solution of integral equations
SIAM Journal on Scientific and Statistical Computing
Stability analysis of algorithms for solving confluent Vandermonde-like systems
SIAM Journal on Matrix Analysis and Applications
A look-ahead Levinson algorithm for indefinite Toeplitz systems
SIAM Journal on Matrix Analysis and Applications
Mixed componentwise and structured condition numbers
SIAM Journal on Matrix Analysis and Applications
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Fast inversion of Chebyshev-Vandermonde matrices
Numerische Mathematik
A Look-Ahead Block Schur Algorithm for Toeplitz-Like Matrices
SIAM Journal on Matrix Analysis and Applications
On the Stability of the Bareiss and Related Toeplitz Factorization Algorithms
SIAM Journal on Matrix Analysis and Applications
Fast Gaussian elimination with partial pivoting for matrices with displacement structure
Mathematics of Computation
Displacement structure: theory and applications
SIAM Review
Cauchy-like preconditioners for 2-dimensional ill-posed problems
Cauchy-like preconditioners for 2-dimensional ill-posed problems
Stable and Efficient Algorithms for Structured Systems of Linear Equations
SIAM Journal on Matrix Analysis and Applications
A displacement approach to efficient decoding of algebraic-geometric codes
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Displacement structure and array algorithms
Fast reliable algorithms for matrices with structure
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Reed-Solomon Codes and Their Applications
Reed-Solomon Codes and Their Applications
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Fast algorithms for multivariable systems.
Fast algorithms for multivariable systems.
Algorithms for finite shift-rank processes
Algorithms for finite shift-rank processes
Computations with quasiseparable polynomials and matrices
Theoretical Computer Science
Hi-index | 0.00 |
Gaussian elimination is a standard tool for computing triangular factorizations for general matrices, and thereby solving associated linear systems of equations. As is well-known, when this classical method is implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundoffs accompanying each elementary floating point operation. This problem motivated a number of interesting and important studies in modern numerical linear algebra; for our purposes in this paper we only mention that starting with the breakthrough work of Wilkinson, several pivoting techniques have been proposed to stabilize the numerical behavior of Gaussian elimination.Interestingly, matrix interpretations of many known and new algorithms for various applied problems can be seen as a way of computing triangular factorizations for the associated structured matrices, where different patterns of structure arise in the context of different physical problems. The special structure of such matrices [e.g., Toeplitz, Hankel, Cauchy, Vandermonde, etc.] often allows one to speed-up the computation of its triangular factorization, i.e., to efficiently obtain fast implementations of the Gaussian elimination procedure. There is a vast literature about such methods which are known under different names, e.g., fast Cholesky, fast Gaussian elimination, generalized Schur, or Schur-type algorithms. However, without further improvements they are efficient fast implementations of a numerically inaccurate [for indefinite matrices] method.In this paper we survey recent results on the fast implementation of various pivoting techniques which allowed us to improve numerical accuracy for a variety of fast algorithms. This approach led us to formulate new more accurate numerical methods for factorization of general and of J-unitary rational matrix functions, for solving various tangential interpolation problems, new Toeplitz-like and Toeplitz-plus-Hankel-like solvers, and new divided differences schemes. We beleive that similar methods can be used to design accurate fast algorithm for the other applied problems and a recent work of our colleagues supports this anticipation.