The WY representation for products of householder matrices
SIAM Journal on Scientific and Statistical Computing - Papers from the Second Conference on Parallel Processing for Scientific Computin
The algebraic eigenvalue problem
The algebraic eigenvalue problem
A storage-efficient WY representation for products of householder transformations
SIAM Journal on Scientific and Statistical Computing
ACM Transactions on Mathematical Software (TOMS)
A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
On a block implementation of Hessenberg multishift QR iteration
International Journal of High Speed Computing
Bounding the error in Gaussian Elimination for Tridiagonal systems
SIAM Journal on Matrix Analysis and Applications
Exploiting fast matrix multiplication within the level 3 BLAS
ACM Transactions on Mathematical Software (TOMS)
Using Strassen's algorithm to accelerate the solution of linear systems
The Journal of Supercomputing
Algorithm 694: a collection of test matrices in MATLAB
ACM Transactions on Mathematical Software (TOMS)
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Auto-blocking matrix-multiplication or tracking BLAS3 performance from source code
PPOPP '97 Proceedings of the sixth ACM SIGPLAN symposium on Principles and practice of parallel programming
Implementation of Strassen's algorithm for matrix multiplication
Supercomputing '96 Proceedings of the 1996 ACM/IEEE conference on Supercomputing
Modified ST algorithms and numerical experiments
Applied Numerical Mathematics - Special issue: 2nd international workshop on numerical linear algebra, numerical methods for partial differential equations and optimization
Computing the sign or the value of the determinant of an integer matrix, a complexity survey
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
On the performance of parallel factorization of out-of-core matrices
Parallel Computing
The aggregation and cancellation techniques as a practical tool for faster matrix multiplication
Theoretical Computer Science - Algebraic and numerical algorithm
Adaptive Strassen's matrix multiplication
Proceedings of the 21st annual international conference on Supercomputing
Stability of block LU factorization for block tridiagonal matrices
Computers & Mathematics with Applications
Adaptive Winograd's matrix multiplications
ACM Transactions on Mathematical Software (TOMS)
Goal-Oriented and Modular Stability Analysis
SIAM Journal on Matrix Analysis and Applications
Journal of Computational and Applied Mathematics
Stability of block LU factorization for block tridiagonal block H-matrices
Journal of Computational and Applied Mathematics
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Block algorithms are becoming increasingly popular in matrix computations. Since their basic unit of data is a submatrix rather than a scalar, they have a higher level of granularity than point algorithms, and this makes them well suited to high-performance computers. The numerical stability of the block algorithms in the new linear algebra program library LAPACK is investigated here. It is shown that these algorithms have backward error analyses in which the backward error bounds are commensurate with the error bounds for the underlying level-3 BLAS (BLAS3). One implication is that the block algorithms are as stable as the corresponding point algorithms when conventional BLAS3 are used. A second implication is that the use of BLAS3 based on fast matrix multiplication techniques affects the stability only insofar as it increases the constant terms in the normwise backward error bounds. For linear equation solvers employing LU factorization, it is shown that fixed precision iterative refinement helps to mitigate the effect of the larger error constants. Despite the positive results presented here, not all plausible block algorithms are stable; we illustrate this with the example of LU factorization with block triangular factors and describe how to check a block algorithm for stability without doing a full error analysis.