Review on stochastic approach to round-off error analysis and its applications
Mathematics and Computers in Simulation
Experience with a matrix norm estimator
SIAM Journal on Scientific and Statistical Computing
Improved error bounds for underdetermined system solvers
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Optimal Backward Perturbation Bounds for Underdetermined Systems
SIAM Journal on Matrix Analysis and Applications
Structured Backward Error and Condition of Generalized Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Error Analysis of Direct Methods of Matrix Inversion
Journal of the ACM (JACM)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
On the Error Analysis and Implementation of Some Eigenvalue Decomposition and Singular Value Decomposition Algorithms
On Floating Point Errors in Cholesky
On Floating Point Errors in Cholesky
Rounding Errors in Algebraic Processes
Rounding Errors in Algebraic Processes
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Error bounds from extra-precise iterative refinement
ACM Transactions on Mathematical Software (TOMS)
Toeplitz And Circulant Matrices: A Review (Foundations and Trends(R) in Communications and Information Theory)
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In this paper, a method to estimate error bounds of algorithms in linear algebra is proposed which is independent of the considered algorithm. The method is based on discrete stochastic arithmetic (DSA) which has been introduced to compute the numerical accuracy of algorithms providing scalar values. In order to extend the DSA concept to algorithms in linear algebra, estimations of numerical error bounds for the 2-norm of vectors and angle between subspaces spanned by computed vectors and corresponding true vectors are derived based on DSA in this paper. To show the quality of these estimations, they are applied to the linear algebra library LAPACK providing tighter error bounds compared to the error bounds of the library itself. These error bounds are especially useful for the implementation of algorithms in linear algebra on low precision (e.g. single precision) arithmetic of massive parallel computing systems (GPUs, FPGAs, Cell processors, multi-core processors). In such systems, single precision arithmetic offers a significant higher performance than double precision arithmetic. In order to avoid numerical inaccurate results, a numerical error control is required which can be provided by the given approach. In a similar way, the error bounds are useful in cases where double precision may be not sufficient and have to be extended to quadruple precision.