Topics in matrix analysis
Overestimation in linear interval equations
SIAM Journal on Numerical Analysis
Bounding the solution of interval linear equations
SIAM Journal on Numerical Analysis
Computer methods for design automation
Computer methods for design automation
Algorithm 737: INTLIB—a portable Fortran 77 interval standard-function library
ACM Transactions on Mathematical Software (TOMS)
A Comparison of some Methods for Solving Linear Interval Equations
SIAM Journal on Numerical Analysis
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
On the Iterative Criterion for Generalized Diagonally Dominant Matrices
SIAM Journal on Matrix Analysis and Applications
Handbook of Floating-Point Arithmetic
Handbook of Floating-Point Arithmetic
Stability of the optimal basis of a linear program under uncertainty
Operations Research Letters
Hi-index | 7.29 |
In Part I and this Part II of our paper we investigate how extra-precise evaluation of dot products can be used to solve ill-conditioned linear systems rigorously and accurately. In Part I only rounding to nearest is used. In this Part II we improve the results significantly by permitting directed rounding. Linear systems with tolerances in the data are treated, and a comfortable way is described to compute error bounds for extremely ill-conditioned linear systems with condition numbers up to about u^-^2/n, where u denotes the relative rounding error unit in a given working precision. We improve a method by Hansen/Bliek/Rohn/Ning/Kearfott/Neumaier. Of the known methods by Krawczyk, Rump, Hansen et al., Ogita and Nguyen we show that our presented Algorithm LssErrBnd seems the best compromise between accuracy and speed. Moreover, for input data with tolerances, a new method to compute componentwise inner bounds is presented. For not too wide input data they demonstrate that the computed inclusions are often almost optimal. All algorithms are given in executable Matlab code and are available from my homepage.