Experience with a matrix norm estimator
SIAM Journal on Scientific and Statistical Computing
Average-case stability of Gaussian elimination
SIAM Journal on Matrix Analysis and Applications
A class of arbitrarily ill conditioned floating-point matrices
SIAM Journal on Matrix Analysis and Applications
The accuracy of floating point summation
SIAM Journal on Scientific Computing
On properties of floating point arithmetics: numerical stability and the cost of accurate computations
Condition Numbers of Random Triangular Matrices
SIAM Journal on Matrix Analysis and Applications
Accurate Singular Value Decompositions of Structured Matrices
SIAM Journal on Matrix Analysis and Applications
On accurate floating-point summation
Communications of the ACM
Algorithm 274: Generation of Hilbert derived test matrix
Communications of the ACM
Design, implementation and testing of extended and mixed precision BLAS
ACM Transactions on Mathematical Software (TOMS)
Computer Methods for Mathematical Computations
Computer Methods for Mathematical Computations
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Accurate and Efficient Floating Point Summation
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A New Distillation Algorithm for Floating-Point Summation
SIAM Journal on Scientific Computing
Error bounds from extra-precise iterative refinement
ACM Transactions on Mathematical Software (TOMS)
A method of obtaining verified solutions for linear systems suited for Java
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
MPFR: A multiple-precision binary floating-point library with correct rounding
ACM Transactions on Mathematical Software (TOMS)
Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices
Journal of Computational and Applied Mathematics
Accurate Computations with Totally Nonnegative Matrices
SIAM Journal on Matrix Analysis and Applications
Accurate Floating-Point Summation Part I: Faithful Rounding
SIAM Journal on Scientific Computing
Ultimately Fast Accurate Summation
SIAM Journal on Scientific Computing
Handbook of Floating-Point Arithmetic
Handbook of Floating-Point Arithmetic
Growth factors of pivoting strategies associated with Neville elimination
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
We investigate how extra-precise accumulation of dot products can be used to solve ill-conditioned linear systems accurately. For a given p-bit working precision, extra-precise evaluation of a dot product means that the products and summation are executed in 2p-bit precision, and that the final result is rounded into the p-bit working precision. Denote by u=2^-^p the relative rounding error unit in a given working precision. We treat two types of matrices: first up to condition number u^-^1, and second up to condition number u^-^2. For both types of matrices we present two types of methods: first for calculating an approximate solution, and second for calculating rigorous error bounds for the solution together with the proof of non-singularity of the matrix of the linear system. In the first part of this paper we present algorithms using only rounding to nearest, in Part II we use directed rounding to obtain better results. All algorithms are given in executable Matlab code and are available from my homepage.