Related papers: Numerical analysis of Givens rotation
Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative…
Balancing a matrix is a preprocessing step while solving the nonsymmetric eigenvalue problem. Balancing a matrix reduces the norm of the matrix and hopefully this will improve the accuracy of the computation. Experiments have shown that…
An improvement on precision of recursive function simulation in IEEE floating point standard is presented. It is shown that the average of rounding towards negative infinite and rounding towards positive infinite yields a better result than…
Transforms using random matrices have been found to have many applications. We are concerned with the projection of a signal onto Gaussian-distributed random orthogonal bases. We also would like to easily invert the process through…
Modern lunar-planetary ephemerides are numerically integrated on the observational timespan of more than 100 years (with the last 20 years having very precise astrometrical data). On such long timespans, not only finite difference…
The Fast Reciprocal Square Root Algorithm is a well-established approximation technique consisting of two stages: first, a coarse approximation is obtained by manipulating the bit pattern of the floating point argument using integer…
This research investigates using a mixed-precision iterative refinement method using posit numbers instead of the standard IEEE floating-point format. The method is applied to solve a general linear system represented by the equation $Ax =…
We show that assuming the availability of the processor with variable precision arithmetic, we can compute matrix-by-matrix multiplications in $O(N^2log_2N)$ computational complexity. We replace the standard matrix-by-matrix multiplications…
We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular,…
We present a detailed study of roundoff errors in probabilistic floating-point computations. We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are…
We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm has an error bound exponent of only log 4 $\gamma$$\infty$,2 $\approx$ 2.386.…
The Richardson-Lucy unfolding approach is simple and excellently performing. It efficiently suppresses artificial high frequency contributions and permits to introduce known features of the true distribution. An algorithm to fix the number…
Logarithmic number systems (LNS) are used to represent real numbers in many applications using a constant base raised to a fixed-point exponent making its distribution exponential. This greatly simplifies hardware multiply, divide and…
Numerical integrations in celestial mechanics often involve the repeated computation of a rotation with a constant angle. A direct evaluation of these rotations yields a linear drift of the distance to the origin. This is due to roundoff in…
We consider linear dynamical systems under floating-point rounding. In these systems, a matrix is repeatedly applied to a vector, but the numbers are rounded into floating-point representation after each step (i.e., stored as a…
This paper provides a rounding-error analysis for two-grid methods that use one relaxation step both before and after coarsening. The analysis is based on floating point arithmetic and focuses on a two-grid scheme that is perturbed on the…
We propose two new algorithms to minimize the constant modulus (CM) criterion in the context of blind source separation. The first algorithm, referred to as Givens CMA (G-CMA) uses unitary Givens rotations and proceeds in two stages:…
We introduce a new method for discovering matrix multiplication schemes based on random walks in a certain graph, which we call the flip graph. Using this method, we were able to reduce the number of multiplications for the matrix formats…
There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of…
On modern architectures, the performance of 32-bit operations is often at least twice as fast as the performance of 64-bit operations. By using a combination of 32-bit and 64-bit floating point arithmetic, the performance of many dense and…