This paper extends the modified affine arithmetic in matrix form method for bivariate polynomial evaluation and algebraic curve plotting in 2D to modified affine arithmetic in tensor form for trivariate polynomial evaluation and algebraic surface plotting in 3D. Experimental comparison shows that modified affine arithmetic in tensor form is not only more accurate but also much faster than stand...

In this paper the arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks proper over a general arithmetic ring. The method used also gives another construction of the product on the usual Chow groups of a regular Deligne-Mumford stack.

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis world in Pythagorean manner. unifies foundations mathematics (Peano and set theory), physics (quantum mechanics information), philosophical transcendentalism (Husserl’s phenomenology) into formal theory mathematical structure literally following Husserl’s tracе “philosophy rigorous science”. In pathway to ...

This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers. We are concerned with software more than hardware - we do not cover computer architecture or the design of computer hardware. Instead we focus on algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to top...

For real symmetric eigenvalue problems, there are a number of algorithms that are mathematically equivalent, for example, the Lanczos algorithm, the Arnoldi method and the unpreconditioned Davidson method. The Lanczos algorithm is often preferred because it uses signiicantly fewer arithmetic operations per iteration. To limit the maximum memory usage, these algorithms are often restarted. In re...

A new method to construct task graphs for $\mathcal{H}$-matrix arithmetic is introduced, which uses the information associated with all tasks of standard recursive algorithms, e.g., block index set matrix blocks involved in computation. Task refinement, i.e., replacement by subcomputations, then used proceed hierarchy until containing actual data are reached. This process a natural extension cl...

Residue Number System is a numerical system which arithmetic operations are performed parallelly. One of the main factors that affects the system’s performance is the complexity of reverse converter. It should be noted that the complexity of this part should not affect the earned speed of parallelly performed arithmetic unit. Therefore in this paper a high speed converter for moduli set {2n-1, ...

Residue Number System is a numerical system which arithmetic operations are performed parallelly. One of the main factors that affects the system’s performance is the complexity of reverse converter. It should be noted that the complexity of this part should not affect the earned speed of parallelly performed arithmetic unit. Therefore in this paper a high speed converter for moduli set {2n-1, ...