Related papers: Derivation of a low multiplicative complexity algo…
In recent years, with the rapid development of electro-optic modulators, optical computing has become a potential excellent candidate for various computing tasks. New structures and devices for optical computing are emerging one after…
We introduce new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops. Our compact formulas for the hafnian and loop hafnian of $n \times n $ complex…
In 1969 Strassen showed surprisingly that it is possible to multiply two 2 x 2 matrices using seven multiplications and 18 additions, instead of the naive eight multiplications and four additions. The number of additions was later reduced…
In this note, the octonion multiplication table is recovered from a regular tesselation of the "equilateral" two dimensional torus by seven hexagons, also known as Heawood's map.
There are four division algebras over $\mathbb{R}$, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions.…
With a view to applying the Generator Coordinate Method to large configuration spaces, we propose a simple approximate formula to compute diabatic many-body matrix elements without having to evaluate two-body interaction matrix elements.…
We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space…
The algebra of octonions is non-associative (as well as non-commutative). This makes it very difficult to derive algebraic results, and to perform computation with octonions. Given a product of more than two octonions, in general, the order…
Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole…
We show how one can use non-prime-power, composite moduli for computing representations of the product of two $n\times n$ matrices using only $n^{2+o(1)}$ multiplications.
The octonionic X-product gives the octonions a flexibility not found in the other real division algebras. The pattern of that flexibility is investigated here.
We present a compact mnemonic device for computing the product of two (split) octonions written in Cayley--Dickson form q+l p with q,p in H. The rule appears as a simple (R+L) pattern of right-ordered and left-ordered (quaternionic)…
We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs…
A new algorithm is derived for computing the actions $f(tA)B$ and $f(tA^{1/2})B$, where $f$ is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with…
This paper deals with circulant matrices. It is shown that a circulant matrix can be multiplied by a vector in time O(n log(n)) in a ring with roots of unity without making use of an FFT algorithm. With our algorithm we achieve a speedup of…
In this paper, we present a probabilistic algorithm to multiply two sparse polynomials almost as efficiently as two dense univariate polynomials with a result of approximately the same size. The algorithm depends on unproven heuristics that…
The Hermitian decomposition of a linear operator is generalized to the case of two or more operations. An additive expansion of the product of three octonions into three parts is constructed, wherein each part either preserve or change the…
Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This…
As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras…
A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be…