We introduce a generalization of intervals over compact matrix Lie groups. Interval group elements are defined using a midpoint-radius representation. Furthermore, we define the respective group operations and the action of interval group elements on vector spaces. We developed structures and operations are implemented for the special case of orthogonal matrices in the matrix library of the COC...

We introduced matrices as a computational tool for working with linear transformations between finitely generated vector spaces. The general properties of linear transformations and the algebraic operations on them — addition, multiplication by scalars and composition — enabled us to define corresponding algebraic operations on matrices — matrix addition, multiplication by scalars and matrix mu...

Inspired by pointfree relational data processing, this paper addresses the foundations of an alternative roadmap for parallel online analytical processing (OLAP) based on a separation of concerns: rather than depending on standard database technology and heavy machinery, OLAP operations are performed by encoding data in matrix format and relying thereupon solely on LA operations. The paper inve...

A new randomized algorithm is presented for computing the Frobenius form of an n×n matrix over a sufficiently large field K. Let 2 < ω ≤ 3 be such that two matrices in Kn×n can be multiplied together with O(n) operations in K. If #K ≥ 2n, the new algorithm uses an expected number of O(n) operations in K to compute the Frobenius form of A, matching the lower bound for the cost of this problem. A...

We present a fast condition estimation algorithm for the eigenvalues of a class of structured matrices. These matrices are low rank modifications to Hermitian, skew-Hermitian, and unitary matrices. Fast structured operations for these matrices are presented, including Schur decomposition, eigenvalue block swapping, matrix equation solving, compact structure reconstruction, etc. Compact semisepa...

Abstract: Quantum computers have been considered as powerful computing apparatus in the future. Various quantum gates and quantum circuits have been presented to solve classical computational problems using quantum mechanical systems. Quantum gates and quantum circuits can be expressed using the tensor product notation. However, the mathematical model of tensor products is usually limited to su...

The multiplication of a vector by a matrix and the solution of triangular linear systems are the most demanding operations in the majority of iterative techniques for the solution of linear systems. Data-driven VLSI networks which perform these two operations, efficiently, for certain sparse matrices are introduced. In order to avoid computations that involve zero operands, the non-zero element...

A real symmetric matrix of order n has a full set of orthogonal eigenvectors. The most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one, i.e., either a tridiagonal matrix [2, 3] or a semiseparable matrix [4]. This step is accomplished in O(n) operations. Once the latter symmetric structured matrix is available, its s...

In this project we explore different ways in which we can optimize the computation of training a Tree-structured RNTN, in particular batching techniques in combining many matrix-vector multiplications into matrix-matrix multiplications, and many tensor-vector operations into tensor-matrix operations. We assume that training is performed using mini-batch AdaGrad algorithm, and explore how we can...

The SR algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix B in butterfly form is uniquely determined by 4n− 1 parameters. Using these 4n− 1 parameters, we show how one step of the symplectic SR algorithm for B can be carried out in O(n) arithmetic operations compa...