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Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial…
We extend the geometrical inverse approximation approach for solving linear least-squares problems. For that we focus on the minimization of $1-\cos(X(A^TA),I)$, where $A$ is a given rectangular coefficient matrix and $X$ is the approximate…
The present paper gives a review of our recent progress and latest results for novel linear-algebraic algorithms and its application to large-scale quantum material simulations or electronic structure calculations. The algorithms are…
The aim of this paper is twofold. First, we introduce a new class of linearizations, based on the generalization of a construction used in polynomial algebra to find the zeros of a system of (scalar) polynomial equations. We show that one…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
Dense and sparse tensors allow the representation of most bulk data structures in computational science applications. We show that sparse tensor algebra can also be used to express many of the transformations on these datasets, especially…
The Scalable ZX-calculus is a compact graphical language used to reason about linear maps between quantum states. These diagrams have multiple applications, but they frequently have to be constructed in a case-by-case basis. In this work we…
To be able to solve operator equations numerically a discretization of those operators is necessary. In the Galerkin approach bases are used to achieve discretized versions of operators. In a more general set-up, frames can be used to…
In the past two decades, some major efforts have been made to reduce exact (e.g. integer, rational, polynomial) linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide…
This article presents matrix backpropagation algorithms for the QR decomposition of matrices $A_{m, n}$, that are either square (m = n), wide (m < n), or deep (m > n), with rank $k = min(m, n)$. Furthermore, we derive novel matrix…
A concise analytical formula is developed for the inverse of an invertible 3 x 3 matrix using a telescoping method, and is generalized to larger square matrices. The formula is confirmed using randomly generated matrices in Matlab
The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using `rewrite rules' to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic…
Quasi-separable matrices are a class of rank-structured matriceswidely used in numerical linear algebra and of growing interestin computer algebra, with applications in e.g. the linearization ofpolynomial matrices. Various representation…
We suggest that a certain one-to-one parametrization of completely positive maps on the matrix algebra might be useful in the study of quantum channels. This is illustrated in the case of binary quantum channels. While the algorithm is…
In our article we consider some algebraical methods which may be useful in some inverse spectral problems. The reconstraction of the matrix from its minors is considered.
Matrices arising in scientific applications frequently admit linear low-rank approximations due to smoothness in the physical and/or temporal domain of the problem. In large-scale problems, computing an optimal low-rank approximation can be…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
In this paper we construct a new factorized representation of the $R$-matrix related to the affine algebra $U_{q}(\widehat{sl_{n}})$ for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit…
A model of representations of a Lie algebra is a representation which a direct sum of all irreducible finite dimensional representations taken with multiplicity $1$. In the paper an explicit construction of a model of representation for all…
Following the ideas of [AGG11] about Zt x Z2,2-cocyclic Hadamard matrices, we introduce the notion of diagram, which visually represents any set of coboundaries. Diagrams are a very useful tool for the description and the study of paths and…