A new formulation for LU decomposition allows efficient representation of intermediate matrices while eliminating blocks of various sizes, i.e. during “undulantblock” elimination. Its efficiency arises from its design for block encapsulization, implicit in data structures that are convenient both for process scheduling and for memory management. Row/column permutations that can destroy such enc...

We present efficient implementations of a number of operations for quantum computers. These include controlled phase adjustments of the amplitudes in a superposition, permutations, approximations of transformations and generalizations of the phase adjustments to block matrix transformations. These operations generalize those used in proposed quantum search algorithms.

This paper proposes a new, large diffusion layer for the AES block cipher. This new layer replaces the ShiftRows and MixColumns operations by a new involutory matrix in every round. The objective is to provide complete diffusion in a single round, thus sharply improving the overall cipher security. Moreover, the new matrix elements have low Hamming-weight in order to provide equally good perfor...

Let M be a complex matrix of rank one, i.e. M = pq′ where p, q are complex nby-1 matrices and ′ denotes Hermitean conjugation. Let μC(M) and μR(M) denote the structured singular values μ(M,∆) of M defined with ∆ = ∆C and ∆ = ∆R respectively, where ∆C is the cone of all diagonal matrices with complex entries, and ∆R is the cone of all diagonal matrices with real entries. (a) Express μC(M) and μR...

In this paper, we define the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We show how this can be applied to prove and improve the convergence of a fixed point problem associated to the matrix iteration scheme, including for distributed ...

We describe a datatype for (dense) matrices whose primitive operations are decomposition and composition (of submatrices), as opposed to indexed element access which is the primitive operation on conventional arrays. Using the composition and decomposition operations it is for example possible to express both recursive and traditional block matrix algorithms (e.g., Cholesky factorization, QR-fa...

In this project, we had to implement a parallel solver for linear equation systems, using a technique known as Gaussian elimination (GE). As with many other algorithms for solving linear equation systems, GE is performed on the matrix representation of the system, Ax = b, where A is the coefficient matrix and b is the vector of known values. GE works by applying a set of elementary row operatio...

The Hermite Normal Form is a canonical matrix analogue of Reduced Echelon Form, but involving matrices over more general rings. In this work we formalise an algorithm to compute the Hermite Normal Form of a matrix by means of elementary row operations, taking advantage of the Echelon Form AFP entry. We have proven the correctness of such an algorithm and refined it to immutable arrays. Furtherm...

Many programs for scienti c computing in Python are based on NumPy and therefore make heavy use of numerical linear algebra (NLA) functions, vectorized operations, slicing and broadcasting. AlgoPy provides the means to compute derivatives of arbitrary order and Taylor approximations of such programs. The approach is based on a combination of univariate Taylor polynomial arithmetic and matrix ca...