27 Quantum computing has enormous potential for introducing fundamentally new capabilities to computational science and engineering, primarily through exponential parallelism.1,2 One of the many challenges in building practical quantum computers is to reduce a general quantum computation to some set of elementary operations that simple quantum devices can implement. By analogy, in the case of c...

Matrices with low-rank off-diagonal blocks are a versatile tool to perform matrix compression and to speed up various matrix operations, such as the solution of linear systems. Often, the underlying block partitioning is described by a hierarchical partitioning of the row and column indices, thus giving rise to hierarchical low-rank structures. The goal of this chapter is to provide a brief int...

Here we tackle a problem from biology in terms of discrete mathematics. We are interested in a complex DNA manipulation process happening in eukaryotic organisms of a subclass of ciliate species called Stichotrichia during so-called gene assembly. This process is in particular interesting since one can interpret gene assembly in ciliates as sorting of permutations. We survey here results relate...

A quantum compiling algorithm is an algorithm for decomposing (“compiling”) an arbitrary unitary matrix into a sequence of elementary operations (SEO). Suppose Uin is an NB-bit unstructured unitary matrix (a unitary matrix with no special symmetries) that we wish to compile. For NB > 10, expressing Uin as a SEO requires more than a million CNOTs. This calls for a method for finding a unitary ma...

We discuss parallel algorithms for solving eight common standard and generalized triangular Sylvester-type matrix equation. Our parallel algorithms are based on explicit blocking, 2D block-cyclic data distribution of the matrices and wavefront-like traversal of the right hand side matrices while solving small-sized matrix equations at different nodes and updating the rest of the right hand side...

We propose a divide-and-conquer algorithm for computing arbitrary functions of upper triangular matrices, which requires approximately the same number of arithmetic operations as Parlett’s algorithm. However, the new algorithm has better performance on computers with two levels of memory due to its block structure and thus, less memory-cache traffic requirements. Like Parlett’s algorithm, the n...

The error backpropagation (EBP) algorithm for training feedforward multilayer perceptron (FMLP) has been used in many applications because it is simple and easy to implement. However, its gradient descent method prevents EBP algorithm from converging fast. To overcome the slow convergence of EBP algorithm, the second order methods have adapted. Levenberg-Marquardt (LM) algorithm is estimated to...

The LAPACK software project currently under development is intended to provide a portable linear algebra library for high performance computers. LAPACK will make use of the Level 1, 2, and 3 BLAS to carry out basic operations. A principal focus of this project is to implement blocked versions of a number of algorithms to take advantage of the greater parallelism and improved data locality of th...

This paper addresses the secure outsourcing problem for large-scale matrix computation to a public cloud. We propose a novel public-key weave ElGamal encryption (WEE) scheme for encrypting a matrix over the field Zp. The scheme has the echelon transformation property. We can apply a series of elementary row/column operations to transform an encrypted matrix under our WEE scheme into the row/col...

‎In this paper‎, ‎we find matrix representation of a class of sixth order Sturm-Liouville problem (SLP) with separated‎, ‎self-adjoint boundary conditions and we show that such SLP have finite spectrum‎. ‎Also for a given matrix eigenvalue problem $HX=lambda VX$‎, ‎where $H$ is a block tridiagonal matrix and $V$ is a block diagonal matrix‎, ‎we find a sixth order boundary value problem of Atkin...