In many trust region algorithms for optimization calculations, each iteration seeks a vector d2R n that solves the linear system of equations (B+ I)d=?g, where B is a symmetric estimate of a second derivative matrix, I is the unit matrix, g is a known gradient vector, and is a parameter that controls the length of d. Several values of may be tried on each iteration, and, when there is no helpfu...

This paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis, when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low-rank property. A preliminary interpolative butterfly factorization is constructed based on interpolative low-ra...

The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i, j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was ...

This paper proposes a reduct construction method based on discernibility matrix simplification. The method works in a similar way to the classical Gaussian elimination method for solving a system of linear equations. Elementary matrix simplification operations are introduced. Each operation transforms a matrix into a simpler form. By applying these operations a finite number of times, one can t...

Current scientific applications must analyze enormous amounts of array data using complex mathematical data processing methods. This paper describes a distributed query processing framework for large-scale scientific data analysis that captures array-based computations using SQL-like queries and optimizes and evaluates these computations using state-of-the-art parallel processing algorithms. In...

We present a data structure and several operations it supports for recursively lowrank compressed matrices; that is, the diagonal blocks of the matrix are recursively partitioned, and the off-diagonal blocks in each partition level admit a low-rank approximation. Such a compression is embraced in many linearor near-linear-time methods for kernel matrices, including the fast multipole method, th...

Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are prohibitive when the scale of data is large. In real-world applications, since the data themselves are noisy, machine-precision matrix operations are not necessary a...

This paper describes two new matrix transform algorithms for the Max-Log-MAP decoding of turbo codes. In the proposed algorithms, the successive decoding procedures carried out in the conventional Max-Log-MAP algorithm are performed in parallel, and well formulated into a set of simple and regular matrix operations, which can therefore considerably speed up the decoding operations and reduce th...

High-performance computing systems today include a variety of compute devices such as multi-core CPUs, GPUs and many-core accelerators. OpenCL allows programming different types of compute devices using a single API and kernel language. However, there is no standard matrix operations library in OpenCL for operations such as matrix multiplication that works well on a variety of hardware from mul...

This paper describes some initial work on a formalization of matrices in ACL2. The current work is focused on creating an executable implementation that is simple to mechanically reason with and complete enough to be capable of analyzing real problems in linear algebra. A number of basic operations have been defined including matrix addition, transposition and multiplication by a scalar, a vect...