Search results for "Multiplication"
showing 10 items of 83 documents
Generalized centro-invertible matrices with applications
2014
Centro-invertible matrices are introduced by R.S. Wikramaratna in 2008. For an involutory matrix R, we define the generalized centro-invertible matrices with respect to R to be those matrices A such that RAR = A^−1. We apply these matrices to a problem in modular arithmetic. Specifically, algorithms for image blurring/deblurring are designed by means of generalized centro-invertible matrices. In addition, if R1 and R2 are n × n involutory matrices, then there is a simple bijection between the set of all centro-invertible matrices with respect to R1 and the set with respect to R2.
Fast Matrix Multiplication
2015
Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time O(n2.3755). Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le~Gall has led to an improved algorithm running in time O(n2.3729). These algorithms are obtained by analyzing higher and higher tensor powers of a certain identity of Coppersmith and Winograd. We show that this exact approach cannot result in an algorithm with running time O(n2.3725), and identify a wide class of variants of this approach which cannot result in an algorithm with running time $O(n^{2.3078}); in particular, this approach cannot prove the conjecture that for every e > 0, …
Multiplications of Distributions in One Dimension and a First Application to Quantum Field Theory
2002
In a previous paper we introduced a class of multiplications of distributions in one dimension. Here we furnish different generalizations of the original definition and we discuss some applications of these procedures to the multiplication of delta functions and to quantum field theory. © 2002 Elsevier Science (USA).
An Scalable matrix computing unit architecture for FPGA and SCUMO user design interface
2019
High dimensional matrix algebra is essential in numerous signal processing and machine learning algorithms. This work describes a scalable square matrix-computing unit designed on the basis of circulant matrices. It optimizes data flow for the computation of any sequence of matrix operations removing the need for data movement for intermediate results, together with the individual matrix operations’ performance in direct or transposed form (the transpose matrix operation only requires a data addressing modification). The allowed matrix operations are: matrix-by-matrix addition, subtraction, dot product and multiplication, matrix-by-vector multiplication, and matrix by scalar multiplication.…
The Sliced COO Format for Sparse Matrix-Vector Multiplication on CUDA-enabled GPUs
2012
Abstract Existing formats for Sparse Matrix-Vector Multiplication (SpMV) on the GPU are outperforming their corresponding implementations on multi-core CPUs. In this paper, we present a new format called Sliced COO (SCOO) and an effcient CUDA implementation to perform SpMV on the GPU. While previous work shows experiments on small to medium-sized sparse matrices, we perform evaluations on large sparse matrices. We compared SCOO performance to existing formats of the NVIDIA Cusp library. Our resutls on a Fermi GPU show that SCOO outperforms the COO and CSR format for all tested matrices and the HYB format for all tested unstructured matrices. Furthermore, comparison to a Sandy-Bridge CPU sho…
Mean ergodicity of weighted composition operators on spaces of holomorphic functions
2016
[EN] Let phi be a self-map of the unit disc D of the complex plane C and let psi be a holomorphic function on D. We investigate the mean ergodicity and power boundedness of the weighted composition operator C-phi,C-psi(f) = psi(f o phi) with symbol phi and multiplier psi on the space H(D). We obtain necessary and sufficient conditions on the symbol phi and on the multiplier psi which characterize when the weighted composition operator is power bounded and (uniformly) mean ergodic. One necessary condition is that the symbol phi has a fixed point in D. If phi is not a rational rotation, the sufficient conditions are related to the modulus of the multiplier on the fixed point of phi. Some of o…
A new mathematical tool for an exact treatment of open quantum system dynamics
2005
A new method to obtain an operatorial exact solution of a wide class of Markovian master equations is presented. Its key point is the existence of a constant of motion partitioning the Hilbert space into finite-dimensional subspaces. The consequent possibility of representing the reduced density operator as a block diagonal matrix is shown. Each “block operator” evolves under the action of a non-unitary operator explicitly derived. Our mathematical approach is illustrated applying it to simple physical systems.
Area-efficient FPGA-based FFT processor
2003
A novel architecture for computing the fast Fourier transform on programmable devices is presented. Main results indicate that the use of one CORDIC operator to perform the multiplication by all the ‘twiddle factors’ sequentially leads to an area saving up to 35% with respect to other cores.
Versatile Direct and Transpose Matrix Multiplication with Chained Operations: An Optimized Architecture Using Circulant Matrices
2016
With growing demands in real-time control, classification or prediction, algorithms become more complex while low power and small size devices are required. Matrix multiplication (direct or transpose) is common for such computation algorithms. In numerous algorithms, it is also required to perform matrix multiplication repeatedly, where the result of a multiplication is further multiplied again. This work describes a versatile computation procedure and architecture: one of the matrices is stored in internal memory in its circulant form, then, a sequence of direct or transpose multiplications can be performed without timing penalty. The architecture proposes a RAM-ALU block for each matrix c…
Reverse-Safe Text Indexing
2021
We introduce the notion of reverse-safe data structures. These are data structures that prevent the reconstruction of the data they encode (i.e., they cannot be easily reversed). A data structure D is called z - reverse-safe when there exist at least z datasets with the same set of answers as the ones stored by D . The main challenge is to ensure that D stores as many answers to useful queries as possible, is constructed efficiently, and has size close to the size of the original dataset it encodes. Given a text of length n and an integer z , we propose an algorithm that constructs a z -reverse-safe data structure ( z -RSDS) that has size O(n) and answers decision and counting pattern matc…