A Las Vegas probabilistic algorithm is presented that finds the Smith normal form S ∈ Q[x] of a nonsingular input matrix A ∈ Z [x]. The algorithm requires an expected number of O (̃nd(d + n log ||A||)) bit operations (where ||A|| bounds the magnitude of all integer coefficients appearing in A and d bounds the degrees of entries of A). In practice, the main cost of the computation is obtaining a ...

1.1. Method 1: Copy transpose DGEMM. Strided accesses pose a critical bottleneck in the naive implementation of matrix multiply. Both A and B are stored in the same order (column-major), but the multiply operation requires that entries of either A or B be loaded with stride M . (Without loss of generality, assume the A matrix.) Large strides result in ineffective use of cache lines, since (for ...

Cell formation problem (CFP) is one of the main problems in cellular manufacturing systems. Minimizing exceptional elements and voids is one of the common objectives in the CFP. The purpose of the present study is to propose a new model for cellular manufacturing systems to group parts and machines in dedicated cells using a part-machine incidence matrix to minimize the voids. After identifying...

This paper presents an improved algorithm for computing the Combinatorial Canonical Form (CCF) of a layered mixed matrix A = Q T , which consists of a numerical matrix Q and a generic matrix T . The CCF is the (combinatorially unique) nest block-triangular form obtained by the row operations on the Q-part, followed by permutations of rows and columns of the whole matrix. The main ingredient of ...

Matrix inversion is a mathematical algorithm that is widely used and applied in many real time engineering applications. It is one of the most computational intensive and time consuming operations especially when it is performed in software. Gauss-Jordan Elimination is one of the many matrix inversion algorithms which has the advantage of using simpler mathematical operations to get the result....

The mass matrix for Gauss-Lobatto grid points is usually approximated by GaussLobatto quadrature because this leads to a diagonal matrix that is easy to invert. The exact mass matrix and its inverse are full. We show that the exact mass matrix and its inverse differ from the approximate diagonal ones by a simple rank-1 update (outer product). They can thus be applied to an arbitrary vector in O...

The usual procedure to compute the determinant is the so-called Gaussian elimination. We can view this as the transformation of the matrix into a lower triangular matrix with column operations. These transformations do not change the determinant but in the triangular matrix, the computation of the determinant is more convenient: we must only multiply the diagonal elements to obtain it. (It is a...

Let G be a nite group and f any complex-valued function deened on G and an irreducible complex matrix representation of G. The Fourier transform of f at is deened to be the matrix P s2G f(s)(s). The Fourier transforms of f at all the irreducible representations of G determine f via the Fourier inversion formula f(s) = 1 j Gj P d trace(b f()(s ?1)): Direct computation of all Fourier transforms o...

Graphs can be naturally represented as sparse matrices. The relationship between graph algorithms and linear algebra algorithms is well understood and many graph problems can be abstracted as Sparse General Matrix-Matrix Multiplication (SpGEMM) operations. While quite some matrix storage formats, including bitmap-based ones, have been proposed for sparse matrices, they are mostly evaluated on t...

Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient, but have off-diagonal blocks that are; specifically, the class of so called Hierarchically Semi-Separable (HSS) matric...