Matrix multiplication via arithmetic progressions

인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

Complexity and Real Computation

We present a new method for accelerating matrix multiplication asymptotically. This work builds on recent ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product. We make novel use of the Salem-Spencer Theorem, which gives a fairly dense set of integers with no three-term arithmetic progression. Our resulting matrix exponent is 2.376.

Polynomial and Matrix Computations

Structure matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms.

Structured Matrices and Polynomials: Unified Superfast Algorithms

The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present-day computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.

Solving a Polynomial Equation: Some History and Recent Progress

The eigenproblem for an n-by-n matrix A is the problem of the approximation (within a relative error bound 2-‘) of all the eigenvalues of the matrix A and computing the associated eigenspaces of all these eigenvalues. We show that the arithmetic complexity of this problem is bounded by O(n3 + (nlog’n)log b). If the characteristic and minimum polynomials of the matrix A coincide with each other (which is the ca8e for generic matrices of all classes of general and special matrices that we consider), then the latter deterministic cost bound can be replaced by the randomized bound O(K.a(2n) + n* + (n log’n) log b) where Ka(2n) denotes the cost of the computation of the 2n 1 vectors A’v, i = 1,. ,2n 1, maximized over all n-dimensional vectors v; Ka(2n) = O(M(n)logn), for M(n) = o(n2.3’6) denoting the arithmetic complexity of n x n matrix multiplication. This bound on the complexity of the eigenproblem is optimal up to a logarithmic factor and implies much faster solution of the eigenprohlem for the important special classes of matrices. In particular, we prove the bound O(n’ log n + (n log’ n) log b) on the randomized arithmetic complexity of the eigenproblem for generic matrices of the classes of n x n Toeplitz, Hank& Toeplitzlike, Hank&like and Toeplits-likeplus-Hank&like matrices. Then again, this bound is optimal (up to a logarithmic factor) for each of the latter classes of input matrices. We also prove similar nearly optimal upper bounds for the generic Caucby-like, Vandermonde-like and sparse matrices. All our complexity estimates for the eigenproblem improve the known ones by order of magnitude.

The complexity of the matrix eigenproblem

http://theory.stanford.edu/~virgi/cs367/papers/huangpan.pdf

Victor Y. Pan

Papers

Polynomial and Matrix Computations

Structured Matrices and Polynomials: Unified Superfast Algorithms

Solving a Polynomial Equation: Some History and Recent Progress

The complexity of the matrix eigenproblem

Fast Rectangular Matrix Multiplication and Applications