STURM SEQUENCES AND RANDOM EIGENVALUE DISTRIBUTIONS (2009)
James T. Albrecht, Cy P. Chan, Alan Edelman
Abstract. This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first...
Cy Chan, Vesselin Drensky, Alan Edelman, Raymond Kan, Plamen Koev, C. Chan, ...
Abstract We present two new algorithms for computing all Schur functions sκ(x1,...,xn) for partitions κ such that |κ | ≤ N. Both algorithms have the property that for nonnegative arguments...
Non-Fourier Encoded Parallel MRI Using Multiple Receiver Coils (2009)
Dimitris Mitsouras, W. Scott Hoge, Frank J. Rybicki, Walid E. Kyriakos, Alan Edelman, Gary P. Zientara
This paper describes a general theoretical framework combining non-Fourier spatially encoded MR imaging with multi-channel acquisition parallel MR imaging. The two spatial encoding mechanisms are...
Cy Chan, Vesselin Drensky, Alan Edelman, Raymond Kan, Plamen Koev
Abstract We present two new algorithms for computing all Schur functions sκ(x1,...,xn) for partitions κ such that |κ | ≤ N. Both algorithms have the property that for nonnegative arguments...
FROM RANDOM MATRICES TO STOCHASTIC OPERATORS (2008)
Abstract. We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise,...
Tails of condition number distributions (2008)
Abstract. Let κ be the condition number of an m-by-n matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C...
MATLAB*P 2.0: USER FRIENDLY, INTERACTIVE ENVIRONMENT FOR PARALLEL SCIENTIFIC COMPUTING (2008)
MATLAB is one of the most widely used mathematical software in technical computing. Because of its ease of use and strong visualization features, it is very often used in numerical experimentations....
Fast Sorting on a Distributed-Memory Architecture (2008)
David R. Cheng, Viral Shah, John R. Gilbert, Alan Edelman
Abstract — We consider the often-studied problem of sorting, for a parallel computer. Given an input array distributed evenly over p processors, the task is to compute the sorted output array, also...
Fast Magnetic Resonance Imaging via Adaptive Broadband Encoding of the MR Signal Content (2008)
Dimitris Mitsouras, Frank J. Rybicki, Alan Edelman, Gary P. Zientara
Abstract — Our goal is to increase the time-efficiency of continuous
A LINEAR-TIME ALGORITHM FOR EVALUATING SERIES OF SCHUR FUNCTIONS (2008)
Cy Chan, Vesselin Drensky, Alan Edelman, Plamen Koev
Abstract. We present a new algorithm for computing all Schur functions sλ(x1, x2,..., xn) for all partitions λ of integers not exceeding N in time O(n 2 KN), where KN ≡ #{λ | |λ | ≤ N} is the...
Fast Sorting on a Distributed-Memory Architecture (2008)
David R. Cheng, Viral Shah, John R. Gilbert, Alan Edelman
Abstract — The abstract goes here.
Abstract. In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the...
Support Vector Machine Lagrange Multipliers and Simplex Volume Decompositions (2007)
The Support Vector Machine (SVM) idea has attracted recent attention in solving classification and regression problems. As an example based method, SVMs distinguish two point classes by finding a...
Interactive Supercomputing with MITMatlab Parry Husbands (2007)
Charles L. Isbell, Alan Edelman
This paper describes MITMatlab, a system that enables users of supercomputers to transparently work on large data sets within Matlab. MITMatlab communicates with an external server that is...
On the Determinant of a Uniformly Distributed Complex Matrix (2007)
We derive the joint density for the singular values of a random complex matrix A uniformly distributed on kAkF = 1. This joint density allows us to obtain the conditional expectation of det(A H A) =...
On Parlett's Matrix Norm Inequality for the Cholesky Decomposition (2007)
Alan Edelman, Walter F. Mascarenhas, Barao Geraldo
We show that a certain matrix norm ratio studied by Parlett has a supremum that is O( p n) when the chosen norm is the Frobenius norm, while it is O(log n) for the 2-norm. This ratio arises in...
Alan Edelman, Robert Schreiber, Hewlett Packard, Shang-hua Teng, U. Minnesota
These lecture notes #under development and constant revision, like the #eld itself # have been used at MIT in a graduate course #rst o#ered by Alan Edelman and Shang-Hua Teng during the spring of...
Parry Husbands, Alan Edelman, Charles L. Isbell
This paper describes MITMatlab, a system that enables users of supercomputers to transparently work on large data sets within Matlab. MITMatlab communicates with an external server that is...
To appear in the SIGGRAPH conference proceedings Modeling and Rendering of Weathered Stone (2007)
Julie Dorsey, Alan Edelman, Henrik Wann, Jensen Justin, Legakis Hans, Køhling Pedersen
Stone is widespread in its use as a building material and artistic medium. One of its most remarkable qualities is that it changes appearance as it interacts with the environment. These changes are...
We present a mathematically justifiable, computationally simple, sample eigenvalue based procedure for estimating the number of high-dimensional signals in white noise using relatively few samples....
Sample size cognizant detection of signals in white noise (2007)
The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a computationally simple, sample eigenvalue based...
Statistical eigen-inference from large Wishart matrices (2007)
Rao, N. Raj, Mingo, James A., Speicher, Roland, Edelman, Alan
We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary...
The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems (2007)
Abstract. We provide a solution to the β-Jacobi matrix model problem posed by Dumitriu and the first author. The random matrix distribution introduced here, called a matrix model, is related to the...
A Novel Parallel Sorting Algorithm for Contemporary Architectures (2007)
David R. Cheng, Viral B. Shah, John R. Gilbert, Alan Edelman
Traditionally, the field of scientific computing has been dominated by numerical methods. However, modern scientific codes often combine numerical methods with combinatorial methods. Sorting, a...
From Random Matrices to Stochastic Operators (2006)
Edelman, Alan, Sutton, Brian D.
We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each...
The polynomial method for random matrices (2006)
We define a class of "algebraic" random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a...
Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models (2006)
ensembles via matrix models
Free Probability, Sample Covariance Matrices and Stochastic Eigen-Inference (2005)
Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This talk is a survey specifically oriented towards the needs and interests of a...
Free Probability, Sample Covariance Matrices and Stochastic Eigen-Inference (2005)
Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This talk is a survey specifically oriented towards the needs and interests of a...
Dumitriu, Ioana, Edelman, Alan
We study the global spectrum fluctuations for $\beta$-Hermite and $\beta$-Laguerre ensembles via the tridiagonal matrix models introduced in \cite{dumitriu02}, and prove that the fluctuations...
The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument (2005)
We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial...
Numerical Methods for Eigenvalue Distributions of Random Matrices (2005)
Edelman, Alan, Persson, Per-Olof
We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on very large matrices,...
David R. Cheng, A Uth Or, Alan Edelman, Arthur C. Smith, David R. Cheng
This thesis studies three problems in the field of parallel computing. The first result provides a deterministic parallel sorting algorithm that empirically shows an improvement over two sample sort...
Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics (2005)
Abstract In this paper we examine the zero and first order eigenvalue fluctuations for the fi-Hermite and fi-Laguerre ensembles, using the matrix models we described in [5], in the limit as fi! 1. We...
Parallel MATLAB: Doing it right (2005)
Ron Choy, Alan Edelman, Cleve Moler Of
MATLAB [20] is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides high performance computational routines and an...
Mesh Generation for Implicit Geometries (2005)
Alan Edelman, Gilbert Strang, Rodolfo Ruben Rosales, Pavel Etingof, Per-olof Persson, Per-olof Persson
The efficient evaluation of the hypergeometric function of a matrix argument (2005)
Abstract. We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the...
Fast Sorting on a Distributed-Memory Architecture (2004)
Cheng, David R., Shah, Viral, Gilbert, John R., Edelman, Alan
We consider the often-studied problem of sorting, for a parallel computer. Given an input array distributed evenly over p processors, the task is to compute the sorted output array, also distributed...
Fast Sorting on a Distributed-Memory Architecture (2004)
Cheng, David R., Shah, Viral, Gilbert, John R., Edelman, Alan
We consider the often-studied problem of sorting, for a parallel computer. Given an input array distributed evenly over p processors, the task is to compute the sorted output array, also distributed...
MOPS: Multivariate Orthogonal Polynomials (symbolically) (2004)
Dumitriu, Ioana, Edelman, Alan, Shuman, Gene
In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi...
Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics (2004)
Dumitriu, Ioana, Edelman, Alan
In this paper we examine the zero and first order eigenvalue fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles, using the matrix models we described in \cite{dumitriu02}, in the...
Star-P: High productivity parallel computing (2004)
Ron Choy, Alan Edelman, John R. Gilbert, Viral Shah, David Cheng
Star-P ‡ is an interactive parallel scientific computing environment. It aims to make parallel programming more accessible. Star-P borrows ideas from Matlab*P [3], but is a new development....
Solving Multiple Classes of Problems in Parallel with MATLAB*P (2003)
MATLAB [7] is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides high performance computational routines and an...
Solving Multiple Classes of Problems in Parallel with MATLAB*P (2003)
MATLAB [7] is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides high performance computational routines and an...
18.337J / 6.338J Applied Parallel Computing (SMA 5505), Spring 2003 (2003)
Advanced interdisciplinary introduction to modern scientific computing on parallel supercomputers. Numerical topics include dense and sparse linear algebra, N-body problems, and Fourier transforms....
18.337J / 6.338J Applied Parallel Computing (SMA 5505), Spring 2003 (2003)
Advanced interdisciplinary introduction to modern scientific computing on parallel supercomputers. Numerical topics include dense and sparse linear algebra, N-body problems, and Fourier transforms....
MATLAB*P 2.0: A unified parallel MATLAB (2003)
MATLAB is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides high performance computational routines and an...
MATLAB*P 2.0: A unified parallel MATLAB (2003)
MATLAB is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides high performance computational routines and an...
Matrix Models for Beta Ensembles (2002)
Dumitriu, Ioana, Edelman, Alan
This paper constructs tridiagonal random matrix models for general ($\beta>0$) $\beta$-Hermite (Gaussian) and $\beta$-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and...
Staircase Failures Explained By Orthogonal Versal Forms (2000)
. Treating matrices as points in n 2 dimensional space, we apply geometry to study and explain algorithms for the numerical determination of the Jordan structure of a matrix. Traditional notions such...
The computation and sensitivity of double eigenvalues \Lambda (1999)
\Lambda This paper is a slightly modified form of the first chapter in the PhD thesis of Ross Lippert [11] This paper is an extension of an article of the same name by Lippert and Edelman included in...
The Future Fast Fourier Transform? (1999)
Alan Edelman, Peter Mccorquodale, Sivan Toledo
.<F3.862e+05> It seems likely that improvements in arithmetic speed will continue to outpace advances in communication bandwidth. Furthermore, as more and more problems are working on huge...
Interactive Supercomputing with MIT Matlab (1998)
Husbands, Parry, Edelman, Alan
This paper describes MITMatlab, a system that enables users of supercomputers or networked PCs to work on large data sets within Matlab transparently. MITMatlab is based on the Parallel Problems...
Interactive Supercomputing with MIT Matlab (1998)
Husbands, Parry, Edelman, Alan
This paper describes MITMatlab, a system that enables users of supercomputers or networked PCs to work on large data sets within Matlab transparently. MITMatlab is based on the Parallel Problems...
The Geometry of Algorithms with Orthogonality Constraints (1998)
Edelman, Alan, Arias, T. A., Smith, Steven T.
In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric...
Multiscale Computation with Interpolating Wavelets (1998)
Lippert, Ross A., Arias, T. A., Edelman, Alan
Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact...
The geometry of algorithms with orthogonality constraints (1998)
Alan Edelman, Tom Ás, A. Arias, T. Smith
Abstract. In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the...
The Geometry Of Algorithms With Orthogonality Constraints (1998)
Alan Edelman, Tomás A. Arias, Steven T. Smith, Tom As, A. Arias, ...
. In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric...
The Computation and Sensitivity of Double Eigenvalues (1998)
In this paper, we address a problem left open by Wilkinson concerning the computation of the shortest (least squares) distance of any matrix A to a matrix A with at least one repeated eigenvalue. We...
Interactive Supercomputing with MITMatlab (1998)
Parry Husbands, Charles L. Isbell, Alan Edelman
This paper describes MITMatlab, a system that enables users of supercomputers to transparently work on large data sets within Matlab. MITMatlab communicates with an external server that is...
A Counterexample to a Hadamard Matrix Pivot Conjecture (1998)
In the study of the growth factor of completely pivoted Hadamard matrices, it becomes natural to study the possible pivots. Very little is known or provable about these pivots other than a few cases...
The mathematics of the Pentium division bug (1997)
Abstract. Despite all of the publicity surrounding the Pentium bug of 1994, the mathematical details of the bug are poorly understood. We discuss these details and supply a new proof of the...
The Mathematics of the Pentium Division Bug (1997)
Despite all of the publicity surrounding the Pentium bug of 1994, the mathematical details of the bug are poorly understood. We discuss these details, supply a new proof of the Coe--Tang result that...
Alan Edelman, Erik Elmroth, Bo Kågström
Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the...
Non-generic Eigenvalue Perturbations of Jordan Blocks (1997)
We show that if an n \Theta n Jordan block is perturbed by an O(ffl) upper k-Hessenberg matrix (k subdiagonals including the main diagonal), then generically the eigenvalues split into p rings of...
The Future Fast Fourier Transform? (1997)
Alan Edelman, Peter Mccorquodale, Sivan Toledo
. It seems likely that improvements in arithmetic speed will continue to outpace advances in communications bandwidth. Furthermore, as more and more problems are working on huge datasets, it is...
Alan Edelman, Erik Elmroth, Pii S
Abstract. We derive versal deformations of the Kronecker canonical form by deriving the tangent space and orthogonal bases for the normal space to the orbits of strictly equivalent matrix pencils....
Alan Edelman, Erik Elmroth, Pii S
Abstract. Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem. We propose that knowledge of the closure relations, i.e., the stratification,...
Alan Edelman, Erik Elmroth, Bo Kågström
. We derive versal deformations of the Kronecker canonical form by deriving the tangent space and orthogonal bases for the normal space to the orbits of strictly equivalent matrix pencils. These...
How Many Zeros of a Random Polynomial are Real? (1995)
. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coe#cients. We...
Polynomial Roots from Companion Matrix Eigenvalues (1995)
In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the...
Alan Edelman, Erik Elmroth, Bo Kågström
We derive versal deformations of the Kronecker canonical form by deriving the tangent space and orthogonal bases for the normal space to the orbits of strictly equivalent matrix pencils. These...
How many zeros of a random polynomial are real? (1994)
We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We...
Index Transformation Algorithms in a Linear Algebra Framework (1994)
Alan Edelman, Alan Edelman, Steve Heller, Steve Heller, S. Lennart Johnsson, S. Lennart Johnsson
We present a linear algebraic formulation for a class of index transformations such as Gray code encoding and decoding, matrix transpose, bit reversal, vector reversal, shuffles, and other index or...
Large Numerical Linear Algebra in 1994: The Continuing Influence of Parallel Computing (1994)
This note covers two aspects of the state of the art of large numerical linear algebra problems. Firstly, we look at the current records for sparse and dense linear systems and eigenvalue problems on...
Large Dense Numerical Linear Algebra in 1993: The Parallel Computing Influence (1994)
This paper surveys the current state of applications of large dense numerical linear algebra, and the influence of parallel computing. Furthermore, we attempt to crystalize many important ideas that...
Alan Edelman, Tomás A. Arias, Tom'as A. Arias, Steven T. Smith
We illustrate the importance of using curvature information when performing constrained conjugate gradient minimization on manifolds and identify certain common and useful submanifolds in numerical...
Large Dense Numerical Linear Algebra in 1993: The Parallel Computing Influence (1994)
This article surveys the current state of applications of large dense numerical linear algebra and the influence of parallel computing. Furthermore, it attempts to crystallize many important ideas...
Index Transformation Algorithms in a Linear Algebra Framework (1994)
Alan Edelman, Steve Heller, S. Lennart Johnsson
We present a linear algebraic formulation for a class of index transformations such as Gray code encoding and decoding, matrix transpose, bit reversal, vector reversal, shuffles, and other index or...
How Many Zeros of a Random Polynomial are Real? (1994)
We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We...
The road from Kac's matrix to Kac's random polynomials (1994)
This paper tells the story of a matrix and a problem both of which are associated with the name Mark Kac, though most likely he never made the connection. This matrix appears as the Clement matrix in...
During The, Alan Edelman, Gamma Ffl
ss, two students Ioanid Rosu and Dimitriy Betaneli succeeded. This note is a simplification of my original solution from 1991 and their solutions. Department of Mathematics Room 2-380, Massachusetts...
The Dimension of Matrices (Matrix Pencils) with Given Jordan (Kronecker) Canonical Forms (1993)
The set of n by n matrices with a given Jordan canonical form defines a subset of matrices in complex n 2 dimensional space. We analyze one classical approach and one new approach to count the...
The Circular Law and the Probability that a Random Matrix Has k Real Eigenvalues (1993)
Let A be an n by n matrix whose elements are independent random variables with standard normal distributions. Girko's controversial circular law states that the distribution of appropriately...
How Many Eigenvalues of a Random Matrix are Real? (1993)
Alan Edelman, Eric Kostlan, Michael Shub
Let A be an n \Theta n matrix whose elements are independent random variables with standard normal distributions. As n ! 1, the expected number of real eigenvalues is asymptotic to p 2n=ß. We obtain...
Scaling for Orthogonality (1992)
In updating algorthms where orthogonal transformations are accumulated, it is important to preserve the orthogonality of the product in the presence of rounding error. Moonen, Van Dooren, and...
Scaling for Orthogonality (1992)
In updating algorthms where orthogonal transformations are accumulated, it is important to preserve the orthogonality of the product in the presence of rounding error. Moonen, Van Dooren, and...
Scaling for Orthogonality (1992)
Alan Edelman, Alan Edelman, G. W. Stewart, G. W. Stewart
In updating algorthms where orthogonal transformations are accumulated, it is important to preserve the orthogonality of the product in the presence of rounding error. Moonen, Van Dooren, and...
The Complete Pivoting Conjecture for Gaussian Elimination is False (1992)
A famous conjecture concerning Gaussian Elimination was recently "settled" as false, by a counterexample found on a Cray supercomputer. Mathematica did not yield the same conclusion when...
Scaling for Orthogonality (1992)
Alan Edelman, Alan Edelman, G. W. Stewart, G. W. Stewart
In updating algorthms where orthogonal transformations are accumulated, it is important to preserve the orthogonality of the product in the presence of rounding error. Moonen, Van Dooren, and...
On the Distribution of a Scaled Condition Number (1992)
In this note, we give the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is difficult. Specifically, consider a random matrix A and the...
Matrix multiplication on hypercubes using full bandwidth and constant storage (1991)
S. Lennart Johnsson, Alan Edelman, Ching-tien Ho, Ching-tien Ho, S. Lennart, Johnsson Alan Edelman
For matrix multiplication on hypercube multiprocessors with the product matrix accumulated in place a processor must receive about P 2
In a hypercube multiprocessor with distributed memory, messages have a street address and an apartment number, i.e., a hypercube node address and a local memory address. Here we describe an optimal...
The First Annual Large Dense Linear System Survey (1991)
In the March 24, 1991 issue of NA Digest, I submitted a questionnaire asking who was solving large dense linear systems of equations. Based on the responses, nearly all large dense linear systems...
Eigenvalues and condition numbers of random matrices (1989)
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989.
Eigenvalues and condition numbers of random matrices /--by Alan Edelman. (1989)
Supervised by Lloyd N. Trefethen.
Eigenvalues and condition numbers of random matrices (1989)
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989.
Eigenvalues and condition numbers of random matrices / (1989)
Thesis (Ph. D.)--Massachusetts Institute of Technology, 1989.
LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized...