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96
Linear Programming, Complexity Theory and Elementary Functional Analysis
 Mathematical Programming
, 1995
"... This paper was conceived in part while the author was sponsored by the visiting scientist program at the IBM T.J. Watson Research Center. Special thanks to Mike Shub, Roy Adler and Shmuel Winograd for their generosity. 1 Introduction ..."
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Cited by 86 (1 self)
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This paper was conceived in part while the author was sponsored by the visiting scientist program at the IBM T.J. Watson Research Center. Special thanks to Mike Shub, Roy Adler and Shmuel Winograd for their generosity. 1 Introduction
Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices
, 1980
"... When computing eigenvalues of sym metric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound proportional to the product of machine precision and the norm of the matrix. In particular, we do not expect to comp ..."
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Cited by 79 (14 self)
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When computing eigenvalues of sym metric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound proportional to the product of machine precision and the norm of the matrix. In particular, we do not expect to compute tiny eigenvalues and singular values to high relative accuracy. There are some important classes of matrices where we can do much better, including bidiagonal matrices, scaled diagonally dominant matrices, and scaled diagonally dominant definite pencils. These classes include many graded matrices, and all sym metric positive definite matrices which can be consistently ordered (and thus all symmetric positive definite tridiagonal matrices). In particular, the singular values and eigenvalues are determined to high relative precision independent of their magnitudes, and there are algorithms to compute them this accurately. The eigenvectors are also determined more accurately than for general matrices, and may be computed more accurately as well. This work extends results of Kahan and Demmel for bidiagonal and tridiagonal matrices.
TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 73 (10 self)
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There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rankr approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Brègman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank3 tensor has an optimal rank2 approximation. The notable exceptions to this misbehavior are rank1 tensors and order2 tensors (i.e. matrices). In a more positive spirit, we propose a natural way of overcoming the illposedness of the lowrank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete lowdimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2 × 2 × 2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular we make extensive use of the hyperdeterminant ∆ on R 2×2×2.
Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I
, 1993
"... The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools ar ..."
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Cited by 63 (13 self)
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The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools are meant to be used in different combinations on different problems and architectures. In this paper, we will describe these tools which include basic block matrix computations, the matrix sign function, 2dimensional bisection, and spectral divide and conquer using the matrix sign function to find selected eigenvalues. We also outline how we deal with illconditioning and potential instability. Numerical examples are included. A future paper will discuss error analysis in detail and extensions to the generalized eigenproblem.
Complexity of Bézout’s Theorem IV : Probability of Success, Extensions
 SIAM J. Numer. Anal
, 1996
"... � � � We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n +1 complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the ..."
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Cited by 60 (9 self)
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� � � We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n +1 complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed. 1. Introduction. 1A. Bezout’s Theorem Revisited. Let f: � n+1 → � n be a system of homogeneous polynomials f =(f1,...,fn), deg fi = di, i=1,...,n. The linear space of such f is denoted by H (d) where d = (d1,...,dn). Consider the
Numerical Computation of an Analytic Singular Value Decomposition of a Matrix Valued Function
 Numer. Math
, 1991
"... This paper extends the singular value decomposition to a path of matrices E(t). An analytic singular value decomposition of a path of matrices E(t) is an analytic path of factorizations E(t) = X(t)S(t)Y (t) T where X(t) and Y (t) are orthogonal and S(t) is diagonal. To maintain differentiability ..."
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Cited by 45 (6 self)
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This paper extends the singular value decomposition to a path of matrices E(t). An analytic singular value decomposition of a path of matrices E(t) is an analytic path of factorizations E(t) = X(t)S(t)Y (t) T where X(t) and Y (t) are orthogonal and S(t) is diagonal. To maintain differentiability the diagonal entries of S(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic path E(t) always admits a real analytic SVD, a fullrank, smooth path E(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Eulerlike and extrapolated Eulerlike numerical methods for approximating an analytic SVD and prove that the Eulerlike method converges. 1 Introduction A singular value decomposition (SVD) of a constant matrix E 2 R m\Thetan , m n, is a factorization E = U...
Computing An Eigenvector With Inverse Iteration
 SIAM REVIEW
, 1997
"... The purpose of this paper is twofold: to analyse the behaviour of inverse iteration for computing a single eigenvector of a complex, square matrix; and to review Jim Wilkinson's contributions to the development of the method. In the process we derive several new results regarding the converge ..."
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Cited by 41 (1 self)
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The purpose of this paper is twofold: to analyse the behaviour of inverse iteration for computing a single eigenvector of a complex, square matrix; and to review Jim Wilkinson's contributions to the development of the method. In the process we derive several new results regarding the convergence of inverse iteration in exact arithmetic. In the case of normal matrices we show that residual norms decrease strictly monotonically. For eighty percent of the starting vectors a single iteration is enough. In the case of nonnormal matrices, we show that the iterates converge asymptotically to an invariant subspace. However the residual norms may not converge. The growth in residual norms from one iteration to the next can exceed the departure of the matrix from normality. We present an example where the residual growth is exponential in the departure of the matrix from normality. We also explain the often significant regress of the residuals after the first iteration: it occurs when the no...
Matrix nearness problems and applications
 Applications of Matrix Theory
, 1989
"... A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, ..."
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Cited by 33 (6 self)
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A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rankdeficiency and instability. Theoretical results and computational methods are described. Applications of nearness problems in areas including control theory, numerical analysis and statistics are outlined.