Topic
Numerical range
About: Numerical range is a research topic. Over the lifetime, 1245 publications have been published within this topic receiving 20112 citations.
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Papers
Book•
1 Jan 1997
Abstract: Exercise 1 (4 points) (a) Let unitary matrices Q 1 ,. .. , Q k ∈ C m×m be fixed and consider the problem of computing, for A ∈ C m×n , the product B = Q k · · · Q 1 A. Let the computation be carried out on a computer satisfying axioms (A1) and (A2). Show that this algorithm is backward stable. (Here A is thought of as data that can be perturbed; the matrices Q j are fixed and not to be perturbed.) (b) Give an example to show that this result no longer holds if the unitary matrices Q j are replaced with arbitrary matrices X j ∈ C m×m. Recall the axioms from exercise sheet 9, exercise 2: (A1) For all x ∈ R, there exists with || ≤ machine such that fl(x) = x(1 +). (A2) For all x, y ∈ F, there exists with || ≤ machine such that x y = (x * y)(1 +). Here * denotes an arithmetic operation and the floating point equivalent. Exercise 2 (4 points) Consider the example A = 1 1 1 1.0001 1 1.0001 , b = 2 0.0001 4.0001 . Do the computations in part (a) and (b) by hand. Show intermediate results where appropriate. Give exact results. From part (c) on, numerical results computed by MATLAB are acceptable. (a) What are the matrices A + and P = AA + for this example? Hint: Compute A + by solving the linear system (A * A)A + = A *. (b) Find the exact solutions x and y = Ax for the least squares problem Ax ≈ b. (c) What are κ(A), θ, and η? (d) What are the four relative condition numbers? (e) Give examples of perturbations δb and δA that approximately attain these four condition numbers.
3,190 citations
1 Jan 2009
TL;DR: An accelerated proximal gradient algorithm is proposed, which terminates in O(1= p †) iterations with an †-optimal solution, to solve this unconstrained nonsmooth convex optimization problem, and in particular, the nuclear norm regularized linear least squares problem.
Abstract: The a‐ne rank minimization problem, which consists of flnding a matrix of minimum rank subject to linear equality constraints, has been proposed in many areas of engineering and science. A speciflc rank minimization problem is the matrix completion problem, in which we wish to recover a (low-rank) data matrix from incomplete samples of its entries. A recent convex relaxation of the rank minimization problem minimizes the nuclear norm instead of the rank of the matrix. Another possible model for the rank minimization problem is the nuclear norm regularized linear least squares problem. This regularized problem is a special case of an unconstrained nonsmooth convex optimization problem, in which the objective function is the sum of a convex smooth function with Lipschitz continuous gradient and a convex function on a set of matrices. In this paper, we propose an accelerated proximal gradient algorithm, which terminates in O(1= p †) iterations with an †-optimal solution, to solve this unconstrained nonsmooth convex optimization problem, and in particular, the nuclear norm regularized linear least squares problem. We report numerical results for solving large-scale randomly generated matrix completion problems. The numerical results suggest that our algorithm is e‐cient and robust in solving large-scale random matrix completion problems. In particular, we are able to solve random matrix completion problems with matrix dimensions up to 10 5 each in less than 10 minutes on a modest PC.
1,144 citations
Book•
7 Nov 1996
TL;DR: In this article, the authors present mapping theorems for W(A) computations, including the following: 1.1 Numerical range, 2.1 radius mapping, 3.2 analytical functions, 4.3 rational functions, 5.4 Hadamard product, 6.5 generalized range, 7.6 beyond spectraloid, and 8.6 pseudo eigenvalues.
Abstract: 1 Numerical Range.- 1.1 Elliptic Range.- 1.2 Spectral Inclusion.- 1.3 Numerical Radius.- 1.4 Normal Operators.- 1.5 Numerical Boundary.- 1.6 Other W-Ranges.- Endnotes for Chapter 1.- 2 Mapping Theorems.- 2.1 Radius Mapping.- 2.2 Analytic Functions.- 2.3 Rational Functions.- 2.4 Operator Products.- 2.5 Commuting Operators.- 2.6 Dilation Theory.- Endnotes for Chapter 2.- 3 Operator Trigonometry.- 3.1 Operator Angles.- 3.2 Minmax Equality.- 3.3 Operator Deviations.- 3.4 Semigroup Generators.- 3.5 Accretive Products.- 3.6 Antieigenvalue Theory.- Endnotes for Chapter 3.- 4 Numerical Analysis.- 4.1 Optimization Algorithms.- 4.2 Conjugate Gradient.- 4.3 Discrete Stability.- 4.4 Fluid Dynamics.- 4.5 Lax-Wendroff Scheme.- 4.6 Pseudo Eigenvalues.- Endnotes for Chapter 4.- 5 Finite Dimensions.- 5.1 Value Field.- 5.2 Gersgorin Sets.- 5.3 Radius Estimates.- 5.4 Hadamard Product.- 5.5 Generalized Ranges.- 5.6 W(A) Computation.- Endnotes for Chapter 5.- 6 Operator Classes.- 6.1 Resolvent Growth.- 6.2 Three Classes.- 6.3 Spectral Sets.- 6.4 Normality Conditions.- 6.5 Finite Inclusions.- 6.6 Beyond Spectraloid.- Endnotes for Chapter 6.
625 citations
Book•
13 May 2021566 citations
TL;DR: The spectrum of the Orr–Sommerfeld operator consists of three branches and it is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the Reynolds number.
Abstract: This paper investigates the pseudospectra and the numerical range of the Orr–Sommerfeld operator for plane Poiseuille flow. A number $z \in {\bf C}$ is in the $\epsilon $-pseudospectrum of a matrix or operator A if $\| ( zI - A )^{ - 1} \| \geq \epsilon ^{ - 1} $, or, equivalently, if z is in the spectrum of $A + E$ for some perturbation E satisfying $\| E \| \leq \epsilon $. The numerical range of A is the set of numbers of the form $( Au,u )$, where $( \cdot , \cdot )$ is the inner product and u is a vector or function with huh $\| u \| = 1$.The spectrum of the Orr–Sommerfeld operator consists of three branches. It is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the Reynolds number. The associated eigenfunctions are nearly linearly dependent, even though they form a complete set.To understand the high sensitivity of the eigenvalues, a model operator is considered, related to the Airy equation tha...
482 citations
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| 2025 | 4 |
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| 2022 | 53 |
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| 2020 | 66 |