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Browsing by Author "Trefethen, Lloyd N."
Now showing items 119 of 19

Calculation of Pseudospectra by the Arnoldi Iteration
Toh, KimChuan; Trefethen, Lloyd N. (Cornell University, 199405)The Arnoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, for in applications involving highly nonnormal ... 
The Chebyshev Polynomials of a Matrix
Toh, KimChuan; Trefethen, Lloyd N. (Cornell University, 199605)A Chebyshev polynomial of a square matrix A is a monic polynomial of specified degree that minimizes p(A)(sub2). The study of such polynomials is motivated by the analysis of Krylov subspace iterations in numerical ... 
The Definition of Numerical Analysis
Trefethen, Lloyd N. (Cornell University, 199209)This is a brief essay discussing perceptions and misperceptions of numerical analysis. A common misperception is that numerical analysis is the study of rounding errors. Even numerical analysts have been misled by this view. 
Do the Pseudospectra of a Matrix Determine its Behavior?
Greenbaum, Anne; Trefethen, Lloyd N. (Cornell University, 199308)Let $A$ and $B$ be square matrices. It is shown that the condition $(R) (zIA)^{1} = (zI B)^{1}$ for all $z \in \complex$ is equivalent to the condition $(P) p(A) = p(B)$ for all polynomials $p$ if ... 
From the Buffon Needle Problem to the Kreiss Matrix Theorem
Wegert, Elias; Trefethen, Lloyd N. (Cornell University, 199203)In this paper we present a theorem concerning the arc length on the Riemann sphere of the image of the unit circle under a rational function. But our larger purpose is to tell a story. We thought at first that the story ... 
GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems
Greenbaum, Anne; Trefethen, Lloyd N. (Cornell University, 199206)The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize p(A)b over polynomials p of degree n. The difference is that p is nor malized at z=0 for GMRES ... 
Lowdimensional models of subcritical transition to turbulence
Baggett, Jeffrey S.; Trefethen, Lloyd N. (Cornell University, 199602)In the past five years, working largely independently, five groups of researchers have proposed lowdimensional models of the behavior of parallel shear flows at high Reynolds numbers. These models are compared, and it ... 
Matrix Iterations: The Six Gaps Between Potential Theory and Convergence
Driscoll, Tobin A.; Toh, KimChuan; Trefethen, Lloyd N. (Cornell University, 199606)The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, BiCGSTAB, ...) is reviewed. For a computation of this kind, an estimated ... 
MultiMATLAB: MATLAB on Multiple Processors
Trefethen, Anne E.; Menon, Vijay S.; Chang, ChiChao; Czajkowski, Grzegorz; Myers, Chris; Trefethen, Lloyd N. (Cornell University, 199605)MATLAB, a commercial product of The MathWorks, Inc., has become one of the principal languages of desktop scientific computing. A system is described that enables one to run MATLAB conveniently on multiple processors. ... 
MultiMATLAB: MATLAB on Multiple Processors
Trefethen, Anne E.; Menon, Vijay S.; Chang, ChiChao; Czajkowski, Grezgorz J.; Myers, Chris; Trefethen, Lloyd N. (Cornell University, 199605)MATLAB(R), a commercial product of The MathWorks, Inc., has become one of the principal languages of desktop scientific computing. A system is described that enables one to run MATLAB conveniently on multiple processors. ... 
A New Direction in Hydrodynamic Stability: Beyond Eigenvalues
Trefethen, Lloyd N.; Trefethen, Anne E.; Reddy, Satish C.; Driscoll, Tobin A. (Cornell University, 199212)Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and looking for unstable ... 
Pseudospectra of Linear Operators
Trefethen, Lloyd N. (Cornell University, 199512)The following contains mathematical formulae and symbols that may become distorted in ASCII text format. The advent of ever more powerful computers has brought with it a new way of conceiving some of the fundamental ... 
Pseudospectra of linear operators
Trefethen, Lloyd N. (Cornell University, 199511)The advent of ever more powerful computers has brought with it a new way of conceiving some of the fundamental eigenvalue problems of applied mathematics. If a matrix or linear operator $A$ is far from normal, its eigenvalues ... 
Pseudospectra of the ConvectionDiffusion Operator
Reddy, Satish C.; Trefethen, Lloyd N.; Pathria, Dimpy (Cornell University, 199304)The spectrum of the simplest 1D convectiondiffusion operator is a discrete subset of the negative real axis, but the pseudospectra are regions in the complex plane that approximate parabolas. Put another way, the norm ... 
Pseudospectra of the Linear NavierStokes Evolution Operator and Instability of Plane Poiseuille and Couette Flows: (preliminary report)
Trefethen, Lloyd N.; Trefethen, Anne E.; Reddy, Satish C. (Cornell University, 199206)This is a rough, interim report on some new results concerning the stability of plane Poiseuille and Couette fluid flows, following upon recent work by Henningson and Reddy, Butler and Farrell, Gustavsson and others. We ... 
Pseudospectra of the Wave Operator with an Absorbing Boundary
Driscoll, Tobin A.; Trefethen, Lloyd N. (Cornell University, 199310)For systems which can be described by u(sub t) = Au with a highly nonnormal matrix or operator A, the spectrum of A may describe the behavior of the system poorly. One such operator arises from the onedimensional wave ... 
Pseudozeros of Polynomials and Pseudospectra of Companion Matrices
Toh, KimChuan; Trefethen, Lloyd N. (Cornell University, 199306)It is well known that the zeros of a polynomial $p$ are equal to the eigenvalues of the associated companion matrix $A$. In this paper, we take a geometric view of the conditioning of these two problems and of the stability ... 
SchwarzChristoffel Mapping in the 1980's
Trefethen, Lloyd N. (Cornell University, 199309)An informal survey is presented of the numerical computation of SchwarzChristoffel maps (i.e., conformal maps from a disk in the complex plane to a polygon) and their applications in science and engineering. It is shown ... 
Stiffness of ODEs
Higham, Desmond J.; Trefethen, Lloyd N. (Cornell University, 199201)It is argued that even for a linear system of ODEs with constant coefficients, stiffness cannot properly be characterized in terms of the eigenvalues of the Jacobian, because stiffness is a transient phenomenon ...