Approximate Calculation of Integrals

Крылов, В. И.; Stroud, A. H.

doi:10.2307/2003796

Cited by 255 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is desirable that weights be positive (w 0) and points lie in the domain of integration (x is an j j element of the interval [a,b]) (KRYLOV, 1962;HABER, 1970). Krylov shows that the sum of the absolute value of the weights positively impacts approximation error.…”

Section: Numerical Integration and Gaussian Quadrature 231 The Univar...mentioning

confidence: 99%

An Introduction to Systematic Sensitivity Analysis via Gaussian Quadrature

Arndt¹

2000

GTAP Technical Paper Series

View full text Add to dashboard Cite

Economists recognize that results from simulation models are dependent, sometimes highly dependent, on values employed for critical exogenous variables. To account for this, analysts sometimes conduct sensitivity analysis with respect to key exogenous variables. This paper presents a practical approach for conducting systematic sensitivity analysis, called Gaussian quadrature. The approach views key exogenous variables as random variables with associated distributions. It produces estimates of means and standard deviations of model results while requiring a limited number of solves of the model. Under mild conditions, all of which hold with respect to the GTAP model, there is strong reason to believe that the estimates of means and standard deviations will be quite accurate.

show abstract

Section: Numerical Integration and Gaussian Quadrature 231 The Univar...mentioning

confidence: 99%

An Introduction to Systematic Sensitivity Analysis via Gaussian Quadrature

Arndt¹

2000

GTAP Technical Paper Series

View full text Add to dashboard Cite

show abstract

“…The integrals (9) may be evaluated very efficiently by a Gauss-type quadrature if the basis functions ui(r) are constructed to obey the 'orthogonality' conditions (Manolopoulos and Wyatt 1988) From this condition and from (6) it is clear that two of the quadrature nodes {rk} must be preassigned to the endpoints of the interval, ro = S and r M + l = 0. This leads to the choice of a Gauss-Lobatto quadrature (Krylov 1962). The basis functions q ( r ) turn out to be the Lagrange interpolation polynomials (of order M + 1) corresponding to the set of nodes {rk} (Manolopoulos and Wyatt 1988).…”

Section: (9)mentioning

confidence: 99%

“…A remarkable property of the Lobatto basis is its ability to exactly represent the kinetic energy, once t8he nodes { r k } a.nd t8heir corresponding weights { w k } have been obtained (Krylov 1962). These matrices, being independent of the electron energy E have t o be computed only once whereas the inverse of the matrix…”

Section: (9)mentioning

confidence: 99%

Atomic photo cross sections based on the logarithmic derivative variational principle: application of Lobatto shape functions

Ackermann

Görling

Rösch

1990

J. Phys. B: At. Mol. Opt. Phys.

View full text Add to dashboard Cite

A b s t r a c t . Lobatto shape functions, recently proposed as a translational basis for atomic scattering investigations based on the logarithmic derivative variational method, are shown to provide a convenient and accurate description of electronic continuum functions. To evaluate the method, applications to photoionisation of various rare gas atoms are presented. An effective local one-electron potential is used which is derived from density-functional theory. The present results allow some critical remarks on the accuracy of this type of potential which is frequently used in photoionisation calculations of larger molecules.

show abstract

Spectra, pseudospectra, and localization for random bidiagonal matrices

Trefethen

Contedini

Embree

2001

Comm. Pure Appl. Math.

View full text Add to dashboard Cite

There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random, non-Hermitian, periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a "bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the nonperiodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the infinite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of finite bidiagonal matrices, infinite bidiagonal matrices ("stochastic Toeplitz operators"), finite periodic matrices, and doubly infinite bidiagonal matrices ("stochastic Laurent operators").

show abstract

Approximate Calculation of Integrals

Cited by 255 publications

References 0 publications

An Introduction to Systematic Sensitivity Analysis via Gaussian Quadrature

An Introduction to Systematic Sensitivity Analysis via Gaussian Quadrature

Atomic photo cross sections based on the logarithmic derivative variational principle: application of Lobatto shape functions

Spectra, pseudospectra, and localization for random bidiagonal matrices

Contact Info

Product

Resources

About