scholarly journals A note on the Ritz method with an application to overtone stellar pulsation theory

1968 ◽  
Vol 8 (2) ◽  
pp. 275-286 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Numerical Solution ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Linear Operators ◽  
Matrix Equations ◽  
Infinite Dimensional ◽  
Stellar Pulsation ◽  
Finite Matrix ◽  
The Matrix ◽  
Matrix Eigenvalue Problems

The Ritz method reduces eigenvalue problems involving linear operators on infinite dimensional spaces to finite matrix eigenvalue problems. This paper shows that for a certain class of linear operators it is possible to choose the coordinate functions so that numerical solution of the matrix equations is considerably simplified, especially when the matrices are large. The method is applied to the problem of overtone pulsations of stars.

scholarly journals LARGE N MATRIX MECHANICS ON THE LIGHT-CONE

2002 ◽  
Vol 17 (11) ◽  
pp. 1517-1542 ◽  
Author(s):  
M. B. HALPERN ◽  
CHARLES B. THORN
Keyword(s):  
Light Cone ◽  
Eigenvalue Problems ◽  
Hamiltonian Formulation ◽  
Cuntz Algebra ◽  
Conserved Quantities ◽  
Basic Tool ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Matrix Mechanics ◽  
Large N

We report a simplification in the large N matrix mechanics of light-cone matrix field theories. The absence of pure creation or pure annihilation terms in the Hamiltonian formulation of these theories allows us to find their reduced large N Hamiltonians as explicit functions of the generators of the Cuntz algebra. This opens up a free-algebraic playground of new reduced models — all of which exhibit new hidden conserved quantities at large N and all of whose eigenvalue problems are surprisingly simple. The basic tool we develop for the study of these models is the infinite dimensional algebra of all normal-ordered products of Cuntz operators, and this algebra also leads us to a special number-conserving subset of these models, each of which exhibits an infinite number of new hidden conserved quantities at large N.

scholarly journals A Subspace Iteration Algorithm for Fredholm Valued Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Christian Engström ◽  
Luka Grubišić
Keyword(s):  
Numerical Experiments ◽  
Eigenvalue Problems ◽  
Spectral Component ◽  
Contour Integration ◽  
Iteration Algorithm ◽  
Nonlinear Eigenvalue Problems ◽  
Infinite Dimensional ◽  
Subspace Iteration ◽  
Fredholm Theorem ◽  
Nonlinear Eigenvalue

We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algorithm for polynomial and rational eigenvalue problems.

REGULARIZED CALCULUS: AN APPLICATION OF ZETA REGULARIZATION TO INFINITE DIMENSIONAL GEOMETRY AND ANALYSIS

2004 ◽  
Vol 01 (01n02) ◽  
pp. 107-157 ◽  
Author(s):  
ASADA AKIRA
Keyword(s):  
Hilbert Space ◽  
Zeta Function ◽  
Path Integral ◽  
Eigenvalue Problems ◽  
Volume Form ◽  
Regularization Procedure ◽  
Infinite Dimensional ◽  
Method Of Regularization ◽  
Spectral Zeta Function ◽  
Mathematical Justification

A method of regularization in infinite dimensional calculus, based on spectral zeta function and zeta regularization is proposed. As applications, a mathematical justification of appearance of Ray–Singer determinant in Gaussian Path integral, regularized volume form of the sphere of a Hilbert space with the determinant bundle, eigenvalue problems of regularized Laplacian, are investigated. Geometric counterparts of regularization procedure are also discussed applying arguments from noncommutative geometry.

Sign in / Sign up

Export Citation Format

Share Document