On a direct‐iterative eigensolution technique

Dong, S. B.; Wolf, Joseph A.; Peterson, Frederick E.

doi:10.1002/nme.1620040202

Cited by 54 publications

(4 citation statements)

References 10 publications

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“…The subscript j is introduced in anticipation of an iterative process, so that j = 1 holds presently for equation (6). The transformation T, must contain the essential capability of giving a relatively complete description in a mathematical sense of the subspace eigenbasis that is sought.…”

Section: Block-stodola Technique For Au = Abu With a And B Symmetricmentioning

confidence: 99%

A Block–Stodola eigensolution technique for large algebraic systems with non‐symmetrical matrices

Dong

1977

Numerical Meth Engineering

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SUMMARYA Block-Stodola eigensolution method is presented for large algebraic eigensystems of the form AU = ABU where A is real but non-symmetric. The steps in this method parallel those of a previous technique for the case when both A and B were real and symmetric. The essence of the technique is simultaneous iteration using a group of trial vectors instead of only one vwor as is the case in the classical Stodola-Vianello iteration method. The problem is then transformed into a subspace where a direct solution of the reduced algebraic eigenvalue problem is sought. The main advantage is the significant reduction of computational effort in extracting a subset of eigenvalues and corresponding eigenvectors. Theorems from linear algebra serve to underlie the basis of the present technique. Complex eigendata that emerge during iteration can be handled without doubling the size of the problem. Higher order eigenvalue problems are reducible to first order form for which this technique is applicable. The treatment of the quadratic eigenvalue problem illustrates the details of this extension.

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Section: Block-stodola Technique For Au = Abu With a And B Symmetricmentioning

confidence: 99%

A Block–Stodola eigensolution technique for large algebraic systems with non‐symmetrical matrices

Dong

1977

Numerical Meth Engineering

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“…(6) is equivalent to the solution of the standard eigenvalue problem Assume that M is positive definite, then if M = SST for any non-singular matrix S, the problem in equation g+ = u2+ (7) i( = S-lKS-T, $ = ST+ (8) where It is computationally efficient to use as S the Cholesky factor EM of My i.e. M = Z , E ' & .…”

Section: Transformation O F Generalized Eigenvalue Problem To Standarmentioning

confidence: 99%

Solution methods for eigenvalue problems in structural mechanics

Bathe

Wilson

1973

Numerical Meth Engineering

177

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SUMMARYA survey of probably the most efficient solution methods currently in use for the problems K+ = w2M+ and K+ = XK,\lr is presented. In the eigenvalue problems the stiffness matrices K and KG and the mass matrix M can be full or banded; the mass matrix can be diagonal with zero diagonal elements. The choice is between the wellknown QR method, a generalized Jacobi iteration, a new determinant search technique and an automated subspace iteration. The system size, the bandwidth and the number of required eigenvalues and eigenvectors determine which method should be used on a particular problem. The numerical advantages of each solution technique, operation counts and storage requirements are given to establish guidelines for the selection of the appropriate algorithm. A large number of typical solution times are presented.

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“…When compounded, these can be represented by an n x m matrix U,, the orthogonality condition being Premultiplication u p 1 = I V, = AU, produces a set of vectors in which the components of all the dominant eigenvectors are magnified. The dominant eigenvalues are predicted from an interaction matrix of order m x m which could be B, = q V l = UTAU, (13) This matrix is also used to realign the trial vectors which, after being re-orthogonalized, can be adopted as the trial vectors for the second round of iteration.…”

Section: Simultaneous Iterationmentioning

confidence: 99%