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# Divide-and-conquer eigenvalue algorithm

About: Divide-and-conquer eigenvalue algorithm is a(n) research topic. Over the lifetime, 2877 publication(s) have been published within this topic receiving 81838 citation(s).

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01 Jan 1965

Abstract: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography Index.

7,420 citations

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Abstract: The present investigation designs a systematic method for finding the latent roots and the principal axes of a matrix, without reducing the order of the matrix. It is characterized by a wide field of applicability and great accuracy, since the accumulation of rounding errors is avoided, through the process of \"minimized iterations\". Moreover, the method leads to a well convergent successive approximation procedure by which the solution of integral equations of the Fredholm type and the solution of the eigenvalue problem of linear differential and integral operators may be accomplished.

3,672 citations

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TL;DR: The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms and developers of new algorithms and perturbation theories will benefit from the theory.

Abstract: In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

2,531 citations

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Abstract: An interpretation of Dr. Cornelius Lanczos' iteration method, which he has named \"minimized iterations\", is discussed in this article, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form. This interpretation leads to a variation of the Lanczos procedure which may frequently be advantageous by virtue of reducing the volume of numerical work in practical applications. Both methods employ essentially the same algorithm, requiring the generation of a series of orthogonal functions through which a simple matrix equation of reduced order is established. The reduced matrix equation may be solved directly in terms of certain polynomial functions obtained in conjunction with the generated orthogonal functions, and the convergence of the solution may be observed as the order of the reduced matrix is successively increased with the order of the original matrix as a limit. The method of minimized iterations is recommended as a rapid means for determining a small number of the larger eigenvalues and modal columns of a large matrix and as a desirable alternative for various series expansions of the Fredholm problem. 1. The conventional iterative procedures. It is frequently required that real latent roots, or eigenvalues, and modal columns be determined for a real numerical matrix, u, of order, n, in the characteristic homogeneous equation,*

1,698 citations

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Abstract: In this paper we investigate the structure of the solution set for a large class of nonlinear eigenvalue problems in a Banach space. Our main results demonstrate the existence of continua, i.e., closed connected sets, of solutions of these equations. Although the emphasis is on the case when bifurcation occurs, the nonbifurcation situation is also treated. Applications are given to ordinary and partial differential equations and to integral equations.

1,662 citations