Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1970"
TL;DR: In this article, the complete eigenvalue problem of a degenerate (that is defective and/or derogatory) matrix is studied theoretically and numerically, using successive QR-factorizations to determine annihilated subspaces.
Abstract: An algorithm, proposed by V. N. Kublanovskaya, for solving the complete eigenvalue problem of a degenerate (that is defective and/or derogatory) matrix, is studied theoretically and numerically. It uses successiveQR-factorizations to determine annihilated subspaces.
77 citations
TL;DR: This paper describes a method for solving ordinary differential eigenvalue problems of the form N(u) + λM( u) = 0, where N and M are linear differential operators and u(x) is a scaler variable.
71 citations
TL;DR: In this article, the authors derived bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix.
Abstract: When a matrix is close to a matrix with a multiple eigenvalue, the arithmetic mean of a group of eigenvalues is a good approximation to this multiple eigenvalue. A theorem of Gershgorin type for means of eigenvalues is proved and applied as a perturbation theorem for a degenerate matrix. For a multiple eigenvalue we derive bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix. These bounds assure that, provided the dimensionalities are chosen appropriately, the angles of rotation of the subspaces are of the same order of magnitude as the perturbation of the matrix. A numerical example is given.
45 citations
TL;DR: In this paper, it was shown that there exists a related hermitian matrix H, satisfying the standard eigenvalue problem Hs = hs, whose solution directly yields the solution of the original problem.
22 citations
21 citations
12 citations
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8 citations
TL;DR: In this paper, the application of the escalator method and diakoptics to large eigenvalue problems is described in detail and the particular advantages of the method and its usefulness for a wide variety of general engineering problems are discussed.
Abstract: The eigenvalue method is well-known as an aid to the solution of a wide range of engineering problems. In power system analysis it has relevance to travelling wave phenomena,1 sensitivity and dynamic stability in synchronous multimachine2 systems, critical speed calculation, vibration of structures and other related topics. The method of numerical solution of eigenvalues depends on the size and nature of the problem and often where large matrices are involved presents great difficulties.
This paper describes in detail the application of the escalator method and diakoptics to large eigenvalue problems. The escalator method is a well-established one which consists of a systematic way of escalating from a 2 × 2 matrix up to any desired order in steps of one row and column at a time. The diakoptical method reduces computer storage, calculation time and improves accuracy. This results in the more economical solution of large and complicated problems which cannot be conveniently solved by any orthodox method. The paper deals with problems of degeneracy and means of overcoming these by simple routine processes. It concludes by indicating the particular advantages of the method and its usefulness for a wide variety of general engineering problems.
7 citations
TL;DR: In this paper, a method for computer programming of the solution of the eigenvalue equation GFL = LΛ is presented, which is a transformation by a triangular matrix V which simultaneously symmetrizes the product GF (VV † = G ) and eliminates the redundancies.
6 citations
TL;DR: In this paper, the problem of maximizing a positive quadratic functional of the states of a linear system, subject to a bound on the integral of the square of the control, is considered and the solution is characterized in terms of the maximum eigenvalue and associated eigenfunction of a non-negative, definite, self-adjoint, integral kernel.
Abstract: The problem is considered of maximizing a positive quadratic functional of the states of a linear system, subject to a bound on the integral of the square of the control. The solution is characterized in terms of the maximum eigenvalue and associated eigenfunction of a non-negative, definite, self-adjoint, integral kernel, and computational techniques for solving the associated eigenvalue problem are discussed.
17 Aug 1970