Browsing by Author "Brebner, M.A."
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Item Metadata only EIGENVALUES OF $Ax~=~lambda$$Bx$ FOR REAL SYMMETRIC MATRICES $A$ AND $B$ COMPUTED BY THE HR PROCESS(1976-11-25) Brebner, M.A.; Grad, JThis paper presents a method for solving the eigenvalue problem $Ax~=~lambda$$Bx$, where $A$ and $B$ are real symmetric but not necessarily positive definite matrices, and $B$ is nonsingular. The method reduces the general case into a form $Cz~=~lambda$$z$ where $C$ is a pseudo-symmetric matrix. A further reduction of $C$ produces a tridiagonal pseudo-symmetric form to which the interactive HR process is applied. The tridiagonal pseudo-symmetric form is invariant under the HR transformations. The amount of computation is significantly less than treating the problem by a general method.Item Metadata only A REDUCTION OF AN INDEFINITE SYMMETRIC MATRIX A TO THE FORM $LJL sup T$ BYROTATION AND DECOMPOSITIONS(1977-01-01) Brebner, M.A.; Grad, M.J.The paper discusses the reduction of a non-singular symmetric matrix $A$ by decomposition and similarity rotations to the form $LJL sup T$ where $L$ is a lower triangular matrix and $J$ is a diagonal matrix with diagonal elements plus or minus unity. In effect $PAP sup T~=~LJL sup T$, where $P$ is the product of plane rotations.Item Metadata only SIMILARITY TRANSFORMATIONS FOR PSEUDO- SYMMETRIC MATRICES WITH PARTICULARREFERENCE TO THE HR METHOD(1977-01-01) Brebner, M.A.; Grad, J.The paper defines the term pseudo-symmetric matrix and discusses similarity transformations which preserve pseudo-symmetry. In particular the use of these similarity transformations in the $HR$ analogues of the single and double step $QR$ methods is described.