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TheQ R algorithm for real hessenberg matrices
The QR algorithm of Francis [1] and Kublanovskaya [4] with shifts of origin is described by the relations $$ \matrix{ {{Q_s}({A_s} - {k_s}I) = {R_s},} & {{A_{s + 1}} = {R_s}Q_s^T + {k_s}I,} &Expand
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The QR and QL Algorithms for Symmetric Matrices
The QR algorithm as developed by Francis [2] and Kublanovskaya [4] is conceptually related to the LR algorithm of Rutishauser [7]. It is based on the observation that if $$A = QR{\text{ andExpand
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Reduction of the symmetric eigenproblemAx=λBx and related problems to standard form
In many fields of work the solution of the eigenproblems Ax = λBx and A B x = λ x (or related problems) is required, where A and B are symmetric and B is positive definite. Each of these problems canExpand
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Symmetric decomposition of positive definite band matrices
The method is based on the following theorem. If A is a positive definite matrix of band form such that $${a_{ij}} = 0{\rm{ (|}}i - j| >m{\rm{)}}$$ (1) then there exists a real non-singularExpand
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Iterative Refinement of the Solution of a Positive Definite System of Equations
In an earlier paper in this series [1] the solution of a system of equations Ax=b with a positive definite matrix of coefficients was described; this was based on the Cholesky factorization of A. IfExpand
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The implicit QL algorithm
In [1] an algorithm was described for carrying out the QL algorithm for a real symmetric matrix using shifts of origin. This algorithm is described by the relations $$\matrix{ {{Q_s}({A_s} -Expand
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Householder's tridiagonalization of a symmetric matrix
In an early paper in this series [4] Householder’s algorithm for the tridiagonalization of a real symmetric matrix was discussed. In the light of experience gained since its publication and in viewExpand
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Solution of real and complex systems of linear equations
If A is a non-singular matrix then, in general, it can be factorized in the form A = LU, where L is lower-triangular and U is upper-triangular. The factorization, when it exists, is unique to withinExpand
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