The implicit Cholesky algorithm has been developed by several authors during the last 10 years but under different names. We identify the algorithm with a special version of Rutishauser's lr algorithm. Intermediat...
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The implicit Cholesky algorithm has been developed by several authors during the last 10 years but under different names. We identify the algorithm with a special version of Rutishauser's lr algorithm. Intermediate quantities in the transformation furnish several attractive approximations to the smallest singular value. The paper extols the advantages of using shifts with the algorithm. The nonorthogonal transformations improve accuracy.
A conjugate gradient type method for solving Ax = b or, more precisely, an orthogonal error method OE(B, C, A), is determined by a ''formal inner product'' matrix B and a (left) preconditioning matrix ...
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A conjugate gradient type method for solving Ax = b or, more precisely, an orthogonal error method OE(B, C, A), is determined by a ''formal inner product'' matrix B and a (left) preconditioning matrix C. Relations are pointed out between quantities such as iterates, residuals, direction vectors, and recurrence coefficients of orthogonal error methods with different but, in a particular way, related matrices B, namely, for OE(B(CA)k, C, A), k = 0, 1, ..., and for OE((A(H)C(H))(k)B,C,A), k = 0,1,.... The relations for the first sequence of methods have to do with Rutishauser's lr algorithm;those for the second one are based on a generalization of the Schonauer-Weiss smoothing algorithm. Relevant for practice are the cases k = 0, 1 for B = CA and B = (CA)H.
Hybrid codes that combine elements of the QR and lr algorithms are described. The codes can calculate the eigenvalues and, optionally, eigenvectors of real, nonsymmetric matrices. Extensive tests are presented as evid...
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Hybrid codes that combine elements of the QR and lr algorithms are described. The codes can calculate the eigenvalues and, optionally, eigenvectors of real, nonsymmetric matrices. Extensive tests are presented as evidence that, for certain choices of parameters, the hybrid codes possess the same high reliability as the QR algorithm and are significantly faster. The greatest success has been achieved with the codes that calculate eigenvalues only. These can do the task in 15% to 50% less time than the QR algorithm.
Certain variants of the Toda flow are continuous analogues of the $QR$ algorithm and other algorithms for calculating eigenvalues of matrices. This was a remarkable discovery of the early eighties. Until very recently...
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Certain variants of the Toda flow are continuous analogues of the $QR$ algorithm and other algorithms for calculating eigenvalues of matrices. This was a remarkable discovery of the early eighties. Until very recently contemporary researchers studying this circle of ideas have been unaware that continuous analogues of the quotient-difference and $lr$ algorithms were already known to Rutishauser in the fifties. Rutishauser’s continuous analogue of the quotient-difference algorithm contains the finite, nonperiodic Toda flow as a special case. A nice feature of Rutishauser’s approach is that it leads from the (discrete) eigenvalue algorithm to the (continuous) flow by a limiting process. Thus the connection between the algorithm and the flow does not come as a surprise. In this paper it is shown how Rutishauser’s approach can be generalized to yield large families of flows in a natural manner. The flows derived include continuous analogues of the $lr$, $QR$, $SR$, and $HR$ algorithms.
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