By use of the three-term recurrence relation, an elementary and constructive proof is given for the global convergence of the symmetric tridiagonal qr algorithm with Wilkinson's shift. It is further illustrated wh...
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By use of the three-term recurrence relation, an elementary and constructive proof is given for the global convergence of the symmetric tridiagonal qr algorithm with Wilkinson's shift. It is further illustrated why the asymptotic rate of convergence is essentially cubic, as has long been observed in numerical experiments. A general mixed shift strategy with global convergence and cubic rate is also presented. (C) 2001 Published by Elsevier Science Inc. All rights reserved.
Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: i...
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Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. We present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and discuss some recent advances in the theory of validated methods for IVPs for ODEs. In particular, we discuss an interval Hermite-Obreschkoff (IHO) scheme for computing rigorous bounds on the solution of an IVP for an ODE, the stability of ITS and IHO methods, and a new perspective on the wrapping effect, where we interpret the problem of reducing the wrapping effect as one of finding a more stable scheme for advancing the solution. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.
Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: i...
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Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. We present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and discuss some recent advances in the theory of validated methods for IVPs for ODEs. In particular, we discuss an interval Hermite-Obreschkoff (IHO) scheme for computing rigorous bounds on the solution of an IVP for an ODE, the stability of ITS and IHO methods, and a new perspective on the wrapping effect, where we interpret the problem of reducing the wrapping effect as one of finding a more stable scheme for advancing the solution. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.
In the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm. This paper discusses the family of GR algorithms, with emphasis on the qr algorithm. Included are historical remar...
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In the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm. This paper discusses the family of GR algorithms, with emphasis on the qr algorithm. Included are historical remarks, an outline of what GR algorithms are and why they work, and descriptions of the latest, highly parallelizable, versions of the qr algorithm. Now that we know how to parallelize it, the qr algorithm seems likely to retain its dominance for many years to come. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 65F15.
A parameterized ordering of Givens rotations and guidelines for choosing parameter values is presented in the context of qr decomposition. Although a standard selection of parameter values retrieves an ordering that c...
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A parameterized ordering of Givens rotations and guidelines for choosing parameter values is presented in the context of qr decomposition. Although a standard selection of parameter values retrieves an ordering that corresponds to a well-known algorithm, we show that non-standard values decrease the execution time. We implement the new ordering on an Intel Pentium Pro system, a single thin POWER2 processor of the IBM SP2, and a single R8000 processor of the SGI POWER Challenge XL. On each machine, we observe performance that is more than twice that of the original ordering.
The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the qr algorithm, but, unlike the qr algorithm, it is well adapted to c...
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The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the qr algorithm, but, unlike the qr algorithm, it is well adapted to computing the eigenvalues of the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the qr algorithm by a factor of 30-60 in computing time and a factor of over 100 in matrix storage space.
For an Hermitian matrix the qr transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the qr transform may be written in rational form.
For an Hermitian matrix the qr transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the qr transform may be written in rational form.
The qr algorithm and its variants are among the most popular methods for calculating eigenvalues of matrices. Typical implementations chase bulges from top to bottom of an upper Hessenberg matrix. It is also possible ...
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The qr algorithm and its variants are among the most popular methods for calculating eigenvalues of matrices. Typical implementations chase bulges from top to bottom of an upper Hessenberg matrix. It is also possible to chase bulges from bottom to top. There are some situations in which it may be advantageous to chase bulges in both directions at once, in which case one needs a procedure for passing bulges through each other without mixing up the information that the bulges convey. This paper derives a procedure for passing bulges of arbitrary degree through each other. Experiments with a Fortran 90 program show that the procedure works well in practice for bulges of degree two.
This paper presents a technique that allows using level 3 BLAS in a number of rotation-based algorithms. In particular, the update of an orthogonal transformation matrix which often involves the vast majority of opera...
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This paper presents a technique that allows using level 3 BLAS in a number of rotation-based algorithms. In particular, the update of an orthogonal transformation matrix which often involves the vast majority of operations can be done with a matrix-matrix product. As a case study, the technique is applied to the qr and QL algorithms for computing the eigensystem of a symmetric tridiagonal matrix. The modifications do not affect the convergence properties of the algorithms nor do they significantly increase the overall number of operations. Thus, the computations can be sped up by more than 50% on machines with a distinct memory hierarchy, like the Intel i860 or IBM RS/6000, provided the block size is set appropriately. We also present a simple theoretical analysis that allows selecting an almost-optimal block size.
Over the last few years, it has been suggested that the popular qr algorithm for the unsymmetric Schur decomposition does not parallelize. In this paper, we present both positive and negative results on this subject. ...
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Over the last few years, it has been suggested that the popular qr algorithm for the unsymmetric Schur decomposition does not parallelize. In this paper, we present both positive and negative results on this subject. In theory, asymptotically perfect speedup can be obtained. In practice, reasonable speedup can be obtained on an MIMD distributed memory computer for a relatively small number of processors. However, we also show theoretically that it is impossible for the standard qr algorithm to be scalable. performance of a parallel implementation of the LAPACK DLAHqr routine on the Intel Paragon(TM) system is reported.
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