D. Skoogh, "An implementation of a parallel rational Krylov algorithm," PhD dissertation, Goteborg University and Chalmers University of Technology, Goteborg, Swed., 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of V and Z can be discarded through a singular value decomposition. Assume that the ranks of V and Z are M Gamma ffi V and M Gamma ffi Z , respectively, so that the rank of ( E) is at most the lesser of (M Gamma ffi v ; M Gamma ffi z ) for any s. Then, by the singular value decomposition [3, 81], there exist orthogonal matrices T l = T l M jT l ] and T r = T r M jT r ] such that for a given matrix ( E) 6 l M T l 7 5 ( A Gamma j E) T r M T r 6 Sigma j 0 0 0 7 5 : 4:13) The matrix Sigma j is nonsingular and square with a rank M that is less ....

....of choosing pm to be the index of the next to last iteration employing oe m . Replacing pm with m Gamma 1 is actually implemented in the RL Algorithm 4.5. The superiority of the m Gamma 1 choice for methods that construct their approximations by assuming (bi)orthogonality was also observed in [81]. Related comments on the topic of the pm choice may also be found in [81] A second practical concern in the RL implementation is the so called serious breakdown. A breakdown occurs in a Lanczos type method when w m 1 vm 1 = 0. The assumptions of Theorem 4.1 are violated in this event, ....

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D. Skoogh, "An implementation of a parallel rational Krylov algorithm," PhD dissertation, Goteborg University and Chalmers University of Technology, Goteborg, Swed., 1996.

....convergence to eigenvalues close to these shifts. The first form describes a sequential variant, where information gained up to step j determines the next shift j 1 , this will be the primary object of this exposition. The second form indicates how a parallel variant can be implemented, see [19]. In that variant, j shifted and inverted matrices with the predetermined shifts 1 : j are applied in parallel to the same starting vector, after which all the resulting vectors are used to span up a subspace where approximations are found. This way we get a jth degree polynomial after just ....

D. Skoogh, An implementation of a parallel rational Krylov algorithm, Licentiate thesis, Chalmers University of Technology, Goteborg, 1996.

....Z can be discarded through a singular value decomposition. Assume that the ranks of V and Z are M Gamma ffi V and M Gamma ffi Z , respectively, so that the rank of ( A Gamma s E) is at most the lesser of (M Gamma ffi v ; M Gamma ffi z ) for any s. Then, by the singular value decomposition [13, 21], there exist orthogonal matrices T l = T l M jT l ] and T r = T r M jT r ] such that for a given matrix ( A Gamma j E) T T l M T l ( A Gamma j E) Theta T r M T r = Sigma j 0 0 0 : 41) The matrix Sigma j is nonsingular and square with a rank M ....

....of choosing pm to be the index of the next to last iteration employing oe m . Replacing pm with m Gamma 1 is actually implemented in the RL Algorithm 5. The superiority of the m Gamma 1 choice for methods that construct their approximations by assuming (bi)orthogonality was also observed in [21]. Related comments on the topic of the pm choice may also be found in [21] A second practical concern in the RL implementation is the so called serious breakdown. A breakdown occurs in a Lanczos type method when w T m 1 vm 1 = 0. The assumptions of Theorem 3 are violated in this event, because ....

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D. Skoogh, An implementation of a parallel rational Krylov algorithm. PhD thesis, Goteborg University and Chalmers University of Technology, 1996.

....changed, the matrix (G s i C) Gamma1 operates on b, and then the result is orthogonalised against the basis vectors. If (G s i C) Gamma1 b is nearly linearly independent of the basis vectors, then this leads to numerical problems similar to those in the parallel rational Krylov algorithm [34]. The more stable algorithm discussed in section 2 comes at an extra cost. the calculation of b i = V H k (G s i C) Gamma1 b needs to be done explicitly, whereas it is included in the process of building a basis in the restarting rational Krylov algorithm. 10 Error Estimates 10.1 ....

Daniel Skoogh. An implementation of a parallel rational Krylov algorithm. Licentiate Thesis, Chalmers University of Technology, G#teborg, Sweden, 1996.

....iterative, model, reduction, passive AMS subject classication 65F15, 65F50, 65Y05, 65F10, 93A30, 93B40 This thesis consists of an introduction and the following three reports: Daniel Skoogh A parallel rational Krylov algorithm for eigenvalue computation. Condensed and updated version of [23]. Part of the material presented at PARA98 and published in [24] Daniel Skoogh A Krylov subspace method to solve a sequence of linear systems with dioeerent right hand sides. Part of the material presented at Dalian [22] Daniel Skoogh A rational Krylov method for model order reduction. Part of ....

Daniel Skoogh. An implementation of a parallel rational Krylov algorithm. Licentiate Thesis, Chalmers University of Technology, G#teborg, Sweden, 1996.

....a rational function of the matrix times the starting vector. In theory it is possible to generate the basis vectors in parallel by operating with shifted and inverted matrices with dioeerent shifts, one on each processor. The purpose of this work is to investigate if this can be done in practice [12]. Our algorithm computes one matrix factorisation on each processor. In ARPACK iARnoldi PACKagej [5] one the other hand, where at most one shift is used, the matrix vector multiplication is parallelised and each basis vector is distributed among the processors; for details see PARPACK [6] ....

....algorithm. They dioeer in the way orthogonalisation is done. Both algorithms use p dioeerent processors to compute r p = A Gamma p B) Gamma1 Br p . The SPMD algorithm which is described below lets each processor orthogonalise its own vector. The Master Slave algorithm which is described in [12] uses an additional processor to do the orthogonalisation. In exact arithmetic, given the same conditions, the dioeerent algorithms generate identical matrices H , K and 4 V k 1 . We have made two implementations of the SPMD algorithm, which dioeer in how the communication is done. The ....

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Daniel Skoogh. An implementation of a parallel rational Krylov algorithm. Licentiate Thesis, Chalmers University of Technology, G#teborg, Sweden, 1996. 35

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