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A new shift strategy for the implicitly restarted generalized second-order Arnoldi method

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FangHui Gong, Yuquan Sun

Numerical Analysis · Submitted 11 Jan 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The new method vastly enhances the overall efficiency of the algorithm.

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arXiv ID1701.03042 (version 1)
Pith IDSCTGPMSN, Pith's permanent record of this paper and its review
CitationFanghui Gong, Yuquan Sun. "A new shift strategy for the implicitly restarted generalized second-order Arnoldi method." arXiv:1701.03042 (2017). https://pith.science/paper/SCTGPMSN
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What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.20 · human participants 0.02 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.16 · compares with prior results 0.20 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

In this paper, a new shift strategy for the implicitly restarted generalized second-order Arnoldi (GSOAR) method is proposed. In implicitly restarted processes, we can get a $k$-step GSOAR decomposition from a $m$-step GSOAR decomposition by performing $p = m-k$ implicit shifted QR iterations. The problem of the implicitly restarted GSOAR is the mismatch between the number of shifts and the dimension of the subspace. There are $2p$ shifts for $p$ QR iterations. We use the shifts to filter out the unwanted information in the current subspace; when more shifts are used, one obtains a better updated subspace. But, if we use more than $p$ shifts, the structure of the GSOAR decomposition will be destroyed. We propose a novel method which can use all $2p$ candidates and preserve the special structure. The new method vastly enhances the overall efficiency of the algorithm. Numerical experiments illustrate the efficiency of every restart process.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, a new shift strategy for the implicitly restarted generalized second-order Arnoldi (GSOAR) method is proposed. contribution 1.00 0.90 0.21 asserted 1.00
s1 In implicitly restarted processes, we can get a $k$-step GSOAR decomposition from a $m$-step GSOAR decomposition by performing $p = m-k$ implicit shifted QR iterations. background 0.94 0.19 0.04 asserted 0.74
s2 The problem of the implicitly restarted GSOAR is the mismatch between the number of shifts and the dimension of the subspace. background 0.74 0.14 0.04 asserted 1.00
s3 There are $2p$ shifts for $p$ QR iterations. background 0.99 0.22 0.04 asserted 1.00
s4 We use the shifts to filter out the unwanted information in the current subspace; when more shifts are used, one obtains a better updated subspace. background 0.87 0.21 0.06 asserted 0.88
s5 But, if we use more than $p$ shifts, the structure of the GSOAR decomposition will be destroyed. background 0.66 0.30 0.05 asserted 0.95
s6 We propose a novel method which can use all $2p$ candidates and preserve the special structure. contribution 0.98 0.89 0.43 asserted 1.00
s7 The new method vastly enhances the overall efficiency of the algorithm. result 0.85 0.76 0.59 asserted 1.00
s8 Numerical experiments illustrate the efficiency of every restart process. result 0.88 0.91 0.09 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.97.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper, a new shift strategy for the implicitly restarted generalized second-order Arnoldi (GSOAR) method is proposed. In implicitly restarted processes, we can get a $k$-step GSOAR decomposition from a $m$-step GSOAR decomposition by performing $p = m-k$ implicit shifted QR iterations. The problem of the implicitly restarted GSOAR is the mismatch between the number of shifts and the dimension of the subspace. There are $2p$ shifts for $p$ QR iterations. We use the shifts to filter out the unwanted information in the current subspace; when more shifts are used, one obtains a better updated subspace. But, if we use more than $p$ shifts, the structure of the GSOAR decomposition will be destroyed. We propose a novel method which can use all $2p$ candidates and preserve the special structure. The new method vastly enhances the overall efficiency of the algorithm. Numerical experiments illustrate the efficiency of every restart process.

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