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doi: 10.3934/naco.2021012

## Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems

 1 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, China 2 College of Science, Nanjing Forestry University, Nanjing, 210023, China

* Corresponding author: Li Wang

Received  January 2021 Revised  March 2021 Published  March 2021

Fund Project: The authors are supported by the National Natural Science Foundation of China under grant 41571380 and 10971102, Major project 16KJA110001 of the Natural Science Foundation of the Jiangsu Higher Education Institution

An efficiently preconditioned Newton-like method for the computation of the eigenpairs of large and sparse nonsymmetric matrices is proposed. A sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system. The properties of the preconditioned matrix are investigated. Numerical results are given which reveal that the new proposed algorithms are efficient.

Citation: Hong-Yi Miao, Li Wang. Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021012
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##### References:
The results for the largest real eigenvalue of $A$
 Algorithm eigenvalue CPU time (s) GMRES-INewton 7.9817 9.8430 PGMRES-INewton 7.9817 6.9330
 Algorithm eigenvalue CPU time (s) GMRES-INewton 7.9817 9.8430 PGMRES-INewton 7.9817 6.9330
The results of GMRES-INewton and PGMRES-INewton method
 GMRES-INewton PGMRES-INewton k $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(\mathcal{A})$ $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(P^{-1}\mathcal{A})$ 1 2.8670e+03 1.4750e+00 3.1275e+09 2.8670e+03 1.4750e+00 3.1275e+09 2 5.6282e+01 7.2040e-01 2.8736e+05 2.4063e+01 4.8650e-02 4.0326e+04 3 1.2415e+00 3.8120e-02 6.1342e+04 3.4120e-03 6.7520e-03 3.3108e+04 4 1.3586e-04 3.3470e-05 3.1625e+04 5.4758e-05 1.2050e-05 3.0213e+03
 GMRES-INewton PGMRES-INewton k $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(\mathcal{A})$ $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(P^{-1}\mathcal{A})$ 1 2.8670e+03 1.4750e+00 3.1275e+09 2.8670e+03 1.4750e+00 3.1275e+09 2 5.6282e+01 7.2040e-01 2.8736e+05 2.4063e+01 4.8650e-02 4.0326e+04 3 1.2415e+00 3.8120e-02 6.1342e+04 3.4120e-03 6.7520e-03 3.3108e+04 4 1.3586e-04 3.3470e-05 3.1625e+04 5.4758e-05 1.2050e-05 3.0213e+03
The results for the largest real eigenvalue of $A$
 Algorithm eigenvalue CPU time (s) GMRES-INewton 0.4338 12.5310 PGMRES-INewton 0.4338 10.4990
 Algorithm eigenvalue CPU time (s) GMRES-INewton 0.4338 12.5310 PGMRES-INewton 0.4338 10.4990
The results of GMRES-INewton and PGMRES-INewton method
 GMRES-INewton PGMRES-INewton k $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(\mathcal{A})$ $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(P^{-1}\mathcal{A})$ 1 2.6740e+02 5.8630e+01 1.2640e+10 2.6740e+02 5.8630e+01 1.2640e+10 2 4.3572e+01 1.0270e+01 6.0304e+07 7.6230e+00 9.6810e-02 5.6210e+05 3 3.0450e+00 7.8140e-01 2.4639e+05 3.2540e-02 4.2830e-04 3.2790e+04 4 6.3580e-02 4.6720e-02 5.1852e+04 4.9130e-05 1.4270e-06 4.5870e+04 5 2.3710e-04 1.0390e-03 1.3089e+05
 GMRES-INewton PGMRES-INewton k $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(\mathcal{A})$ $||r_{k}||$ $|\lambda_{k}-\lambda^{*}|$ $cond(P^{-1}\mathcal{A})$ 1 2.6740e+02 5.8630e+01 1.2640e+10 2.6740e+02 5.8630e+01 1.2640e+10 2 4.3572e+01 1.0270e+01 6.0304e+07 7.6230e+00 9.6810e-02 5.6210e+05 3 3.0450e+00 7.8140e-01 2.4639e+05 3.2540e-02 4.2830e-04 3.2790e+04 4 6.3580e-02 4.6720e-02 5.1852e+04 4.9130e-05 1.4270e-06 4.5870e+04 5 2.3710e-04 1.0390e-03 1.3089e+05
The results for the largest real eigenvalue of a random matrix
 Algorithm eigenvalue CPU time (s) GMRES-INewton 298.2441 15.7350 PGMRES-INewton 298.2441 9.1470
 Algorithm eigenvalue CPU time (s) GMRES-INewton 298.2441 15.7350 PGMRES-INewton 298.2441 9.1470
The results for the largest imaginary eigenvalue of a random matrix
 Algorithm eigenvalue CPU time (s) GMRES-INewton -0.0001+0.0054$i$ 16.2370 PGMRES-INewton -0.0001+0.0054$i$ 10.4790
 Algorithm eigenvalue CPU time (s) GMRES-INewton -0.0001+0.0054$i$ 16.2370 PGMRES-INewton -0.0001+0.0054$i$ 10.4790
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