{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T05:10:33Z","timestamp":1776921033948,"version":"3.51.2"},"reference-count":60,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12401544"],"award-info":[{"award-number":["12401544"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Hong Kong RGC","award":["22300921"],"award-info":[{"award-number":["22300921"]}]},{"name":"the Hong Kong RGC","award":["12301480"],"award-info":[{"award-number":["12301480"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>The complex-shifted Laplacian systems arising in a wide range of applications. In this work, we propose an absolute-value based preconditioner for solving the complex-shifted Laplacian system. In our approach, the complex-shifted Laplacian system is equivalently rewritten as a <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$2\\times 2$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> block real linear system. With the Toeplitz structure of uniform-grid discretization of the constant-coefficient Laplacian operator, the absolute value of the block real matrix is fast invertible by means of fast sine transforms. For more general coefficient function, we then average the coefficient function and take the absolute value of the averaged matrix as our preconditioner. With assumptions on the complex shift, we theoretically prove that the eigenvalues of the preconditioned matrix in absolute value are upper and lower bounded by constants independent on the matrix size, indicating a matrix-size independent linear convergence rate of the MINRES solver. Interestingly, numerical results show that the proposed preconditioner is still efficient even if the assumptions on the complex shift are not met. The fast invertibility of the proposed preconditioner and the robust convergence rate of the preconditioned MINRES solver lead to a linearithmic (nearly optimal) complexity of the proposed solver. The proposed preconditioner is compared with several state-of-the-art preconditioners via several numerical examples to demonstrate the efficiency of the proposed preconditioner.<\/jats:p>","DOI":"10.1007\/s10915-025-02859-4","type":"journal-article","created":{"date-parts":[[2025,4,4]],"date-time":"2025-04-04T01:57:26Z","timestamp":1743731846000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Absolute-Value Based Preconditioner for Complex-Shifted Laplacian Systems"],"prefix":"10.1007","volume":"103","author":[{"given":"Xuelei","family":"Lin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8654-2614","authenticated-orcid":false,"given":"Congcong","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sean","family":"Hon","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,1]]},"reference":[{"key":"2859_CR1","doi-asserted-by":"crossref","unstructured":"Lin, X.L., Lu, X., Ng, M.K., Sun, H.W.: A fast accurate approximation method with multigrid solver for two-dimensional fractional sub-diffusion equation. J. Comput. Phys., 323, 204\u2013218 (2016)","DOI":"10.1016\/j.jcp.2016.07.031"},{"issue":"1","key":"2859_CR2","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1016\/j.jcp.2007.05.013","volume":"226","author":"T Airaksinen","year":"2007","unstructured":"Airaksinen, T., Heikkola, E., Pennanen, A., Toivanen, J.: An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation. J. Comput. Phys. 226(1), 1196\u20131210 (2007)","journal-title":"J. Comput. Phys."},{"key":"2859_CR3","unstructured":"Axelsson, O.: Iterative Solution Methods. Cambridge university press, Cambridge (1996)"},{"issue":"2","key":"2859_CR4","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.camwa.2019.09.024","volume":"80","author":"O Axelsson","year":"2020","unstructured":"Axelsson, O.: Optimality properties of a square block matrix preconditioner with applications. Comput. Math. Appl. 80(2), 286\u2013294 (2020)","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"2859_CR5","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s00211-020-01143-x","volume":"146","author":"O Axelsson","year":"2020","unstructured":"Axelsson, O., Kar\u00e1tson, J.: Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks. Numerische Mathematik 146(2), 335\u2013368 (2020)","journal-title":"Numerische Mathematik"},{"issue":"4","key":"2859_CR6","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1002\/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S","volume":"7","author":"O Axelsson","year":"2000","unstructured":"Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7(4), 197\u2013218 (2000)","journal-title":"Numer. Linear Algebra Appl."},{"key":"2859_CR7","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s11075-013-9764-1","volume":"66","author":"O Axelsson","year":"2014","unstructured":"Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66, 811\u2013841 (2014)","journal-title":"Numer. Algorithms"},{"key":"2859_CR8","doi-asserted-by":"crossref","unstructured":"Axelsson, O., Pourbagher, M., Salkuyeh, D.K.: Robust iteration methods for complex systems with an indefinite matrix term. arXiv preprint arXiv:2110.00537 (2021)","DOI":"10.1007\/s10092-022-00461-w"},{"key":"2859_CR9","doi-asserted-by":"crossref","unstructured":"Axelsson, O., Pourbagher, M., Salkuyeh, D.K.: Efficient iteration methods for complex systems with an indefinite matrix term. Calcolo, 59(2), 15 (2022)","DOI":"10.1007\/s10092-022-00461-w"},{"issue":"2","key":"2859_CR10","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s10543-018-0741-x","volume":"59","author":"O Axelsson","year":"2019","unstructured":"Axelsson, O., Salkuyeh, D.K.: A new version of a preconditioning method for certain two-by-two block matrices with square blocks. BIT Nume. Math. 59(2), 321\u2013342 (2019)","journal-title":"BIT Nume. Math."},{"key":"2859_CR11","unstructured":"Axelsson, O., Slakuyeh, D.K.: An efficient preconditioner for a class of non-Hermitian two-by-two block complex system of linear equations. arXiv preprint arXiv:2402.11184, 2024"},{"issue":"4","key":"2859_CR12","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s00607-010-0125-9","volume":"91","author":"Z-Z Bai","year":"2011","unstructured":"Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91(4), 379\u2013395 (2011)","journal-title":"Computing"},{"issue":"2","key":"2859_CR13","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s11075-010-9441-6","volume":"56","author":"Z-Z Bai","year":"2011","unstructured":"Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56(2), 297\u2013317 (2011)","journal-title":"Numer. Algorithms"},{"issue":"4","key":"2859_CR14","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1002\/nla.517","volume":"14","author":"Z-Z Bai","year":"2007","unstructured":"Bai, Z.-Z., Golub, G.H., Ng, M.K.: On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14(4), 319\u2013335 (2007)","journal-title":"Numer. Linear Algebra Appl."},{"key":"2859_CR15","doi-asserted-by":"crossref","unstructured":"Benedusi, P., Ferrari, P., Rognes, M.E., Serra-Capizzano, S.: Modeling excitable cells with the EMI equations: spectral analysis and iterative solution strategy. J. Sci. Comput. 98(3), 23 (2024)","DOI":"10.1007\/s10915-023-02449-2"},{"key":"2859_CR16","doi-asserted-by":"crossref","unstructured":"Bini, D., Di\u00a0Benedetto, F.: A new preconditioner for the parallel solution of positive definite Toeplitz systems. In: Proceedings of the Second Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 220\u2013223 (1990)","DOI":"10.1145\/97444.97688"},{"key":"2859_CR17","doi-asserted-by":"crossref","unstructured":"Bolten, M., Donatelli, M., Ferrari, P., Furci, I.: A symbol-based analysis for multigrid methods for block-circulant and block-Toeplitz systems. SIAM J. Matrix Anal. Appl. 43(1), 405\u2013438 (2022)","DOI":"10.1137\/21M1390554"},{"issue":"2","key":"2859_CR18","doi-asserted-by":"crossref","first-page":"A438","DOI":"10.1137\/15M102085X","volume":"39","author":"P-H Cocquet","year":"2017","unstructured":"Cocquet, P.-H., Gander, M.J.: How large a shift is needed in the shifted Helmholtz preconditioner for its effective inversion by multigrid? SIAM J. Sci. Comput. 39(2), A438\u2013A478 (2017)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2859_CR19","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1002\/nla.1881","volume":"20","author":"S Cools","year":"2013","unstructured":"Cools, S., Vanroose, W.: Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems. Numer. Linear Algebra Appl. 20(4), 575\u2013597 (2013)","journal-title":"Numer. Linear Algebra Appl."},{"key":"2859_CR20","series-title":"Numerical Mathematics and Scientific Computation","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199678792.001.0001","volume-title":"Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics","author":"HC Elman","year":"2014","unstructured":"Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford (2014)"},{"issue":"403\u2013424","key":"2859_CR21","first-page":"3","volume":"31","author":"YA Erlangga","year":"2008","unstructured":"Erlangga, Y.A., Nabben, R.: On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian. Electronic Trans. Numer. Anal. 31(403\u2013424), 3 (2008)","journal-title":"Electronic Trans. Numer. Anal."},{"issue":"5","key":"2859_CR22","doi-asserted-by":"crossref","first-page":"3417","DOI":"10.1137\/080731657","volume":"31","author":"YA Erlangga","year":"2009","unstructured":"Erlangga, Y.A., Nabben, R.: Algebraic multilevel Krylov methods. SIAM J. Sci. Comput. 31(5), 3417\u20133437 (2009)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2859_CR23","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1137\/040615195","volume":"27","author":"YA Erlangga","year":"2006","unstructured":"Erlangga, Y.A., Vuik, C., Oosterlee, C.W.: A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27(4), 1471\u20131492 (2006)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"2859_CR24","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1016\/j.apnum.2005.04.039","volume":"56","author":"YA Erlangga","year":"2006","unstructured":"Erlangga, Y.A., Vuik, C., Oosterlee, C.W.: Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl. Numer. Math. 56(5), 648\u2013666 (2006)","journal-title":"Appl. Numer. Math."},{"issue":"3\u20134","key":"2859_CR25","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.apnum.2004.01.009","volume":"50","author":"YA Erlangga","year":"2004","unstructured":"Erlangga, Y.A., Vuik, C., Oosterlee, C.W.: On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math. 50(3\u20134), 409\u2013425 (2004)","journal-title":"Appl. Numer. Math."},{"key":"2859_CR26","doi-asserted-by":"crossref","unstructured":"Erlangga, Y.A., Vuik, K., Oosterlee, K., Plessix, R.-E., Mulder, W.A.: A robust iterative solver for the two-way wave equation based on a complex shifted-Laplace preconditioner. In: SEG Technical Program Expanded Abstracts 2004, pp. 1897\u20131900. Society of Exploration Geophysicists (2004)","DOI":"10.1190\/1.1845184"},{"issue":"3","key":"2859_CR27","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1137\/18M1207399","volume":"40","author":"P Ferrari","year":"2019","unstructured":"Ferrari, P., Furci, I., Hon, S., Ayman-Mursaleen, M., Serra-Capizzano, S.: The eigenvalue distribution of special 2-by-2 block matrix-sequences with applications to the case of symmetrized Toeplitz structures. SIAM J. Matrix Anal. Appl. 40(3), 1066\u20131086 (2019)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"2859_CR28","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1007\/s00211-015-0700-2","volume":"131","author":"MJ Gander","year":"2015","unstructured":"Gander, M.J., Graham, I.G., Spence, E.A.: Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? Numerische Mathematik 131(3), 567\u2013614 (2015)","journal-title":"Numerische Mathematik"},{"issue":"1","key":"2859_CR29","doi-asserted-by":"crossref","first-page":"A220","DOI":"10.1137\/17M1148347","volume":"41","author":"MJ Gander","year":"2019","unstructured":"Gander, M.J., Halpern, L., Rannou, J., Ryan, J.: A direct time parallel solver by diagonalization for the wave equation. SIAM J. Sci. Comput. 41(1), A220\u2013A245 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"2859_CR30","doi-asserted-by":"crossref","unstructured":"Gander, M.J., Halpern, L., Ryan, J., Tran, T.T.B.: A direct solver for time parallelization. In: Domain Decomposition Methods in Science and Engineering XXII, pp. 491\u2013499. Springer (2016)","DOI":"10.1007\/978-3-319-18827-0_50"},{"key":"2859_CR31","unstructured":"Gander, M.J., Liu, J., Wu, S.L., Yue, X., Zhou, T.: Paradiag: parallel-in-time algorithms based on the diagonalization technique. arXiv preprint arXiv:2005.09158 (2020)"},{"key":"2859_CR32","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1553\/etna_vol51s135","volume":"51","author":"A Goddard","year":"2019","unstructured":"Goddard, A., Wathen, A.: A note on parallel preconditioning for all-at-once evolutionary PDEs. Electronic Trans. Numer. Anal. 51, 135\u2013150 (2019)","journal-title":"Electronic Trans. Numer. Anal."},{"key":"2859_CR33","unstructured":"Golub, G.H., Van\u00a0Loan, C.F.: Matrix Computations. The Johns Hopkins University Press, Fourth edn. (2013)"},{"issue":"307","key":"2859_CR34","doi-asserted-by":"crossref","first-page":"2089","DOI":"10.1090\/mcom\/3190","volume":"86","author":"IG Graham","year":"2017","unstructured":"Graham, I.G., Spence, E.A., Vainikko, E.: Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Math. Comput. 86(307), 2089\u20132127 (2017)","journal-title":"Math. Comput."},{"key":"2859_CR35","doi-asserted-by":"crossref","unstructured":"Greif, C., He, Y.: A closed-form multigrid smoothing factor for an additive Vanka-type smoother applied to the Poisson equation. Numer. Linear Algebra Appl., p. e2500 (2023)","DOI":"10.1002\/nla.2500"},{"key":"2859_CR36","volume":"132","author":"Y He","year":"2022","unstructured":"He, Y., Liu, J.: A Vanka-type multigrid solver for complex-shifted Laplacian systems from diagonalization-based parallel-in-time algorithms. Appl. Math. Lett. 132, 108125 (2022)","journal-title":"Appl. Math. Lett."},{"key":"2859_CR37","doi-asserted-by":"crossref","unstructured":"Hocking, L.R., Greif, C.: Optimal complex relaxation parameters in multigrid for complex-shifted linear systems. SIAM J. Matrix Anal. Appl., 42(2):475\u2013502 (2021)","DOI":"10.1137\/20M1342161"},{"issue":"4","key":"2859_CR38","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1137\/23M1547251","volume":"44","author":"S Hon","year":"2023","unstructured":"Hon, S., Dong, J., Serra-Capizzano, S.: A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory. SIAM J. Matrix Anal. Appl. 44(4), 1477\u20131509 (2023)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"2859_CR39","doi-asserted-by":"crossref","first-page":"A2766","DOI":"10.1137\/20M1316354","volume":"43","author":"X-L Lin","year":"2021","unstructured":"Lin, X.-L., Ng, M.: An all-at-once preconditioner for evolutionary partial differential equations. SIAM J. Sci. Comput. 43(4), A2766\u2013A2784 (2021)","journal-title":"SIAM J. Sci. Comput."},{"key":"2859_CR40","doi-asserted-by":"crossref","first-page":"827","DOI":"10.4208\/nmtma.2018.s09","volume":"11","author":"X-L Lin","year":"2018","unstructured":"Lin, X.-L., Ng, M.K., Sun, H.-W.: A separable preconditioner for time-space fractional Caputo-Riesz diffusion equations. Numer. Math. Theory Methods Appl. 11, 827\u2013853 (2018)","journal-title":"Numer. Math. Theory Methods Appl."},{"key":"2859_CR41","volume":"434","author":"X-L Lin","year":"2021","unstructured":"Lin, X.-L., Ng, M.K., Zhi, Y.: A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel. J. Comput. Phys. 434, 110221 (2021)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"2859_CR42","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s10444-022-09928-4","volume":"48","author":"J Liu","year":"2022","unstructured":"Liu, J., Wang, X.-S., Wu, S.-L., Zhou, T.: A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations. Adv. Comput. Math. 48(3), 16 (2022)","journal-title":"Adv. Comput. Math."},{"issue":"4","key":"2859_CR43","doi-asserted-by":"crossref","first-page":"1912","DOI":"10.1137\/19M1309869","volume":"41","author":"J Liu","year":"2020","unstructured":"Liu, J., Wu, S.-L.: A fast block $$\\alpha $$-circulant preconditoner for all-at-once systems from wave equations. SIAM J. Matrix Anal. Appl. 41(4), 1912\u20131943 (2020)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"5","key":"2859_CR44","doi-asserted-by":"crossref","first-page":"866","DOI":"10.1002\/nla.1972","volume":"22","author":"L Xin","year":"2015","unstructured":"Xin, L., Pang, H.-K., Sun, H.-W.: Fast approximate inversion of a block triangular Toeplitz matrix with applications to fractional sub-diffusion equations. Numer. Linear Algebra Appl. 22(5), 866\u2013882 (2015)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"3","key":"2859_CR45","doi-asserted-by":"crossref","first-page":"1548","DOI":"10.1137\/070687232","volume":"30","author":"SP MacLachlan","year":"2008","unstructured":"MacLachlan, S.P., Oosterlee, C.W.: Algebraic multigrid solvers for complex-valued matrices. SIAM J. Sci. Comput. 30(3), 1548\u20131571 (2008)","journal-title":"SIAM J. Sci. Comput."},{"key":"2859_CR46","doi-asserted-by":"crossref","unstructured":"Maday, Y., R\u00f8nquist, E.M.: Parallelization in time through tensor-product space\u2013time solvers. Comptes Rendus. Math\u00e9matique, 346(1\u20132):113\u2013118 (2008)","DOI":"10.1016\/j.crma.2007.09.012"},{"key":"2859_CR47","doi-asserted-by":"crossref","unstructured":"McDonald, E., Hon, S., Pestana, J., Wathen, A.: Preconditioning for nonsymmetry and time-dependence. In: Domain Decomposition Methods in Science and Engineering XXIII, pp. 81\u201391, Springer (2017)","DOI":"10.1007\/978-3-319-52389-7_7"},{"issue":"2","key":"2859_CR48","doi-asserted-by":"crossref","first-page":"A1012","DOI":"10.1137\/16M1062016","volume":"40","author":"E McDonald","year":"2018","unstructured":"McDonald, E., Pestana, J., Wathen, A.: Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations. SIAM J. Sci. Comput. 40(2), A1012\u2013A1033 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"2859_CR49","first-page":"123","volume":"37","author":"Y Notay","year":"2010","unstructured":"Notay, Y.: An aggregation-based algebraic multigrid method. Electronic Trans. Numer. Anal. 37(6), 123\u2013146 (2010)","journal-title":"Electronic Trans. Numer. Anal."},{"issue":"4","key":"2859_CR50","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1016\/j.apnum.2009.09.003","volume":"60","author":"D Osei-Kuffuor","year":"2010","unstructured":"Osei-Kuffuor, D., Saad, Y.: Preconditioning Helmholtz linear systems. Appl. Numer. Math. 60(4), 420\u2013431 (2010)","journal-title":"Appl. Numer. Math."},{"key":"2859_CR51","doi-asserted-by":"crossref","unstructured":"Pourbagher, M., Salkuyeh, D.K.: A new two-parameter iteration method for indefinite complex symmetric linear systems. Japan J. Ind. Appl. Math. 39(1), 145\u2013163 (2022)","DOI":"10.1007\/s13160-021-00479-7"},{"issue":"2","key":"2859_CR52","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1137\/0914028","volume":"14","author":"Y Saad","year":"1993","unstructured":"Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461\u2013469 (1993)","journal-title":"SIAM J. Sci. Comput."},{"key":"2859_CR53","doi-asserted-by":"crossref","unstructured":"Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Second edn. (2003)","DOI":"10.1137\/1.9780898718003"},{"issue":"4","key":"2859_CR54","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1137\/S089547989427141X","volume":"17","author":"S Serra","year":"1996","unstructured":"Serra, S.: Preconditioning strategies for Hermitian Toeplitz systems with nondefinite generating functions. SIAM J. Matrix Anal. Appl. 17(4), 1007\u20131019 (1996)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"6","key":"2859_CR55","doi-asserted-by":"crossref","first-page":"2219","DOI":"10.1137\/S0036142902401074","volume":"40","author":"V Simoncini","year":"2002","unstructured":"Simoncini, V., Szyld, D.B.: Flexible inner-outer Krylov subspace methods. SIAM J. Numer. Anal. 40(6), 2219\u20132239 (2002)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"2859_CR56","doi-asserted-by":"crossref","first-page":"1942","DOI":"10.1137\/060661491","volume":"29","author":"MB van Gijzen","year":"2007","unstructured":"van Gijzen, M.B., Erlangga, Y.A., Vuik, C.: Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29(5), 1942\u20131958 (2007)","journal-title":"SIAM J. Sci. Comput."},{"key":"2859_CR57","unstructured":"van\u00a0Rienen, U.: Numerical Methods in Computational Electrodynamics: Linear Systems in Practical Applications, vol.\u00a012. Springer Science & Business Media (2012)"},{"key":"2859_CR58","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.cam.2016.09.028","volume":"313","author":"S-L Wu","year":"2017","unstructured":"Wu, S.-L., Li, C.-X.: A splitting method for complex symmetric indefinite linear system. J. Comput. Appl. Math. 313, 343\u2013354 (2017)","journal-title":"J. Comput. Appl. Math."},{"issue":"5","key":"2859_CR59","volume":"25","author":"S-L Wu","year":"2018","unstructured":"Wu, S.-L., Zhang, H., Zhou, T.: Solving time-periodic fractional diffusion equations via diagonalization technique and multigrid. Numer. Linear Algebra Appl. 25(5), e2178 (2018)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"2","key":"2859_CR60","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1137\/100814767","volume":"32","author":"W Zulehner","year":"2011","unstructured":"Zulehner, W.: Nonstandard norms and robust estimates for saddle point problems. SIAM J. Matrix Anal. Appl. 32(2), 536\u2013560 (2011)","journal-title":"SIAM J. Matrix Anal. Appl."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02859-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-02859-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02859-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T19:10:56Z","timestamp":1745694656000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-02859-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,1]]},"references-count":60,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["2859"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-02859-4","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,1]]},"assertion":[{"value":"1 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 April 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"I confirm that I have read, understand and agreed to the submission guidelines, policies and submission declaration of the journal. I confirm that all authors of the manuscript have no Conflict of interest to declare. I confirm that the manuscript is the authors\u2019 original work and the manuscript has not received prior publication and is not under consideration for publication elsewhere. On behalf of all Co-Authors, I shall bear full responsibility for the submission. I confirm that all authors listed on the title page have contributed significantly to the work, have read the manuscript, attest to the validity and legitimacy of the data and its interpretation, and agree to its submission. I confirm that the paper now submitted is not copied or plagiarized version of some other published work. I declare that I shall not submit the paper for publication in any other Journal or Magazine till the decision is made by journal editors. If the paper is finally accepted by the journal for publication, I confirm that I will either publish the paper immediately or withdraw it according to withdrawal policies. I understand that submission of false or incorrect information\/undertaking would invite appropriate penal actions as per norms\/rules of the journal.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"53"}}