{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:41:23Z","timestamp":1740123683591,"version":"3.37.3"},"reference-count":58,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T00:00:00Z","timestamp":1636329600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T00:00:00Z","timestamp":1636329600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004729","name":"Technische Universit\u00e4t Wien","doi-asserted-by":"publisher","award":["TU-D"],"award-info":[{"award-number":["TU-D"]}],"id":[{"id":"10.13039\/501100004729","id-type":"DOI","asserted-by":"publisher"}]},{"name":"TU Wien"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential e<jats:sup><jats:italic>t<\/jats:italic><jats:italic>A<\/jats:italic><\/jats:sup><jats:italic>v<\/jats:italic>, is extended to the case of associated <jats:italic>\u03c6<\/jats:italic>-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of <jats:italic>A<\/jats:italic>. Ritz values yield properties of the spectrum of <jats:italic>A<\/jats:italic> (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector <jats:italic>v<\/jats:italic> and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.<\/jats:p>","DOI":"10.1007\/s11075-021-01190-x","type":"journal-article","created":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T06:04:42Z","timestamp":1636351482000},"page":"323-361","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A study of defect-based error estimates for the Krylov approximation of \u03c6-functions"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5720-3613","authenticated-orcid":false,"given":"Tobias","family":"Jawecki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,11,8]]},"reference":[{"issue":"10","key":"1190_CR1","doi-asserted-by":"publisher","first-page":"2293","DOI":"10.1016\/j.laa.2008.06.029","volume":"429","author":"M Afanasjew","year":"2008","unstructured":"Afanasjew, M., Eiermann, M., Ernst, O., G\u00fcttel, S.: Implementation of a restarted Krylov subspace method for the evaluation of matrix functions. Linear Algebra Appl. 429(10), 2293\u20132314 (2008). https:\/\/doi.org\/10.1016\/j.laa.2008.06.029","journal-title":"Linear Algebra Appl."},{"issue":"2","key":"1190_CR2","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1137\/100788860","volume":"33","author":"A Al-Mohy","year":"2011","unstructured":"Al-Mohy, A., Higham, N.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488\u2013511 (2011). https:\/\/doi.org\/10.1137\/100788860","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"1190_CR3","doi-asserted-by":"publisher","first-page":"3849","DOI":"10.1137\/080741744","volume":"47","author":"B Beckermann","year":"2009","unstructured":"Beckermann, B., Reichel, L.: Error estimates and evaluation of matrix functions via the Faber transform. SIAM J. Numer. Anal. 47(5), 3849\u20133883 (2009). https:\/\/doi.org\/10.1137\/080741744","journal-title":"SIAM J. Numer. Anal."},{"key":"1190_CR4","first-page":"46","volume":"1","author":"C de Boor","year":"2005","unstructured":"de Boor, C.: Divided differences. Surv. Approx. Theory 1, 46\u201369 (2005)","journal-title":"Surv. Approx. Theory"},{"issue":"3","key":"1190_CR5","doi-asserted-by":"publisher","first-page":"A1376","DOI":"10.1137\/110820191","volume":"35","author":"M Botchev","year":"2013","unstructured":"Botchev, M., Grimm, V., Hochbruck, M.: Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376\u2013A1397 (2013). https:\/\/doi.org\/10.1137\/110820191","journal-title":"SIAM J. Sci. Comput."},{"key":"1190_CR6","doi-asserted-by":"publisher","unstructured":"Botchev, M., Knizhnerman, L.: ART: Adaptive Residual-time restarting for Krylov subspace matrix exponential evaluations. J. Comput. Appl. Math. https:\/\/doi.org\/10.1016\/j.cam.2019.06.027 (2019)","DOI":"10.1016\/j.cam.2019.06.027"},{"issue":"1","key":"1190_CR7","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/S0024-3795(99)00100-7","volume":"309","author":"T Braconnier","year":"2000","unstructured":"Braconnier, T., Langlois, P., Rioual, J.: The influence of orthogonality on the Arnoldi method. Linear Algebra Appl. 309(1), 307\u2013323 (2000). https:\/\/doi.org\/10.1016\/S0024-3795(99)00100-7","journal-title":"Linear Algebra Appl."},{"issue":"3","key":"1190_CR8","doi-asserted-by":"publisher","first-page":"A1639","DOI":"10.1137\/15M1027620","volume":"38","author":"M Caliari","year":"2016","unstructured":"Caliari, M., Kandolf, P., Ostermann, A., Rainer, S.: The Leja method revisited: backward error analysis for the matrix exponential. SIAM J. Sci. Comput. 38(3), A1639\u2013A1661 (2016). https:\/\/doi.org\/10.1137\/15M1027620","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"1190_CR9","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1016\/S0168-9274(97)00033-0","volume":"24","author":"E Celledoni","year":"1997","unstructured":"Celledoni, E., Moret, I.: A Krylov projection method for systems of ODEs. Appl. Numer. Math. 24(2), 365\u2013378 (1997). https:\/\/doi.org\/10.1016\/S0168-9274(97)00033-0","journal-title":"Appl. Numer. Math."},{"issue":"4","key":"1190_CR10","doi-asserted-by":"publisher","first-page":"1546","DOI":"10.1137\/070688924","volume":"30","author":"F Diele","year":"2009","unstructured":"Diele, F., Moret, I., Ragni, S.: Error estimates for polynomial Krylov approximations to matrix functions. SIAM J. Matrix Anal. Appl. 30(4), 1546\u20131565 (2009). https:\/\/doi.org\/10.1137\/070688924","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"1","key":"1190_CR11","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1137\/S1064827596303661","volume":"19","author":"V Druskin","year":"1998","unstructured":"Druskin, V., Greenbaum, A., Knizhnerman, L.: Using nonorthogonal Lanczos vectors in the computation of matrix functions. SIAM J. Sci. Comput. 19(1), 38\u201354 (1998). https:\/\/doi.org\/10.1137\/S1064827596303661","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"1190_CR12","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1016\/S0041-5553(89)80020-5","volume":"29","author":"V Druskin","year":"1989","unstructured":"Druskin, V., Knizhnerman, L.: Two polynomial methods of calculating functions of symmetric matrices. USSR Comput Math. Math. Phys. 29 (6), 112\u2013121 (1989). https:\/\/doi.org\/10.1016\/S0041-5553(89)80020-5","journal-title":"USSR Comput Math. Math. Phys."},{"issue":"7","key":"1190_CR13","first-page":"20","volume":"31","author":"V Druskin","year":"1992","unstructured":"Druskin, V., Knizhnerman, L.: Error bounds in the simple Lanczos procedure for computing functions of symmetric matrices and eigenvalues. Comput. Math. Math. Phys. 31(7), 20\u201330 (1992)","journal-title":"Comput. Math. Math. Phys."},{"issue":"3","key":"1190_CR14","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1137\/S0895479895292400","volume":"19","author":"V Druskin","year":"1998","unstructured":"Druskin, V., Knizhnerman, L.: Extended Krylov subspaces: Approximation of the matrix square root and related functions. SIAM. J. Matrix Anal. Appl. 19(3), 755\u2013771 (1998). https:\/\/doi.org\/10.1137\/S0895479895292400","journal-title":"J. Matrix Anal. Appl."},{"key":"1190_CR15","doi-asserted-by":"publisher","first-page":"2481","DOI":"10.1137\/050633846","volume":"44","author":"M Eiermann","year":"2006","unstructured":"Eiermann, M., Ernst, O.: A restarted Krylov subspace method for the evaluation of matrix functions. SIAM J. Numer. Anal. 44, 2481\u20132504 (2006). https:\/\/doi.org\/10.1137\/050633846","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"1190_CR16","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1137\/090774665","volume":"32","author":"M Eiermann","year":"2011","unstructured":"Eiermann, M., Ernst, O., G\u00fcttel, S.: Deflated restarting for matrix functions. SIAM J. Matrix Anal. Appl. 32(2), 621\u2013641 (2011). https:\/\/doi.org\/10.1137\/090774665","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"1190_CR17","doi-asserted-by":"publisher","first-page":"1438","DOI":"10.1137\/040605461","volume":"27","author":"J van den Eshof","year":"2006","unstructured":"van den Eshof, J., Hochbruck, M.: Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438\u20131457 (2006). https:\/\/doi.org\/10.1137\/040605461","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"1190_CR18","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/BF01060992","volume":"4","author":"R Friesner","year":"1989","unstructured":"Friesner, R., Tuckerman, L., Dornblaser, B., Russo, T.: A method for exponential propagation of large systems of stiff nonlinear differential equations. J. Sci. Comput. 4 (4), 327\u2013354 (1989). https:\/\/doi.org\/10.1007\/BF01060992","journal-title":"J. Sci. Comput."},{"issue":"2","key":"1190_CR19","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1137\/13093491X","volume":"35","author":"A Frommer","year":"2014","unstructured":"Frommer, A., G\u00fcttel, S., Schweitzer, M.: Efficient and stable Arnoldi restarts for matrix functions based on quadrature. SIAM J. Matrix Anal. Appl. 35(2), 661\u2013683 (2014). https:\/\/doi.org\/10.1137\/13093491X","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"5","key":"1190_CR20","doi-asserted-by":"publisher","first-page":"1236","DOI":"10.1137\/0913071","volume":"13","author":"E Gallopoulos","year":"1992","unstructured":"Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236\u20131264 (1992). https:\/\/doi.org\/10.1137\/0913071","journal-title":"SIAM J. Sci. Statist. Comput."},{"issue":"4","key":"1190_CR21","doi-asserted-by":"publisher","first-page":"2189","DOI":"10.1137\/12089226X","volume":"51","author":"T G\u00f6ckler","year":"2013","unstructured":"G\u00f6ckler, T., Grimm, V.: Convergence analysis of an extended Krylov subspace method for the approximation of operator functions in exponential integrators. SIAM J. Numer. Anal. 51(4), 2189\u20132213 (2013). https:\/\/doi.org\/10.1137\/12089226X","journal-title":"SIAM J. Numer. Anal."},{"key":"1190_CR22","unstructured":"G\u00fcttel, S.: Rational Krylov methods for operator functions. Ph.D. thesis, Technische universit\u00e4t Bergakademie Freiberg, Germany. http:\/\/eprints.ma.man.ac.uk\/2586\/. Dissertation available as MIMS Eprint 2017.39 (2010)"},{"key":"1190_CR23","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718027","volume-title":"Accuracy and Stability of Numerical Algorithms","author":"N Higham","year":"2002","unstructured":"Higham, N.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, USA (2002). https:\/\/doi.org\/10.1137\/1.9780898718027","edition":"2nd edn."},{"key":"1190_CR24","doi-asserted-by":"publisher","unstructured":"Higham, N.: Functions of matrices. Society for industrial and applied mathematics, philadelphia, PA USA. https:\/\/doi.org\/10.1137\/1.9780898717778 (2008)","DOI":"10.1137\/1.9780898717778"},{"key":"1190_CR25","unstructured":"Hochbruck, M., Hochstenbach, M.: Subspace extraction for matrix functions. Tech. rep., Dept. of Math., Case Western Reserve University. http:\/\/na.math.kit.edu\/download\/papers\/funext.pdf (2005)"},{"issue":"5","key":"1190_CR26","doi-asserted-by":"publisher","first-page":"1911","DOI":"10.1137\/S0036142995280572","volume":"34","author":"M Hochbruck","year":"1997","unstructured":"Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911\u20131925 (1997). https:\/\/doi.org\/10.1137\/S0036142995280572","journal-title":"SIAM J. Numer. Anal."},{"key":"1190_CR27","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","volume":"19","author":"M Hochbruck","year":"2010","unstructured":"Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209\u2013286 (2010). https:\/\/doi.org\/10.1017\/S0962492910000048","journal-title":"Acta Numerica"},{"issue":"5","key":"1190_CR28","doi-asserted-by":"publisher","first-page":"1552","DOI":"10.1137\/S1064827595295337","volume":"19","author":"W Hochbruck","year":"1998","unstructured":"Hochbruck, W., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552\u20131574 (1998). https:\/\/doi.org\/10.1137\/S1064827595295337","journal-title":"SIAM J. Sci. Comput."},{"key":"1190_CR29","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.cpc.2018.07.010","volume":"234","author":"A Iserles","year":"2019","unstructured":"Iserles, A., Kropielnicka, K., Singh, P.: Compact schemes for laser-matter interaction in schr\u00f6dinger equation based on effective splittings of Magnus expansion. J. Comput. Phys. Comm. 234, 195\u2013201 (2019). https:\/\/doi.org\/10.1016\/j.cpc.2018.07.010","journal-title":"J. Comput. Phys. Comm."},{"key":"1190_CR30","doi-asserted-by":"publisher","unstructured":"Jawecki, T., Auzinger, W., Koch, O.: Computable upper error bounds for Krylov approximations to matrix exponentials and associated \u03c6-functions BIT. https:\/\/doi.org\/10.1007\/s10543-019-00771-6 (2019)","DOI":"10.1007\/s10543-019-00771-6"},{"issue":"1","key":"1190_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11075-014-9878-0","volume":"69","author":"Z Jia","year":"2015","unstructured":"Jia, Z., Lv, H.: A posteriori error estimates of Krylov subspace approximations to matrix functions. Numer. Algorithms 69(1), 1\u201328 (2015). https:\/\/doi.org\/10.1007\/s11075-014-9878-0","journal-title":"Numer. Algorithms"},{"key":"1190_CR32","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1002\/nla.652","volume":"17","author":"L Knizhnerman","year":"2010","unstructured":"Knizhnerman, L., Simoncini, V.: A new investigation of the extended Krylov subspace method for matrix function evaluations. Numer. Linear Algebra Appl. 17, 615\u2013638 (2010). https:\/\/doi.org\/10.1002\/nla.652","journal-title":"Numer. Linear Algebra Appl."},{"issue":"4","key":"1190_CR33","doi-asserted-by":"publisher","first-page":"044111","DOI":"10.1063\/1.1961341","volume":"123","author":"A Kuleff","year":"2005","unstructured":"Kuleff, A., Breidbach, J., Cederbaum, L.: Multielectron wave-packet propagation: General theory and application. J. Chem. Phys. 123(4), 044111 (2005). https:\/\/doi.org\/10.1063\/1.1961341","journal-title":"J. Chem. Phys."},{"key":"1190_CR34","doi-asserted-by":"crossref","unstructured":"Lubich, C.: From Quantum to Classical Molecular Dynamics; Reduced Models and Numerical Analysis. Zurich lectures in advanced mathematics. European Math. Soc. z\u00fcrich (2008)","DOI":"10.4171\/067"},{"key":"1190_CR35","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1090\/S0025-5718-1984-0758198-0","volume":"43","author":"A McCurdy","year":"1984","unstructured":"McCurdy, A., Ng, K., Parlett, B.: Accurate computation of divided differences of the exponential function. Math. Comp. 43, 501\u2013528 (1984). https:\/\/doi.org\/10.2307\/2008291","journal-title":"Math. Comp."},{"issue":"1","key":"1190_CR36","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1137\/S00361445024180","volume":"45","author":"C Moler","year":"2003","unstructured":"Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3\u201349 (2003). https:\/\/doi.org\/10.1137\/S00361445024180","journal-title":"SIAM Rev."},{"issue":"1","key":"1190_CR37","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1016\/S0377-0427(00)00261-2","volume":"131","author":"I Moret","year":"2001","unstructured":"Moret, I., Novati, P.: An interpolatory approximation of the matrix exponential based on Faber polynomials. J. Comput. Appl. Math. 131(1), 361\u2013380 (2001). https:\/\/doi.org\/10.1016\/S0377-0427(00)00261-2","journal-title":"J. Comput. Appl. Math."},{"key":"1190_CR38","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1023\/B:BITN.0000046805.27551.3b","volume":"44","author":"I Moret","year":"2004","unstructured":"Moret, I., Novati, P.: RD-rational approximations of the matrix exponential. BIT 44, 595\u2013615 (2004). https:\/\/doi.org\/10.1023\/B:BITN.0000046805.27551.3b","journal-title":"BIT"},{"key":"1190_CR39","doi-asserted-by":"publisher","first-page":"2238","DOI":"10.1103\/PhysRevLett.51.2238","volume":"51","author":"A Nauts","year":"1983","unstructured":"Nauts, A., Wyatt, R.: New approach to many-state quantum dynamics: the recursive-residue-generation method. Phys. Rev. Lett. 51, 2238\u20132241 (1983). https:\/\/doi.org\/10.1103\/PhysRevLett.51.2238","journal-title":"Phys. Rev. Lett."},{"key":"1190_CR40","unstructured":"Niehoff, J.: Projektionsverfahren Zur Approximation Von Matrixfunktionen Mit Anwendungen Auf Die Implementierung Exponentieller Integratoren. Ph.D. thesis, Heinrich-Heine-Universit\u00e4t D\u00fcsseldorf (2007)"},{"issue":"3","key":"1190_CR41","doi-asserted-by":"publisher","first-page":"22:1","DOI":"10.1145\/2168773.2168781","volume":"38","author":"J Niesen","year":"2012","unstructured":"Niesen, J., Wright, W.: Algorithm 919: A Krylov subspace algorithm for evaluating the \u03d5-functions appearing in exponential integrators. ACM Trans. Math. Softw. 38(3), 22:1\u201322:19 (2012). https:\/\/doi.org\/10.1145\/2168773.2168781","journal-title":"ACM Trans. Math. Softw."},{"issue":"S1","key":"1190_CR42","doi-asserted-by":"publisher","first-page":"T52","DOI":"10.1002\/zamm.19640441321","volume":"44","author":"G Opitz","year":"1964","unstructured":"Opitz, G.: Steigungsmatrizen. Z. Angew. Math. Mech. 44(S1), T52\u2013T54 (1964). https:\/\/doi.org\/10.1002\/zamm.19640441321","journal-title":"Z. Angew. Math. Mech."},{"issue":"3","key":"1190_CR43","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1093\/imamat\/18.3.341","volume":"18","author":"C Paige","year":"1976","unstructured":"Paige, C.: Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix. IMA J. Appl. Math. 18(3), 341\u2013349 (1976). https:\/\/doi.org\/10.1093\/imamat\/18.3.341","journal-title":"IMA J. Appl. Math."},{"key":"1190_CR44","doi-asserted-by":"publisher","first-page":"5870","DOI":"10.1063\/1.451548","volume":"85","author":"T Park","year":"1986","unstructured":"Park, T., Light, J.: Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870\u20135876 (1986). https:\/\/doi.org\/10.1063\/1.451548","journal-title":"J. Chem. Phys."},{"key":"1190_CR45","doi-asserted-by":"publisher","unstructured":"Parlett, B.: The symmetric eigenvalue problem. Society for industrial and applied mathematics, philadelphia, PA USA. https:\/\/doi.org\/10.1137\/1.9781611971163 (1998)","DOI":"10.1137\/1.9781611971163"},{"issue":"1","key":"1190_CR46","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1137\/0729014","volume":"29","author":"Y Saad","year":"1992","unstructured":"Saad, Y.: Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209\u2013228 (1992). https:\/\/doi.org\/10.1137\/0729014","journal-title":"SIAM J. Numer. Anal."},{"key":"1190_CR47","doi-asserted-by":"crossref","unstructured":"Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA USA (2003)","DOI":"10.1137\/1.9780898718003"},{"key":"1190_CR48","unstructured":"Schweitzer, M.: Restarting and error estimation in polynomial and extended Krylov subspace methods for the approximation of matrix functions. Ph.D. thesis, Bergische Universit\u00e4t Wuppertal, Germany. http:\/\/nbn-resolving.de\/urn\/resolver.pl?urn=urn%3Anbn%3Ade%3Ahbz%3A468-20160212-112106-7 (2015)"},{"issue":"1","key":"1190_CR49","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1145\/285861.285868","volume":"24","author":"R Sidje","year":"1998","unstructured":"Sidje, R.: Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Software 24(1), 130\u2013156 (1998). https:\/\/doi.org\/10.1145\/285861.285868","journal-title":"ACM Trans. Math. Software"},{"key":"1190_CR50","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/0024-3795(84)90025-9","volume":"61","author":"H Simon","year":"1984","unstructured":"Simon, H.: Analysis of the symmetric Lanczos algorithm with reorthogonalization methods. Linear Algebra Appl. 61, 101\u2013131 (1984). https:\/\/doi.org\/10.1016\/0024-3795(84)90025-9","journal-title":"Linear Algebra Appl."},{"issue":"15","key":"1190_CR51","doi-asserted-by":"publisher","first-page":"154111","DOI":"10.1063\/1.5065902","volume":"150","author":"P Singh","year":"2019","unstructured":"Singh, P.: Sixth-order schemes for laser-matter interaction in the Schr\u00f6dinger equation. J. Chem. Phys. 150(15), 154111 (2019). https:\/\/doi.org\/10.1063\/1.5065902","journal-title":"J. Chem. Phys."},{"issue":"2","key":"1190_CR52","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0377-0427(96)00006-4","volume":"72","author":"D Stewart","year":"1996","unstructured":"Stewart, D., Leyk, T.: Error estimates for Krylov subspace approximations of matrix exponentials. J. Comput. Appl. Math. 72(2), 359\u2013369 (1996). https:\/\/doi.org\/10.1016\/0377-0427(96)00006-4","journal-title":"J. Comput. Appl. Math."},{"issue":"6","key":"1190_CR53","doi-asserted-by":"publisher","first-page":"2426","DOI":"10.1137\/040617868","volume":"29","author":"H Tal-Ezer","year":"2007","unstructured":"Tal-Ezer, H.: On restart and error estimation for Krylov approximation of W = F(A)V. SIAM J. Sci. Comput. 29 (6), 2426\u20132441 (2007). https:\/\/doi.org\/10.1137\/040617868","journal-title":"SIAM J. Sci. Comput."},{"issue":"9","key":"1190_CR54","doi-asserted-by":"publisher","first-page":"3967","DOI":"10.1063\/1.448136","volume":"81","author":"H Tal-Ezer","year":"1984","unstructured":"Tal-Ezer, H., Kosloff, R.: An accurate and efficient scheme for propagating the time dependent schr\u00f6dinger equation. J. Chem. Phys. 81(9), 3967\u20133971 (1984). https:\/\/doi.org\/10.1063\/1.448136","journal-title":"J. Chem. Phys."},{"issue":"6","key":"1190_CR55","doi-asserted-by":"publisher","first-page":"971","DOI":"10.1137\/0714065","volume":"14","author":"C Van Loan","year":"1977","unstructured":"Van Loan, C.: The sensitivity of the matrix exponential. SIAM J. Numer. Anal. 14(6), 971\u2013981 (1977). https:\/\/doi.org\/10.1137\/0714065","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1190_CR56","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1137\/16M1063733","volume":"38","author":"H Wang","year":"2017","unstructured":"Wang, H., Ye, Q.: Error bounds for the Krylov subspace methods for computations of matrix exponentials. SIAM J. Matrix Anal. Appl. 38 (1), 155\u2013187 (2017). https:\/\/doi.org\/10.1137\/16M1063733","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"1190_CR57","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1007\/s10444-015-9433-0","volume":"42","author":"G Wu","year":"2016","unstructured":"Wu, G., Zhang, L., Xu, T.: A framework of the harmonic Arnoldi method for evaluating \u03d5-functions with applications to exponential integrators. Adv. Comput. Math. 42 (3), 505\u2013541 (2016). https:\/\/doi.org\/10.1007\/s10444-015-9433-0","journal-title":"Adv. Comput. Math."},{"key":"1190_CR58","doi-asserted-by":"publisher","unstructured":"Zemke, J.: Krylov Subspace Methods in Finite Precision : a Unified Approach. Ph.D. thesis, Technische Universit\u00e4t Hamburg. https:\/\/doi.org\/10.15480\/882.8(2003)","DOI":"10.15480\/882.8"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01190-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-021-01190-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01190-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,18]],"date-time":"2022-04-18T17:37:50Z","timestamp":1650303470000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-021-01190-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,8]]},"references-count":58,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,5]]}},"alternative-id":["1190"],"URL":"https:\/\/doi.org\/10.1007\/s11075-021-01190-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2021,11,8]]},"assertion":[{"value":"31 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 August 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}