Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Evaluating the action of a matrix function on a vector, that is  x = f ( M ) v , is an ubiquitous task in applications. When  M  is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating  x  wh...

Full description

Saved in:
Bibliographic Details
Published inBIT Vol. 61; no. 1; pp. 237 - 273
Main Authors Massei, Stefano, Robol, Leonardo
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.03.2021
Springer Nature B.V
Subjects
Approximation
Computational Mathematics and Numerical Analysis
Convergence
Error analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrix methods
Numeric Computing
Subspaces
Tensors
Zolotarev problem
65F60
30E20
Pole selection
Rational Krylov
Function of matrices
65E05
Kronecker sum
Stieltjes functions
Online AccessGet full text

Cover

Abstract Evaluating the action of a matrix function on a vector, that is  x = f ( M ) v , is an ubiquitous task in applications. When  M  is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating  x  when  f ( z ) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and  M  is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case  M = I ⊗ A - B T ⊗ I , and  v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of  x . Pole selection strategies with explicit convergence bounds are given also in this case.
AbstractList Evaluating the action of a matrix function on a vector, that is x=f(M)v, is an ubiquitous task in applications. When M is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating x when f(z) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and M is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case M=I⊗A-BT⊗I, and v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of x. Pole selection strategies with explicit convergence bounds are given also in this case.
Evaluating the action of a matrix function on a vector, that is  x = f ( M ) v , is an ubiquitous task in applications. When  M  is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating  x  when  f ( z ) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and  M  is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case  M = I ⊗ A - B T ⊗ I , and  v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of  x . Pole selection strategies with explicit convergence bounds are given also in this case.
Evaluating the action of a matrix function on a vector, that is  $$x=f({\mathcal {M}})v$$ x = f ( M ) v , is an ubiquitous task in applications. When  $${\mathcal {M}}$$ M  is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating  x  when  f ( z ) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and  $${\mathcal {M}}$$ M  is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case  $${\mathcal {M}}=I \otimes A - B^T \otimes I$$ M = I ⊗ A - B T ⊗ I , and  v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of  x . Pole selection strategies with explicit convergence bounds are given also in this case.
Author Robol, Leonardo
Massei, Stefano
Author_xml – sequence: 1
  givenname: Stefano
  orcidid: 0000-0003-1813-4181
  surname: Massei
  fullname: Massei, Stefano
  email: s.massei@tue.nl
  organization: TU Eindhoven
– sequence: 2
  givenname: Leonardo
  surname: Robol
  fullname: Robol, Leonardo
  organization: Dipartimento di Matematica, Università di Pisa
BookMark eNp9kE1LAzEQQIMo2Fb_gKeA59VJspvNepPiFxYF7T2k2dmyZbupSVpsf73briB46Gku7w0zb0hOW9ciIVcMbhhAfhsYZKlIgEMCoLhMdidkwLKcJwXj2SkZAIBMhBLZORmGsADghWRiQN4-TKxdaxr66reN29DKefoZa2ziAgNdmujrb1qtW7vHwh21rt2gn2NrkZq2pCvXIA3Y4AG4IGeVaQJe_s4RmT4-TMfPyeT96WV8P0mskCImZcmVERWwiomZtDJVMlMzlRtELARP04opm2a5KlluOStT4NYWqRIohZ2hGJHrfu3Ku681hqgXbu27L4LmGXRckYHsKNVT1rsQPFba1vHwbvSmbjQDvY-n-3i6i6cP8fSuU_k_deXrpfHb45LopdDB7Rz931VHrB8Q94Tm
CitedBy_id crossref_primary_10_1080_23799927_2022_2114381
crossref_primary_10_1093_imanum_drad089
crossref_primary_10_1007_s00211_023_01368_6
crossref_primary_10_1007_s00211_022_01293_0
crossref_primary_10_1007_s10543_021_00881_0
crossref_primary_10_1090_mcom_3788
crossref_primary_10_1002_nla_2488
crossref_primary_10_1007_s00211_023_01372_w
crossref_primary_10_3390_axioms12020105
crossref_primary_10_1007_s11075_022_01256_4
crossref_primary_10_1007_s10543_023_00974_y
crossref_primary_10_1016_j_camwa_2023_05_016
crossref_primary_10_1137_23M1559439
crossref_primary_10_1007_s44146_024_00155_5
crossref_primary_10_1080_03081087_2024_2404455
crossref_primary_10_1137_22M1499674
Cites_doi 10.1145/361573.361582
10.1137/080742403
10.1016/j.jcp.2015.06.031
10.1137/080741744
10.1137/19M1244433
10.1137/S0895479895292400
10.1080/10652469.2011.613830
10.1137/130912839
10.1137/17M1161038
10.1137/100800634
10.1007/s10543-013-0420-x
10.1016/j.cam.2009.08.108
10.13001/1081-3810.1010
10.1007/BF02547400
10.1137/110824590
10.1017/S0962492910000048
10.1093/comnet/cnt007
10.1007/s00211-016-0799-9
10.1137/13093491X
10.1137/18M1180803
10.1137/090774082
10.1070/SM2007v198n03ABEH003842
10.1002/gamm.201310002
10.1016/j.sysconle.2011.04.013
10.1137/17M1157155
ContentType Journal Article
Copyright The Author(s) 2020
The Author(s) 2020. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Copyright_xml – notice: The Author(s) 2020
– notice: The Author(s) 2020. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
DBID C6C
AAYXX
CITATION
DOI 10.1007/s10543-020-00826-z
DatabaseName Springer Nature OA Free Journals
CrossRef
DatabaseTitle CrossRef
DatabaseTitleList

CrossRef
Database_xml – sequence: 1
  dbid: C6C
  name: SpringerOpen Free (Free internet resource, activated by CARLI)
  url: http://www.springeropen.com/
  sourceTypes: Publisher
DeliveryMethod fulltext_linktorsrc
Discipline Applied Sciences
Mathematics
Computer Science
EISSN 1572-9125
EndPage 273
ExternalDocumentID 10_1007_s10543_020_00826_z
GrantInformation_xml – fundername: Eindhoven University of Technology
GroupedDBID -52
-BR
-~X
1N0
23N
40D
40E
95-
95.
95~
ABDPE
ABMNI
ACIWK
AGWIL
ALMA_UNASSIGNED_HOLDINGS
ASPBG
AVWKF
BBWZM
C6C
CAG
COF
CS3
H~9
KOW
N2Q
RHV
SDD
SOJ
TN5
WH7
~EX
AAYXX
CITATION
ID FETCH-LOGICAL-c363t-dd28a3f01f13b6c648658b87aeee93244f18c4578d17c21d402cc9483e63cbe3
IEDL.DBID U2A
ISSN 0006-3835
IngestDate Mon Jun 30 09:03:03 EDT 2025
Tue Jul 01 02:03:19 EDT 2025
Thu Apr 24 23:05:10 EDT 2025
Fri Feb 21 02:49:44 EST 2025
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 1
Keywords Zolotarev problem
65F60
30E20
Pole selection
Rational Krylov
Function of matrices
65E05
Kronecker sum
Stieltjes functions
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c363t-dd28a3f01f13b6c648658b87aeee93244f18c4578d17c21d402cc9483e63cbe3
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ORCID 0000-0003-1813-4181
OpenAccessLink https://link.springer.com/10.1007/s10543-020-00826-z
PQID 2504839506
PQPubID 2043657
PageCount 37
ParticipantIDs proquest_journals_2504839506
crossref_citationtrail_10_1007_s10543_020_00826_z
crossref_primary_10_1007_s10543_020_00826_z
springer_journals_10_1007_s10543_020_00826_z
ProviderPackageCode CITATION
AAYXX
PublicationCentury 2000
PublicationDate 2021-03-01
PublicationDateYYYYMMDD 2021-03-01
PublicationDate_xml – month: 03
  year: 2021
  text: 2021-03-01
  day: 01
PublicationDecade 2020
PublicationPlace Dordrecht
PublicationPlace_xml – name: Dordrecht
PublicationTitle BIT
PublicationTitleAbbrev Bit Numer Math
PublicationYear 2021
Publisher Springer Netherlands
Springer Nature B.V
Publisher_xml – name: Springer Netherlands
– name: Springer Nature B.V
References MasseiSPalittaDRobolLSolving rank-structured Sylvester and Lyapunov equationsSIAM J. Matrix Anal. Appl.201839415641590386761710.1137/17M11571551404.65036
TownsendAOlverSThe automatic solution of partial differential equations using a global spectral methodJ. Comput. Phys.2015299106123338471910.1016/j.jcp.2015.06.0311352.65579
BeckermannBAn error analysis for rational Galerkin projection applied to the Sylvester equationSIAM J. Numer. Anal.201149624302450285460310.1137/1108245901244.65057
BenziMSimonciniVApproximation of functions of large matrices with Kronecker structureNumer. Math.20171351126359210510.1007/s00211-016-0799-91365.65134
Susnjara, A., Perraudin, N., Kressner, D., Vandergheynst, P.: Accelerated filtering on graphs using Lanczos method. arXiv preprint arXiv:1509.04537 (2015)
DruskinVLiebermanCZaslavskyMOn adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problemsSIAM J. Sci. Comput.201032524852496268472410.1137/090774082
BeckermannBReichelLError estimates and evaluation of matrix functions via the Faber transformSIAM J. Numer. Anal.200947538493883257652310.1137/0807417441204.65041
AbramowitzMStegunIAHandbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables1965ChelmsfordCourier Corporation0171.38503
DruskinVKnizhnermanLZaslavskyMSolution of large scale evolutionary problems using rational Krylov subspaces with optimized shiftsSIAM J. Sci. Comput.200931537603780255656110.1137/0807424031204.65042
Schweitzer, M.: Restarting and error estimation in polynomial and extended Krylov subspace methods for the approximation of matrix functions. Ph.D. thesis, Universitätsbibliothek Wuppertal (2016)
BreitenTSimonciniVStollMLow-rank solvers for fractional differential equationsElectron. Trans. Numer. Anal.20164510713234981431338.65071
FrommerAGüttelSSchweitzerMEfficient and stable Arnoldi restarts for matrix functions based on quadratureSIAM J. Matrix Anal. Appl.2014352661683320972610.1137/13093491X1309.65050
KressnerDMasseiSRobolLLow-rank updates and a divide-and-conquer method for linear matrix equationsSIAM J. Sci. Comput.2019412A848A876392835110.1137/17M11610381448.65034
DruskinVKnizhnermanLExtended Krylov subspaces: approximation of the matrix square root and related functionsSIAM J. Matrix Anal. Appl.1998193755771161658410.1137/S0895479895292400
BraessDNonlinear Approximation Theory2012BerlinSpringer0656.41001
BennerPLiRCTruharNOn the ADI method for Sylvester equationsJ. Comput. Appl. Math.2009233410351045255729310.1016/j.cam.2009.08.1081176.65050
DruskinVSimonciniVAdaptive rational Krylov subspaces for large-scale dynamical systemsSyst. Control Lett.2011608546560285834010.1016/j.sysconle.2011.04.0131236.93035
OseledetsIVLower bounds for separable approximations of the Hilbert kernelSb. Math.20071983137144235428210.1070/SM2007v198n03ABEH0038421151.41016
BernsteinSSur les fonctions absolument monotonesActa Math.1929521166155526910.1007/BF0254740055.0142.07
HochbruckMOstermannAExponential integratorsActa Numer.201019209286265278310.1017/S09624929100000481242.65109
BeckermannBTownsendABounds on the singular values of matrices with displacement structureSIAM Rev.2019612319344394728310.1137/19M12444331441.15005Revised reprint of “On the singular values of matrices with displacement structure” [ MR3717820]
GüttelSRational Krylov approximation of matrix functions: numerical methods and optimal pole selectionGAMM-Mitt.2013361831309591210.1002/gamm.2013100021292.65043
MasseiSMazzaMRobolLFast solvers for two-dimensional fractional diffusion equations using rank structured matricesSIAM J. Sci. Comput.2019414A2627A2656399530410.1137/18M11808031420.65096
HornRAKittanehFTwo applications of a bound on the Hadamard product with a Cauchy matrixElectron. J. Linear Algebra19983412160872910.13001/1081-3810.10100890.15020Dedicated to Hans Schneider on the occasion of his 70th birthdayDedicated to Hans Schneider on the occasion of his 70th birthday
BartelsRHStewartGWAlgorithm 432: solution of the matrix equation AX+XB=C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AX+XB=C$$\end{document}Commun. ACM19721582082610.1145/361573.361582
SimonciniVComputational methods for linear matrix equationsSIAM Rev.2016583377441353279410.1137/1309128391386.65124
BenziMKlymkoCTotal communicability as a centrality measureJ. Complex Netw.20131212414910.1093/comnet/cnt007
YangQTurnerILiuFIlićMNovel numerical methods for solving the time-space fractional diffusion equation in two dimensionsSIAM J. Sci. Comput.201133311591180280056810.1137/1008006341229.35315
Berg, C.: Stieltjes–Pick–Bernstein–Schoenberg and their connection to complete monotonicity. In: Positive Definite Functions: From Schoenberg to Space-Time Challenges, pp. 15–45 (2008)
GüttelSKnizhnermanLA black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functionsBIT2013533595616309525710.1007/s10543-013-0420-x1276.65026
KaluginGAJeffreyDJCorlessRMBorweinPBStieltjes and other integral representations for functions of Lambert WIntegral Transf. Spec. Funct.2012238581593295945710.1080/10652469.2011.613830
WidderDVThe Laplace Transform1941PrincetonPrinceton University Press0063.08245
ZolotarevEApplication of elliptic functions to questions of functions deviating least and most from zeroZap. Imp. Akad. Nauk. St. Petersb.1877305159
M Hochbruck (826_CR20) 2010; 19
S Güttel (826_CR19) 2013; 53
GA Kalugin (826_CR22) 2012; 23
A Townsend (826_CR30) 2015; 299
D Kressner (826_CR23) 2019; 41
M Abramowitz (826_CR1) 1965
V Druskin (826_CR16) 2011; 60
S Güttel (826_CR18) 2013; 36
RA Horn (826_CR21) 1998; 3
Q Yang (826_CR32) 2011; 33
A Frommer (826_CR17) 2014; 35
T Breiten (826_CR12) 2016; 45
826_CR29
B Beckermann (826_CR4) 2009; 47
S Massei (826_CR25) 2018; 39
V Druskin (826_CR14) 2009; 31
S Massei (826_CR24) 2019; 41
D Braess (826_CR11) 2012
826_CR27
RH Bartels (826_CR2) 1972; 15
DV Widder (826_CR31) 1941
E Zolotarev (826_CR33) 1877; 30
S Bernstein (826_CR10) 1929; 52
B Beckermann (826_CR3) 2011; 49
826_CR9
P Benner (826_CR6) 2009; 233
V Druskin (826_CR13) 1998; 19
V Simoncini (826_CR28) 2016; 58
IV Oseledets (826_CR26) 2007; 198
B Beckermann (826_CR5) 2019; 61
M Benzi (826_CR8) 2017; 135
M Benzi (826_CR7) 2013; 1
V Druskin (826_CR15) 2010; 32
References_xml – reference: FrommerAGüttelSSchweitzerMEfficient and stable Arnoldi restarts for matrix functions based on quadratureSIAM J. Matrix Anal. Appl.2014352661683320972610.1137/13093491X1309.65050
– reference: BenziMKlymkoCTotal communicability as a centrality measureJ. Complex Netw.20131212414910.1093/comnet/cnt007
– reference: BeckermannBTownsendABounds on the singular values of matrices with displacement structureSIAM Rev.2019612319344394728310.1137/19M12444331441.15005Revised reprint of “On the singular values of matrices with displacement structure” [ MR3717820]
– reference: KaluginGAJeffreyDJCorlessRMBorweinPBStieltjes and other integral representations for functions of Lambert WIntegral Transf. Spec. Funct.2012238581593295945710.1080/10652469.2011.613830
– reference: KressnerDMasseiSRobolLLow-rank updates and a divide-and-conquer method for linear matrix equationsSIAM J. Sci. Comput.2019412A848A876392835110.1137/17M11610381448.65034
– reference: BenziMSimonciniVApproximation of functions of large matrices with Kronecker structureNumer. Math.20171351126359210510.1007/s00211-016-0799-91365.65134
– reference: HochbruckMOstermannAExponential integratorsActa Numer.201019209286265278310.1017/S09624929100000481242.65109
– reference: TownsendAOlverSThe automatic solution of partial differential equations using a global spectral methodJ. Comput. Phys.2015299106123338471910.1016/j.jcp.2015.06.0311352.65579
– reference: GüttelSRational Krylov approximation of matrix functions: numerical methods and optimal pole selectionGAMM-Mitt.2013361831309591210.1002/gamm.2013100021292.65043
– reference: GüttelSKnizhnermanLA black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functionsBIT2013533595616309525710.1007/s10543-013-0420-x1276.65026
– reference: AbramowitzMStegunIAHandbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables1965ChelmsfordCourier Corporation0171.38503
– reference: BeckermannBAn error analysis for rational Galerkin projection applied to the Sylvester equationSIAM J. Numer. Anal.201149624302450285460310.1137/1108245901244.65057
– reference: Susnjara, A., Perraudin, N., Kressner, D., Vandergheynst, P.: Accelerated filtering on graphs using Lanczos method. arXiv preprint arXiv:1509.04537 (2015)
– reference: DruskinVLiebermanCZaslavskyMOn adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problemsSIAM J. Sci. Comput.201032524852496268472410.1137/090774082
– reference: OseledetsIVLower bounds for separable approximations of the Hilbert kernelSb. Math.20071983137144235428210.1070/SM2007v198n03ABEH0038421151.41016
– reference: MasseiSMazzaMRobolLFast solvers for two-dimensional fractional diffusion equations using rank structured matricesSIAM J. Sci. Comput.2019414A2627A2656399530410.1137/18M11808031420.65096
– reference: BraessDNonlinear Approximation Theory2012BerlinSpringer0656.41001
– reference: BartelsRHStewartGWAlgorithm 432: solution of the matrix equation AX+XB=C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AX+XB=C$$\end{document}Commun. ACM19721582082610.1145/361573.361582
– reference: BernsteinSSur les fonctions absolument monotonesActa Math.1929521166155526910.1007/BF0254740055.0142.07
– reference: Berg, C.: Stieltjes–Pick–Bernstein–Schoenberg and their connection to complete monotonicity. In: Positive Definite Functions: From Schoenberg to Space-Time Challenges, pp. 15–45 (2008)
– reference: BreitenTSimonciniVStollMLow-rank solvers for fractional differential equationsElectron. Trans. Numer. Anal.20164510713234981431338.65071
– reference: WidderDVThe Laplace Transform1941PrincetonPrinceton University Press0063.08245
– reference: DruskinVKnizhnermanLExtended Krylov subspaces: approximation of the matrix square root and related functionsSIAM J. Matrix Anal. Appl.1998193755771161658410.1137/S0895479895292400
– reference: DruskinVSimonciniVAdaptive rational Krylov subspaces for large-scale dynamical systemsSyst. Control Lett.2011608546560285834010.1016/j.sysconle.2011.04.0131236.93035
– reference: SimonciniVComputational methods for linear matrix equationsSIAM Rev.2016583377441353279410.1137/1309128391386.65124
– reference: HornRAKittanehFTwo applications of a bound on the Hadamard product with a Cauchy matrixElectron. J. Linear Algebra19983412160872910.13001/1081-3810.10100890.15020Dedicated to Hans Schneider on the occasion of his 70th birthdayDedicated to Hans Schneider on the occasion of his 70th birthday
– reference: BeckermannBReichelLError estimates and evaluation of matrix functions via the Faber transformSIAM J. Numer. Anal.200947538493883257652310.1137/0807417441204.65041
– reference: BennerPLiRCTruharNOn the ADI method for Sylvester equationsJ. Comput. Appl. Math.2009233410351045255729310.1016/j.cam.2009.08.1081176.65050
– reference: MasseiSPalittaDRobolLSolving rank-structured Sylvester and Lyapunov equationsSIAM J. Matrix Anal. Appl.201839415641590386761710.1137/17M11571551404.65036
– reference: YangQTurnerILiuFIlićMNovel numerical methods for solving the time-space fractional diffusion equation in two dimensionsSIAM J. Sci. Comput.201133311591180280056810.1137/1008006341229.35315
– reference: DruskinVKnizhnermanLZaslavskyMSolution of large scale evolutionary problems using rational Krylov subspaces with optimized shiftsSIAM J. Sci. Comput.200931537603780255656110.1137/0807424031204.65042
– reference: Schweitzer, M.: Restarting and error estimation in polynomial and extended Krylov subspace methods for the approximation of matrix functions. Ph.D. thesis, Universitätsbibliothek Wuppertal (2016)
– reference: ZolotarevEApplication of elliptic functions to questions of functions deviating least and most from zeroZap. Imp. Akad. Nauk. St. Petersb.1877305159
– volume: 15
  start-page: 820
  year: 1972
  ident: 826_CR2
  publication-title: Commun. ACM
  doi: 10.1145/361573.361582
– volume: 31
  start-page: 3760
  issue: 5
  year: 2009
  ident: 826_CR14
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/080742403
– volume: 299
  start-page: 106
  year: 2015
  ident: 826_CR30
  publication-title: J. Comput. Phys.
  doi: 10.1016/j.jcp.2015.06.031
– ident: 826_CR29
– ident: 826_CR27
– volume: 30
  start-page: 1
  issue: 5
  year: 1877
  ident: 826_CR33
  publication-title: Zap. Imp. Akad. Nauk. St. Petersb.
– volume: 47
  start-page: 3849
  issue: 5
  year: 2009
  ident: 826_CR4
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/080741744
– volume: 61
  start-page: 319
  issue: 2
  year: 2019
  ident: 826_CR5
  publication-title: SIAM Rev.
  doi: 10.1137/19M1244433
– volume: 19
  start-page: 755
  issue: 3
  year: 1998
  ident: 826_CR13
  publication-title: SIAM J. Matrix Anal. Appl.
  doi: 10.1137/S0895479895292400
– volume: 23
  start-page: 581
  issue: 8
  year: 2012
  ident: 826_CR22
  publication-title: Integral Transf. Spec. Funct.
  doi: 10.1080/10652469.2011.613830
– volume: 58
  start-page: 377
  issue: 3
  year: 2016
  ident: 826_CR28
  publication-title: SIAM Rev.
  doi: 10.1137/130912839
– volume: 41
  start-page: A848
  issue: 2
  year: 2019
  ident: 826_CR23
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/17M1161038
– volume: 33
  start-page: 1159
  issue: 3
  year: 2011
  ident: 826_CR32
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/100800634
– volume: 53
  start-page: 595
  issue: 3
  year: 2013
  ident: 826_CR19
  publication-title: BIT
  doi: 10.1007/s10543-013-0420-x
– volume: 233
  start-page: 1035
  issue: 4
  year: 2009
  ident: 826_CR6
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2009.08.108
– ident: 826_CR9
– volume: 3
  start-page: 4
  year: 1998
  ident: 826_CR21
  publication-title: Electron. J. Linear Algebra
  doi: 10.13001/1081-3810.1010
– volume: 52
  start-page: 1
  issue: 1
  year: 1929
  ident: 826_CR10
  publication-title: Acta Math.
  doi: 10.1007/BF02547400
– volume-title: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables
  year: 1965
  ident: 826_CR1
– volume: 49
  start-page: 2430
  issue: 6
  year: 2011
  ident: 826_CR3
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/110824590
– volume: 19
  start-page: 209
  year: 2010
  ident: 826_CR20
  publication-title: Acta Numer.
  doi: 10.1017/S0962492910000048
– volume-title: Nonlinear Approximation Theory
  year: 2012
  ident: 826_CR11
– volume: 1
  start-page: 124
  issue: 2
  year: 2013
  ident: 826_CR7
  publication-title: J. Complex Netw.
  doi: 10.1093/comnet/cnt007
– volume: 135
  start-page: 1
  issue: 1
  year: 2017
  ident: 826_CR8
  publication-title: Numer. Math.
  doi: 10.1007/s00211-016-0799-9
– volume: 35
  start-page: 661
  issue: 2
  year: 2014
  ident: 826_CR17
  publication-title: SIAM J. Matrix Anal. Appl.
  doi: 10.1137/13093491X
– volume: 45
  start-page: 107
  year: 2016
  ident: 826_CR12
  publication-title: Electron. Trans. Numer. Anal.
– volume: 41
  start-page: A2627
  issue: 4
  year: 2019
  ident: 826_CR24
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/18M1180803
– volume: 32
  start-page: 2485
  issue: 5
  year: 2010
  ident: 826_CR15
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/090774082
– volume: 198
  start-page: 137
  issue: 3
  year: 2007
  ident: 826_CR26
  publication-title: Sb. Math.
  doi: 10.1070/SM2007v198n03ABEH003842
– volume-title: The Laplace Transform
  year: 1941
  ident: 826_CR31
– volume: 36
  start-page: 8
  issue: 1
  year: 2013
  ident: 826_CR18
  publication-title: GAMM-Mitt.
  doi: 10.1002/gamm.201310002
– volume: 60
  start-page: 546
  issue: 8
  year: 2011
  ident: 826_CR16
  publication-title: Syst. Control Lett.
  doi: 10.1016/j.sysconle.2011.04.013
– volume: 39
  start-page: 1564
  issue: 4
  year: 2018
  ident: 826_CR25
  publication-title: SIAM J. Matrix Anal. Appl.
  doi: 10.1137/17M1157155
SSID ssj0029613
ssj0014816
ssj0000615
Score 2.4179895
Snippet Evaluating the action of a matrix function on a vector, that is  x = f ( M ) v , is an ubiquitous task in applications. When  M  is large, one usually relies...
Evaluating the action of a matrix function on a vector, that is  $$x=f({\mathcal {M}})v$$ x = f ( M ) v , is an ubiquitous task in applications. When ...
Evaluating the action of a matrix function on a vector, that is x=f(M)v, is an ubiquitous task in applications. When M is large, one usually relies on Krylov...
SourceID proquest
crossref
springer
SourceType Aggregation Database
Enrichment Source
Index Database
Publisher
StartPage 237
SubjectTerms Approximation
Computational Mathematics and Numerical Analysis
Convergence
Error analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrix methods
Numeric Computing
Subspaces
Tensors
Title Rational Krylov for Stieltjes matrix functions: convergence and pole selection
URI https://link.springer.com/article/10.1007/s10543-020-00826-z
https://www.proquest.com/docview/2504839506
Volume 61
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV3dS8MwED_c9qIPfkzF-THy4JsWljVNW9_mcIpje9AN9Kk0aQrK7MRWkf31XmK6TVHBp0KahnK5S36X3P0O4Fj6cSqYLxz0fkKHCZ87QaoSR3oBOiBMpdTkVwyG_GrMru-8O5sUlpfR7uWVpFmpl5LdPKbvHHUmNIJiZ1aBmqd9d9TicbuztP7SOQJGsK_hj3XAQk5tZTWu_82zSTQ_j_51o1qgz28XpmYf6m3CugWQpPM541uworI6bFgwSayp5thU1mso2-qwNphztObbMLyx54Ck_4Je-xtB-Epuiwc1KR5VTp40d_870fueUc0zYgLUTa6mInGWkOfpRJHc1NHBDjsw6l2MuleOra7gSJe7hZMk7SB20xZNqSu45CxAMCICP1ZKIahjLKWBZGjQCfVlmyboaEoZssBV3JVCubtQzaaZ2gOiCWU0R43JjPJFK0xikUg0bB-xAbpDDaClJCNpmcd1AYxJtOBM1tKPUPqRkX40a8DJ_JvnT96NP3sflhMUWRvMI03OhgrotXgDTstJW7z-fbT9_3U_gNW2DnQxgWmHUC1eXtURIpVCNKHW6Z2fD_Xz8r5_0YRKl3ebRl0_AMfz3sg
linkProvider Springer Nature
linkToHtml http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV09T8MwED1BGYCBjwKiUMADG0SqG8dJ2VBFVejHAEVii2LHkUAlrZqAUH89Z9dpoQIkVse2orMvfhffewdwLv0oEcwXDkY_DYcJnztBomJHegEGIEwl1PAren3efmR3T96TlcnRXJil-3tNcfOYvmnU_GeEws50FdYYRso6fa_Jm1--unSOexHia9Bjw64Gp7aeGtdv5FnqzM-zfz-eFphz6ZrUnD6tHdiysJFcz9Z5F1ZUWoZtCyGJddAMm4oqDUVbGTZ7c2XWbA_69_bvH-lMMFZ_JwhayUP-rIb5i8rIq1bs_yD6tDMb8oqYtHTD0FQkSmMyHg0VyUz1HOywD4PWzaDZdmxNBUe63M2dOK4HkZvUaEJdwSVnAUIQEfiRUgqhHGMJDSRDN46pL-s0xvBSygYLXMVdKZR7AKV0lKpDIFpGRivTGD6UL2qNOBKxRHf2ERFgEFQBWlgylFZvXJe9GIYLpWRt_RCtHxrrh9MKXMzHjGdqG3_2rhYLFFrPy0ItyYbbzqvxClwWi7Z4_PtsR__rfgbr7UGvG3Zv-51j2KjrVBeTmlaFUj55UyeIVXJxajbpJwbB2a0
linkToPdf http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LS8NAEB58gOjBR1Wszz1402C32WxSb1It9dEiWsFbyD4CSo3FRBF_vbPbTVtFBa_JZgmzO9lvMvN9A7AvwyQVLBQeRj8Nj4mQe1GqlSeDCAMQplNq-RWdLm_fsYv74H6CxW-r3cuU5JDTYFSasuJooNKjCeJbwEz-0bCiESB7H9Mwi5GKTdQ2eXPiW0xHaBiBv4FCLhhrcOq6rHHznoEj1Pw8-9dDa4xEvyVP7ZnUWoZFBybJyXD1V2BKZxVYcsCSOLfN8VLZu6G8VoGFzkivNV-F7o37J0guXzCCfyMIZclt8aD7xaPOyZPR8X8n5gy02_SY2GJ1y9vUJMkUGTz3NcltTx0csAa91lmv2fZcpwVP-twvPKXqUeKnNZpSX3DJWYTARERhorVGgMdYSiPJ0LkVDWWdKgw6pWywyNfcl0L76zCTPWd6A4gRlzF6NZYlFYpaQyVCSXTyEHEChkZVoKUlY-lUyE0zjH481k821o_R-rG1fvxRhYPRM4OhBsefo7fLBYqdP-axEWrDzRjUeBUOy0Ub3_59ts3_Dd-DuevTVnx13r3cgvm6qX-x9WrbMFO8vOodBDCF2LV79BNRc-H0
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Rational+Krylov+for+Stieltjes+matrix+functions%3A+convergence+and+pole+selection&rft.jtitle=BIT+Numerical+Mathematics&rft.au=Massei%2C+Stefano&rft.au=Robol%2C+Leonardo&rft.date=2021-03-01&rft.issn=0006-3835&rft.eissn=1572-9125&rft.volume=61&rft.issue=1&rft.spage=237&rft.epage=273&rft_id=info:doi/10.1007%2Fs10543-020-00826-z&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s10543_020_00826_z
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0006-3835&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0006-3835&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0006-3835&client=summon