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@ARTICLE{Frommer:137505,
      author       = {Frommer, A. and Kahl, K. and Lippert, Th. and Rittich, H.},
      title        = {2-{N}orm {E}rror {B}ounds and {E}stimates for {L}anczos
                      {A}pproximations to {L}inear {S}ystems and {R}ational
                      {M}atrix {F}unctions},
      journal      = {SIAM journal on matrix analysis and applications},
      volume       = {34},
      number       = {3},
      issn         = {1095-7162},
      address      = {Philadelphia, Pa.},
      publisher    = {Soc.},
      reportid     = {FZJ-2013-03942},
      pages        = {1046 - 1065},
      year         = {2013},
      abstract     = {The Lanczos process constructs a sequence of orthonormal
                      vectors $v_m$ spanning a nested sequence of Krylov subspaces
                      generated by a hermitian matrix $A$ and some starting vector
                      $b$. In this paper we show how to cheaply recover a
                      secondary Lanczos process starting at an arbitrary Lanczos
                      vector $v_m$. This secondary process is then used to
                      efficiently obtain computable error estimates and error
                      bounds for the Lanczos approximations to the action of a
                      rational matrix function on a vector. This includes, as a
                      special case, the Lanczos approximation to the solution of a
                      linear system $Ax = b$. Our approach uses the relation
                      between the Lanczos process and quadrature as developed by
                      Golub and Meurant. It is different from methods known so far
                      because of its use of the secondary Lanczos process. With
                      our approach, it is now possible in particular to
                      efficiently obtain upper bounds for the error in the 2-norm,
                      provided a lower bound on the smallest eigenvalue of $A$ is
                      known. This holds in particular for a large class of
                      rational matrix functions, including best rational
                      approximations to the inverse square root and the sign
                      function. We compare our approach to other existing error
                      estimates and bounds known from the literature and include
                      results of several numerical experiments.},
      cin          = {JSC},
      ddc          = {510},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {411 - Computational Science and Mathematical Methods
                      (POF2-411)},
      pid          = {G:(DE-HGF)POF2-411},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000325092700010},
      doi          = {10.1137/110859749},
      url          = {https://juser.fz-juelich.de/record/137505},
}