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Finite Section Method and Stability

  • Albrecht Böttcher
  • Bernd Silbermann
Chapter
  • 598 Downloads
Part of the Universitext book series (UTX)

Abstract

Let \( A = \left( {{a_{jk}}} \right)_{j,k = 1}^\infty \) be an infinite matrix and suppose A generates a bounded operator on l2. In order to solve the equation Ax = y, i.e., the infinite linear system
$$ \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{a_{11}}}{{a_{12}}}{{a_{13}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} {{a_{21}}}{{a_{22}}}{{a_{23}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} {{a_{31}}}{{a_{32}}}{{a_{33}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} \ldots \ldots \ldots \ldots \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \\ \vdots \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ {{y_3}} \\ \vdots \end{array}} \right) $$
(2.1)
we consider the truncated systems
$$ \left( {\begin{array}{*{20}c} {a_{11} } & \ldots & {a_{1n} } \\ \vdots & {} & \vdots \\ {a_{n1} } & \cdots & {a_{nn} } \\ \end{array}} \right)\left( \begin{array}{l} x_1^{(n)} \\ \vdots \\ x_n^{(n)} \\ \end{array} \right) = \left( \begin{array}{l} y_1 \\ \vdots \\ y_n \\ \end{array} \right). $$
(2.2)

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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Albrecht Böttcher
    • 1
  • Bernd Silbermann
    • 1
  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

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