Abstract
The spectral portrait of a matrix is the picture of its ɛ-spectra forε ∈[ε 1,ε 2], where an ɛ-spectrum ofA is the union of all the eigenvalues of all the matricesA+Δ with ∥Δ∥2≤ε∥A∥2. The spectral portrait is, for example, useful to study the stability of various problems, or, as we illustrate in this paper, to visualize the condition number of an eigenvalue. Some methods to estimate the spectral portrait already exist, but only for small matrices. We propose here a new algorithm for non hermitian large sparse matrices.
Zusammenfassung
Die Spektrale Darstellung einer Matrix ist das Bild ihres ɛ-Spektrum mitε ∈[ε 1,ε 2], wo das ɛ-Spektrum vonA als Vereinigung der Eigenwerte der MatrizenA+Δ mit ∥Δ∥2≤ε∥A∥2 zu verstehen ist. Spektrale Darstellung werden, zum Beispiel, im Fall Stabilitätsuntersuchung bestimmter Probleme oder bei der Konditionsdarstellung einer Eigenwert angewandt. Mehrere Methoden die, die Spektrale Darstellung für kleine Matrizen berechnen, sind schon vorhanden. Hier wird ein neuer Algorithms für große nicht hemitesche dünnbesetzte Matrizen vorgestellt.
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Carpraux, J.F., Erhel, J. & Sadkane, M. Spectral portrait for non-hermitian large sparse matrices. Computing 53, 301–310 (1994). https://doi.org/10.1007/BF02307381
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DOI: https://doi.org/10.1007/BF02307381