Calculating a function of a matrix with a real spectrum |
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Authors: | Kubelík P Kurbatov V G Kurbatova I V |
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Institution: | 1.J. Heyrovsky Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolej?kova 3, 18223, Prague 8, Czech Republic ;2.Department of System Analysis and Control, Voronezh State University 1, Universitetskaya Square, Voronezh, 394018,, Russia ;3.Department of Software Development and Information Systems Administration, Voronezh State University 1, Universitetskaya Square, Voronezh, 394018,, Russia ; |
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Abstract: | Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries tii are arranged in increasing order. To avoid calculations using the differences tii ? tjj with close (including equal) tii and tjj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits. |
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