The generalized null space decomposition (GNSD) is a unitary reductionof a general matrix $A$ of order $n$ to a block upper triangular formthat reveals the structure of the Jordan blocks of $A$ correspondingto a zero eigenvalue. The reduction was introduced by Kublanovskaya.It was extended first by Ruhe and then by Golub and Wilkinson, whobased the reduction on the singular value decomposition.Unfortunately, if $A$ has large Jordan blocks, the complexity of thesealgorithms can approach the order of $n^4$. This paper presents analternative algorithm, based on repeated updates of a QR decompositionof $A$, that is guaranteed to be of order $n^{3}$. Numericalexperiments confirm the stability of this algorithm, which turns outto produce essentially the same form as that of Golub and Wilkinson.The effect of errors in $A$ on the ability to recover the originalstructure is investigated empirically. Several applications are discussed,including the computation of the Drazin inverse.

An efficient algorithm for computing the generalized null space decomposition

GUGLIELMI, NICOLA;
2015

Abstract

The generalized null space decomposition (GNSD) is a unitary reductionof a general matrix $A$ of order $n$ to a block upper triangular formthat reveals the structure of the Jordan blocks of $A$ correspondingto a zero eigenvalue. The reduction was introduced by Kublanovskaya.It was extended first by Ruhe and then by Golub and Wilkinson, whobased the reduction on the singular value decomposition.Unfortunately, if $A$ has large Jordan blocks, the complexity of thesealgorithms can approach the order of $n^4$. This paper presents analternative algorithm, based on repeated updates of a QR decompositionof $A$, that is guaranteed to be of order $n^{3}$. Numericalexperiments confirm the stability of this algorithm, which turns outto produce essentially the same form as that of Golub and Wilkinson.The effect of errors in $A$ on the ability to recover the originalstructure is investigated empirically. Several applications are discussed,including the computation of the Drazin inverse.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/2816
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