This paper concerns the study of a unitary transformation from a generic symmetric matrix A into a semiseparable matrix. The problem is studied both theoretically and from an algorithmic point of view. In particular, we first give a proof of the existence of such a transformation and then we discuss the uniqueness of such transformation proving an Implicit-Q Theorem for semiseparable matrices. Finally, we study structural properties of the factors of the QR-decomposition of a semiseparable matrix. These properties allows us to design a method based on the QR iterations applied to a semiseparable matrix for reducing a symmetric matrix to semiseparable form. This method has the same asymptotic cost of the reduction of a symmetric matrix to tridiagonal form. Once the transformation has been accomplished, if one is interested in computing the eigenvalues each further QR iteration can be done in linear time.
Existence, uniqueness and algorithms for matrix unitary reduction to semiseparable form
BEVILACQUA, ROBERTO;DEL CORSO, GIANNA MARIA
2003-01-01
Abstract
This paper concerns the study of a unitary transformation from a generic symmetric matrix A into a semiseparable matrix. The problem is studied both theoretically and from an algorithmic point of view. In particular, we first give a proof of the existence of such a transformation and then we discuss the uniqueness of such transformation proving an Implicit-Q Theorem for semiseparable matrices. Finally, we study structural properties of the factors of the QR-decomposition of a semiseparable matrix. These properties allows us to design a method based on the QR iterations applied to a semiseparable matrix for reducing a symmetric matrix to semiseparable form. This method has the same asymptotic cost of the reduction of a symmetric matrix to tridiagonal form. Once the transformation has been accomplished, if one is interested in computing the eigenvalues each further QR iteration can be done in linear time.File | Dimensione | Formato | |
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