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##

The determinant of a semiseparable matrix in order
flops.

In this section, we will design an order algorithm for
calculating the determinant of a semiseparable matrix represented with
the Givens-vector representation.
We will use the fact that the Givens transformations for representing
the matrix in fact contain all the information needed for the -factorization of the corresponding matrix. Using this information, we
can very easily calculate the diagonal elements of the factor of
the semiseparable matrix. Multiplying these diagonal elements will
give us the wanted determinant of the semiseparable matrix.

We have, because of the special structure of the representation, the
Givens transformations
such that applying
on the last two rows will annihilate all the elements
except for the diagonal element, applying on the third
last and second last row, will annihilate all the elements in the
second last row, except the last two elements (note that the Givens
transformations by construction have the determinant equal to ). In
this fashion we can continue very easily to annihilate all the
elements in the strictly lower triangular part. In fact we are only
interested in the diagonal elements. Performing these transformations
will change the diagonal elements in the following way. Denote with
the diagonal elements of the semiseparable matrix, and with
the super diagonal elements. The diagonal elements
change by performing:

where the elements denote the diagonal elements of the
upper triangular matrix . Using this information one can easily
deduce an order algorithm for calculating the determinant.
More information about the -factorization, from a computational
point of view, can be found in Chapter 9 where the
more general class of semiseparable plus diagonal matrices is
considered. More information about the structure of the factor and
the factor, when calculating the -factorization of a
semiseparable plus diagonal matrix can be found in Chapter 1.

** Next:** Conclusions
** Up:** Some algorithms connected to
** Previous:** A fast matrix vector
** Contents**
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Raf Vandebril
2004-05-03