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:
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.