Successively Ordered Elementary Bidiagonal Factorization

Further applications of the Cauchon algorithm to rank determination and bidiagonal factorization

“…The existence of a bidiagonal factorization for a given matrix has been linked to certain types of rank conditions associated with the ranks of sets of submatrices, and we make use of this connection here by incorporating them both with Cauchon matrices (or diagrams) and the condensed Cauchon algorithm. The definitions in this section can be found in [10].…”

“…If l jk or u jk is 0 in any factor, then that factor is not included in the ESEB factorization. It was shown in [10,Theorem 9] that if an ESEB factorization exists, then it is unique. A very interesting by-product of this characterization (see Theorem 4.8) is the complexity saving associated with determining if a matrix satisfies the row and/or column rank descending conditions.…”

“…Elementary bidiagonal factorization itself is linked to certain types of rank conditions associated with submatrices called descending rank conditions [10]; see [8] for an earlier paper discussing these conditions. We directly link Cauchon diagrams (or matrices) to the descending rank conditions hereby reducing the complexity by some orders of magnitude.…”

Erratum: Successively Ordered Elementary Bidiagonal Factorization

“…An error in the statement of Theorem 15 in [1] was recently discovered. That statement is correct if the matrix A is a unit lower triangular matrix, as was assumed in an earlier version, but is not true in general.…”

“…Another correction in [1] is as follows: ( n 2 ) in line 6 of the paragraph after Example 1 on page 1080 should be ( n+1 2 ).…”