Bidiagonal decomposition

The rectangular matrix A could be represented as A=Q·B·P^T, where Q and P are orthogonal matrices, and B is a bidiagonal matrix.

Like other algorithms of orthogonal factorization (for example, QR and LQ decomposition algorithms), this algorithm uses a sequence of elementary reflections to transform the matrix A. The matrix is transformed from the left and from the right. The transformations gradually remove off-diagonal elements until all elements will be nulled except those located on 2 diagonals. If the number of rows in the matrix A is no less than the number of columns, we'll get an upper bidiagonal matrix. Otherwise, we'll get a lower bidiagonal matrix.

As a result of subroutine RMatrixBD, the matrix A is replaced by the bidiagonal matrix B and a sequence of reflection transformations stored in compact form. The form in which the transformations are stored and the subroutine parameters are described in more details in the subroutine comments.

The method is similar to QR decomposition which uses the lower part of matrix R to store matrix Q, but here two matrices are stored: matrix Q is stored in the lower part of matrix B, and matrix P is stored in the upper part.

As with QR decomposition, the subroutines for "unpacking" matrices Q and P are presented: RMatrixBDUnpackQ and RMatrixBDUnpackPT. The first subroutine allows to get matrix Q or some of its columns, the second subroutine allows to get matrix P^T or some of its rows. The subroutine RMatrixBDUnpackDiagonals "unpacks" the diagonals of a bidiagonal matrix.

This algorithm is transferred from the LAPACK library.

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