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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C#

C# 1.0 source.
bidiagonal.csharp.zip - Reduction of a general matrix to bidiagonal form

C++

C++ source.
bidiagonal.cpp.zip - Reduction of a general matrix to bidiagonal form
ablas.zip - optimized basic linear algebra subroutines with SSE2 support (for C++ sources only)

C++, multiple precision arithmetic

C++ source. MPFR/GMP is used.
GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.
bidiagonal.mpfr.zip - Reduction of a general matrix to bidiagonal form
mpfr.zip - precompiled Win32 MPFR/GMP binaries

Delphi

Delphi source.
Can be compiled under FPC (in Delphi compatibility mode).
bidiagonal.delphi.zip - Reduction of a general matrix to bidiagonal form

Visual Basic 6

Visual Basic 6 source.
bidiagonal.vb6.zip - Reduction of a general matrix to bidiagonal form

Zonnon ^beta

Zonnon source.
Zonnon is an experimental language developed at ETH Zurich.
See www.zonnon.ethz.ch for more information.
bidiagonal.zonnon.zip - Reduction of a general matrix to bidiagonal form