The square matrix A could be represented as A=Q·H·Q ^T, where Q is an orthogonal matrix, and H is a matrix in upper Hessenberg form.

The matrix in upper Hessenberg form is as follows (non-zero elements are marked with an "X"):

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 the elements.

As a result of subroutine RMatrixHessenberg, the matrix A is replaced with matrix H 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.

As with QR decomposition, the subroutines for "unpacking" matrices Q and H are presented: RMatrixHessenbergUnpackQ and RMatrixHessenbergUnpackH. The first subroutine allows to get matrix Q, the second subroutine allows to get matrix H.

This algorithm is transferred from the LAPACK library.

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C#

C# 1.0 source.
hessenberg.csharp.zip - Reduction of a square matrix to upper Hessenberg form

C++

C++ source.
hessenberg.cpp.zip - Reduction of a square matrix to upper Hessenberg 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.
hessenberg.mpfr.zip - Reduction of a square matrix to upper Hessenberg form
mpfr.zip - precompiled Win32 MPFR/GMP binaries

Delphi

Delphi source.
Can be compiled under FPC (in Delphi compatibility mode).
hessenberg.delphi.zip - Reduction of a square matrix to upper Hessenberg form

Visual Basic 6

Visual Basic 6 source.
hessenberg.vb6.zip - Reduction of a square matrix to upper Hessenberg form

Zonnon ^beta

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