Root class TDecompQRH - how to use

The question is actually: How to ubderstand a completely misleading description of the class?

The question has been raised in RootTalk -- TDecompQRH - how to use the householder decomposition? -- but was not answered properly by the author, Eddy Offermann .

Here is a snpashot of the description from the Root User's Guide v5.26:

Having inspected the code, I found that the code may be ok, but the description is wrong and requires the following corrections line by line:

original	comment / corrected version
Decompose an (m x n)-matrix A with m ≥ n.	ok
A = QRH	A = QR no matrix H !
Q : orthogonal (m x n) - matrix, stored in fQ;	Everything wrong! "Orthogonal" - yes, but : (m x m) instead of (m x n) not stored in fQ Q is not explicitly stored anywhere Q is defined as , where where is a Householder matrix of dimension ( with ) which performs a Householder transformation by reflecting about a hyperplane that is perpendicular to a Householder vector can be expessed via as where That allows the matrix Q to be stored implicitly via a set of (in fQ and fUp, see below) and - for faster computations in future - a set of (in fW). Now we can describe what does fQ contain? before the decomposition it contains the matrix A after the decomposition: it contains the upper triangular matrix R in the diagonal and above the latter it also contains a part of vectors , namely, the k-th column contains under the diagonal i.e. the without its first element
R : upper triangular (n x n)-matrix, stored in fR;	It is found as , calculated iteratively: (to be noted: ) First, R is computed and placed into upper part of fQ, then copied into fR
H : (n x n)-Householder matrix, stored through;	The H matrix is a nonsence: as we see, the involved matrices are of dimension (m x m), and the Householder matrices of dimensions (m x m), (m-1 x m-1), ... .
fUp : n-vector with Householder up‘s;	Correct if one deciphers the "Householder up‘s" as upper components of Householder vectors , i.e.
fW : n-vector with Householder beta‘s.	Similarly, an explicit definition is much better:
The decomposition fails if in the formation of reflectors a zero appears, i.e. singularity.	? (I did not check this statement)

• 3ed.pdf
• 2ed_ru.djvu