Root class TDecompQRH  how to use 
The question is actually: How to understand 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 (which differs very little from what can be found in Reference Manual or in the comments in source files):
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

Q : orthogonal (m x n)  matrix, stored in fQ;  Everything is wrong! "Orthogonal"  yes, but :
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?

R : upper triangular (n x n)matrix, stored in fR;  Right. R 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 nonsense: as we see, the involved matrices are of dimension (m x m), and the matrices of dimensions (m x m), (m1 x m1), ... . 
fUp : nvector with Householder up‘s;  This is correct if one deciphers [3] the "Householder up‘s" as upper components of Householder vectors , i.e. 
fW : nvector 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) 
[1] Definition of Householder vectors for the decomposition.
The vector is chosen such that the corresponding
Householder transform would reflect the first column of matrix A
[2] Actually, matrices also represent Householder reflections in with Householder vectors
[3] Do terms "Householder's up's and beta's" come from the book "Matrix Computations" by Golub and Van Loan ?
There is a comment in the text of TDecompQRH::Decompose
function:
I  Attachment  History  Action  Size  Date  Who  Comment 

gif  QRH_userguide526.gif  r1  manage  15.3 K  20110503  20:05  AlexanderFedotov  
djvu  Golub_VanLoan.Matr_comp_2ed_ru.djvu  r1  manage  7867.5 K  20110502  04:37  AlexanderFedotov  
Golub_VanLoan.Matr_comp_3ed.pdf  r1  manage  11833.8 K  20110502  04:25  AlexanderFedotov  
Golub_VanLoan.Matr_comp_2ed_ru.pdf  r1  manage  56447.9 K  20110504  19:50  AlexanderFedotov 