-- AlexanderFedotov - 03-May-2011

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:

QRH_userguide526.gif

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 ups; Correct if one deciphers the "Householder ups" as upper components of Householder vectors , i.e.
fW : n-vector with Householder betas. 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)

Topic attachments
I Attachment History Action Size Date Who Comment
Unknown file formatdjvu Golub_VanLoan.Matr_comp_2ed_ru.djvu r1 manage 7867.5 K 2011-05-02 - 04:37 AlexanderFedotov  
PDFpdf Golub_VanLoan.Matr_comp_3ed.pdf r1 manage 11833.8 K 2011-05-02 - 04:25 AlexanderFedotov  
GIFgif QRH_userguide526.gif r1 manage 15.3 K 2011-05-03 - 20:05 AlexanderFedotov  
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Topic revision: r3 - 2011-05-04 - AlexanderFedotov
 
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