We can compute the QR decomposition by
- Householder transformations ✅
- Gram Schmidt orthogonalization ✅
- Givens transformations ✅
Theorem:
If has full column rank, i.e, Rank(
)=
then
has the QR decomposition:
where is orthogonal and
is nonsingular upper triangular.
python main.py ######################################################################################
We regrad the results calculated by SciPy as Ground Truth (GT).
*. [Q_gt, R_gt] are the right answers.
Press "1". [Q1, R1] are the answers calculated by Householder.
Press "2". [Q2, R2] are the answers calculated by Gram-Schimidt.
Press "3". [Q3, R3] are the answers calculated by Givens.
######################################################################################
please choose the mode:1
SciPy GT.
Q_gt:
[[-0.22449845 -0.14701545 0.13135285 0.95432349]
[ 0.17229869 -0.97252432 -0.12679039 -0.09183577]
[-0.10862132 0.09695988 -0.98148158 0.12447528]
[ 0.95295069 0.15225535 -0.05800457 0.25561448]]
R_gt:
[[-0.54177081 0.88741156 0.61971468 -0.5685338 ]
[ 0. 1.2231714 -1.27234912 -2.05923665]
[ 0. 0. -1.86608816 0.0738163 ]
[ 0. 0. 0. -0.35487044]]
Verify A=Q_gt*R_gt
True
Householder.
Q:
[[ 0.22449845 -0.14701545 -0.13135285 0.95432349]
[-0.17229869 -0.97252432 0.12679039 -0.09183577]
[ 0.10862132 0.09695988 0.98148158 0.12447528]
[-0.95295069 0.15225535 0.05800457 0.25561448]]
R:
[[ 0.54177081 -0.88741156 -0.61971468 0.5685338 ]
[ 0. 1.2231714 -1.27234912 -2.05923665]
[ 0. -0. 1.86608816 -0.0738163 ]
[-0. 0. 0. -0.35487044]]
Verify A=Q*R
True
We regrad the results calculated by SciPy as Ground Truth (GT).
*. [Q_gt, R_gt] are the right answers.
Press "1". [Q1, R1] are the answers calculated by Householder.
Press "2". [Q2, R2] are the answers calculated by Gram-Schimidt.
Press "3". [Q3, R3] are the answers calculated by Givens.
######################################################################################
please choose the mode:2
SciPy GT.
Q_gt:
[[-0.44866273 -0.55346929 0.40539809 0.57273545]
[ 0.33963341 -0.33279768 0.69089137 -0.54457687]
[-0.76208911 -0.13804063 -0.1717496 -0.60882431]
[ 0.32026721 -0.75090761 -0.57343141 -0.06887009]]
R_gt:
[[ 1.69018432 0.44829956 0.39125491 -0.10082661]
[ 0. -0.77033621 -1.11526531 -1.18449026]
[ 0. 0. -1.13237619 2.03735831]
[ 0. 0. 0. 0.3770548 ]]
Verify A=Q_gt*R_gt
True
Gram_Schimidt.
Q:
[[-0.44866273 0.55346929 -0.40539809 0.57273545]
[ 0.33963341 0.33279768 -0.69089137 -0.54457687]
[-0.76208911 0.13804063 0.1717496 -0.60882431]
[ 0.32026721 0.75090761 0.57343141 -0.06887009]]
R:
[[ 1.69018432 0.44829956 0.39125491 -0.10082661]
[ 0. 0.77033621 1.11526531 1.18449026]
[ 0. 0. 1.13237619 -2.03735831]
[ 0. 0. 0. 0.3770548 ]]
Verify A=Q*R
True
######################################################################################
We regrad the results calculated by SciPy as Ground Truth (GT).
*. [Q_gt, R_gt] are the right answers.
Press "1". [Q1, R1] are the answers calculated by Householder.
Press "2". [Q2, R2] are the answers calculated by Gram-Schimidt.
Press "3". [Q3, R3] are the answers calculated by Givens.
######################################################################################
please choose the mode:3
SciPy GT.
Q_gt:
[[-0.19226559 -0.03612689 -0.85748615 -0.47586374]
[-0.3035267 -0.72079242 -0.22423525 0.58141928]
[ 0.25503492 0.55778387 -0.44341219 0.65362066]
[-0.89769854 0.40991497 0.1334979 0.09102393]]
R_gt:
[[ 2.74233718 -1.24420653 0.50836465 -0.44123894]
[ 0. 1.77774359 -0.35345856 -2.15600945]
[ 0. 0. 0.97608152 0.0262946 ]
[ 0. 0. 0. -0.89003521]]
Verify A=Q_gt*R_gt
True
Givens.
Q:
[[-0.19226559 -0.03612689 -0.85748615 0.47586374]
[-0.3035267 -0.72079242 -0.22423525 -0.58141928]
[ 0.25503492 0.55778387 -0.44341219 -0.65362066]
[-0.89769854 0.40991497 0.1334979 -0.09102393]]
R:
[[ 2.74233718 -1.24420653 0.50836465 -0.44123894]
[ 0. 1.77774359 -0.35345856 -2.15600945]
[ 0. 0. 0.97608152 0.0262946 ]
[ 0. 0. 0. 0.89003521]]
Verify A=Q*R
True
eval `ssh-agent -s`
ssh-add ~/.ssh/id_rsa