Numerical Methods Contents - SAM

Example 2.8.7 (Complexity of Householder QR-factorization).
If
m > n, rank(R) = rank(A) = n (full rank)
Code 2.8.8: timing MATLAB QR-factorizations
1 % Timing QR factorizations
2
3 K = 3 ; r = [ ] ;
4 for n = 2 . ^ ( 2 : 6 )
5 m = n∗n ;
6
7 A = ( 1 :m) ’ ∗ ( 1 : n ) + [ eye ( n ) ; ones (m−n , n ) ] ;
8 t1 = 1000; for k =1:K, t i c ; [Q,R] = qr (A) ; t1 = min ( t1 , toc ) ; clear Q,R;
end
9 t2 = 1000; for k =1:K, t i c ; [Q,R] = qr (A, 0 ) ; t2 = min ( t2 , toc ) ; clear
Q,R; end
10 t3 = 1000; for k =1:K, t i c ; R = qr (A) ; t3 = min ( t3 , toc ) ; clear R; end
11 r = [ r ; n , m , t1 , t2 , t3 ] ;
12 end
➤ {q·,1 ,...,q·,n } is orthonormal basis of Im(A) with
Span { q·,1 , ...,q·,k
}
= Span
{
a·,1 , ...,a·,k
}
,1 ≤ k ≤ n .
Remark 2.8.10 (Keeping track of unitary transformations).
☛
How to store
For Householder reflections
G i1 j 1
(a 1 , b 1 ) · · · · · G ik j k
(a k ,b k ) ,
H(v 1 ) · · · · · H(v k )
H(v 1 ) · · · · · H(v k ):
For in place QR-factorization of A ∈ K m,n :
!) in lower triangle of A
store v 1 ,...,v k
store "‘Householder vectors” v j (decreasing size
?
△
Ôº¾¼ ¾º
Ôº¾¼ ¾º
10 2
[Q,R] = qr(A)
[Q,R] = qr(A,0)
R = qr(A)
O(n 4 )
tic-toc-timing of different variants of QRfactorization
in MATLAB
✄
time [s]
10 1
10 0
10 −1
O(n 6 )
R
R
10 3 n
Use [Q,R] = qr(A,0), if output sufficient
!
10 −2
10 −3
10 −4
↑ Case m < n
10 −5
10 0 10 1 10 2
Fig. 24 ✸
= Householder vectors
Remark 2.8.9 (QR-orthogonalization).
⎛ ⎞ ⎛
A
=
Q
⎜ ⎟ ⎜
⎝ ⎠ ⎝
⎞
⎛
⎜
⎝
⎟
⎠
R
⎞
⎟
⎠ , A,Q ∈ Km,n ,R ∈ K n,n .
Ôº¾¼ ¾º
Case m > n →
☛ Convention for Givens rotations (K = R)
⎧
( )
⎪⎨ 1 , if γ = 0 ,
γ σ
G =
⇒ store ρ := 1
−σ γ
2 sign(γ)σ , if |σ| < |γ| ,
⎪⎩
2 sign(σ)/γ , if |σ| ≥ |γ| .
⎧
⎨ ρ = 1 ⇒ γ = 0 , σ = 1
|ρ| < 1 ⇒ σ = 2ρ , γ = √ 1 − σ
⎩
2
Ôº¾¼ ¾º
|ρ| > 1 ⇒ γ = 2/ρ , σ = √ 1 − γ 2 .