Numerical Methods Contents - SAM

Computation of Choleskyfactorization
I + αww H = R H 1 R 1 .
Motivation: “recursion” (2.9.8).
➜ asymptotic complexity O(n 2 )
(O(n), if only d,s computed
→ (2.9.9))
Code 2.9.6: Cholesky factorization of rank-1-modified identity
matrix
1 function [ d , s ] = r o i d ( alpha ,w)
2 n = length (w) ;
3 d = [ ] ; s = [ ] ;
4 for i =1:n
5 t = alpha∗w( i ) ;
6 d = [ d ; sqrt (1+ t ∗w( i ) ) ] ;
7 s = [ s ; t / d ( i ) ] ;
8 alpha = alpha − s ( i ) ^2;
9 end
2.9.0.2 Adding a column
Let us adopt an academic point of view: Before we have seen how to update a QR-factorization in
the case of rank-1-modification of a square matrix.
However, the QR-factorization makes sense for an arbitrary rectangular matrix. A possible modification
of rectangular matrices is achieved by adding a row or a column. How can QR-factors updated
efficiently for these kinds of modifications.
An application of these modification techniques will be given in Chapter 6.
3 Special structure of R 1 :
⎛
d 1 .
R 1 = ⎜
..
⎝ ...
⎞ ⎛
s 1 ⎟
⎠ + .
⎜
..
⎝
d n
...
⎛
⎞
⎞ 0 w 2 w 3 · · · · · · w n
0 0 w 3 · · · · · · w n
⎟
. . .. .
⎠
⎜. 0 w n−1 w n
⎟
s n ⎝0 · · · · · · 0 w n ⎠
0 · · · · · · 0
(2.9.9)
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Known:
A ∈ K m,n ↦→ Ã = [ (A :,1 ,...,(A) :,k−1 ,v, (A) :,k ,...,(A) :,n
]
, v ∈ K m . (2.9.10)
QR-factorization A = QR, Q ∈ K m,m unitary R ∈ K m,n upper triangular matrix.
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Idea: Easy, if k = n + 1 (adding last column)
Task: Efficient computation of QR-factorization à = ˜Q˜R of à from (2.9.10) , ˜Q ∈ K m,m unitary,
˜R ∈ K m,n+1 upper triangular
⎛
⎞
⎛ ⎞ ⎛ ⎞ 0 1 1 · · · · · · 1 ⎛ ⎞
d 1 s 1 0 0 1 · · · · · · 1
w 1 . .
R 1 = ⎜ .
⎟
⎝ . .. ⎠ + .
⎜
..
⎟
. ... .
. ⎜
..
⎟
⎝ . .. ⎠
⎜. 0 1 1
⎝
⎟
... ⎠
d n s n ⎝0 · · · · · · 0 1⎠
w n
0 · · · · · · 0
Smart
multiplication
˜R := R 1 R
Code 2.9.8: Update of Cholesky factorization in the case of s.p.d. preserving
rank-1-modification
1 function Rt = roudchol (R, alpha , v )
2 w = R ’ \ v ;
➜ Complexity O(n 2 3 [ d , s ] = r o i d ( alpha ,w) ;
)
4 T = zeros ( 1 , n ) ;
5 for j =n−1:−1:1
A + αvv H = ˜R H ˜R .
6 T = [w( j +1)∗R( j + 1 , : ) +T ( 1 , : ) ; T ] ;
7 end
8 Rt = spdiags ( d , 0 , n , n ) ∗R+spdiags ( s , 0 , n , n ) ∗T ;
MATLAB-function: R = cholupdate(R,v);
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∃ column permutation
k ↦→ n + 1 , i ↦→ i − 1 , i = k + 1, ...,n + 1
∼ permutation matrix
⎛
⎞
1 0 · · · · · · 0
0 . ..
1 0
P =
. 0 1
∈ R n+1,n+1
⎜.
...
⎟
⎝
1⎠
0 · · · 1 0
à −→
1 If m > n + 1:
A 1 = ÃP = [a·1,...,a·n ,v] = Q [ R Q H v ] = Q
⎜
column Q H v ⎝
case m > n + 1
∃ orthogonal transformation Q 1 ∈ K m,m (Householder reflection) with
column k
⎛
R
⎞
⎟
⎠
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