THESE de DOCTORAT - cerfacs

3.3 Solving the system Ax = b 49
where ̂Q m+1 ∈ C n×m+1 is an orthonormal matrix and ˜H m ∈ C m+1×m is an upper-Hessenberg
matrix 4 .
̂D y ˜H y
̂Q m+1
m
m +1× m
m
n × m
n × m +1
Figure 3.6: Hessenberg decomposition
The least square problem is now given by
Find y such that ‖r 0 − Q m+1 ˜H m y‖ 2 = ‖r m ‖ 2 is minimized (3.21)
Let us introduce r 0 = q 1 ‖r 0 ‖ (see Eq. 3.43). The vector q 1 can be expressed as Q m+1 e 1 = q 1
where e 1 is the canonical unit vector e 1 = (1, 0, 0, ...). Furthermore, the invariance of inner
products means that the angles between vectors are preserved, and so are their lengths:
‖Qx‖ = ‖x‖. The final expression results in
‖Q m+1
(
‖r0 ‖e 1 − ˜H m y ) ‖ 2 = ‖ ( ‖r‖e 1 − ˜H m y ) ‖ 2 (3.22)
leading to
Find y such that ‖ ( ‖r‖e 1 − ˜H m y ) ‖ 2 = ‖r m ‖ 2 is minimized (3.23)
Once y is found through the least square methodology, a solution for the Ax m ≈ b problem can
be found by solving x m = x 0 + ̂Q m y. There are still two crucial issues to resolve:
1. How to construct such matrices ̂Q m and ˜H m ?
2. What is the ‘least square problem’ about ?
These two issues will be addressed in sections (3.3.1) and (3.3.2) respectively.
4 An upper- Hessenberg matrix H is a matrix with zeros below the first subdiagonal