THESE de DOCTORAT - cerfacs

3.3 Solving the system Ax = b 51
v n = a n −
n
∑
j=1
(q ∗ j a n)q j → q n = v n
‖v n ‖
(3.29)
This procedure of obtention of q n orthonormal matrices is known as the Gram-Schmidt Orthogonalization.
This algorithm is useful to understand the basic procedure of orthogonalization.
Moreover, in addition to the matrix Q, the upper triangular matrix R is obtained straightforward
by doing
r ij = q ∗ i a j for every i ̸= j and r ii = ‖v i ‖ (3.30)
In practice, the Gram-Schmidt formula are not applied since it has been demonstrated that this
algorithm turns out to be unstable. The Modified Gram-Schmidt algorithm computes the same
result but with a more stable algorithm as follows.
v (1)
j
= a j (3.31)
v (2)
j
= v (1)
j
− q 1 q1 ∗ v(1) j
(3.32)
v (3)
j
= v (2)
j
− q 2 q ∗ 2v (2)
j
(3.33)
v (j)
j
. (3.34)
= v (j−1)
j
− q j−1 q ∗ j−1 v(j−1) j
(3.35)
recalling that ∑ n j=1 (q∗ j v)q j = ∑ n j=1 (q jq ∗ j
)v. This algorithm should be read as:
v (1)
1
→ v (1)
2 , v(2) 2
→ · · · → v (1)
i
, v (2)
i
, · · · , v (i)
i
→ · · · (3.36)
It has been explained how to perform a QR factorization with an efficient algorithm. However,
as stated before, the QR factorization is not the best strategy to decompose a matrix such as
̂D (Eq. 3.18). A better method is to factorize by ̂Q m+1 and ˜H. The procedure performed to
this purpose is similar to the Modified Gram-Schmidt orthogonalization and is known as the
Arnoldi Algorithm [2].
A ̂Q m = ̂Q m+1 ˜H m (3.37)
The 1st to mth column of this equation can be written as follows: