THESE de DOCTORAT - cerfacs

50 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f
3.3.1 On the Arnoldi algorithm
The most important idea to draw from concepts of inner products and orthogonality is this:
inner products can be used to decompose arbitrary vectors into orthogonal components [101].
For example, suppose that {q 1 , q 2 , · · · , q n } is an orthonormal set, and let a be an arbitrary vector.
The quantity q ∗ a is then a scalar 5 . Utilizing these scalars as coordinates in an expansion, a
vector v is defined.
v = a − (q ∗ 1 a)q 1 − (q ∗ 2a)q 2 − · · · − (q ∗ na)q n (3.24)
It is clear that the vector v is orthogonal to {q 1 , q 2 , · · · , q n }. This can be verified by multiplying
all terms by q ∗ i
q ∗ i v = q∗ i a − (q∗ 1 a)(q∗ i q 1) − (q ∗ 2a)(q ∗ i q 2) − · · · − (q ∗ na)(q ∗ i q n) = q ∗ i a − (q∗ 1 a)(q∗ i q i) = 0 (3.25)
Equation 3.24 is the basis of orthogonal decomposition. Let us consider now a matrix A composed
by n columns a j .
⎡ ∣ ∣ ∣ ⎤
∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣ A =
a
⎢ 1 a 2 a 3 · · ·
a n ⎥
⎣
⎦
∣
(3.26)
At the very beginning no q j exists. It must be built iteratively as follows:
v 1 = a 1 → q 1 = v 1
‖v 1 ‖
(3.27)
since q 1 must be also normalized. Subsequently, q 2 should be then
v 2 = a 2 − (q ∗ 1 a 2)q 1 → q 2 = v 2
‖v 2 ‖
(3.28)
And after n steps, the q n vector is computed
5 The symbol () ∗ represents the complex conjugate of given a quantity