THESE de DOCTORAT - cerfacs

52 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f
Aq 1 = h 11 q 1 + h 21 q 2 (3.38)
Aq 2 = h 12 q 1 + h 22 q 2 + h 32 q 3 (3.39)
Aq 3 = h 13 q 1 + h 23 q 2 + h 33 q 3 + h 43 q 4 (3.40)
. (3.41)
Aq m = h 1m q 1 + · · · + h mm q n + h m+1,m q m+1 (3.42)
And the vectors v j are obtained as
q 1 = r 0 /‖r 0 ‖ (3.43)
v (1)
j
= Aq j (3.44)
v (2)
j
= v (1)
j
− q 1 q1 ∗ v(1) j
(3.45)
v (3)
j
= v (2)
j
− q 2 q ∗ 2v (2)
j
(3.46)
v (j)
j
. (3.47)
= v ji−1)
j
− q j−1 q ∗ j−1 v(j−1) j
(3.48)
Finally, all the elements q j and h ij are computed by
for j = 1 : m − 1 → q j+1 = v j /‖v j ‖ and h ij = q ∗ i Aq j for every i ̸= j + 1 and h j+1,j = ‖v j ‖
It should be evident from Eqs. (3.38) and (3.42) that the vectors {q j } form bases of the successive
Krylov subspaces generated by A and r defined as follows:
K m = 〈r 0 , Ar 0 , ..., A n−1 r 0 〉 = 〈q 1 , q 2 , ..., q m 〉 ⊆ C n (3.49)
where 〈·〉 means the span of. The Arnoldi process can be described as the systematic construction
of orthonormal bases for successive Krylov subspaces knowing that the Krylov n × m
matrix K n is written as