S. Soit and . Vecteur, S(i) contient les champs de vitesses et de flottabilité Soit S, le sous-espace de Krylov de dimension N contenant les vecteurs d'états de l'instant n à n+N Nous définissons A le sous-opérateur tel que S(i + 1) = AS(i) Q est une base orthogonale de S et R est l'image de S dans la base Q, soit S(i) = QR(i). L'algorithme d'ortho-normalisation QR consiste à projeter sur l'hyperplan orthogonal au premier vecteur d'état, l'ensemble des vecteurs d'états suivants, à normaliser les vecteur d'états projetés puis à réitérer avec la deuxième vecteur d'état (qui a déjà été projeté une fois), puis le troisième? jusqu'à obtenir une base ortho-normale de notre sous-espace, Nous avons alors : S(i) = QR(i), AS(i) = AQR(i), S(i + 1) = AQR(i), mais S(i + 1) = QR(i + 1)

H. Nous-définissons, H. La-matrice-de-hessenberg-telle-que, and . Aq, décomposition de Schur) et nous avons H = R(i + 1)R ?1 (i) Nous perdons la dernière colonne puisque pour N vecteurs d'états nous avons N ? 1 images de

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