Considérons maintenant un problème matriciel s’écrivant de la manière suivante :
c1 d1 e1 0 0 b2 ... ... ... a3 ... 0 ... 0 eL−2 dL−1 0 0 aL bL cL · x1 x2 ... xL = y1 y2 ... yL
avec cj non-nul pour tout j compris entre 1 et L.
Pour résoudre ce type d’équation, et en suivant l’idée de l’algorithme de double descente, il suffit de modifier les coefficients de la matrice par un premier passage pour écrire l’ensemble des inconnues par une relation de récurrence. Puis, lors du second passage, en déduire l’ensemble des variables.
Données : {a, b, c, d, e, y} Résultat : x
pour tous les j ∈ [[2; L − 1]] faire
tmp ← bj/cj−1; cj ← cj− tmp ∗ dj−1; dj← dj− tmp ∗ ej−1; yj ← yj− tmp ∗ yj−1; tmp ← aj+1/cj−1; bj+1← bj− tmp ∗ dj−1; cj+1← cj− tmp ∗ ej−1; yj+1← yj+1− tmp ∗ yj−1; fin tmp ← bL/cL−1; cL← cL− tmp ∗ dL−1); xL← (yL− tmp ∗ yL−1)/cL; xL−1← (yL−1− dL−1∗ xL)/cL−1; pour tous les j ∈ [[1; L − 2]] faire
xL−1−j← (yL−1−j− eL−1−j∗ xL+1−j− dL−1−j∗ xL−j)/cL−1−j; fin
Comparé à la méthode de double descente classique, cet algorithme est également d’ordre O(L), mais nécessite plus de calculs. Elle permet une résolution directe du problème, sans avoir à réaliser de décomposition de type LU.
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