Deuxi\303\250me cours sur l'algorithme de Cholesky
On va commencer par quelques exercices un peu th\303\251oriques pendant que vous \303\252tes frais :-)
On reprendra l'algo qu'on avait esquiss\303\251 au cours pr\303\251c\303\251dent en fin de s\303\251ance.
Je rappelle la d\303\251finition. Soit A une matrice sym\303\251trique (donc carr\303\251e !). A est d\303\251finie positive si, quel que soit le vecteur (colonne) x non nul, on a x^T . A . x > 0.
Exercice 1. Montrer que toute matrice sym. d\303\251f. pos. est inversible.
Preuve . On va faire une d\303\251monstration par l'absurde. On fait deux hypoth\303\250ses H1 et H2. On doit conclure (C) \303\240 une contradiction. Pensez \303\240 utiliser le th\303\251or\303\250me 1 page 18, 19 du poly. Je vous laisse 10 minutes mais je peux r\303\251pondre sur le chat. H1 : A est sym. d\303\251f. pos. H2 : A n'est pas inversible C : une contradiction
L'implication 4 => 1 du Thm 1 dit que: [le seul vecteur x tel que A . x = 0 est le vecteur nul (x = 0)] => A inversible. La contrapos\303\251e de l'implication (\303\251nonc\303\251 \303\251quivalent \303\240 l'implication) c'est : non 1 => non 4. Donc le Thm 1 nous dit que : A pas inversible => il existe un vecteur x diff\303\251rent de 0 tel que A . x = 0. La matrice A n'est pas inversible (H1). Donc il existe x diff\303\251rent de 0 tel que A . x = 0. Pour ce vecteur x, on a donc x^t . A . x = 0. Cette contradiction avec (H2) prouve que toute matrice sym. d\303\251f. pos. est inversible.
Autre preuve, probablement plus simple .
On veut montrer : A sym. d\303\251f. pos. => A inversible.
On montre la contrapos\303\251e : A pas inversible => A pas sym. d\303\251f. pos. H : A pas inversible C : A pas sym. d\303\251f. pos. L'implication 4 => 1 du Thm 1 (c'est la contrapos\303\251e de l'implication qui est \303\251crite ici) nous dit que : A pas inversible => il existe un vecteur x diff\303\251rent de 0 tel que A . x = 0. La matrice A n'est pas inversible (H). Donc il existe x diff\303\251rent de 0 tel que A . x = 0. Pour ce vecteur x, on a donc x^t . A . x = 0. Donc A n'est pas d\303\251finie positive.
Exercice 2 Montrer que les valeurs propres d'une matrice sym\303\251trique d\303\251finie positive sont strictement positives. Preuve (je cadre un peu) : H1 : A est sym\303\251trique d\303\251finie positive H2 : lambda est une valeur propre de A C : lambda > 0
Soit x un vecteur propre associ\303\251 \303\240 la valeur propre lambda. Alors x est non nul. Alors A . x = lambda * x (d\303\251f. d'un vecteur propre). Donc x^T . A . x = x^T . (lambda * x) Comme lambda est un scalaire (un r\303\251el, un complexe), on a x^T . (lambda * x) = lambda * x^T . x Comme x^T . x est strictement positif (c'est une somme de carr\303\251s), le signe de x^T . A . x est le m\303\252me que le signe de lambda. Comme A est sym\303\251trique d\303\251finie positive (H1), on voit que lambda est positif.
Exercice 3 Je construis une matrice sym\303\251trique d\303\251finie positive. Vous retouvez la factorisation de Cholesky.
with (LinearAlgebra):
Je triche. Je pars de L :-)
L := <<2,4,1/2> | <0, 3, 5> | <0, 0, 4>>;
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
A := L . Transpose (L);
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
Avant de calculer, on peut v\303\251rifier ce qu'on a d\303\251montr\303\251 sur les valeurs propres de A.
Et v\303\251rifier que A est inversible. Par exemple, pour l'inversibilit\303\251 :
Determinant (A);
IiR3Jg==
Il est non nul donc A est inversible (Thm 1, implication 8 => 1).
Eigenvalues (evalf (A));
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
La fonction Eigenvalues calcule les valeurs propres de A.
Sans le 'evalf', on obtient l'expression symbolique des valeurs propres, en calculant les racines du polyn\303\264me caract\303\251ristique. C'est inutilisable. Avec 'evalf', on demande le calcul des valeurs propres de A, vue comme une matrice de nombres \303\240 virgule flottante. On remarque que l'algorithme utilis\303\251 est un algorithme g\303\251n\303\251ral qui ne tire pas parti du fait que A est sym\303\251trique. Il cherche donc des valeurs propres complexes. Il se trouve que ces valeurs propres complexes sont en fait r\303\251elles. Et m\303\252me positives, as predicted by the theory .
J'indique \303\240 Maple que A est sym\303\251trique, puis je recalcule les valeurs propres. On va voir qu'il change d'algorithme puisque les valeurs propres d'une matrice sym\303\251trique sont n\303\251cessairement r\303\251elles.
A := Matrix(A, shape=symmetric);
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
Eigenvalues(evalf(A));
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
On voit que l'algo utilis\303\251 a chang\303\251 : les '+ 0. I' qui d\303\251signent les parties imaginaires ont disparu.
Allez-y, appliquez l'algo. Vous savez ce que vous devez trouver !
Pour commencer, je vous montre comment calculer la factorisation de Cholesky avec LinearAlgebra.
LUDecomposition (A, method='Cholesky');
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
Trois it\303\251rations.
A1 := A;
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
alpha1 := A1[1,1];
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYTFHRigiIiU3I0Yu
Nombre positif donc on peut continuer.
a1 := A1[2..3,1];
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
M1 := A1[2..3,2..3];
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
beta1 := sqrt(alpha1);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhMUdGKCIiIzcjRi4=
b1 := (1/beta1)*a1;
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
Dans la formule suivante, on utilise b1 . b1^T C'est un produit tensoriel. Le r\303\251sultat est une matrice 2 x 2. Ne pas confondre avec b1^T . b1, le produit scalaire, dont le r\303\251sultat est un r\303\251el.
A2 := M1 - b1 . Transpose(b1);
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
Et on recommence (it\303\251ration 2). alpha2 est positif donc on peut continuer.
alpha2 := A2[1,1];
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYTJHRigiIio3I0Yu
a2 := A2[2..2,1];
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
M2 := A2[2..2,2..2];
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
beta2 := sqrt(alpha2);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhMkdGKCIiJDcjRi4=
b2 := (1/beta2)*a2;
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
A3 := M2 - b2 . Transpose(b2);
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
On recommence. Derni\303\250re it\303\251ration o\303\271 la seule op\303\251ration \303\240 faire consiste \303\240 prendre la racine carr\303\251e.
alpha3 := A3[1,1];
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYTNHRigiIzs3I0Yu
beta3 := sqrt(alpha3);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhM0dGKCIiJTcjRi4=
\303\200 chaque it\303\251ration, on a calcul\303\251 une colonne de la matrice L. Le r\303\251sultat est donc :
< <beta1, b1 > | <0, beta2, b2 > | <0, 0, beta3> >;
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
L;
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