Introduction<\/h2>\n\n\n\n
Cet article est assez technique et est destin\u00e9 \u00e0 documenter le calcul de la position d’un salaire dans la distribution des salaires du secteur priv\u00e9 dans notre Observatoire des salaires des enseignants<\/a>. Afin d’estimer la distribution des salaires, l’INSEE fournit un certain nombre de quantiles, qui sont en g\u00e9n\u00e9ral les premier et dernier d\u00e9ciles, les quartiles, la m\u00e9diane, ainsi que les 95\u00e8me et 99\u00e8me centiles. Cette approximation sera r\u00e9alis\u00e9e en utilisant :<\/p>\n\n\n\n Supposons connus On cherche une fonction <\/span> <\/span> La fonction Un raccordement Le syst\u00e8me \u00e0 r\u00e9soudre est alors le suivant : <\/span> <\/span> Ce syst\u00e8me \u00e9tant non lin\u00e9aire, nous proc\u00e9derons \u00e0 sa r\u00e9solution approch\u00e9e \u00e0 l’aide de la m\u00e9thode de Newton<\/a>. Pour cela, la matrice jacobienne correspondante sera n\u00e9cessaire.<\/p>\n\n\n\n Dans ce qui suit, on note : Voici les <\/span> <\/span> Avec :<\/p>\n\n\n\n <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> Pour tout <\/span> <\/span> Dans ce qui suit, on note Voici les <\/span> <\/span> Avec :<\/p>\n\n\n\n <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> Nous avons test\u00e9 cette m\u00e9thode en utilisant les donn\u00e9es des salaires nets \u00e9quivalent temps plein de 2020 (figure compl\u00e9mentaire 4 du fichier xls fourni par l’INSEE sur cette page<\/a>). Pour cela, nous avons utilis\u00e9 les 10\u00e8me, 25\u00e8me, 50\u00e8me, 75\u00e8me, 90\u00e8me et 95\u00e8me centiles (qui sont disponibles pour la plupart des ann\u00e9es) pour servir de base \u00e0 l’optimisation, et les autres centiles disponibles (du 5\u00e8me au 99\u00e8me) pour tester la justesse de notre fonction d’estimation. On peut tout d’abord constater une convergence assez rapide. Seules 6 it\u00e9rations de la m\u00e9thode de Newton sont n\u00e9cessaires pour que la norme de la diff\u00e9rence entre deux valeurs successives du vecteur de param\u00e8tres On peut maintenant comparer l’image de l’ensemble des centiles disponibles dans le tableau de l’INSEE aux valeurs calcul\u00e9es \u00e0 l’aide de la fonction de r\u00e9partition obtenue avec notre m\u00e9thode (Fig 3.). On constate que l’erreur n’exc\u00e8de pas 0,2% jusqu’au 96\u00e8me centile inclus, et reste inf\u00e9rieure \u00e0 0,5% pour les centiles restants (Fig 4).<\/p>\n\n\n\n [ MAJ du 28\/06\/2023 : le mod\u00e8le pr\u00e9sent\u00e9 dans cet article a \u00e9t\u00e9 \u00e9toff\u00e9 ici.] Introduction Cet article est assez technique et est destin\u00e9 \u00e0 documenter le calcul de la position d’un salaire dans la distribution des salaires du secteur priv\u00e9 dans notre Observatoire des salaires des enseignants.Si vous \u00eates int\u00e9ress\u00e9 par les animations de … Continuer la lecture de « Interpoler une fonction de r\u00e9partition des salaires \u00e0 partir de quantiles connus »<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":904,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,4,10,11,13,9,14,5],"tags":[],"class_list":["post-881","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-analyse","category-courbes","category-interpolation","category-mathematiques","category-optimisation","category-probabilites","category-salaires","category-statistiques"],"_links":{"self":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts\/881","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=881"}],"version-history":[{"count":49,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts\/881\/revisions"}],"predecessor-version":[{"id":1233,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts\/881\/revisions\/1233"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/media\/904"}],"wp:attachment":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=881"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=881"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Si vous \u00eates int\u00e9ress\u00e9 par les animations de convergence et la pr\u00e9cision obtenue, rendez-vous directement dans la derni\u00e8re partie de l’article, mais si vous avez quelques bases en analyse num\u00e9rique et \u00eates int\u00e9ress\u00e9 par le type d’interpolation utilis\u00e9, vous pourrez trouver dans ce qui suit quelques indications sur la m\u00e9thode que nous avons utilis\u00e9e.<\/p>\n\n\n\n\n\n\n\n
L’objectif est ici de d\u00e9terminer une approximation de la fonction de r\u00e9partition des salaires bas\u00e9e sur les quantiles fournis.<\/p>\n\n\n\n\n
La forte croissance de la fonction de r\u00e9partition des salaires pour les petits quantiles est probablement due \u00e0 l’existence d’un salaire minimum emp\u00eachant aux salaires d’avoir une valeur arbitrairement petite. \u00c9tant donn\u00e9 qu’il s’agit d’un trait sp\u00e9cifique au cas \u00e9tudi\u00e9 ici, les fonctions \u00e0 utiliser dans d’autres cas de figure d\u00e9pendront donc des caract\u00e9ristiques de la fonction de r\u00e9partition \u00e0 approximer.<\/li>\n<\/ul>\n\n\n\nFonction utilis\u00e9e pour approximer la fonction de r\u00e9partition<\/h2>\n\n\n\n
![\"n\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-37ebcaed9023ec813c4cd8937fb78ae3_l3.png\" "\\"Rendered")
quantiles d’un \u00e9chantillon statistique.
Notons ![\"(x_0; y_0)\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-9884d63fa68ed8c095d5cbcb7041780a_l3.png\" "\\"Rendered")
, ![\"(x_1; y_1)\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-b74ac154e3e83df834d0278cda38028a_l3.png\" "\\"Rendered")
, ![\"\dots\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-ea9c0e4da2858d54d98ff9306c98fe99_l3.png\" "\\"Rendered")
![\"(x_{n-1}; y_{n-1})\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-febb73ac0d391aa0495581afc11470b6_l3.png\" "\\"Rendered")
les points correspondants par lesquels doit passer la fonction de r\u00e9partition (les ordonn\u00e9es sont comprises entre 0 et 1 et les abscisses, tout comme les ordonn\u00e9es, sont class\u00e9es par ordre croissant).<\/p>\n\n\n\n![\"f\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-376a4cb6a8858812296a3f1cede44773_l3.png\" "\\"Rendered")
de classe ![\"\mathcal{C}^2(\mathbb{R})\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-0e40f3c861c0dbd11b555c14fedded91_l3.png\" "\\"Rendered")
passant par ces points, et param\u00e9tr\u00e9e par le vecteur ![\"a = (a_0 \dots a_{4n})^\top\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-c0e0ca61983d8d6c6a11bdb076231f7f_l3.png\" "\\"Rendered")
contenant ![\"4n+1\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-a5e5a6908bb2a33c4bb99cdf48c60bfd_l3.png\" "\\"Rendered")
coefficients, de la forme :<\/p>\n\n\n\n![\"\[f(x;a)=\left\{\begin{array}{ll}\exp(a_0x^2+a_1x+a_2) & \text{si } x \leq x_0 \\a_3x^3+a_4x^2+a_5x+a_6 & \text{si } x_0 < x \leq x_1 \\a_7x^3+a_8x^2+a_9x+a_{10}& \text{si } x_1 < x \leq x_2 \\\vdots & \vdots \\a_{4k-1}x^3+a_{4k}x^2+a_{4k+1}x+a_{4k+2} & \text{si } x_{k-1} < x \leq x_k \\\vdots & \vdots \\a_{4n-5}x^3+a_{4n-4}x^2+a_{4n-3}x+a_{4n-2} & \text{si } x_{n-2} < x \leq x_{n-1} \\1-\exp(a_{4n-1}x+a_{4n}) & \text{si } x_{n-1} < x \\\end{array}\right. .\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-528e0186413f3eb3c4c6df69d8d2ea82_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n \u00e9tant de classe , sa fonction de densit\u00e9 ![\"f'\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-7bb2ca34992b24b02e1d001e1d6c05c3_l3.png\" "\\"Rendered")
est de classe ![\"\mathcal{C}^1(\mathbb{R})\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-ac4d92d4c9373acbde878d6035b51282_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\n![\"\mathcal{C}^2\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-1529edc4aefbd0ac09a5ac86783f3b06_l3.png\" "\\"Rendered")
en chacun des points permet d’obtenir ![\"4n\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-601e8be31485ca2775897f8d9c1b0277_l3.png\" "\\"Rendered")
\u00e9quations (![\"2n\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-94525b283d6203a78ff09178cd8dfdaa_l3.png\" "\\"Rendered")
pour les raccordements ![\"\mathcal{C}^0\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-58541f867c77fbe9bfb806bc0274a963_l3.png\" "\\"Rendered")
\u00e0 droite et \u00e0 gauche de chaque point, pour le raccordement ![\"\mathcal{C}^1\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-2c27ef47068f8e4af30e85cd85117ee5_l3.png\" "\\"Rendered")
, et pour le raccordement ).
Afin d’obtenir un syst\u00e8me carr\u00e9, on ajoute une \u00e9quation de raccordement ![\"\mathcal{C}^3\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-583f49c9349f989a9fd8f04a2b11b025_l3.png\" "\\"Rendered")
sur le point ![\"(x_0, y_0)\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-4ada7c1eff54e3499f4d9d1ea2d6b0b8_l3.png\" "\\"Rendered")
ce qui nous donne un total de \u00e9quations, \u00e9gal au nombre de param\u00e8tres \u00e0 d\u00e9terminer.<\/p>\n\n\n\nSyst\u00e8me d’\u00e9quations \u00e0 r\u00e9soudre<\/h2>\n\n\n\n
![\"\[\left\{\begin{array}{l}\begin{array}{lclcl}\text{raccordements } \mathcal{C}^0 \text{ : } & & \\\exp(a_0x_0^2+a_1x_0+a_2) & - & y_0 & = & 0 \\a_3x_0^3+a_4x_0^2+a_5x_0+a_6 & - & y_0 & = & 0 \\a_3x_1^3+a_4x_1^2+a_5x_1+a_6 & - & y_1 & = & 0 \\a_7x_1^3+a_8x_1^2+a_9x_1+a_{10} & - & y_1 & = & 0 \\& \vdots & & \vdots & \\a_{4k-5}x_{k-1}^3+a_{4k-4}x_{k-1}^2+a_{4k-3}x_{k-1}+a_{4k-2} & - & y_{k-1} & = & 0 \\a_{4k-1}x_{k-1}^3+a_{4k}x_{k-1}^2+a_{4k+1}x_{k-1}+a_{4k+2} & - & y_{k-1} & = & 0 \\a_{4k-1}x_{k}^3+a_{4k}x_{k}^2+a_{4k+1}x_{k}+a_{4k+2} & - & y_{k} & = & 0 \\a_{4k+3}x_{k}^3+a_{4k+4}x_{k}^2+a_{4k+5}x_{k}+a_{4k+6} & - & y_{k} & = & 0 \\& \vdots & & \vdots & \\a_{4n-5}x_{n-1}^3+a_{4n-4}x_{n-1}^2+a_{4n-3}x_{n-1}+a_{4n-2} & - & y_{n-1} & = & 0 \\1-\exp(a_{4n-1}x_{n-1}+a_{4n}) & - & y_{n-1} & = & 0 \\\end{array}\\\\\begin{array}{lclcl}\text{raccordements } \mathcal{C}^1 \text{ : } & & \\(2a_0x_0+a_1)\exp(a_0x_0^2+a_1x_0+a_2) & - & (3a_3x_0^2+2a_4x_0+a_5) & = & 0 \\3a_3x_1^2+2a_4x_1+a_5 & - & (3a_7x_1^2+2a_8x_1+a_9) & = & 0 \\& \vdots & & \vdots & \\3a_{4k-5}x_{k-1}^2+2a_{4k-4}x_{k-1}+a_{4k-3} & - & (3a_{4k-1}x_{k-1}^2+2a_{4k}x_{k-1}+a_{4k+1}) & = & 0 \\3a_{4k-1}x_{k}^2+2a_{4k}x_{k}+a_{4k+1} & - & (3a_{4k+3}x_{k}^2+2a_{4k+4}x_{k}+a_{4k+5}) & = & 0 \\& \vdots & & \vdots & \\3a_{4n-5}x_{n-1}^2+2a_{4n-4}x_{n-1}+a_{4n-3} & + & a_{4n-1}\exp(a_{4n-1}x_{n-1}+a_{4n}) & = & 0 \\\end{array}\\\\\begin{array}{lclcl}\text{raccordements } \mathcal{C}^2 \text{ : } & & \\((2a_0x_0+a_1)^2+2a_0)\exp(a_0x_0^2+a_1x_0+a_2) & - & (6a_3x_0+2a_4) & = & 0 \\6a_3x_1+2a_4 & - & (6a_7x_1+2a_8) & = & 0 \\& \vdots & & \vdots & \\6a_{4k-5}x_{k-1}+2a_{4k-4} & - & (6a_{4k-1}x_{k-1}+2a_{4k}) & = & 0 \\6a_{4k-1}x_{k}+2a_{4k} & - & (6a_{4k+3}x_{k}+2a_{4k+4}) & = & 0 \\& \vdots & & \vdots & \\6a_{4n-5}x_{n-1}+2a_{4n-4} & + & a_{4n-1}^2\exp(a_{4n-1}x_{n-1}+a_{4n}) & = & 0 \\\end{array}\\\\\begin{array}{lcl}\text{raccordement } \mathcal{C}^3 \text{ : } & & \\(8a_0^3x_0^3+6a_0(2a_0x_0+a_1)(a_1x_0+1)+a_1^3)\exp(a_0x_0^2+a_1x_0+a_2) - 6a_3 & = & 0 \\\end{array}\end{array}\right. .\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-3dd75d7fc82d1543932aa0f36916187f_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nMatrice jacobienne<\/h2>\n\n\n\n
![\"P_0(x) = a_0x^2+a_1x+a_2\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-277709cac65f0df978c6a63c409cac5c_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\n![\"7\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-8be530c00effc51a58a97342968d7e3e_l3.png\" "\\"Rendered")
premi\u00e8res colonnes de la matrice jacobienne, elles contiennent les d\u00e9riv\u00e9es partielles selon les param\u00e8tres ![\"a_0\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-1cd443d04e7bb5292c0e0dd446784832_l3.png\" "\\"Rendered")
\u00e0 ![\"a_6\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-818ecee88b5cf525947745c4fc720de1_l3.png\" "\\"Rendered")
:<\/p>\n\n\n\n![\"\[\left(\begin{array}{*3{>{\displaystyle}c}|*4{>{\displaystyle}c}|*1{>{\displaystyle}c}}E_1 & E_2 & E_3 & 0 & 0 & 0 & 0 & \multirow{4}{*}{\text{$(\ast)$}} \\\multicolumn{3}{c|}{\multirow{3}{*}{\text{$(0)$}}} & x_0^3 & x_0^2 & x_0 & 1 \\\multicolumn{3}{c|}{} & x_1^3 & x_1^2 & x_1 & 1 \\\multicolumn{3}{c|}{} & \multicolumn{4}{c|}{\text{$(0)$}} \\\hline & & & & & & & \\E_4 & E_5 & E_6 & -3x_0^2 & -2x_0 & -1 & 0 & \multirow{3}{*}{\text{$(\ast)$}} \\\multicolumn{3}{c|}{\multirow{2}{*}{\text{$(0)$}}} & 3x_1^2 & 2x_1 & 1 & 0 \\\multicolumn{3}{c|}{} & \multicolumn{4}{c|}{\text{$(0)$}} \\\hline & & & & & & & \\E_7 & E_8 & E_9 & -6x_0 & -2 & 0 & 0 & \multirow{3}{*}{\text{$(\ast)$}} \\\multicolumn{3}{c|}{\multirow{2}{*}{\text{$(0)$}}} & 6x_1 & 2 & 0 & 0 \\\multicolumn{3}{c|}{} & \multicolumn{4}{c|}{\text{$(0)$}} \\\hline & & & & & & & \\E_{10} & E_{11} & E_{12} & -6 & 0 & 0 & 0 & \text{$(\ast)$} \\\end{array}\right).\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-6b24ce0ea1a4a3380724ee7dfd1ece49_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_1 = x_0^2 \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-20b07e87f6535f387771a1f67d86f486_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_2 = x_0 \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-3ff8bba6f4cebeab48a6c6ec6d97fbbe_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_3 = \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-c8f306246fe5b7246a72b65bfe769e90_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_4 = x_0 (2 a_0 x_0^2 + a_1 x_0 + 2) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-c63549c85f9ae964cee5b64454b278b5_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_5 = (2 a_0 x_0^2 + a_1 x_0 + 1) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-ccfe7025ea3389c2755c06b73cf626a2_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_6 = (2 a_0 x_0 + a_1 ) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-c30dab9eb6c0ee2aae8e408556cac4fe_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_7 = (4 a_0^2 x_0^4 + 2 a_0 x_0^2 (2 a_1 x_0 + 5) + a_1^2 x_0^2 + 4 a_1 x_0 + 2) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-a46f9fda2a7e56b9386c251eb5a2493b_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_8 = (4 a_0^2 x_0^3 + 2 a_0 x_0 (2 a_1 x_0 + 3) + a_1 (a_1 x_0 + 2)) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-9b755dda335bcc2a38fea09f75bde854_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_9 = (4 a_0 x_0 (a_0 x_0 + a_1) + 2 a_0 + a_1^2) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-720c7ff65564e7e147b941d2697c2485_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_{10} = (8 a_0^3 x_0^5 + 12 a_0^2 x_0^3 (a_1 x_0 + 3) + 6 a_0 x_0 (a_1^2 x_0^2 + 5 a_1 x_0 + 4) + a_1 (a_1^2 x_0^2 + 6 a_1 x_0 + 6)) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-a3b301e7778eaf75dccafd28760bca14_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_{11} = (8 a_0^3 x_0^4 + 12 a_0^2 a_1 x_0^3 + 24 a_0^2 x_0^2 + 6 a_0 a_1^2 x_0^2 + 18 a_0 a_1 x_0 + 6 a_0 + a_1^3 x_0 + 3 a_1^2) \exp(P_0(x_0)) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-96c418add4bac5405b2e82749357d6ff_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E_{12} = (2 a_0 x_0 + a_1) (4 a_0^2 x_0^2 + 4 a_0 a_1 x_0 + 6 a_0 + a_1^2) \exp(P_0(x_0)).\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-fad84d858957f90d2a6dd8c9a596a39d_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"k\in\llbracket 2 ; n-2 \rrbracket\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-ca18b76bce824de887d8dbf5a3d5f5c2_l3.png\" "\\"Rendered")
, voici les colonnes ![\"{4k-5}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-2df8e047b180a3f98ed5c0c46918954f_l3.png\" "\\"Rendered")
\u00e0 ![\"{4k+6}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-9e9196545cb052388428b45689448f8d_l3.png\" "\\"Rendered")
de la matrice jacobienne, elles contiennent les d\u00e9riv\u00e9es partielles selon les param\u00e8tres ![\"a_{4k-5}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-b6480a637b39db6ff2bb492f7f69dc91_l3.png\" "\\"Rendered")
\u00e0 ![\"a_{4k+6}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-6d16c8a8fc5d13846c238423ff3fa253_l3.png\" "\\"Rendered")
:<\/p>\n\n\n\n![\"\[\left(\begin{array}{*1{>{\displaystyle}c}|*4{>{\displaystyle}c}|*4{>{\displaystyle}c}|*4{>{\displaystyle}c}|*1{>{\displaystyle}c}}\multirow{9}{*}{\text{$(\ast)$}} & \multicolumn{4}{c|}{\text{$(0)$}} & \multicolumn{4}{c|}{\multirow{3}{*}{\text{$(0)$}}} & \multicolumn{4}{c|}{\multirow{5}{*}{\text{$(0)$}}} & \multirow{9}{*}{\text{$(\ast)$}} \\& x_{k-2}^3 & x_{k-2}^2 & x_{k-2} & 1 & \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & \\& x_{k-1}^3 & x_{k-1}^2 & x_{k-1} & 1 & \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & \\& \multicolumn{4}{c|}{\multirow{5}{*}{\text{$(0)$}}} & x_{k-1}^3 & x_{k-1}^2 & x_{k-1} & 1 & \multicolumn{4}{c|}{} & \\& \multicolumn{4}{c|}{} & x_{k}^3 & x_{k}^2 & x_{k} & 1 & \multicolumn{4}{c|}{} & \\& \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{\multirow{2}{*}{\text{$(0)$}}} & x_{k}^3 & x_{k}^2 & x_{k} & 1 & \\& \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & x_{k+1}^3 & x_{k+1}^2 & x_{k+1} & 1 & \\& \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{\text{$(0)$}} & \\\hline & & & & & & & & & & & & & \\\multirow{6}{*}{\text{$(\ast)$}} & \multicolumn{4}{c|}{\text{$(0)$}} & \multicolumn{4}{c|}{\multirow{2}{*}{\text{$(0)$}}} & \multicolumn{4}{c|}{\multirow{3}{*}{\text{$(0)$}}} & \multirow{6}{*}{\text{$(\ast)$}} \\& -3x_{k-2}^2 & -2x_{k-2} & -1 & 0 & \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & \\& 3x_{k-1}^2 & 2x_{k-1} & 1 & 0 & -3x_{k-1}^2 & -2x_{k-1} & -1 & 0 & \multicolumn{4}{c|}{} & \\& \multicolumn{4}{c|}{\multirow{3}{*}{\text{$(0)$}}} & 3x_{k}^2 & 2x_{k} & 1 & 0 & -3x_{k}^2 & -2x_{k} & -1 & 0 & \\& \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{\multirow{2}{*}{\text{$(0)$}}} & 3x_{k+1}^2 & 2x_{k+1} & 1 & 0 & \\& \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{\text{$(0)$}} & \\\hline\multirow{6}{*}{\text{$(\ast)$}} & \multicolumn{4}{c|}{\text{$(0)$}} & \multicolumn{4}{c|}{\multirow{2}{*}{\text{$(0)$}}} & \multicolumn{4}{c|}{\multirow{3}{*}{\text{$(0)$}}} & \multirow{6}{*}{\text{$(\ast)$}} \\& -6x_{k-2} & -2 & 0 & 0 & \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & \\& 6x_{k-1} & 2 & 0 & 0 & -6x_{k-1} & -2 & 0 & 0 & \multicolumn{4}{c|}{} & \\& \multicolumn{4}{c|}{\multirow{3}{*}{\text{$(0)$}}} & 6x_{k} & 2 & 0 & 0 & -6x_{k} & -2 & 0 & 0 & \\& \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{\multirow{2}{*}{\text{$(0)$}}} & 6x_{k+1} & 2 & 0 & 0 & \\& \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{} & \multicolumn{4}{c|}{\text{$(0)$}} & \\\hline & & & & & & & & & & & & & \\{\text{$(\ast)$} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \text{$(\ast)$} \\\end{array}\right).\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-4a49895a5e85e560acc70b3105b9a5c1_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"P_{4n-1}(x) = a_{4n-1}x+a_{4n}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-5bfba77ed71369d3013e73ed6e83baa6_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\n![\"6\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-d1a6fce7ee899b2b3922f491c5dad249_l3.png\" "\\"Rendered")
derni\u00e8res colonnes de la matrice jacobienne, elles contiennent les d\u00e9riv\u00e9es partielles selon les param\u00e8tres ![\"a_{4n-5}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-638d0131ba29af8f4aca3244ec569bff_l3.png\" "\\"Rendered")
\u00e0 ![\"a_{4n}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-914d2b2d2798ca98b4af2c610b99d590_l3.png\" "\\"Rendered")
:<\/p>\n\n\n\n![\"\[\left(\begin{array}{*1{>{\displaystyle}c}|*4{>{\displaystyle}c}|*2{>{\displaystyle}c}}\multirow{4}{*}{\text{$(\ast)$}} & \multicolumn{4}{c|}{\text{$(0)$}} & \multicolumn{2}{c}{\multirow{3}{*}{\text{$(0)$}}} \\& x_{n-2}^3 & x_{n-2}^2 & x_{n-2} & 1 & \multicolumn{2}{c}{} \\& x_{n-1}^3 & x_{n-1}^2 & x_{n-1} & 1 & \multicolumn{2}{c}{} \\& 0 & 0 & 0 & 0 & E'_{1} & E'_{2} \\\hline & & & & & & \\\multirow{3}{*}{\text{$(\ast)$}} & \multicolumn{4}{c|}{\text{$(0)$}} & \multicolumn{2}{c}{\multirow{2}{*}{\text{$(0)$}}} \\& -3x_{n-2}^2 & -2x_{n-2} & -1 & 0 & \multicolumn{2}{c}{} \\& 3x_{n-1}^2 & 2x_{n-1} & 1 & 0 & E'_{3} & E'_{4} \\\hline & & & & & & \\\multirow{3}{*}{\text{$(\ast)$}} & \multicolumn{4}{c|}{\text{$(0)$}} & \multicolumn{2}{c}{\multirow{2}{*}{\text{$(0)$}}} \\& -6x_{n-2} & -2 & 0 & 0 & \multicolumn{2}{c}{} \\& 6x_{n-1} & 2 & 0 & 0 & E'_{5} & E'_{6} \\\hline & & & & & & \\\text{$(\ast)$} & 0 & 0 & 0 & 0 & 0 & 0 \\\end{array}\right).\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-99a82fefa267f8ea500fd29c34c56f2b_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E'_1 = -x_{n-1} \exp(P_{4n-1}(x_{n-1})) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-8f3734c29223e362690e89cd1c927a99_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E'_2 = -\exp(P_{4n-1}(x_{n-1})) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-538b1c9bddd0c4b271609c0fef4f4fcd_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E'_3 = (a_{4n-1} + 1)\exp(P_{4n-1}(x_{n-1})) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-cea001cc105a6f1c181a8eaa040b9592_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E'_4 = a_{4n-1} \exp(P_{4n-1}(x_{n-1})) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-4f6c362122d944bd682c378c33c90786_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E'_5 =(a_{4n-1}^2x_{n-1} + 2a_{4n-1}) \exp(P_{4n-1}(x_{n-1})) ,\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-70251b747019aab1bcd88fe2c8716964_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[E'_6 = a_{4n-1}^2 \exp(P_{4n-1}(x_{n-1})).\]\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-d15f7c0e9d72a6e0e5c39dd53f7455eb_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nR\u00e9sultats<\/h2>\n\n\n\n
La valeur initiale du vecteur de param\u00e8tres est d\u00e9termin\u00e9e de telle fa\u00e7on que les points servant de base \u00e0 l’interpolation soient reli\u00e9s par des segments de droite, et que les exponentielles des intervalles extr\u00eames soient raccord\u00e9es de mani\u00e8re au segment le plus proche.<\/p>\n\n\n\n![\"a\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-ba99d12f07a534cadd0fe88e1d2d337e_l3.png\" "\\"Rendered")
devienne inf\u00e9rieure \u00e0 ![\"10^{-15}\"](\"https:\/\/blog.epicycle.fr\/wp-content\/ql-cache\/quicklatex.com-245f79268e6a3a7717b10cbcd27ecc97_l3.png\" "\\"Rendered")
. Le processus de convergence est illustr\u00e9 par les figures 1 et 2.<\/p>\n\n\n\n![\"\"](\"https:\/\/blog.epicycle.fr\/wp-content\/uploads\/2022\/12\/f.gif\")
![\"\"](\"https:\/\/blog.epicycle.fr\/wp-content\/uploads\/2022\/12\/d.gif\")
![\"\"](\"https:\/\/blog.epicycle.fr\/wp-content\/uploads\/2022\/12\/comparaison-1.png\")
![\"\"](\"https:\/\/blog.epicycle.fr\/wp-content\/uploads\/2022\/12\/erreur.png\")