init
This commit is contained in:
commit
cc75934361
SSH key fingerprint:
SHA256:kZEpW8cMJ54PDeCvOhzreNr4FSh6R13CMGH/POoO8DI
87
Algo_Gauss_Newton.m
Executable file
87
Algo_Gauss_Newton.m
Executable file
|
|
@ -0,0 +1,87 @@
|
|||
function [beta, norm_grad_f_beta, f_beta, norm_delta, nb_it, exitflag] ...
|
||||
= Algo_Gauss_Newton(residu, J_residu, beta0, option)
|
||||
|
||||
%*****************************************************************
|
||||
% Fichier ~gergaud/ENS/Optim1a/TP-optim-20-21/TP-ref/GN_ref.m *
|
||||
% Novembre 2020 *
|
||||
% Université de Toulouse, INP-ENSEEIHT *
|
||||
%*****************************************************************
|
||||
%
|
||||
% GN resout par l'algorithme de Gauss-Newton les problemes aux moindres carres
|
||||
% Min 0.5||r(beta)||^2
|
||||
% beta \in \IR^p
|
||||
%
|
||||
% Paramètres en entrés
|
||||
% --------------------
|
||||
% residu : fonction qui code les résidus
|
||||
% r : \IR^p --> \IR^n
|
||||
% J_residu : fonction qui code la matrice jacobienne
|
||||
% Jr : \IR^p --> real(n,p)
|
||||
% beta0 : point de départ
|
||||
% real(p)
|
||||
% option(1) : Tol_abs, tolérance absolue
|
||||
% real
|
||||
% option(2) : Tol_rel, tolérance relative
|
||||
% real
|
||||
% option(3) : n_itmax, nombre d'itérations maximum
|
||||
% integer
|
||||
%
|
||||
% Paramètres en sortie
|
||||
% --------------------
|
||||
% beta : beta
|
||||
% real(p)
|
||||
% norm_grad_f_beta : ||gradient f(beta)||
|
||||
% real
|
||||
% f_beta : f(beta)
|
||||
% real
|
||||
% r_beta : r(beta)
|
||||
% real(n)
|
||||
% norm_delta : ||delta||
|
||||
% real
|
||||
% nbit : nombre d'itérations
|
||||
% integer
|
||||
% exitflag : indicateur de sortie
|
||||
% integer entre 1 et 4
|
||||
% exitflag = 1 : ||gradient f(beta)|| < max(Tol_rel||gradient f(beta0)||,Tol_abs)
|
||||
% exitflag = 2 : |f(beta^{k+1})-f(beta^k)| < max(Tol_rel|f(beta^k)|,Tol_abs)
|
||||
% exitflag = 3 : ||delta)|| < max(Tol_rel delta^k),Tol_abs)
|
||||
% exitflag = 4 : nombre maximum d'itérations atteint
|
||||
%
|
||||
% ---------------------------------------------------------------------------------
|
||||
|
||||
norm_grad_f_beta0 = norm(J_residu(beta0).'*residu(beta0));
|
||||
|
||||
nb_it = 0;
|
||||
beta = beta0;
|
||||
|
||||
continuer = 1;
|
||||
|
||||
while continuer
|
||||
|
||||
nb_it = nb_it + 1;
|
||||
|
||||
beta_last = beta;
|
||||
beta = beta - (J_residu(beta).'*J_residu(beta))\(J_residu(beta).'*residu(beta));
|
||||
|
||||
norm_grad_f_beta = norm(J_residu(beta).'*residu(beta));
|
||||
f_beta = norm(residu(beta))/2;
|
||||
delta = beta - beta_last;
|
||||
norm_delta = norm(delta);
|
||||
|
||||
if norm(J_residu(beta).'*residu(beta)) <= max(option(2)*norm_grad_f_beta0, option(1))
|
||||
exitflag = 1;
|
||||
continuer = 0;
|
||||
elseif abs(norm(residu(beta))/2 - norm(residu(beta_last))/2) <= max(option(2)*norm(residu(beta))/2, option(1))
|
||||
exitflag = 2;
|
||||
continuer = 0;
|
||||
elseif norm_delta <= max(option(2)*norm(beta), option(1))
|
||||
exitflag = 3;
|
||||
continuer = 0;
|
||||
elseif nb_it >= option(3)
|
||||
exitflag = 4;
|
||||
continuer = 0;
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
91
Algo_Newton.m
Executable file
91
Algo_Newton.m
Executable file
|
|
@ -0,0 +1,91 @@
|
|||
function [beta, norm_grad_f_beta, f_beta, norm_delta, nb_it, exitflag] ...
|
||||
= Algo_Newton(Hess_f, beta0, option)
|
||||
%************************************************************
|
||||
% Fichier ~gergaud/ENS/Optim1a/TP-optim-20-21/Newton_ref.m *
|
||||
% Novembre 2020 *
|
||||
% Université de Toulouse, INP-ENSEEIHT *
|
||||
%************************************************************
|
||||
%
|
||||
% Newton résout par l'algorithme de Newton les problemes aux moindres carres
|
||||
% Min 0.5||r(beta)||^2
|
||||
% beta \in R^p
|
||||
%
|
||||
% Parametres en entrees
|
||||
% --------------------
|
||||
% Hess_f_C14 : fonction qui code la hessiennne de f
|
||||
% Hess_f_C14 : R^p --> matrice (p,p)
|
||||
% (la fonction retourne aussi le residu et la jacobienne)
|
||||
% beta0 : point de départ
|
||||
% real(p)
|
||||
% option(1) : Tol_abs, tolérance absolue
|
||||
% real
|
||||
% option(2) : Tol_rel, tolérance relative
|
||||
% real
|
||||
% option(3) : nitimax, nombre d'itérations maximum
|
||||
% integer
|
||||
%
|
||||
% Parametres en sortie
|
||||
% --------------------
|
||||
% beta : beta
|
||||
% real(p)
|
||||
% norm_gradf_beta : ||gradient f(beta)||
|
||||
% real
|
||||
% f_beta : f(beta)
|
||||
% real
|
||||
% res : r(beta)
|
||||
% real(n)
|
||||
% norm_delta : ||delta||
|
||||
% real
|
||||
% nbit : nombre d'itérations
|
||||
% integer
|
||||
% exitflag : indicateur de sortie
|
||||
% integer entre 1 et 4
|
||||
% exitflag = 1 : ||gradient f(beta)|| < max(Tol_rel||gradient f(beta0)||,Tol_abs)
|
||||
% exitflag = 2 : |f(beta^{k+1})-f(beta^k)| < max(Tol_rel|f(beta^k)|,Tol_abs)
|
||||
% exitflag = 3 : ||delta)|| < max(Tol_rel delta^k),Tol_abs)
|
||||
% exitflag = 4 : nombre maximum d'itérations atteint
|
||||
%
|
||||
% ---------------------------------------------------------------------------------
|
||||
|
||||
% TO DO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
[H_f0, res0, J_res0] = Hess_f(beta0);
|
||||
norm_grad_f_beta0 = norm(J_res0.'*res0);
|
||||
|
||||
nb_it = 0;
|
||||
beta = beta0;
|
||||
|
||||
continuer = 1;
|
||||
|
||||
while continuer
|
||||
|
||||
nb_it = nb_it + 1;
|
||||
|
||||
beta_last = beta;
|
||||
[H_f_last, res_last, J_res_last] = Hess_f(beta_last);
|
||||
|
||||
beta = beta - H_f_last\(J_res_last.'*res_last);
|
||||
[H_f, res, J_res] = Hess_f(beta);
|
||||
|
||||
norm_grad_f_beta = norm(J_res.'*res);
|
||||
f_beta = norm(res)/2;
|
||||
delta = beta - beta_last;
|
||||
norm_delta = norm(delta);
|
||||
|
||||
if norm_grad_f_beta <= max(option(2)*norm_grad_f_beta0, option(1))
|
||||
exitflag = 1;
|
||||
continuer = 0;
|
||||
elseif abs(norm(res)/2 - norm(res_last)/2) <= max(option(2)*norm(res)/2, option(1))
|
||||
exitflag = 2;
|
||||
continuer = 0;
|
||||
elseif norm_delta <= max(option(2)*norm(beta), option(1))
|
||||
exitflag = 3;
|
||||
continuer = 0;
|
||||
elseif nb_it >= option(3)
|
||||
exitflag = 4;
|
||||
continuer = 0;
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
BIN
C14_figures.png
Executable file
BIN
C14_figures.png
Executable file
Binary file not shown.
|
After Width: | Height: | Size: 130 KiB |
46
C14_results.txt
Executable file
46
C14_results.txt
Executable file
|
|
@ -0,0 +1,46 @@
|
|||
Algorithme de Gauss-Newton
|
||||
--------------------------------------------------------------------------------------------
|
||||
nb_iter A0 lambda ||f'(beta)|| f(beta) ||delta|| exitflag
|
||||
--------------------------------------------------------------------------------------------
|
||||
0 10 0.0001 4.6322e+05 48.07
|
||||
|
||||
1 15.022 0.00010633 15913 0.22921 5.0219 4
|
||||
|
||||
2 15.025 0.00010433 5.9024 0.2105 0.0032964 4
|
||||
|
||||
3 15.025 0.00010432 0.39911 0.2105 0.00068766 4
|
||||
|
||||
4 15.024 0.00010432 0.004769 0.2105 4.9165e-06 2
|
||||
|
||||
4 15.024 0.00010432 0.004769 0.2105 4.9165e-06 2
|
||||
|
||||
4 15.024 0.00010432 0.004769 0.2105 4.9165e-06 2
|
||||
|
||||
4 15.024 0.00010432 0.004769 0.2105 4.9165e-06 2
|
||||
|
||||
4 15.024 0.00010432 0.004769 0.2105 4.9165e-06 2
|
||||
|
||||
--------------------------------------------------------------------------------------------
|
||||
Algorithme de Newton
|
||||
--------------------------------------------------------------------------------------------
|
||||
nb_iter A0 lambda ||f'(beta)|| f(beta) ||delta|| exitflag
|
||||
--------------------------------------------------------------------------------------------
|
||||
0 10 0.0001 4.6322e+05 48.07
|
||||
|
||||
1 13.954 5.5076e-05 3.0226e+05 1.6021 3.9536 4
|
||||
|
||||
2 14.774 0.00010067 6753.7 0.26139 0.82072 4
|
||||
|
||||
3 14.987 0.00010359 450.57 0.2117 0.21288 4
|
||||
|
||||
4 15.019 0.00010424 80.175 0.21053 0.031965 4
|
||||
|
||||
5 15.024 0.00010431 5.8177 0.2105 0.004593 4
|
||||
|
||||
6 15.024 0.00010432 1.184 0.2105 0.0006653 4
|
||||
|
||||
7 15.024 0.00010432 0.066427 0.2105 9.519e-05 2
|
||||
|
||||
7 15.024 0.00010432 0.066427 0.2105 9.519e-05 2
|
||||
|
||||
--------------------------------------------------------------------------------------------
|
||||
32
Hess_f_C14.m
Executable file
32
Hess_f_C14.m
Executable file
|
|
@ -0,0 +1,32 @@
|
|||
function [H_f, res, J_res] = Hess_f_C14(beta, donnees, residu, J_residu)
|
||||
%
|
||||
% Paramètres en entrés
|
||||
% --------------------
|
||||
% beta : vecteur des paramètres
|
||||
% real(p)
|
||||
% donnees : Données
|
||||
% real(n,2)
|
||||
% residu : fonction qui code les résidus
|
||||
% res_beta = residus(beta)
|
||||
% J_residu : fonction qui code la matrice jacobienne
|
||||
% J_res_beta = J_residu(beta);
|
||||
%
|
||||
% Paramètres en sortie
|
||||
% --------------------
|
||||
% H_f : Matrice hessienne
|
||||
% real(p,p)
|
||||
% res : vecteur des résidus
|
||||
% real(n)
|
||||
% J_res : Matrice jacobienne des résiduis
|
||||
% real(n,p)
|
||||
%
|
||||
|
||||
res = residu(beta);
|
||||
J_res = J_residu(beta);
|
||||
% cf formule du cours
|
||||
S = [ sum(0*donnees(:,1)) , sum(donnees(:,1).*exp(-beta(2)*donnees(:, 1)));
|
||||
sum(donnees(:, 1).*exp(-beta(2)*donnees(:, 1))), sum(-beta(1)*(donnees(:, 1).^2).*exp(-beta(2)*donnees(:, 1))) ];
|
||||
H_f = S + J_res.'*J_res;
|
||||
|
||||
end
|
||||
|
||||
28
J_residu_C14.m
Executable file
28
J_residu_C14.m
Executable file
|
|
@ -0,0 +1,28 @@
|
|||
%-------------------------------------------------------------------------% %
|
||||
% Fonction de calcul de la Jacobienne du residu de la fonction de %
|
||||
% desintegration radioactive du carbone 14 %
|
||||
%-------------------------------------------------------------------------%
|
||||
|
||||
function J_res = J_residu_C14(beta, donnees)
|
||||
%
|
||||
% Paramètres en entrés
|
||||
% --------------------
|
||||
% beta : vecteur des paramètres
|
||||
% real(p)
|
||||
% donnees : Données
|
||||
% real(n,2)
|
||||
%
|
||||
% Paramètres en sortie
|
||||
% --------------------
|
||||
% J_res : Matrice jacobienne des résidus
|
||||
% real(n,p)
|
||||
%
|
||||
|
||||
% res = donnees(:, 2) - beta(1)*exp(-beta(2)*donnees(:, 1));
|
||||
|
||||
v1 = -exp(-beta(2)*donnees(:, 1));
|
||||
v2 = beta(1)*donnees(:, 1).*exp( -beta(2)*donnees(:, 1) );
|
||||
|
||||
J_res = [ v1, v2 ];
|
||||
|
||||
end
|
||||
206
Modelisation_C14.m
Executable file
206
Modelisation_C14.m
Executable file
|
|
@ -0,0 +1,206 @@
|
|||
%-------------------------------------------------------------------------%
|
||||
% 1SN - TP Optimisation %
|
||||
% INP Toulouse - ENSEEIHT %
|
||||
% Novembre 2020 %
|
||||
% %
|
||||
% Ce fichier contient le programme principal permettant l'estimation %
|
||||
% des parametres de la fonction de desintegration radioactive du %
|
||||
% carbone 14 par une approche des moindres carres. % %
|
||||
% Modele : A(t)= A0*exp(-lambda*t) %
|
||||
% Les algorithmes utilises pour la minimisation sont : %
|
||||
% - l'algorithme de Gauss-Newton %
|
||||
% - l'algorithme de Newton. %
|
||||
%-------------------------------------------------------------------------%
|
||||
|
||||
clear
|
||||
close all
|
||||
clc
|
||||
format shortG
|
||||
taille_ecran = get(0,'ScreenSize');
|
||||
L = taille_ecran(3);
|
||||
H = taille_ecran(4);
|
||||
|
||||
% Initialisation
|
||||
|
||||
% Donnees
|
||||
if exist('C14_results.txt','file')
|
||||
delete('C14_results.txt');
|
||||
end
|
||||
diary C14_results.txt
|
||||
Ti = [ 500; 1000; 2000; 3000; 4000; 5000; 6300];
|
||||
Ai = [14.5; 13.5; 12.0; 10.8; 9.9; 8.9; 8.0];
|
||||
Donnees = [Ti, Ai];
|
||||
|
||||
% Estimation a priori des parametres du modele : beta0 = [A0, lambda]
|
||||
beta0 = [10; 0.0001]; % Newton, Gauus-Newton, fminunc et leastsq converge
|
||||
% beta0 = [15; 0.001]; % Newton, Gauss-Newton, fminunc et leastsq divergent
|
||||
% beta0 = [15; 0.0005]; % Newton diverge, Gauss-Newton, fminunc et leastsq convergent
|
||||
% beta0 = [10; 0.0005]; % Gauss-Newton converge
|
||||
|
||||
%% Calcul et affichage du modele initial ----------------------------------
|
||||
xmin = 9; xmax = 20;
|
||||
x = linspace(xmin, xmax, 100);
|
||||
|
||||
ymin = -0.0001; ymax = 0.0005;
|
||||
y = linspace(ymin, ymax, 100);
|
||||
|
||||
[A0_plot, lambda_plot] = meshgrid(x, y);
|
||||
[m , n] = size(A0_plot);
|
||||
f_plot = zeros(m, n);
|
||||
|
||||
for i = 1:m
|
||||
for j = 1:n
|
||||
res_plot = residu_C14([A0_plot(i,j); lambda_plot(i,j)], Donnees);
|
||||
f_plot(i,j) = res_plot.'*res_plot /2;
|
||||
end
|
||||
end
|
||||
|
||||
figure('Position',[0.1*L,0.1*H,0.8*L,0.8*H]);
|
||||
subplot(1,3,1) % Fonction f en 3D
|
||||
mesh(A0_plot, lambda_plot, f_plot)
|
||||
axis('square')
|
||||
title('Representation de la fonction f des moindres carres')
|
||||
xlabel('A_0');
|
||||
ylabel('\lambda')
|
||||
zlabel('f(A_0,\lambda)')
|
||||
|
||||
T = linspace(0,6500,100);
|
||||
A = beta0(1)*exp(-beta0(2)*T);
|
||||
txt_legend{1} = 'donnees';
|
||||
txt_legend{2} = 'depart';
|
||||
|
||||
subplot(2, 3, 2) % Donnees + modele initial pour Gauss-Newton
|
||||
title('Desintegration radioactive du carbone 14 (Gauss-Newton)')
|
||||
axis([0 max(T) 0 18])
|
||||
xlabel('dur<EFBFBD>e T')
|
||||
ylabel('radioactivit<EFBFBD> A')
|
||||
hold on
|
||||
grid on
|
||||
plot(Ti, Ai, 'ok')
|
||||
plot(T, A)
|
||||
legend(txt_legend{1:2}, 'Location', 'SouthWest')
|
||||
|
||||
subplot(2, 3, 5) % Courbes de niveaux de f pour Gauss-Newton
|
||||
title('Recherche des parametres (Gauss-Newton)')
|
||||
xlabel('A_0')
|
||||
ylabel('\lambda')
|
||||
hold on
|
||||
contour(A0_plot, lambda_plot, f_plot, 100);
|
||||
plot(beta0(1),beta0(2),'ok')
|
||||
text(beta0(1),beta0(2),' depart \beta^{(0)}')
|
||||
|
||||
subplot(2, 3, 3) % Donnees + modele initial pour Newton
|
||||
title('Desintegration radioactive du carbone 14 (Newton)')
|
||||
axis([0 max(T) 0 18])
|
||||
xlabel('dur<EFBFBD>e T')
|
||||
ylabel('radioactivit<EFBFBD> A')
|
||||
hold on
|
||||
grid on
|
||||
plot(Ti, Ai, 'ok')
|
||||
plot(T, A)
|
||||
legend(txt_legend{1:2}, 'Location', 'SouthWest')
|
||||
|
||||
subplot(2, 3, 6) % Courbes de niveaux de f pour Newton
|
||||
title('Recherche des parametres (Newton)')
|
||||
xlabel('A_0')
|
||||
ylabel('\lambda')
|
||||
hold on
|
||||
contour(A0_plot, lambda_plot, f_plot, 100);
|
||||
plot(beta0(1),beta0(2),'ok')
|
||||
text(beta0(1),beta0(2),' depart \beta^{(0)}')
|
||||
|
||||
pause(0.5)
|
||||
|
||||
%% Algorithmes
|
||||
% Choix du nombre d'iteration maximal
|
||||
|
||||
nb_iterations_max = 8;
|
||||
txt_legend = cell(1,nb_iterations_max+2);
|
||||
txt_legend{1} = 'donnees';
|
||||
txt_legend{2} = 'depart';
|
||||
|
||||
|
||||
%% Gauss-Newton
|
||||
% -------------
|
||||
% Initialisation de l'affichage
|
||||
disp('Algorithme de Gauss-Newton')
|
||||
disp('--------------------------------------------------------------------------------------------')
|
||||
disp(' nb_iter A0 lambda ||f''(beta)|| f(beta) ||delta|| exitflag ')
|
||||
disp('--------------------------------------------------------------------------------------------')
|
||||
|
||||
% Calcul et affichage des valeurs initiales
|
||||
res_beta = residu_C14(beta0, Donnees);
|
||||
f_beta = 0.5*(res_beta.')*res_beta;
|
||||
J_res_beta = J_residu_C14(beta0, Donnees);
|
||||
norm_grad_f_beta = norm((J_res_beta.')*res_beta);
|
||||
disp([0 beta0(1) beta0(2) norm_grad_f_beta f_beta]);
|
||||
|
||||
options = [sqrt(eps) 1.e-12 0];
|
||||
|
||||
for i = 1:nb_iterations_max
|
||||
txt_legend{i+2} = ['iteration ' num2str(i)];
|
||||
options(3) = i;
|
||||
[beta, norm_grad_f_beta, f_beta, norm_delta, k, exitflag] = ...
|
||||
Algo_Gauss_Newton(@(beta) residu_C14(beta, Donnees), ...
|
||||
@(beta) J_residu_C14(beta, Donnees), ...
|
||||
beta0, options);
|
||||
disp([k beta(1) beta(2) norm_grad_f_beta f_beta norm_delta exitflag])
|
||||
% Visualisation
|
||||
A = beta(1)*exp(-beta(2)*T);
|
||||
subplot(2, 3, 2)
|
||||
plot(T,A)
|
||||
legend(txt_legend{1:i+2})
|
||||
% eval(['print -depsc fig_GN_courbe' int2str(i) '_C14'])
|
||||
subplot(2, 3, 5)
|
||||
plot(beta(1),beta(2),'ok')
|
||||
text(beta(1),beta(2),[' \beta^{(' num2str(i) ')}'])
|
||||
|
||||
pause(0.5)
|
||||
end
|
||||
|
||||
% Affichage des itérés de beta et sauvegarde des graphique
|
||||
disp('--------------------------------------------------------------------------------------------')
|
||||
|
||||
%% Newton
|
||||
% -------
|
||||
% Initialisation de l'affichage
|
||||
disp('Algorithme de Newton')
|
||||
disp('--------------------------------------------------------------------------------------------')
|
||||
disp(' nb_iter A0 lambda ||f''(beta)|| f(beta) ||delta|| exitflag ')
|
||||
disp('--------------------------------------------------------------------------------------------')
|
||||
|
||||
% Calcul et affichage des valeurs initiales
|
||||
res_beta = residu_C14(beta0, Donnees);
|
||||
f_beta = 0.5*(res_beta.')*res_beta;
|
||||
J_res_beta = J_residu_C14(beta0, Donnees);
|
||||
norm_grad_f_beta = norm((J_res_beta.')*res_beta);
|
||||
disp([0 beta0(1) beta0(2) norm_grad_f_beta f_beta]);
|
||||
|
||||
options = [sqrt(eps) 1.e-12 0];
|
||||
|
||||
for i = 1:nb_iterations_max
|
||||
options(3) = i;
|
||||
% Algorithme de Newton
|
||||
[beta, norm_grad_f_beta, f_beta, norm_delta, k, exitflag] = ...
|
||||
Algo_Newton(@(beta) Hess_f_C14(beta, Donnees, ...
|
||||
@(beta) residu_C14(beta, Donnees), ...
|
||||
@(beta) J_residu_C14(beta, Donnees)), ...
|
||||
beta0, options);
|
||||
% Affichage des valeurs
|
||||
disp([k beta(1) beta(2) norm_grad_f_beta f_beta norm_delta exitflag])
|
||||
% Visualisation
|
||||
A = beta(1)*exp(-beta(2)*T);
|
||||
subplot(2, 3, 3)
|
||||
plot(T,A)
|
||||
legend(txt_legend{1:i+2})
|
||||
subplot(2, 3, 6)
|
||||
plot(beta(1),beta(2),'ok')
|
||||
text(beta(1),beta(2),[' \beta^{(' num2str(i) ')}'])
|
||||
|
||||
pause(0.5)
|
||||
end
|
||||
|
||||
% Affichage des iteres de beta et sauvegarde des courbes
|
||||
disp('--------------------------------------------------------------------------------------------')
|
||||
print('C14_figures','-dpng')
|
||||
diary
|
||||
BIN
TP_Opti1A.pdf
Executable file
BIN
TP_Opti1A.pdf
Executable file
Binary file not shown.
23
residu_C14.m
Executable file
23
residu_C14.m
Executable file
|
|
@ -0,0 +1,23 @@
|
|||
%%------------------------------------------------------------------------% %
|
||||
% Fonction de calcul du residu de la fonction de desintegration %
|
||||
% radioactive du carbone 14 %
|
||||
%-------------------------------------------------------------------------%
|
||||
|
||||
function res = residu_C14(beta, donnees)
|
||||
%
|
||||
% Paramètres en entrés
|
||||
% --------------------
|
||||
% beta : vecteur des paramètres
|
||||
% real(p)
|
||||
% donnees : Données
|
||||
% real(n,2)
|
||||
%
|
||||
% Paramètres en sortie
|
||||
% --------------------
|
||||
% res : vecteur des résidus
|
||||
% real(n)
|
||||
%
|
||||
|
||||
res = donnees(:, 2) - beta(1)*exp(-beta(2)*donnees(:, 1));
|
||||
|
||||
end
|
||||
Loading…
Reference in a new issue