Esercitazioni matlab

LABORATORIO 6 CALCOLO%

per n = 5% f(xi) = yi

x = linspace(-5, 5, 6); %ho messo 6 perché ci sono n+1 nodi
wf = 1./(1 + w.^2);
y = subs(f, w, x)
format short e
y = [1/26, 1/10, 1/2, 1/2, 1/10, 1/26] %perché se no mi dava che era ancora simbolico
p = polyfit(x, y, 5)
x1 = -5:0.01:5;
y1 = polyval(p, x1);

per n = 10% f(xi) = yi

x = linspace(-5, 5, 11); %ho messo 11 perché ci sono n+1 nodi
wf = 1./(1 + w.^2);
y = subs(f, w, x);
y = [1/26, 1/17, 1/10, 1/5, 1/2, 1, 1/2, 1/5, 1/10, 1/17, 1/26];
p = polyfit(x, y, 10);
x2 = -5:0.01:5;
y2 = polyval(p, x2);

GRAFICI DI F E I DUE INTERPOLANTI

plot(x1, y1, 'k-', 'linewidth', 2)
hold on
fplot(f, 'm', 'linewidth', 2)
hold on
plot(x2, y2, 'k', 'linewidth', 2)
syms winterpolante_err_5 = subs(f, w, x1);
%poi devi farlo diventare double e non syms
err_5 = abs(y1 - interpolante_err_5)
plot(err_5, 'r', 'linewidth', 2)
syms

winterpolante_err_10 = subs(f, w, x2)%poi devi farlo diventare double e non syms
err_10 = abs(y2 - interpolante_err_10);
hold on
plot(err_10, 'b', 'linewidth', 2)

RI C HI ESTA 1

POLINOMIO LAGRANGE
sigma = [0 0.06 0.14 0.25 0.31 0.47 0.6 0.7];
epsilon = [0 0.08 0.14 0.20 0.23 0.25 0.28 0.29];
p = polyfit(sigma, epsilon, 7);
sigma1 = 0:0.01:0.7;
epsilon1 = polyval(p, sigma1);
plot(sigma1, epsilon1, 'k', 'linewidth', 2)
hold on
plot(sigma, epsilon, 'o', 'linewidth', 2)

COMPOSITA LINEARE
sigma2 = 0:0.01:0.7;
epsilon2 = interp1(sigma, epsilon, sigma2);
plot(sigma2, epsilon2, 'k', 'linewidth', 2)
hold on
plot(sigma, epsilon, 'o', 'linewidth', 2)

MINIMI QUADRATI
epsilonIMQ(1, :) = polyval(polyfit(sigma, epsilon, 1), sigma1);
epsilonIMQ(2, :) = polyval(polyfit(sigma, epsilon, 2), sigma1);
epsilonIMQ(3, :) = polyval(polyfit(sigma, epsilon, 4), sigma1);
plot(sigma1,

epsilonIMQ( 1 , : ) , 'k' , 'linewidth' ,2)
hold on
plot( sigma1, epsilonIMQ( 2 , : ) , 'm' , 'linewidth' ,2)
hold on
plot( sigma1, epsilonIMQ( 3 , : ) , 'g' , 'linewidth' ,2)
hold on
plot( sigma, epsilon , 'o' , 'linewidth' ,2)
legend('MQ 1', 'MQ 2' , 'MQ 4', 'dati')

RI C HI ESTA 2

figure ()
plot( sigma1 , epsilon1 , 'k' , 'linewidth' , 2);
hold on
plot( sigma2 , epsilon2 , 'm' , 'linewidth' ,2)
hold on
plot( sigma1, epsilonIMQ( 3 , : ) , 'g' , 'linewidth' ,2)
hold on
plot( sigma, epsilon , 'o' , 'linewidth' ,2)
legend('Lagrange', 'composita', 'minimi quadrati' , 'dati')

RI C HI ESTA 3

La_1 = polyval( p , 0.4)
La_2 = polyval( p , 0.75)
%La_1 = 2.5275e-01
%La_2 = 8.0963e-02

com_1= interp1(sigma, epsilon, 0.4)
com_2= interp1(sigma, epsilon, 0.75)
%com_1 = 2.4125e-01
%com_2 =

x = -2:0.01:6;
y3 = polyval(p, x3);
x = -2:6;
y = x.*sin(x);
plot(x, y)
hold on
plot(x1, y1, 'm', 'linewidth', 2)
hold on
plot(x2, y2, 'g', 'linewidth', 2)
hold on
plot(x3, y3, 'r', 'linewidth', 2)
legend('dati', 'n = 2', 'n = 4', 'n = 6')
syms winter_err_2 = subs(f, w, x1) %lo metti in double
err_2 = abs(y1 - inter_err_2);
syms winter_err_4 = subs(f, w, x2) %lo metti in double
err_4 = abs(y2 - inter_err_4);
syms winter_err_6 = subs(f, w, x3) %lo metti in double
err_6 = abs(y3 - inter_err_6);
plot(err_2, 'k', 'linewidth', 2)
hold on
plot(err_4, 'm', 'linewidth', 2)
hold on
plot(err_6, 'g', 'linewidth', 2)
legend('n = 2', 'n = 4', 'n = 6')
norm(err_2, inf)
norm(err_4, inf)
norm(err_6, inf)
ans = 4.5425
ans = 2.0946
ans = 0.5724
%x nod è quello con linspace
%xdis è x1
% n = 5
% x = (a + b)

./ 2 + ( b -a )*c ./ 2n = 5;a = -5;b = 5;%for j = 0 : 5c = cos( - (pi * (0:n)) ./ n);%endx = ( a + b ) ./ 2 + ( b -a )*c ./ 2;syms wf = 1./ ( 1 + w.^2) ;y = subs(f , w , x);y = double(y);p = polyfit( x , y , 5);x1 = a : 0.01 : b ;y1 = polyval (p , x1);plot(x1 , y1 , 'k' , 'linewidth' , 2 )hold onfplot ( f , 'm' , 'linewidth' , 2)% n = 10n = 10;a = -5;b = 5;c = cos( - (pi * (0:n)) ./ n);x = ( a + b ) ./ 2 + ( b -a )*c ./ 2;syms wf = 1./ ( 1 + w.^2) ;y = subs(f , w , x);y = double(y);p = polyfit( x , y , 10);x2 = a : 0.01 : b ;y2 = polyval (p , x2);hold on %così aggiunge sul precedente graficoplot( x2 , y2 , 'g' , 'linewidth' , 2)legend( ' n = 5 ' , 'f' , ' n = 10 ' )ERRORIsyms win_er_5 = subs (f , w , x1);in_er_5 = double(in_er_5);er_5 = abs( y1 - in_er_5);syms win_er_10 = subs (f , w , x2);in_er_10 = double(in_er_10);er_10 = abs( y2 - in_er_10);plot( x1, er_5 , 'k',

sin(1./ ( 1 + w.^2)) ;y = subs(f , w , x);y = double(y);p = polyfit( x , y , 4);x1 = a : 0.01 : b ;y1 = polyval (p , x1);syms winter4 = subs ( f , w , x1);inter4 = double(inter4);err4 = abs(y1 - inter4);norm(err4, inf)% ans = 0.4248Con 8 l’errore max è 0.2007. con 10 è 0.1309x = linspace(-1 , 1);f = abs(x - pi/12);plot (x,f)n = 5;a = -1;b= 1;x = linspace(a , b , n+1);syms wf = abs(w - pi/12);y = subs(f , w ,x);y = double(y);p = polyfit (x , y ,n);x1 = a : 0.01 : b;y1 = polyval( p , x1);(nero è 5 , rosa 10 , verde 15)N = [2 : 25];a = -1;b = 1;ris = [];err = [];syms wf = abs(w - pi/12) ;figure()x1 = (a : 0.01 : b);for n = Nc = cos( - (pi * (0:n)) ./ n);x = ( a + b) ./ 2 + ( b -a )*c ./ 2;y = subs(f , w , x);y = double (y);p = polyfit( x , y , n);y1 = polyval(p , x1);inter = subs( f, w , x1);inter = double(inter);err = [err ; abs(y1 - inter)];ris = [ris ; norm(abs(y1 - inter), inf)];plot(x1,y1)hold onendloglog ( N , ris) %(io l’avevo fatto solo

così)M = 2.^[1:7];a = -1;b = 1;err = [];ris = [];syms wf = abs(w - pi/12) ;figure()x1 = a : 0.01 : b;ampiezza= [];for n = MH = (b - a)./n ;ampiezza = [ampiezza ; H];x2 = a : H : b ;y = subs(f , w ,x2);y = double(y);y1 = interp1(x2, y, x1); %(x1 sarebbe la seconda x che trovo%solitamente)inter = subs( f, w , x1);inter = double(inter);err = [err ; abs(y1 - inter)];ris=[ris ; norm(abs(y1 - inter),inf)];plot(x1,y1)hold onendloglog(ampiezza , ris)%al grafico che hai fatto al punto 4 , ci metti sopra questoM = 2.^[1:5];a = -1;b = 1;err = [];ris = [];syms wf = abs(w - pi/12) ;% figure()x1 = a : 0.01 : b;ampiezza= [];for n = MH = (b - a)./n ;ampiezza = [ampiezza ; H];x2 = a : H : b ;y = subs(f , w ,x2);y = double(y);y1 = interp1(x2, y, x1); %(x1 sarebbe la seconda x che troco%solitamente)inter = subs( f, w , x1);inter = double(inter);err = [err ; abs(y1 - inter)];ris=[ris ; norm(abs(y1 - inter),inf)];% plot(x1,y1)% hold onendloglog(M , ris)PUNTO 2>> lab6_7_1>> hold