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Eser050719T.m

function [Q,E]=luglioT(fun,a,b)
H=(b-a)/8;
x=a:H:b;
fx=fun(x);
Q=H*(fx(1)/2+sum(fx(2:8))+fx(9)/2);
Qaux=2*H*(fx(1)/2+sum(fx(3:2:7))+fx(9)/2);
E=abs(Q-Qaux)/3;


Eser050719S.m

function [Q,E]=luglioS(fun,a,b)
h=(b-a)/8;
x=a:h:b;
fx=fun(x);
H=2*h;
Q=H/6*(fx(1)+4*sum(fx(2:2:8))+2* sum(fx(3:2:7))+fx(9));
Qaux=(2*H)/6*(fx(1)+4*fx(3) +2 *fx(5)+4*fx(7)+fx(9));
E=abs(Q-Qaux)/15;

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Eser050916.m

function [x,nit]=gradiente(A,b,x)
toll=norm(b)*1.e-8;
for nit=1:1000
    r=b-A*x;
    if norm(r) <toll, return, end
    alpha=(r'*r)/(r'*A*r);
    x=x+alpha*r;
end
end

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Eser050918.m

function [Q1,Q2]=Eser050918(x,y)
N1=length(x);
N2=length(y);
if N1~=N2, error('Non sono di uguale lunghezza'), end
xx=linspace(x(1),x(end));
p=polyfit(x,y,N1-1);
pxx=polyval(p,xx);
sxx=spline(x,y,xx);
plot(xx,pxx,x,y,xx,sxx);
legend('Polinomio intepolatore','Interpolazione composita','Spline')
q=polyint(p);
Q1=polyval(q,x(end))-polyval(q,x(1));
Q2=sum((x(2:end)-x(1:end-1)).*(y(2:end)+y(1:end-1))/2);

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Eser060218.m

function [x,n]=Richardson(A,b,alpha,x0)
x=x0;
P=tril(A);
for n=1:1000
    r=b-A*x;
    if norm(r) < 1.e-8, return, end
    d=forward(P,r);
    x=x+alpha*d;
end


function x=forward(A,b)
n=length(b);
x=zeros(n,1);
x(1) = b(1)/A(1,1);
for i=2:n
    x(i)=(b(i)-A(i,:)*x)/A(i,i);
end


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Eser100118.m

function [intT,N]=esercizioT(fun,a,b,toll)
N=1;
intT=esercizioN(fun,a,b,N);
toll=toll/2;
for i=1:10
    intTold=intT;
    N=2*N;
    intT=esercizioN(fun,a,b,N);
    err=abs(intT-intTold)/15;
    if err<toll, return, end
end
    

function intN=esercizioN(fun,a,b,N)
H=(b-a)/N;
m=a+H/2:H:b-H/2;
aux=H/(2*sqrt(3));
p1=m-aux;
p2=m+aux;
intN=H/2*(sum(fun(p1))+sum(fun(p2)));

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Eser100717.m

function [t,u]=pc13(fun,t0,y0,T,N)
h=T/N;
t=t0:h:t0+T;
u=zeros(1,N+1);
f=zeros(1,N);
u(1)=y0;
f(1)=fun(t(1),u(1));
u(2)=u(1)+h*f(1);
for i=2:N
    f(i)=fun(t(i),u(i));
    aux=u(i)+h*f(i);
    u(i+1)=u(i)+h/12*(5*fun(t(i+1),aux)+ 8*f(i)-f(i-1));
end

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Eser120617.m

f=@(x)exp(-x);
g=@(x) 2-x.^2;
xx=linspace(-2,2);
fxx=f(xx);
gxx=g(xx);
plot(xx,fxx,xx,gxx);
h=@(x)2-x.^2-exp(-x);
p1=fzero(h,-2)
q1=fzero(h,2)
int1=quad(h,p1,q1)
x=linspace(p1,q1,5);
y=h(x);
Pi=polyfit(x,y,4);
primPi=polyint(Pi);
int2=polyval(primPi,q1)-polyval(primPi,p1)

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Eser140616.m

N=input('Numero sottointervalli: ')
h=1/N;
t=[0:h:1];
u=zeros(1,N+1);
aux=-3*t.^2+2*t-1;
u(1)=1;
for i=1:N
    u(i+1)=(2+h*aux(i))/(2-h*aux(i+1))*u(i);
end
tt=linspace(0,1);
y=@(t) exp(-t.^3+t.^2-t);
plot(tt,y(tt),t,u,'*');
legend('Esatta','Approssimata');
err=max(abs(y(t)-u))

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Eser180213A.m

function [x,n]=pfisso(fun,x0,nmax)
for n=1:nmax
    x=fun(x0);
    inc=abs(x-x0);
    if inc < 1.e-8, return,end;
    x0=x;
end


Eser180213B.m

xx=linspace(-5,5);
f=@(x) 1./(1+x.^2);
g=@(x)x;
plot(xx,f(xx),xx,g(xx));
[Q,n]=pfisso(f,0,100)
h=@(x)f(x)-g(x);
P=fzero(h,0)

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Eser220119A.m

function [t,u]=RK3b(fun,t0,y0,T,n)
h=T/n;
t=t0:h:t0+T;
d=length(y0);
u(1)=y0;
for i=1:n
    K1=fun(t(i),u(i));
    K2=fun(t(i)+2/3*h,u(i)+2/3*h*K1);
    K3=fun(t(i),u(i)+h*(-K1+K2));    
    u(i+1)=u(i)+h/4*(3*K2+K3);
end


Eser220119B.m

N=input('Numero sottointervalli: ');
fun=@(t,z) -t.*(z+1).^2;
sol=@(t) 2./(t.^2+1)-1;
[t,u]=RK3b(fun,0,1,1,N);
err=max(abs(sol(t)-u));

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Eser220618.m

f=@(x) 3-x.^2;
g=@(x)x.*log(abs(x)+1);
x=linspace(-3,3);
plot (x,f(x),x,g(x))
legend('f(x)', 'g(x)')
h=@(x) f(x)-g(x);
P=fzero(h,-3);
Q=fzero(h,3);
Area=quad(h,P,Q)

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