Esercizi svolti in MATLAB, Calcolo numerico

function [ xn resn ] = bisezione(f,a,b,toll,nmax )

a=input('dammi il valore estremo sinistro');

b=input('dammi il valore estremo destro');

toll=input('dammi il valore tolleranza');

nmax=input('dammi il valore max iterazioni');

f=inline('x^3-x-5');

n=-1;

amp=toll+1;

xn=[];

resn=[];

while amp>=toll && n < nmax

n=n+1;

fa=feval(f,a);

fb=feval(f,b);

amp=abs(b-a);

x=a+(amp)*0.5;

fx=feval(f,x);

xn=[xn;x];

resn=[resn;fx];

if fa*fx>0

a=x ;

elseif fa*fx<0

b=x;

end

%BISEZIONE Summary of this function goes here

end

plot(xn,resn,'r*')

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function [ x res ] = bisez(f,a,b,toll,nmax )

a=input('dammi il valore di estremo sinitro a= ');

b=input('dammi il valore di estremo destro b= ');

toll=input('dammi il valore di toll= ');

nmax=input('dammi il valore di iterazioni nmax= ');

f=inline('x^3-x-5');

n=0;

chk=toll+1 ;

fa=feval(f,a);

while chk>=toll && n < nmax

n=n+1;

x(n)=a+(abs(b-a))*0.5 ;

fx=feval(f,x(n));

res(n)=fx;

if fa*fx>0

a=x(n);

elseif fa*fx<0

b=x(n);

end

%BISEZ Summary of this function goes here

% Detailed explanation goes here

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function [ L,x ] = CHOLO( A,b,n )

%CHOLO Summary of this function goes here

% Detailed explanation goes here

n=length(A);

L=zeros(n,n);

if A(1,1)<0

disp('no solution');

return

else L(1,1)=sqrt(A(1,1));

end for i=2:n

L(i,1)=A(i,1)/L(1,1);

end

for j=2:n

L(j,j)=A(j,j);

for k=1:j-1

L(j,j)=L(j,j)-L(j,k)*L(j,k);

end

if L(j,j)<0

disp('no solution');

return

else L(j,j)=sqrt(L(j,j));

end

for i=j+1:n

L(i,j)=A(i,j);

for k=1:j-1

L(i,j)=L(i,j)-L(i,k)*L(j,k);

end

L(i,j)=L(i,j)/L(j,j);

end

end

% RISOLVO IL SISTEMA Ly=b L'x=y

for i=1:n

y(i)=b(i);

for j=1:i-1

y(i)=y(i)-L(i,j)*y(j);

end

y(i)=y(i)/L(i,i);

end

C=L';

for i=n:-1:1

x(i)=y(i);

for j =i+1:n

x(i)=x(i)-C(i,j)*x(j);

end

x(i)=x(i)/C(i,i);

end

x=x';

end

choleky_test

(permette di richiamare la function dal command window.Basta scrivere il nome

della function "[ L,x ] = CHOLO( A,b,n )" e premere INVIO. E' anche un modo di

verifica della function)

A=[ 4 2 -2;2 10 -7;-2 -7 9]

b=[-8 -7 2]';

[ L,x ] = CHOLO( A,b,n )

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function [ C,p ] = DIFF_DIVISE( x,y,xx,n )

%DIFF_DIVISE Summary of this function goes here

% Detailed explanation goes here

C=zeros(n,n);

for i=1:n

C(i,1)=y(i);

end

for j=2:n

for i=j:n

C(i,j)=(C(i,j-1)-C(i-1,j-1))/(x(i)-x(i-j+1));

end

end

m=length(xx);

p=ones(m,1)*C(n,n)

for j=n-1:-1:1

p=p.*(xx-x(j))+C(j,j);

end

end

DIFF_DIVISE _TEST

x=[-1 2 -2 3 ]';

y=[-1 2 -2 4 ]';

xx=[0 1 2 4 5 6 0.1 ]';

n=4;

[ C,p ] = DIFF_DIVISE( x,y,xx,n )

-----------------------------------------------------------------

function [ C,yy] = diffdivise( x,y,xx )

%DIFFDIVISE Summary of this function goes here

% Detailed explanation goes here

n=length(x);

C=zeros(n,1);

C(:,1)=y(:);

for j=2:n

for i=j:n

C(i,j)=(C(i,j-1)- C(i-1,j-1))/(x(i)-x(i-j+1));

end

end

%HORNER

m=length(xx);

yy=zeros(1,m);

for i=1:m

p=C(1,1);

for j=2:n

p=p+C(j,j)*prod(xx-x(1:j-1));

end

yy(i)=p;

end

end

test

x=[0 1 2 3 ]

y=[5 -2 4 18]

xx=0.5

[ C,yy] = diffdivise( x,y,xx )

----------------------------------------------------------------

function [ x,y ] = EULERO(f,x0,y0,b,h )

%EULERO Summary of this function goes here

% Detailed explanation goes here

m=(b-x0)/h;

x=0;

y(1)=y0;

for n=2:m+1

x(n)=x0+(n-1)*h;

y(n)=y(n-1)+h*feval(f,x(n-1),y(n-1));

end

x=x';

y=y';

T=[x y]

end

test

f=inline('-y*exp(x)');

b=0.2;

x0=0;

y0=1

h=0.1;

[ x,y ] = EULERO(f,x0,y0,b,h )

----------------------------------------------------------------

function [x,y ] = EURLERO_MODIFIC(f,x0,y0,b,h )

%EURLERO_MODIFIC Summary of this function goes here

% Detailed explanation goes here

x=x0;

y(1)=0;

m=(b-x0)/h;

for n=2:m+1

k1=feval(f,x(n-1),y(n-1));

k2=feval(f,x(n-1) + h*0.5,y(n-1)+h*0.5*k1);

y(n)=y(n-1)+h*k2;

x(n)=x0+(n-1)*h;

end

x=x';

y=y';

T=[x, y]

end

test

f=inline('4*x.^3*sqrt(1-y.^2)');

x0=0;

y0=0;

b=0.2;

h=0.1;

[x,y ] = EURLERO_MODIFIC(f,x0,y0,b,h )

----------------------------------------------------------------

function [L Uy ] = GAUS_P