% 1.NALOGA
% funkcija in odvodi
f = @(x) x.^2 + log(x);
df = @(x) 2*x + 1/x;
ddf = @(x) 2 - 1/(x.^2);

% iteracijske funkcije
g1 = @(x) exp(-x.^2);
g2 = @(x) (x + exp(-x.^2))/2;
g3 = @(x) (2*x.^3 + exp(-x.^2))./(1 + 2*x.^2);

% parametri iteracije
tol = 1e-10;
N = 200;
x0 = 2;

% tocna nicla
xp = fzero(f,x0);

% regula falsi in bisekcija
N1 = 15;
a = 0.1;
b = 2;
r = regula_falsi(f,a,b,N1);
b = bisekcija(f,a,b,N1);

% navadna iteracija
[x1,X1,k1] = iteracija(g1,x0,tol,N);
[x2,X2,k2] = iteracija(g2,x0,tol,N);
[x3,X3,k3] = iteracija(g3,x0,tol,N);

% tanegnta in Halleyeva metoda
[xt,Xt,kt] = tangentna(f,df,x0,tol,N);
[xh,Xh,kh] = halley(f,df,ddf,x0,tol,N);

% stevilo korakov
st_korakov = [k1 k2 k3 kt kh N1 N1]';

Stevilo korakov:
144 
11 
10 
5 
4 
15 
15 

% koncni priblizki
priblizki = [x1 x2 x3 xt xh r b]';

Dobljeni priblizki:
0.6529186405 
0.6529186404 
0.6529186404 
0.6529186404 
0.6529186404 
0.6529186466 
0.6528717041 

% napake 
priblizki_napake = abs([x1 x2 x3 xt xh r b]' - xp);

Napake priblizkov:
0.0000000000 
0.0000000000 
0.0000000000 
0.0000000000 
0.0000000000 
0.0000000062 
0.0000469363 


% vsi priblizki

Povprecne vrednosti vseh izracunanih priblizkov (brez regula-falsi in bisekcije, ker tam ne hranimo vseh izracunanih priblizkov):
0.6590322503 %g1
0.7977594143 %g2
1.0696671471 %g3
0.9296950920 %tangentna
0.9301038489 %halley


Se resitve za parametre:
tol = 1e-10;
N = 7; % prej smo gledali za N=200 in so bile napake zelo majhne, za N=7 bodo vecje
x0 = 2;

N1 = 15;

Stevilo korakov:
7 
7 
7 
5 
4 
15 
15 

Dobljeni priblizki:
0.5235309984 
0.6529187092 
0.6544223829 
0.6529186404 
0.6529186404 
0.6529186466 
0.6528717041 

Napake priblizkov:
0.1293876420 
0.0000000688 
0.0015037425 
0.0000000000 
0.0000000000 
0.0000000062 
0.0000469363 

Povprecne vrednosti vseh izracunanih priblizkov (brez regula-falsi in bisekcije, ker tam ne hranimo vseh izracunanih priblizkov):
0.7567275308 %g1
0.8701798006 %g2
1.2259474536 %g3
0.9296950920 %tangentna
0.9301038489 %halley






% 2.NALOGA 
% LU brez pivotiranja
A = [2 -1 1 4; 4 -3 7 14; 0 -3 18 19; 6 -2 7 14];
[~,U] = lubp(A);

U frobeniusova:
10.9087121146 

A norma 1:
51.0

% LU z delnim pivotiranjem
L Inf norma:
2.2222222222 


% Thomasov algoritem, matriki L in U po vrsticah

L(1,:) = 
3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000 3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  3.0000000000  

L(2,:) = 
-0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  -0.2500000000  
 
L(3,:) = 
0.8000000000  0.6896551724  0.6060606061  0.5405405405  0.4878048780  0.4444444444  0.4081632653  0.3773584906  0.3508771930  0.3278688525  0.3076923077  0.2898550725  0.2739726027  0.2597402597  0.2469135802  0.2352941176  0.2247191011  0.2150537634  0.2061855670  0.1980198020  0.1904761905  
 
U(1,:) = 
1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  1.0000000000  
 
U(2,:) = 
-8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  -8.0000000000  
 
U(3,:) = 
1.2500000000  1.4500000000  1.6500000000  1.8500000000  2.0500000000  2.2500000000  2.4500000000  2.6500000000  2.8500000000  3.0500000000  3.2500000000  3.4500000000  3.6500000000  3.8500000000  4.0500000000  4.2500000000  4.4500000000  4.6500000000  4.8500000000  5.0500000000  5.2500000000  
 
U(4,:) = 
-0.2000000000  0.2413793103  0.5757575758  0.8378378378  1.0487804878  1.2222222222  1.3673469388  1.4905660377  1.5964912281  1.6885245902  1.7692307692  1.8405797101  1.9041095890   1.9610389610  2.0123456790  2.0588235294  2.1011235955  2.1397849462  2.1752577320  2.2079207921  2.2380952381  
 

% Cholesky 
A = [1 1 1 1 1;1 2 3 4 5;1 3 6 10 15;1 4 10 20 35;1 5 15 35 70];
[V,~] = CholPivot(A);

det(V-A):
744.3972103718  

det(V1'-V):
0.2795591260