Anul 3 Semestrul 2

2025-07-03 20:56:38 +03:00
parent 184f3bd92e
commit 3b7fb85767
269 changed files with 20955 additions and 0 deletions
@@ -0,0 +1,106 @@
+A = [1,2;3,4]
+det(A)
+inv(A)*A
+A*A
+A.*A
+A^2
+A.^2
+
+v=1:10
+V=1:-0.1:0
+
+v.^2
+
+transpose(v) # or v'
+
+A(1,:)
+
+A(:,2)
+
+
+#L1_NC.pdf
+
+#1.a
+x=-4:0.1:7.2;
+p = x.^5-5*x.^4-16*x.^3+16*x.^2-17.*x+21;
+
+#plot(x,p)
+
+#1.b
+x=-2.5;
+p = [1,-5,-16,+16,-17,21];
+polyval(p,x)
+#1.c
+roots(p)
+
+polyval(p,7)
+
+#2
+x = 0:0.1*pi:2*pi;
+
+f = sin(x);
+g = sin(2*x);
+h = sin(3*x);
+
+
+#subplot(3,1,1)
+#plot(x,f)
+#subplot(3,1,2)
+
+#plot(x,g)
+#subplot(3,1,3)
+
+#plot(x,h)
+clf  #clear plot
+t= 0:0.1*pi:10*pi
+
+R=3.8;
+r=1;
+
+x = (R+r)*cos(t) - r*cos((R/r+1)*t);
+y = (R+r)*sin(t) - r*sin((R/r+1)*t);
+
+plot(x,y)
+
+[x, y] = meshgrid(-2:0.1:2, 0.5:0.1:4.5);
+
+f = sin(e.^x).*cos(log(y));
+clf
+#mesh(x, y, f);
+
+
+#xlabel('X-axis');
+#ylabel('Y-axis');
+#zlabel('Z-axis');
+#title('3D Surface Plot using mesh');
+#colormap('jet');
+#colorbar;
+#grid on;
+
+
+figure;
+plot3(x, y, f);
+xlabel('X-axis');
+ylabel('Y-axis');
+zlabel('Z-axis');
+title('3D Line Plot using plot3');
+grid on;
+
+
+
+function result = funct(n)
+    if n == 0
+        result = 1+1;
+    else
+        result = 1 + 1/ funct(n - 1);
+    end
+end
+% Increase the recursion limit to 5000
+#max_recursion_depth(2025);
+
+funct(2)
+funct(10)
+funct(100)
+funct(2025)
+
+
@@ -0,0 +1,9 @@
+
+f=@(x)1./x;
+
+R = rectangle_int(f,1,2,100)
+
+R = trapz_int(f,1,2,100)
+
+R = homer_int(f,1,2,100)
+
@@ -0,0 +1,25 @@
+
+
+#f=@(x)((2./sqrt(pi)).*(e.^(-x.^2)));
+
+#for b=0.1:0.1:1
+#  I = adquad(f,0,b,0.1,4)
+#  I2 = integral(f,0,b)
+#end
+
+
+f=@(x)(sin(x.^(1/3)));
+g=@(x)(sin(x).*3.*(x.^2));
+
+x = [2,4,8,16,32,64,128,256];
+trueI = integral(f,0,1)
+for n = x
+  n
+  I1=homer_int(f,0,1,n)
+  fprintf('%.*f \n', 10, abs(trueI-I1))
+  I2=homer_int(g,0,1,n)
+  fprintf('%.*f \n', 10, abs(trueI-I2))
+end
+
+
+
@@ -0,0 +1,80 @@
+#pkg load symbolic;
+
+#1.a
+syms x;
+
+f=exp(x)
+
+t1=taylor(f,x,0,'order',1);
+t2=taylor(f,x,0,'order',2);
+t3=taylor(f,x,0,'order',3);
+t4=taylor(f,x,0,'order',4);
+t10 = taylor(f,x,0,'order',10);
+
+#ezplot(t1)
+hold on
+ezplot(t2)
+hold on
+ezplot(t3)
+hold on
+ezplot(t4)
+xlim([-3,3])
+
+#1.b
+vpa(exp(1),7)
+
+vpa(subs(t10,x,1),7)
+
+#2.a
+
+syms x;
+f=sin(x)
+
+t3=taylor(f,x,0,'order',3);
+t5=taylor(f,x,0,'order',5);
+hold off
+ezplot(f)
+hold on
+ezplot(t3)
+hold on
+ezplot(t5)
+xlim([-pi,pi])
+ylim([-2,2])
+
+#2.b
+t10=taylor(f,x,0,'order',10);
+
+vpa(sin(pi/5),5)
+vpa(subs(t10,x,sym(pi)/5),5)
+
+vpa(sin(10*pi/3),5)
+vpa(subs(t10,x,10*sym(pi)/3),5) #it's not precise enought, increase n for Taylor series or move x0=0 to x0= 10pi/3
+
+
+#3.a
+
+syms x;
+f=log(1+x);
+t2= taylor(f,x,0,'order',2);
+t5= taylor(f,x,0,'order',5);
+
+hold off
+ezplot(f)
+hold on
+ezplot(t2)
+hold on
+ezplot(t5)
+
+xlim([-0.9,1])
+ylim([-1,1])
+t= taylor(f,x,0,'order',10);
+vpa(log(2),5)
+vpa(subs(t,x,1),5)
+
+syms x;
+g=log(1-x);
+t2 = taylor(g,x,0,'order',10)
+
+t-t2
+
+vpa(subs(t,x,sym(0.999,'f')) - subs(g,x,sym(0.999,'f')),5)
@@ -0,0 +1,30 @@
+x = [0 1 2];
+f = 1./(1+x);
+
+t=divdiff(x,f);
+
+
+x = [0, 1, 2];
+f = 1./(1+x);
+df = (-1)./((1+x).^2);
+
+[z,t]=divdiff2(x,f,df)
+
+
+
+x=linspace(1,2,11);
+f=1./(1+x);
+df = (-1)./((1+x).^2);
+t=divdiff(x,f)
+[z,t]=divdiff2(x,f,df)
+
+x = [-2, -1, 0, 1, 2, 3, 4];
+f = [-5, 1, 1, 1,7 ,25, 60];
+
+t=divdiff(x,f)
+
+x = [-2, -1, 0, 1, 2, 3, 4];
+f = [-5, 1, 1, 1,7 ,25, 60];
+
+t=forwarddiff(f)
+t=backdiff(f)
@@ -0,0 +1,26 @@
+#U= [2,4,2;0,-1,1;0,0,-1];
+#b = [8;0;-1];
+
+
+
+
+#x = backsub(U,b);
+
+A= [2 1 -1 -2; 4 4 1 3; -6 -1 10 10; -2 1 8 4];
+b = [2;4;-5;1];
+
+x = gauss(A,b);
+
+
+
+n=5
+
+A=5*eye(n) -diag(ones(1,n-1),1)- diag(ones(1,n-1),-1)
+
+b = [4,3* ones(1,n-2),4]'
+
+[L,U,P]= lu(A);
+
+y = forwsub(L,P*b)
+x = backsub(U, y)
+
@@ -0,0 +1,42 @@
+n=7;
+
+A=5*eye(n) -diag(ones(1,n-1),1)- diag(ones(1,n-1),-1);
+
+b = [4,3* ones(1,n-2),4]';
+
+
+x0 = zeros(size(b));
+
+[x, nit] = jacobi(A,b, x0,10^(-5), 1000);
+
+[x, nit] = gaussseidel(A,b, x0,10^(-5), 1000);
+
+A = [10 7 8 7; 7 5 6 5; 8 6 10 9; 7 5 9 10];
+b=[32;23;33;31];
+
+x = A^-1*b
+trueB = b;
+trueX = x;
+trueA = A;
+
+A = [10 7 8 7; 7 5 6 5; 8 6 10 9; 7 5 9 10];
+
+
+
+b=[32.1;22.9;33.1;30.9];
+
+x = A^-1*b
+norm(trueX-x,inf)/norm(trueX,inf)
+norm(trueB-b,inf)/norm(trueB,inf)
+cond(A)
+
+A = [10 7 8.1 7.2; 7.8 5.04 6 5; 8 5.98 9.89 9; 6.99 4.99 9 9.98];
+b=[32;23;33;31];
+x = A^-1*b
+
+cond(A)
+
+norm(trueX-x,inf)/norm(trueX,inf)
+norm(trueA-A,inf)/norm(trueA,inf)
+
+
@@ -0,0 +1,39 @@
+#1.b
+
+
+nodes = linspace(-2,4,10);
+f=@(x)(x+1)./(3*x.^2+2*x+1);
+
+plot(nodes, f(nodes),'o');
+hold on
+
+
+x=linspace(-2,4,500);
+
+
+plot(x,f(x));
+hold on
+
+lnf=lagrange_classic(nodes,f(nodes),x);
+
+plot(x,lnf)
+
+plot(x,abs(f(x)-lnf),'x')
+
+
+nodes = [81,100,121];
+
+f_nodes = [9,10,11];
+
+sqrt(118);
+lnf = lagrange_classic(nodes,f_nodes, [118]);
+
+
+nodes = [1980,1990,2000,2010,2020];
+f_nodes = [4451,5287,6090,6970,7821];
+
+lnf = lagrange_barycentric(nodes, f_nodes, [2005,2015])
+
+abs([6474,7405] - lnf)
+
+
@@ -0,0 +1,32 @@
+
+#truex = linspace(0,1,20);
+#x0=[0,1/3,1/2,1]
+#f=@(x)cos(pi*x);
+
+#plot(x0, f(x0),'o');
+#hold on
+
+#plot(truex, f(truex),'blue');
+
+
+#lnf = newton(x0,f(x0),truex)
+
+#plot(truex, lnf, 'r');
+
+#lnf = newton(x0,f(x0),1/5)
+
+#truex = linspace(1001,1009,9);
+
+#x0=[1000,1010,1020,1030,1040,1050];
+#f0=[3.000000,3.0043214,3.0086002,3.0128372,3.0170333,3.0211893];
+
+#lnf = newton(x0,f0,truex)
+
+
+x0 = linspace(-4,4,9);
+f=@(x)power(2,x);
+
+lnf = aitken(x0,f(x0),1/2)
+
+
+
@@ -0,0 +1,63 @@
+nodes = linspace(-2,4,7);
+
+f=@(x)(x+1)./(3*x.^2+2*x+1);
+
+
+#plot(nodes, f(nodes),'o');
+#hold on
+
+
+x=linspace(-2,4,500);
+
+
+#plot(x,f(x));
+hold on
+
+lnf=lagrange_classic(nodes,f(nodes),x);
+
+#plot(x,lnf)
+
+
+boor = spline(nodes,f(nodes),x);
+
+#plot(x,boor)
+
+df = @(x)(-(3*x.^2+6*x+1)./(3*x.^2+2*x+1).^2);
+
+herm = hermite(nodes,f(nodes), df(nodes),x)
+#plot(x,herm)
+
+
+f=@(x)(x.*sin(pi.*x));
+
+nodes = [-1,-1/2,0,1/2,1,3/2];
+
+x=linspace(-1,3/2,500);
+
+boor = spline(nodes,f(nodes),x);
+boor_comp = spline(nodes,[pi,f(nodes),-1],x)
+piece = pchip(nodes,f(nodes),x)
+
+#plot(nodes, f(nodes),'o');
+#hold on
+#plot(x,boor);
+#plot(x, boor_comp);
+#plot(x, piece);
+
+
+nodes = [0.5,1.5,2,3,3.5,4.5,5,6,7,8];
+
+f_nodes = [5,5.8,5.8,6.8,6.9,7.6,7.8,8.2,9.2,9.9];
+x=linspace(0.5,8,500);
+
+p = polyfit(nodes,f_nodes,1);
+plot(nodes, f_nodes,'o');
+hold on
+plot(x, polyval(p,x))
+
+norm(p)
+
+polyval(p,4)
+
+
+
@@ -0,0 +1,10 @@
+function I = adquad(f, a, b, err, m)
+  I1 = trapz_int(f, a, b, m);
+  I2 = trapz_int(f, a, b, 2*m);
+  if abs(I1 - I2) < err % success
+    I = I2;
+    return
+  else % recursive subdivision
+    I = adquad(f, a, (a+b)/2, err, m) + adquad(f, (a+b)/2, b, err, m);
+  end
+end;
@@ -0,0 +1,12 @@
+function Lnf = aitken(x0,f0,x)
+  p = zeros(length(x0));
+  p(:,1) = f0'
+
+  n = length(x0)
+  for i=2:n
+    for j=2:i
+      p(i,j) = (1/(x0(i)-x0(i-j+1)))*det([x-x0(i-j+1), p(i-1,j-1);x-x0(i),p(i,j-1)]);
+    endfor
+  endfor
+  Lnf=p(n,n);
+end;
@@ -0,0 +1,11 @@
+function t = backdiff(f)
+    n = length(f);
+    t = zeros(n);
+    t(:,1)= f'
+    for j = 2:n
+      t(j:n,j) = diff(t(j-1:n,j-1));
+    endfor
+end;
+
+
+
@@ -0,0 +1,10 @@
+function x = backsub(U, b)
+    n = length(b);
+    x = zeros(size(b));
+    for k = n:-1:1
+      x(k) = (b(k) - (U(k,k+1:n) * x(k+1:n)))/U(k,k);
+    endfor
+end;
+
+
+
@@ -0,0 +1,11 @@
+function t = divdiff(x, f)
+    n = length(x);
+    t=zeros(n);
+    t(:,1)= f'
+    for j = 2:n
+      t(1:n-j+1,j) = (t(2:n-j+2,j-1) - t(1:n-j+1,j-1))./((x(j:n)-x(1:n-j+1))');
+    endfor
+end
+
+
+
@@ -0,0 +1,13 @@
+function [z, t] = divdiff2(x,f,df)
+  z = repelem(x,2);
+  lz = length(z);
+  f2 = repelem(f,2);
+  t=zeros(lz);
+  t(:,1)= f2';
+  t(1:2:lz-1,2) = df';
+  t(2:2:lz-2,2) = (diff(f)./diff(x))';
+  for j = 3:lz
+      t(1:lz-j+1,j) = (t(2:lz-j+2,j-1) - t(1:lz-j+1,j-1))./((z(j:lz)-z(1:lz-j+1))');
+    endfor
+end
+
@@ -0,0 +1,11 @@
+function t = forwarddiff(f)
+    n = length(f);
+    t = zeros(n);
+    t(:,1)= f'
+    for j = 2:n
+      t(1:n-j+1,j) = diff(t(1:n-j+2,j-1));
+    endfor
+end;
+
+
+
@@ -0,0 +1,10 @@
+function x =  forwsub(U, b)
+    n = length(b);
+    x = zeros(size(b));
+    for k = 1:n
+      x(k) = (b(k) - (U(k,1:k-1) * x(1:k-1)))/U(k,k);
+    endfor
+end;
+
+
+
@@ -0,0 +1,18 @@
+function x = gauss(A, b)
+    [r,n] = size(A);
+    x = zeros(size(b));
+    A = [A,b]
+    for i = 1:n
+      [v, p] = max(abs(A(i:n, i)));
+      p = p + i - 1;
+      A([i p],:) = A([p i],:);
+      for j = i+1:n
+        coeff = A(j, i)/ A(i,i);
+        A(j, i:n+1) = A(j, i:n+1) - coeff * A(i,i:n+1)
+      endfor
+    endfor
+    x = backsub(A(:,1:n),A(:,n+1));
+end;
+
+
+
@@ -0,0 +1,20 @@
+function [x, nit] = gausssidel(A, b, x0, err, maxnit)
+  n = length(b);
+  D = diag(diag(A));
+  L = tril(A, -1);
+  U = A - D - L;
+  M = D + L;
+  N = -U;
+  T = M^(-1)*N;
+  c = M^(-1)*b;
+  x = x0;
+  for k = 1:maxnit
+    old_x = x;
+    x = T*x+c;
+    if norm(x-old_x,inf) <= ((1-norm(T,inf))/norm(T,inf))*err
+      nit = k;
+      return;
+    endif
+  endfor
+  error('Too dificult');
+end;
@@ -0,0 +1,13 @@
+function Hnf = hermite(x0,f0,df0,x)
+  [z,dz] = divdiff2(x0,f0,df0);
+  x0=z;
+  ds = dz(1,:);
+  n = length(ds);
+  m = length(x);
+  Hnf = zeros(size(x));
+  for i=1:m
+    for j=1:n
+      Hnf(i) =  Hnf(i) + ds(j) * (prod( x(i) - x0([1:j-1])));
+    endfor
+  endfor
+end;
@@ -0,0 +1,4 @@
+function R = homer_int(f, a, b, n)
+  R = ((b-a)/(2*3*n)) * (f(a)+f(b)+4*sum(f(a+([1:2:2*n-1])*((b-a)/(2*n))))+2*sum(f(a+([2:2:2*n-2])*((b-a)/(2*n)))));
+
+ end;
@@ -0,0 +1,17 @@
+function [x, nit] = jacobi(A, b, x0, err, maxnit)
+  n = length(b);
+  M = diag(diag(A));
+  N = M - A;
+  T = M^(-1)*N;
+  c = M^(-1)*b;
+  x = x0;
+  for k = 1:maxnit
+    old_x = x;
+    x = T*x+c;
+    if norm(x-old_x,inf) <= ((1-norm(T,inf))/norm(T,inf))*err
+      nit = k;
+      return;
+    endif
+  endfor
+  error('Too dificult');
+end;
@@ -0,0 +1,14 @@
+function Lnf = lagrange_barycentric(nodes,f_nodes,x)
+  n = length(nodes);
+  m = length(x);
+  Lnf = zeros(size(x));
+  l = zeros(size(nodes));
+  for i = 1:m
+    ux = prod( x(i) - nodes([1:n]));
+    for j = 1:n
+      wi(j) = 1/prod( nodes(j) - nodes([1:j-1,j+1:n]));
+    endfor
+    Lnf(i) = ux* ((wi./(x(i) - nodes([1:n])))*f_nodes');
+
+  endfor
+end;
@@ -0,0 +1,13 @@
+function Lnf = lagrange_classic(nodes,f_nodes,x)
+  n = length(nodes);
+  m = length(x);
+  Lnf = zeros(size(x));
+  l = zeros(size(nodes));
+  for i = 1:m
+    for j = 1:n
+      l(j) =(prod( x(i) - nodes([1:j-1,j+1:n])))/(prod( nodes(j) - nodes([1:j-1,j+1:n])));
+    endfor
+    Lnf(i) = l*f_nodes';
+
+  endfor
+end;
@@ -0,0 +1,13 @@
+function Lnf = newton(x0,f0,x)
+  d = divdiff(x0,f0);
+  ds = d(1,:);
+  n = length(ds);
+  m = length(x);
+  Lnf = zeros(size(x));
+  for i=1:m
+    for j=1:n
+      Lnf(i) =  Lnf(i) + ds(j) * (prod( x(i) - x0([1:j-1])));
+    endfor
+
+  endfor
+end;
@@ -0,0 +1,4 @@
+function R = rectangle_int(f, a, b, n)
+  R = ((b-a)/n) * sum(f(a+([0:n-1]+1/2)*((b-a)/n)))
+
+ end;
@@ -0,0 +1,4 @@
+function R = trapz_int(f, a, b, n)
+  R = ((b-a)/(2*n)) * (f(a)+f(b)+2*sum(f(a+([1:n-1])*((b-a)/n))));
+
+ end;