├── MNOctave2018.pdf ├── _lista_programas.txt ├── nif.m ├── rectang.m ├── sustprogr.m ├── sustregr.m ├── derinum.m ├── trapecios.m ├── simpson.m ├── derinum2.m ├── plotspline.m ├── diasmes.m ├── scriptfenomenoRunge.m ├── scripttirosbola.m ├── scriptanimacionbola.m ├── difdiv.m ├── scriptPVIrobertson.m ├── scriptdeformacionviga.m ├── scriptPVIEuler.m ├── qrvp.m ├── README.md ├── dhont.m ├── gausspiv.m ├── potencias.m ├── rk4solver.m ├── scriptajustetrigocontinuo.m ├── scriptsplinecuadratico.m ├── scriptbiseccion.m ├── scriptajustepolicontinuo.m ├── gradconj.m ├── scriptPVFpoisson.m ├── scriptdescomposicionlu.m ├── gaussseidel.m ├── scriptsplinecubico.m ├── scriptPVFlineal.m ├── scriptPVIpresadepre.m ├── newton.m ├── biseccion.m ├── scriptmetodotiro.m ├── secante.m └── LICENSE /MNOctave2018.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/oscarsanchezromero/Calculo-Cientifico-Octave/HEAD/MNOctave2018.pdf -------------------------------------------------------------------------------- /_lista_programas.txt: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/oscarsanchezromero/Calculo-Cientifico-Octave/HEAD/_lista_programas.txt -------------------------------------------------------------------------------- /nif.m: -------------------------------------------------------------------------------- 1 | function y = nif(dni) 2 | % Esta función calcula la letra del NIF para el DNI introducido 3 | letras = 'TRWAGMYFPDXBNJZSQVHLCKE'; 4 | n = rem(dni,23)+1; 5 | y = letras(n); 6 | endfunction -------------------------------------------------------------------------------- /rectang.m: -------------------------------------------------------------------------------- 1 | function int = rectang(f,a,b,N = 150) 2 | % Función que aproxima la integral de f en el 3 | % intervalo [a,b] mediante la fórmula compuesta de 4 | % rectángulos (punto medio) con N subintervalos 5 | % 6 | h = (b-a)/N; 7 | int = h*sum(feval(f,a+h*(1:N)-h/2)); 8 | endfunction -------------------------------------------------------------------------------- /sustprogr.m: -------------------------------------------------------------------------------- 1 | function x = sustprogr(T,c) 2 | % Esta función resuelve un SEL Tx = c, con T triangular inferior 3 | % y regular, mediante sustituciones progresivas. 4 | [dim,nc] = size(c); 5 | x = zeros(dim,nc); 6 | for i = 1:dim 7 | x(i,:) = (c(i,:)-T(i,:)*x)/T(i,i); 8 | endfor 9 | endfunction -------------------------------------------------------------------------------- /sustregr.m: -------------------------------------------------------------------------------- 1 | function x = sustregr(T,c) 2 | % Esta función resuelve el SEL Tx = c, con T triangular superior 3 | % y regular, mediante sustituciones regresivas. 4 | [dim,nc] = size(c); 5 | x = zeros(dim,nc); 6 | for i = dim:-1:1 7 | x(i,:) = (c(i,:)-T(i,:)*x)/T(i,i); 8 | endfor 9 | endfunction -------------------------------------------------------------------------------- /derinum.m: -------------------------------------------------------------------------------- 1 | function [x,df] = derinum(f,a,b,N) 2 | % df es la derivada primera numérica de la función f sobre el 3 | % vector x de N valores equiespaciados en el intervalo [a,b]. 4 | h = (b-a)/(N-1); 5 | x = linspace(a,b,N); 6 | df = (feval(f,a+h*(1:N))-feval(f,a+h*(-1:(N-2))))/(2*h); 7 | endfunction 8 | -------------------------------------------------------------------------------- /trapecios.m: -------------------------------------------------------------------------------- 1 | function int = trapecios(f,a,b,N = 150) 2 | % Función que aproxima la integral de f en el 3 | % intervalo [a,b] mediante la fórmula compuesta de 4 | % trapecios con N subintervalos 5 | % 6 | h = (b-a)/N; 7 | x = a+h*(0:N); 8 | y = feval(f,x); 9 | int = h*trapz(y); 10 | endfunction 11 | -------------------------------------------------------------------------------- /simpson.m: -------------------------------------------------------------------------------- 1 | function int = simpson(f,a,b,N = 75) 2 | % Función que aproxima la integral de f en el 3 | % intervalo [a,b] mediante la fórmula compuesta de 4 | % Simpson con 2N+1 nodos 5 | % 6 | h = (b-a)/N; 7 | x = a+h*(0:(N-1)); 8 | int = (sum(feval(f,x)+4*feval(f,x+h/2)+feval(f,x+h)))*h/6; 9 | endfunction 10 | -------------------------------------------------------------------------------- /derinum2.m: -------------------------------------------------------------------------------- 1 | function[x,d2f] = derinum2(f,a,b,N) 2 | % d2f es la segunda derivada numérica de la función f sobre el 3 | % vector x de N valores equiespaciados en el intervalo [a,b]. 4 | h = (b-a)/(N-1); 5 | x = linspace(a,b,N); 6 | d2f = (feval(f,a+h*(-1:(N-2)))-2*feval(f,a+h*(0:N-1))+... 7 | feval(f,a+h*(1:N)))/h^2; 8 | endfunction -------------------------------------------------------------------------------- /plotspline.m: -------------------------------------------------------------------------------- 1 | function [] = plotspline(p,x) 2 | % Esta función representa en cada intervalo [x(i),x(i+1)] 3 | % el polinomio p(i,:) empleando el comando plot. 4 | n = length(x)-1; % Número de subintervalos 5 | for i = 1:n 6 | z = linspace(x(i),x(i+1),20); 7 | plot(z,polyval(p(i,:),z)); 8 | hold on 9 | endfor 10 | hold off 11 | endfunction -------------------------------------------------------------------------------- /diasmes.m: -------------------------------------------------------------------------------- 1 | function y = diasmes(x) 2 | % Número de días del mes x-ésimo. 3 | switch x 4 | case ({1 3 5 7 8 10 12}) 5 | y = 31; 6 | case {4,6,9,11} 7 | y = 30; 8 | case (2) 9 | y = 28; 10 | warning('29 si es bisiesto'); 11 | otherwise 12 | error('x no parece ser un número entre 1 y 12'); 13 | endswitch 14 | endfunction -------------------------------------------------------------------------------- /scriptfenomenoRunge.m: -------------------------------------------------------------------------------- 1 | % Función de Runge 2 | f = @(x) 1./(1+25*x.^2); 3 | % Función que evalúa el polinomio interpolador de grado n. 4 | function v = pz(z,n) 5 | x = linspace(-1,1,n+1); % n+1 nodos equiespaciados 6 | y = 1./(1+25*x.^2); 7 | p = polyfit(x,y,n); 8 | v = polyval(p,z); 9 | endfunction 10 | z = linspace(-1,1,100); 11 | plot(z,f(z),'g',z,pz(z,8),'--b',z,pz(z,14),'-.r') -------------------------------------------------------------------------------- /scripttirosbola.m: -------------------------------------------------------------------------------- 1 | global vx; 2 | v = 50*1000/3600; % velocidad expresada en m/s 3 | hold on 4 | axis([0,15,0,8]); 5 | function [d] = bolafun(z,x) 6 | global vx; 7 | d = [z(2);-9.8/(vx-0.25*x).^2]; 8 | endfunction 9 | for theta = pi/16:pi/16:6*pi/16 10 | vx = v*cos(theta); 11 | [xnum,solnum] = rk4solver('bolafun',0,15,[0;tan(theta)],100); 12 | plot(xnum,solnum(1,:)) 13 | endfor 14 | xlabel("x"); 15 | ylabel("y"); 16 | hold off -------------------------------------------------------------------------------- /scriptanimacionbola.m: -------------------------------------------------------------------------------- 1 | 1; 2 | function [d] = bolafun(z,x) 3 | vx = (50*1000/3600)*cos(pi/4); 4 | d = [z(2);-9.8/(vx-0.25*x).^2]; 5 | endfunction 6 | [xnum,solnum] = rk4solver('bolafun',0,15,[0;tan(pi/4)],300); 7 | clf 8 | axis([0,15,-1,6]); 9 | hold on 10 | for i = 0:100 11 | plot(xnum(1+3*i),solnum(1,1+3*i),'r*'); 12 | saveas(1,strcat('./graftiro/imagen',num2str(i),'.png'),'png'); 13 | % Ojo! La carpeta graftiro ha de estar creada con anterioridad. 14 | endfor 15 | hold off -------------------------------------------------------------------------------- /difdiv.m: -------------------------------------------------------------------------------- 1 | function p = difdiv(x,y) 2 | % Esta función calcula el polinomio interpolador p mediante 3 | % diferencias divididas para datos lagragianos. 4 | n = length(y)-1; % n+1 es el número de nodos 5 | d = y; % Columna con diferencias divididas de orden 0 6 | p = d(1); % Polinomio que interpola en el primer nodo 7 | for k = 1:n 8 | % Columna con diferencias divididas de orden k 9 | d = (d(2:end)-d(1:end-1))./(x(k+1:end)-x(1:end-k)); 10 | % Polinomio que interpola los k+1 primeros datos. 11 | p = [0,p]+d(1)*poly(x(1:k)); 12 | endfor 13 | endfunction -------------------------------------------------------------------------------- /scriptPVIrobertson.m: -------------------------------------------------------------------------------- 1 | b = 10^12; % Extremo superior de integración 2 | t = [0,10.^linspace(-4,12,100)]; 3 | function up = f(u,t) % definición de la EDO 4 | n = 0; k1 = 0.04; k2 = 3*10^7; k3 = 10^4; 5 | up(1) = -k1*u(1)+k3*u(2)*u(3); 6 | up(2) = k1*u(1)-k2*u(2)^2-k3*u(2)*u(3); 7 | up(3) = k2*u(2)^2; 8 | endfunction 9 | [sol,ystate] = lsode('f',[1;0;0],t); % resolución 10 | % representación gráfica 11 | subplot(2,1,1) semilogx(t(2:end),sol(2:end,[1,3])) % gráfica de u(1) y u(3) subplot(2,1,2) semilogx(t(2:end),sol(2:end,2),'r') % gráfica de u(2) 12 | ystate 13 | -------------------------------------------------------------------------------- /scriptdeformacionviga.m: -------------------------------------------------------------------------------- 1 | % Definiciones: longitud, número de nodos interiores y matriz 2 | L = 5; n = 50; 3 | v = [-2,1,zeros(1,n-2)]; A = toeplitz(v); 4 | % Cálculo de valores propios y localización del menor en |.| 5 | [P,D] = eig(A); 6 | [lambda,posicion] = max(diag(D)); % Valores propios negativos 7 | vector = P(:,posicion); 8 | % Representación gráfica de la deformación 9 | y = linspace(0,L,n+2); 10 | deformacion = [0,vector',0]; 11 | plot(deformacion,y) 12 | texto = sprintf("fuerza = %f", lambda*(n+1)^2/L^2); 13 | xlabel(texto); 14 | axis([-3,3,0,L]); -------------------------------------------------------------------------------- /scriptPVIEuler.m: -------------------------------------------------------------------------------- 1 | lambda = 1; 2 | % Representamos la solución analítica 3 | clf 4 | hold on 5 | t = 5:0.01:6; 6 | plot(t,exp(lambda*t-5),'k'); 7 | % Representamos los valores aproximados con 6 nodos 8 | N = 5; 9 | h = 1/N; 10 | y = [1]; 11 | for i = 1:N 12 | y = [y,y(:,i)*(1+lambda*h)]; 13 | endfor 14 | plot(5:h:6,y,'xr') 15 | % Representamos los valores aproximados con 11 nodos 16 | N = 10; 17 | h = 1/N; 18 | y = [1]; 19 | for i = 1:N 20 | y = [y,y(:,i)*(1+lambda*h)]; 21 | endfor 22 | plot(5:h:6,y,'ob') 23 | xlabel("t");ylabel("y"); 24 | legend('Soluc. exacta','Aprox. 6 nodos', 'Aprox. 11 nodos') 25 | hold off 26 | 27 | -------------------------------------------------------------------------------- /qrvp.m: -------------------------------------------------------------------------------- 1 | function d = qrvp(A,tol,maxiter) 2 | % Esta función calcula una aproximación de los valores propios 3 | % de una matriz cuadrada A mediante el método QR. 4 | % nmax determina el número máximo de iteraciones a realizar. 5 | % tol es una tolerancia sobre lo que le sobra a las matrices 6 | % para ser triangulares. 7 | T = A; 8 | k = 0; 9 | while k<=maxiter && norm(tril(T,-1),inf)>=tol 10 | [Q,R] = qr(T); 11 | T = R*Q; 12 | k = k+1; 13 | endwhile 14 | if k>maxiter 15 | warning('No converge tras %i iteraciones \n',maxiter); 16 | else 17 | fprintf('El método converge en %i iteraciones \n',k); 18 | endif 19 | d = diag(T); 20 | end 21 | 22 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Calculo-Cientifico-Octave 2 | Algoritmos básicos de Cálculo Científico programados con Octave 3 | 4 | En este repositorio se encuentran distintos códigos que se usan frecuentemente 5 | en el ámbito del Cálculo Científico. Están acompañados de un amplio manual que 6 | presenta los fundamentos teóricos de estos algoritmos, una breve explicación de su 7 | uso, numerosos ejemplos y ejercicios que pueden ser de utilidad para docentes y 8 | estudiantes de estudios científicos y técnicos. 9 | 10 | 11 | 12 | Este material está bajo continuo mantenimiento, por lo que en el caso de detectar erratas 13 | no duden en ponerse en contacto con los autores para que puedan ser corregidas. 14 | -------------------------------------------------------------------------------- /dhont.m: -------------------------------------------------------------------------------- 1 | function [y,valor] = dhont(votos,n) 2 | % Reparto proporcional de n escaños según la Ley d'Hont 3 | % votos = vector fila conteniendo los votos de cada partido 4 | % n = número de escaños a repartir 5 | % y = número de escaños asignados a cada partido en el mismo 6 | % orden que aparecen en votos. 7 | % valor = número de votos resultante para obtener un escaño. 8 | % NOTA: votos ha de ser >=0 9 | % Este programa básico NO contempla los casos de empates 10 | a = 1./(1:n); 11 | x = vec(votos'*a); 12 | for i = 1:n 13 | [valor,p] = max(x); 14 | x(p) = -1; 15 | endfor 16 | m = length(votos); 17 | x = reshape(x,m,n)'; 18 | x = (x<0); 19 | y = sum(x); 20 | endfunction -------------------------------------------------------------------------------- /gausspiv.m: -------------------------------------------------------------------------------- 1 | function [x,p] = gausspiv(A,b) 2 | % Esta función resuelve el sistema Ax=b mediante Gauss. 3 | % x, solucion del SEL (vector/es columna). 4 | % p, vector con las permutaciones realizadas en el proceso. 5 | [dim,ncb] = size(b); 6 | Ab = [A b]; 7 | p = 1:dim; 8 | for j = 1:dim-1 9 | % Pivoteo parcial dentro de la columna j 10 | [piv,pos] = max(abs(Ab(j:dim,j))); 11 | Ab([j,j+pos-1],:) = Ab([j+pos-1,j],:); 12 | p([j,j+pos-1]) = p([j+pos-1,j]); 13 | % Anulación de los elementos bajo el pivote 14 | Ab(j+1:dim,j:dim+ncb) = Ab(j+1:dim,j:dim+ncb) - ... 15 | Ab(j+1:dim,j)*Ab(j,j:dim+ncb)/Ab(j,j); 16 | endfor 17 | % Resolvemos por sustitución regresiva 18 | x = sustregr(Ab(1:dim,1:dim),Ab(1:dim,dim+1:dim+ncb)); 19 | endfunction 20 | -------------------------------------------------------------------------------- /potencias.m: -------------------------------------------------------------------------------- 1 | function [lambda,y] = potencias(A,x,tol,maxiter) 2 | % Calcula una aproximación del valor propio dominante de A, 3 | % matriz cuadrada, mediante el método de potencias a partir 4 | % de un vector inicial x (columna). 5 | % maxiter determina el número máximo de iteraciones del método. 6 | % tol es la tolerancia sobre el error relativo entre iteraciones. 7 | lambdaviejo = rand(); 8 | k = 0; 9 | error = 1000; 10 | while k<=maxiter && error>=tol 11 | y = x/norm(x); 12 | x = A*y; 13 | lambda = y'*x; 14 | error = abs((lambda-lambdaviejo)./lambda); 15 | lambdaviejo = lambda; 16 | k = k+1; 17 | endwhile 18 | if k>maxiter 19 | warning('No converge tras %i iteraciones \n',maxiter); 20 | else 21 | fprintf('El método converge en %i iteraciones \n',k); 22 | endif 23 | endfunction -------------------------------------------------------------------------------- /rk4solver.m: -------------------------------------------------------------------------------- 1 | function [t,y] = rk4solver(f,t0,tfinal,u0,N) 2 | % Resuelve el PVI: u'(t)=f(u(t),t) en [t0,tfinal] 3 | % empleando el método de Runge-Kutta explícito de orden 4. 4 | % En el caso n-dimensional, u0 es un vector columna. 5 | % Devuelve: 6 | % t = vector con los nodos; 7 | % y(:,i) = vector con las aproximaciones de u(t(i)). 8 | h = (tfinal-t0)/N; % h tamaño de paso temporal 9 | t = linspace(t0,tfinal,N+1); % vector de nodos 10 | y = u0; % condición inicial 11 | % Bucle con evolución temporal 12 | for i = 1:N 13 | K1 = feval(f,y(:,i),t(i)); 14 | K2 = feval(f,y(:,i)+h*K1/2,t(i)+h/2); 15 | K3 = feval(f,y(:,i)+h*K2/2,t(i)+h/2); 16 | K4 = feval(f,y(:,i)+h*K3,t(i)+h); 17 | y = [y,(y(:,i)+h*(K1+2*K2+2*K3+K4)/6)]; 18 | endfor 19 | endfunction -------------------------------------------------------------------------------- /scriptajustetrigocontinuo.m: -------------------------------------------------------------------------------- 1 | % Script para calcular la aproximación trigonométrica por 2 | % mínimos cuadrados continua de la función x^2 en [-1,1]. 3 | % Definición de una función vectorial, que contiene a la base, 4 | % evaluable sobre vectores columna. 5 | 1+1; 6 | function y = basetrig(x) 7 | y = [1+0*x,sin(x),cos(x)]; 8 | endfunction; 9 | % Construcción del sistema. 10 | gramm = quadv(@(x) basetrig(x)'*basetrig(x),-1,1); 11 | % Costrucción del vector de términos independientes. 12 | b = quadv(@(x) basetrig(x)'*x.^2,-1,1); 13 | % Resolución del sistema y expresión de la aprox obtenida. 14 | c = gramm\b 15 | % Gráfica comparativa de la identidad y su aprox trigonométrica. 16 | x = linspace(-1,1,100)'; 17 | plot(x,x.^2,'g',x,basetrig(x)*c,':k'); 18 | legend('Parabola','Aproximacion'); 19 | axis([-1 1 -0.05 1.05]); -------------------------------------------------------------------------------- /scriptsplinecuadratico.m: -------------------------------------------------------------------------------- 1 | f = @(x) 1./(1+25*x.^2); % Definimos la función de Runge 2 | n = 19; % Número de subintervalos (n+1 nodos) 3 | x = linspace(-1,1,n+1); % Nodos equiespaciados 4 | y = f(x); % Valores de la función en los nodos 5 | d0 = 25/338; % Derivada en el primer nodo 6 | p = [0,0,y(1)]+[0,d0*poly(x(1))]; % Término lineal del spline 7 | s = zeros(n,3); % Matriz que guarda, por filas, los coeficientes 8 | % del polinomio de cada trozo 9 | gam = []; % Vector que guarda los valores de las gammas 10 | for j = 1:n % Determinamos las n gammas 11 | pot = poly([x(j),x(j)]); % Potencia truncada cuadrática en x(j) 12 | gam(j) = (y(j+1)-polyval(p,x(j+1)))./polyval(pot,x(j+1)); 13 | p = p+gam(j)*pot; 14 | s(j,:) = p; 15 | endfor 16 | plotspline(s,x) 17 | hold on 18 | plot(x,y,'.*r'); 19 | hold off -------------------------------------------------------------------------------- /scriptbiseccion.m: -------------------------------------------------------------------------------- 1 | 1; 2 | % Declaración de datos 3 | function y = fun(x) 4 | y = x+exp(2*x); 5 | endfunction 6 | a = -1; 7 | b = 0; 8 | tol = 10^(-2); 9 | % Iniciación de variables 10 | maxIter = ceil((log(b-a)-log(tol))/log(2))-1; 11 | n = 0; 12 | c = (a+b)/2; 13 | yc = fun(c); 14 | fprintf('n intervalo c_n f(c_n) \n') 15 | fprintf('%i [%f,%f] %f %f \n', n, a, b, c, yc) 16 | ya = fun(a); 17 | yb = fun(b); 18 | % Bucle para n>=1 19 | for n = 1:maxIter 20 | if yc == 0 % La raíz es c 21 | a = c; 22 | b = c; 23 | fprintf('%i [%f,%f] %f %f \n', n, a, b, c, yc) 24 | break 25 | elseif yb*yc>0 % La raíz está en [a,c] 26 | b = c; 27 | yb = yc; 28 | else % La raíz está en [c,b] 29 | a = c; 30 | ya = yc; 31 | endif 32 | c = (a+b)/2; 33 | yc = fun(c); 34 | fprintf('%i [%f,%f] %f %f \n', n, a, b, c, yc) 35 | endfor -------------------------------------------------------------------------------- /scriptajustepolicontinuo.m: -------------------------------------------------------------------------------- 1 | % Script para calcular la aproximación por mínimos cuadrados 2 | % continua de la función seno, en [0,pi], mediante polinomios 3 | % de grado menor o igual que un valor prefijado. 4 | % Definición del polinomio t^i (tomando i como parámetro). 5 | 1; 6 | function y = pol(x,i) y = x.^i; endfunction; 7 | gr = 3; % Elección del grado del polinomio. 8 | vgr = gr:-1:0; % Vector de grados en el polinomio. 9 | % Construcción de la matriz de coeficientes del sistema. 10 | gramm = quadv(@(x) pol(x,vgr)'*pol(x,vgr),0,pi) 11 | % Costrucción del vector de términos independientes. 12 | b = quadv(@(x) pol(x,vgr)'*sin(x),0,pi) 13 | % Resolución del sistema y expresión del polinomio obtenido. 14 | p = gramm\b; 15 | polyout(p,'x') 16 | % Gráfica comparativa del seno y el polinomio obtenido. 17 | x = linspace(0,pi,100); 18 | plot(x,sin(x),'g',x,polyval(p,x),':k'); 19 | legend('Seno','Aproximacion'); 20 | axis([0 pi 0 1.05]); -------------------------------------------------------------------------------- /gradconj.m: -------------------------------------------------------------------------------- 1 | function [x,iter] = gradconj(A,b,x0,tol = 10^-5) 2 | % Esta función calcula una solución aproximada del sistema 3 | % A x = b mediante el método del gradiente conjugado. 4 | % Datos necesarios para llamar a la función: 5 | % A, matriz de coeficientes (simétrica y definida positiva o 6 | % negativa). 7 | % b, vector de términos independientes (vector columna). 8 | % x0, aproximación inicial de la raíz (vector columna). 9 | % tol, tolerancia preestablecida sobre el módulo del residuo. 10 | % La función devuelve como respuesta: 11 | % x, solución aproximada del SEL. 12 | % iter, número de iteraciones realizadas para obtener x. 13 | % 14 | n = length(A); 15 | x = x0; 16 | r = b-A*x; 17 | p = r; 18 | iter = 0; 19 | while (norm(r)>tol) && (iter<2*n) 20 | iter++; 21 | alpha = (p'*r)/(p'*(A*p)); 22 | x = x+alpha*p; 23 | r = b-A*x; 24 | beta = -(r'*(A*p))/(p'*(A*p)); 25 | p = r+beta*p; 26 | endwhile 27 | endfunction -------------------------------------------------------------------------------- /scriptPVFpoisson.m: -------------------------------------------------------------------------------- 1 | % Calcula una aproximación por diferencias finitas centradas del 2 | % PVF: x''(t) = r(t), t en [a,b], x(a) = alpha, x(b) = beta 3 | % Datos del problema 4 | a = 0; b = 4; 5 | alpha = 1; beta = 2; 6 | r = @(t) -cos(t); % Función r(t) (evaluable sobre vectores) 7 | % Definimos la discretización 8 | N = 30; % Número de nodos interiores (ha de ser >2) 9 | h = (b-a)/(N+1); 10 | t = linspace(a,b,N+2);% Nodos, incluidos a y b 11 | % Definición de la matriz de coeficientes 12 | aux = ones(N,1)/h^2; 13 | A = spdiags([aux -2*aux aux],[-1 0 1],N,N); 14 | % Definición del vector de términos independientes 15 | tint = (t(2:N+1))'; % Lista de N nodos interiores 16 | rhs = r(tint); % rhs será el término independiente 17 | rhs(1) = rhs(1)-alpha/h^2; 18 | rhs(N) = rhs(N)-beta/h^2; 19 | % Resolución del sistema 20 | y = A\rhs; 21 | % Representación de la solución numérica y la exacta 22 | y = [alpha; y; beta]; 23 | plot(t,y,'*',t,alpha -1+(beta-cos(4))*t/4+cos(t)) -------------------------------------------------------------------------------- /scriptdescomposicionlu.m: -------------------------------------------------------------------------------- 1 | % Datos del ejemplo 2 | A = [3 1 2 ; 1 4 3 ; 3 3 2]; 3 | b = [1 ; 3 ; 4]; 4 | % Realizamos la factorización LU de Doolittle de A 5 | n = size(A)(1); 6 | aux = zeros(n,n); 7 | % Calculamos la primera fila de U y la primera columna de L 8 | aux(1,1) = A(1,1); 9 | if aux(1,1) == 0 10 | error('No se puede realizar la factorización LU'); 11 | endif; 12 | for s = 2:n 13 | aux(1,s) = A(1,s); 14 | aux(s,1) = A(s,1)/aux(1,1); 15 | endfor 16 | % Filas y columnas siguientes, partiendo desde la diagonal 17 | for r = 2:n 18 | aux(r,r) = A(r,r)-(aux(r,1:r-1)*aux(1:r-1,r)); 19 | if aux(r,r) == 0 20 | error('No se puede realizar la factorización LU') 21 | endif; 22 | for s = r+1:n 23 | aux(r,s) = A(r,s)-aux(r,1:r-1)*aux(1:r-1,s)); 24 | aux(s,r) = (A(s,r)-(aux(s,1:r-1)*aux(1:r-1,r)))/aux(r,r); 25 | endfor 26 | endfor 27 | L = tril(aux,-1)+eye(n); 28 | U = triu(aux); 29 | % Resolvemos el sistema a partir de dicha descomposición 30 | y = sustprogr(L,b); 31 | x = sustregr(U,y) -------------------------------------------------------------------------------- /gaussseidel.m: -------------------------------------------------------------------------------- 1 | function [x,iter] = gaussseidel(A,b,x0,tol = 10^-12,maxiter = 25) 2 | % Esta función calcula una solución aproximada del sistema 3 | % A x = b mediante el método de Gauss-Seidel. 4 | % Datos necesarios para llamar a la función: 5 | % A, matriz de coeficientes del SEL (regular). 6 | % b, vector de términos independientes (vector columna). 7 | % x0, aproximación inicial de la raíz (vector columna). 8 | % tol, tolerancia preestablecida sobre la norma del residuo. 9 | % maxiter, número máximo de iteraciones del método. 10 | % La función devuelve como respuesta: 11 | % x, solución aproximada del SEL. 12 | % iter, número de iteraciones realizadas para obtener x 13 | % (si coincide con maxiter es que x no verifica la condición 14 | % de tolerancia impuesta). 15 | % 16 | n = length(A); 17 | x = x0; % se inicializa x 18 | normb = norm(b); 19 | iter = 0; 20 | while (norm(A*x-b)>tol*normb) && (iter2) 12 | h = (b-a)/(N+1); 13 | t = linspace(a,b,N+2);% Nodos, incluidos a y b 14 | tint = (t(2:N+1))'; % Vector columna de N nodos interiores 15 | % Definición de la matriz A de coeficientes 16 | B = [1+p(tint)*h/2,-2-q(tint)*h^2,1-p(tint)*h/2]./h^2; 17 | A = spdiags(B, [1 0 -1], N, N)'; 18 | % Definición del vector de términos independientes 19 | rhs = r(tint); 20 | rhs(1) = rhs(1)-alpha*B(1,1); 21 | rhs(N) = rhs(N)-beta*B(N,3); 22 | % Resolución del sistema 23 | y = A\rhs; 24 | % Representación de las soluciones numérica y exacta 25 | soln = [alpha; y; beta]; 26 | plot(t,soln,'*',t,t.^(1/3)+log(t)) 27 | -------------------------------------------------------------------------------- /scriptPVIpresadepre.m: -------------------------------------------------------------------------------- 1 | 1; 2 | % 3 | function udot = presadepre(u,t) 4 | a=1; b=1; c=1; d=1; udot(1,1) = a*u(1)-b*u(1)*u(2); 5 | udot(2,1) = -c*u(2)+d*u(1)*u(2); endfunction 6 | t = linspace(0,10,100)'; 7 | sol = lsode("presadepre",[0.5;0.5],t); 8 | 9 | plot(t,sol') 10 | 11 | fontsize=16; 12 | fontname='Arial'; 13 | %ley=legend('example'); 14 | %set(ley, "fontsize", fontsize, 'FontName', fontname) 15 | %text (2, 8, "arbitrary text"); 16 | set(gca(), 'tickdir','in'); 17 | %set(gca(), 'tickdir','out','xtick',[1:15]); 18 | set([gca; findall(gca, 'Type','text')], 'FontSize', fontsize, 'FontName', fontname); 19 | set([gca; findall(gca, 'Type','line')], 'linewidth', 3); 20 | saveas(1, "../graficas/Cap9-presadepre1.pdf"); 21 | 22 | 23 | plot(sol(:,1),sol(:,2)) 24 | 25 | fontsize=16; 26 | fontname='Arial'; 27 | %ley=legend('example'); 28 | %set(ley, "fontsize", fontsize, 'FontName', fontname) 29 | %text (2, 8, "arbitrary text"); 30 | set(gca(), 'tickdir','in'); 31 | %set(gca(), 'tickdir','out','xtick',[1:15]); 32 | set([gca; findall(gca, 'Type','text')], 'FontSize', fontsize, 'FontName', fontname); 33 | set([gca; findall(gca, 'Type','line')], 'linewidth', 3); 34 | saveas(1, "../graficas/Cap9-presadepre2.pdf"); 35 | hold off -------------------------------------------------------------------------------- /newton.m: -------------------------------------------------------------------------------- 1 | function [c,err,numiter] = newton(f,fdx,x0,tol,maxIter) 2 | % Esta función genera una aproximación de la raíz c de la 3 | % función derivable f(x) mediante el método Newton-Raphson. 4 | % Datos necesarios para llamar a la función: 5 | % f y fdx, expresiones de f(x) y f'(x); 6 | % x0, aproximación inicial de la raíz c; 7 | % tol, tolerancia máxima entre aproximaciones sucesivas; 8 | % maxIter, número máximo de iteraciones a realizar. 9 | % La función devuelve como respuesta tres valores: 10 | % c = valor aproximado de la raíz; 11 | % err = |f(c)|; 12 | % numiter = número de iteraciones realizadas. 13 | % Iniciación de variables 14 | n = 0; xn = x0; 15 | fxn = feval(f,xn); 16 | fdxxn = feval(fdx,xn); 17 | difsuc = fxn/fdxxn; % Diferencia entre iteraciones sucesivas 18 | % Bucle 19 | while abs(difsuc)>=tol && n<=maxIter 20 | n = n+1; 21 | xn = xn-difsuc; 22 | fxn = feval(f,xn); 23 | fdxxn = feval(fdx,xn); 24 | difsuc = fxn/fdxxn; % Guarda datos para la próxima iteración 25 | endwhile 26 | if n>maxIter 27 | warning('Newton-Raphson llegó al máximo de iteraciones \n'); 28 | endif 29 | % Definición de respuestas 30 | c = xn; err = abs(fxn); numiter = n; 31 | endfunction -------------------------------------------------------------------------------- /biseccion.m: -------------------------------------------------------------------------------- 1 | function [caprox,err,numiter] = biseccion(f,a,b,tol) 2 | % Esta función aproxima una raíz c de la función continua f(x), 3 | % localizada en [a,b], mediante el método de bisección. 4 | % Datos necesarios para llamar a la función: 5 | % f, expresión de f(x); 6 | % a y b, extremos del intervalo donde sabemos que existe 7 | % una raíz por el Teorema de Bolzano ya que f(a)*f(b)<0; 8 | % tol, tolerancia máxima en la aproximación de la raíz, 9 | % La función devuelve como respuesta tres números: 10 | % caprox = valor aproximado de la raíz; 11 | % err = |f(caprox)|; 12 | % numiter = número de iteraciones realizadas. 13 | 14 | % Iniciación de variables 15 | n = 0; 16 | c = (a+b)/2; 17 | yc = feval(f,c); 18 | maxIter = ceil((log(b-a)-log(tol))/log(2))-1; 19 | ya = feval(f,a); 20 | yb = feval(f,b); 21 | % Bucle para n>=1 22 | for n = 1:maxIter 23 | if yc == 0 % La raíz es c 24 | a = c; 25 | b = c; 26 | fprintf('Se ha alcanzado el cero exacto \n'); 27 | break 28 | elseif yb*yc>0 % La raíz está en [a,c] 29 | b = c; 30 | yb = yc; 31 | else % La raíz está en [c,b] 32 | a = c; 33 | ya = yc; 34 | endif 35 | c = (a+b)/2; 36 | yc = feval(f,c); 37 | endfor 38 | % Definición de la respuesta 39 | caprox = c; err = abs(yc); numiter = n; 40 | endfunction -------------------------------------------------------------------------------- /scriptmetodotiro.m: -------------------------------------------------------------------------------- 1 | clear(); 2 | global vx v ejex ejey; 3 | v = 50*1000/3600; % velocidad en m/s 4 | % Definición de la ecuación diferencial 5 | function [d] = bolafun(z,x) 6 | global vx; 7 | d = [z(2);-9.8/(vx-0.25*x).^2]; 8 | endfunction 9 | ejey = 3; 10 | ejex = sqrt(11^2-3^2); 11 | % Representamos el punto donde se encuentra el objetivo 12 | clf; 13 | plot(ejex,ejey,'k*') 14 | axis([0,12,0,5]); 15 | hold on 16 | % Primer tiro de prueba 17 | theta = pi/8; 18 | vx = v*cos(theta); 19 | [xn,soln] = rk4solver("bolafun",0,ejex,[0;tan(theta)],100); 20 | plot(xn,soln(1,:),':b') 21 | printf("\n Diferencia tras primer tiro %f \n",ejey-soln(1,end)) 22 | % Segundo tiro de prueba 23 | theta = pi/4; 24 | vx = v*cos(theta); 25 | [xn,soln] = rk4solver("bolafun",0,ejex,[0;tan(theta)],100); 26 | plot(xn,soln(1,:),':b') 27 | printf("Diferencia tras segundo tiro %f\n",3-soln(1,end)) 28 | % Función auxiliar 29 | function [dif,soln] = errortiro(z) 30 | global vx v ejex ejey 31 | theta = z; 32 | vx = v*cos(theta); 33 | [xn,soln] = rk4solver("bolafun",0,ejex,[0;tan(theta)],100); 34 | dif = ejey-soln(1,end); 35 | endfunction 36 | % Resolvemos mediante el método de la secante 37 | [sol,errSec,numiter] = secante("errortiro",pi/8,pi/4,0.01,10) 38 | % Representamos la solución 39 | [t,soln] = rk4solver("bolafun",0,ejex,[0;tan(sol)],100); 40 | plot(t,soln(1,:),'r') 41 | hold off -------------------------------------------------------------------------------- /secante.m: -------------------------------------------------------------------------------- 1 | function [c,err,numiter] = secante(f,x0,x1,tol,maxIter) 2 | % Esta función genera una aproximación de una raíz c de la 3 | % función continua f(x) mediante el método de la secante. 4 | % Datos necesarios para llamar a la función: 5 | % f, expresión de f(x); 6 | % x0 y x1, aproximaciones iniciales de la raíz; 7 | % tol, tolerancia máxima entre aproximaciones sucesivas, 8 | % es decir, |xk - xkmenos1| < tol; 9 | % maxIter-1, número máximo de iteraciones a realizar. 10 | % La función devuelve como respuesta tres valores: 11 | % c = valor aproximado de la raíz; 12 | % err = |f(c)|; 13 | % numiter = número de iteraciones realizadas. 14 | 15 | % Iniciación de variables 16 | n = 1; 17 | difsuc = x1-x0; % Diferencia entre iteraciones sucesivas 18 | xnmenos1 = x0; 19 | xn = x1; 20 | fxnmenos1 = feval(f,xnmenos1); 21 | fxn = feval(f,xn); 22 | % Bucle 23 | while abs(difsuc)>=tol && n<=maxIter 24 | n = n+1; 25 | difsuc = fxn*(xn-xnmenos1)/(fxn-fxnmenos1); 26 | xnmenos1 = xn; fxnmenos1 = fxn; % Se guardan datos para 27 | xn = xn-difsuc; % la próxima iteración 28 | fxn = feval(f,xn); 29 | endwhile 30 | if n>maxIter 31 | warning('Secante ha llegado al máximo de iteraciones \n'); 32 | endif 33 | % Definición de respuestas 34 | c = xn; err = abs(fxn); numiter = n-1; 35 | endfunction -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------