--- /dev/null
+// vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+//
+// Trabajo Práctico II de Análisis Numérico I
+// Este programa resuelve un sistema de ecuaciones diferenciales
+// resultante de un problema físico de oscilación de líquidos.
+// Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+//
+// Este programa es Software Libre; usted puede redistribuirlo
+// y/o modificarlo bajo los términos de la "GNU General Public
+// License" como lo publica la "FSF Free Software Foundation",
+// o (a su elección) de cualquier versión posterior.
+//
+// Este programa es distribuido con la esperanza de que le será
+// útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+// implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+// particular. Vea la "GNU General Public License" para más
+// detalles.
+//
+// Usted debe haber recibido una copia de la "GNU General Public
+// License" junto con este programa, si no, escriba a la "FSF
+// Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+// Boston, MA 02111-1307, USA.
+//
+// $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/77891.cpp $
+// $Date: 2002-11-24 02:48:38 -0300 (dom, 24 nov 2002) $
+// $Rev: 27 $
+// $Author: luca $
+//
+
+#include <iostream.h>
+#include <stdlib.h>
+#include <math.h>
+
+// Tipos de datos.
+typedef unsigned int Indice;
+typedef float Numero;
+struct Datos;
+typedef Numero (*Friccion)( Datos&, Numero, Numero );
+typedef Numero (*Funcion)( Datos&, Numero, Numero, Numero );
+typedef void (*Metodo)( Datos& );
+struct Datos {
+ Numero to; // Tiempo inicial
+ Numero tf; // Tiempo final
+ Numero k; // Paso
+ Numero xo; // X inicial = X(to) = Z(to)
+ Numero yo; // Y inicial = Y(to) = X'(to) = Z'(to)
+ Numero D; // Diametro del tubo
+ Numero n; // Viscosidad cinemática del líquido
+ Numero g; // Aceleración de la gravedad
+ Numero f; // Factor de fricción
+ Numero L; // Longitud del tubo
+ Numero Le; // Factor de pérdida de carga
+ Numero G; // Factor geométrico
+ Funcion fx; // Función asociada a la derivada de x
+ Funcion fy; // Función asociada a la derivada de y
+ Friccion fi; // Factor asociado a pérdidas por fricción
+};
+
+// Factor de fricción nulo.
+Numero friccionNula( Datos&, Numero, Numero );
+
+// Factor de fricción laminar.
+Numero friccionLaminar( Datos&, Numero, Numero );
+
+// Factor de fricción turbulenta.
+Numero friccionTurbulenta( Datos&, Numero, Numero );
+
+// Factor de fricción turbulenta (con depósitos).
+Numero friccionTurbulentaConDepositos( Datos&, Numero, Numero );
+
+// Función asociada a la primera derivada (x'=z').
+Numero funcionX( Datos&, Numero, Numero, Numero );
+
+// Función asociada a la segunda derivada (y'=x''=z'').
+Numero funcionY( Datos&, Numero, Numero, Numero );
+
+// Calcula la ecuación diferencial por el método de Euler.
+void euler( Datos& );
+
+// Calcula la ecuación diferencial por el método de Runge-Kutta de órden 4.
+void rk4( Datos& );
+
+// Calcula la ecuación diferencial por el método de Nystrom.
+void nystrom( Datos& );
+
+// Constantes.
+const Numero DEFAULT_N = 1000, // Cantidad de pasos por defecto
+ DEFAULT_to = 0.0,
+ DEFAULT_tf = 440.0,
+ DEFAULT_xo = 2.9866369,
+ DEFAULT_yo = 0.0,
+ DEFAULT_D = 0.5,
+ DEFAULT_n = 1e-6,
+ DEFAULT_g = 9.8,
+ DEFAULT_f = 0.03,
+ DEFAULT_L = 892.0,
+ DEFAULT_Le = 1070.4,
+ DEFAULT_G = 2.0;
+const Funcion DEFAULT_fx = &funcionX,
+ DEFAULT_fy = &funcionY;
+const Friccion DEFAULT_fi = &friccionNula;
+
+int main( int argc, char* argv[] ) {
+
+ // Se fija que tenga los argumentos necesarios para correr.
+ if ( argc < 2 ) {
+ cerr << "Faltan argumentos. Modo de uso:" << endl;
+ cerr << "\t" << argv[0] << " método fi G tf N to xo yo D n g f L Le" << endl;
+ cerr << "Desde fi en adelante son opcionales. Los valores por defecto son:" << endl;
+ cerr << "\tfi = n" << endl;
+ cerr << "\tG = " << DEFAULT_G << endl;
+ cerr << "\ttf = " << DEFAULT_tf << endl;
+ cerr << "\tN = " << DEFAULT_N << endl;
+ cerr << "\tto = " << DEFAULT_to << endl;
+ cerr << "\txo = " << DEFAULT_xo << endl;
+ cerr << "\tyo = " << DEFAULT_yo << endl;
+ cerr << "\tD = " << DEFAULT_D << endl;
+ cerr << "\tn = " << DEFAULT_n << endl;
+ cerr << "\tg = " << DEFAULT_g << endl;
+ cerr << "\tf = " << DEFAULT_f << endl;
+ cerr << "\tL = " << DEFAULT_L << endl;
+ cerr << "\tLe = " << DEFAULT_Le << endl;
+ return EXIT_FAILURE;
+ }
+
+ // Selecciona el método deseado.
+ Metodo metodo = NULL;
+ switch ( char( argv[1][0] ) ) {
+ case 'e':
+ metodo = &euler;
+ break;
+ case 'r':
+ metodo = &rk4;
+ break;
+ case 'n':
+ metodo = &nystrom;
+ break;
+ default:
+ cerr << "Debe especificar un método válido:" << endl;
+ cerr << "\te: Euler" << endl;
+ cerr << "\tr: Runge-Kutta 4" << endl;
+ cerr << "\tn: Nystrom" << endl;
+ return EXIT_FAILURE;
+ }
+
+ // Se inicializan los datos.
+ Numero N = DEFAULT_N;
+ Datos D;
+ D.to = DEFAULT_to;
+ D.tf = DEFAULT_tf;
+ D.k = ( D.tf - D.to ) / N;
+ D.xo = DEFAULT_xo;
+ D.yo = DEFAULT_yo;
+ D.D = DEFAULT_D;
+ D.n = DEFAULT_n;
+ D.g = DEFAULT_g;
+ D.f = DEFAULT_f;
+ D.L = DEFAULT_L;
+ D.Le = DEFAULT_Le;
+ D.G = DEFAULT_G;
+ D.fx = DEFAULT_fx;
+ D.fy = DEFAULT_fy;
+ D.fi = DEFAULT_fi;
+
+ // Si se pasaron datos como argumento, se los va agregando.
+ switch ( argc ) {
+ case 15: D.Le = Numero( atof( argv[14] ) );
+ case 14: D.L = Numero( atof( argv[13] ) );
+ case 13: D.f = Numero( atof( argv[12] ) );
+ case 12: D.g = Numero( atof( argv[11] ) );
+ case 11: D.n = Numero( atof( argv[10] ) );
+ case 10: D.D = Numero( atof( argv[9] ) );
+ case 9: D.yo = Numero( atof( argv[8] ) );
+ case 8: D.xo = Numero( atof( argv[7] ) );
+ case 7: D.to = Numero( atof( argv[6] ) );
+ case 6: N = Numero( atof( argv[5] ) );
+ case 5: D.tf = Numero( atof( argv[4] ) );
+ // Se recalcula el paso (k) si se cambio to, N o tf.
+ D.k = ( D.tf - D.to ) / N;
+ case 4: D.G = Numero( atof( argv[3] ) );
+ case 3: switch ( char( argv[2][0] ) ) { // Tipo de fricción
+ case 'n':
+ D.fi = &friccionNula;
+ break;
+ case 'l':
+ D.fi = &friccionLaminar;
+ break;
+ case 't':
+ D.fi = &friccionTurbulenta;
+ break;
+ case 'd':
+ D.fi = &friccionTurbulentaConDepositos;
+ break;
+ default:
+ cerr << "Debe especificar un tipo de fricción válido:" << endl;
+ cerr << "\tn: Nula" << endl;
+ cerr << "\tl: Laminar" << endl;
+ cerr << "\tt: Turbulenta" << endl;
+ cerr << "\td: Turbulenta (con depósitos)" << endl;
+ return EXIT_FAILURE;
+ }
+ }
+
+ // Imprime el paso utilizado.
+ cerr << "Paso k = " << D.k << endl;
+
+ // Ejecuta el método correspondiente con los datos correspondientes.
+ metodo( D );
+
+ return EXIT_SUCCESS;
+
+}
+
+Numero friccionNula( Datos& D, Numero x, Numero y ) {
+
+ return 0.0;
+
+}
+
+Numero friccionLaminar( Datos& D, Numero x, Numero y ) {
+
+ return 32 * D.n / ( D.D * D.D );
+
+}
+
+Numero friccionTurbulenta( Datos& D, Numero x, Numero y ) {
+
+ return D.f * fabs( y ) / ( 2 * D.D );
+
+}
+
+Numero friccionTurbulentaConDepositos( Datos& D, Numero x, Numero y ) {
+
+ return D.f * fabs( y ) * D.Le / ( 2 * D.D * D.L);
+
+}
+
+Numero funcionX( Datos& D, Numero t, Numero x, Numero y ) {
+
+ return y;
+
+}
+
+Numero funcionY( Datos& D, Numero t, Numero x, Numero y ) {
+
+ return - D.fi( D, x, y ) * y - D.g * D.G * x / D.L;
+
+}
+
+void euler( Datos& D ) {
+
+ Numero xo = D.xo,
+ yo = D.yo,
+ x = 0.0,
+ y = 0.0,
+ t = D.to;
+
+ while ( t < D.tf ) {
+
+ // Calculo los datos para este punto.
+ x = xo + D.k * D.fx( D, t, xo, yo );
+ y = yo + D.k * D.fy( D, t, xo, yo );
+
+ // Imprimo resultados.
+ cout << t << " " << x << " " << y << endl;
+
+ // Reemplazo valores iniciales.
+ xo = x;
+ yo = y;
+ t += D.k;
+
+ }
+
+}
+
+void rk4( Datos& D ) {
+
+ Numero x = D.xo,
+ y = D.yo,
+ t = D.to,
+ qx1 = 0.0,
+ qx2 = 0.0,
+ qx3 = 0.0,
+ qx4 = 0.0,
+ qy1 = 0.0,
+ qy2 = 0.0,
+ qy3 = 0.0,
+ qy4 = 0.0,
+ unSexto = 1.0 / 6.0;
+
+ // Imprimo datos iniciales.
+ cout << t << " " << x << " " << y << endl;
+
+ while ( t < D.tf ) {
+
+ // Calculo los datos para este punto.
+ qx1 = D.k * D.fx( D, t, x, y );
+ qy1 = D.k * D.fy( D, t, x, y );
+
+ qx2 = D.k * D.fx( D, t + D.k / 2.0, x + qx1 / 2.0, y + qy1 / 2.0 );
+ qy2 = D.k * D.fy( D, t + D.k / 2.0, x + qx1 / 2.0, y + qy1 / 2.0 );
+
+ qx3 = D.k * D.fx( D, t + D.k / 2.0, x + qx2 / 2.0, y + qy2 / 2.0 );
+ qy3 = D.k * D.fy( D, t + D.k / 2.0, x + qx2 / 2.0, y + qy2 / 2.0 );
+
+ qx4 = D.k * D.fx( D, t + D.k, x + qx3, y + qy3 );
+ qy4 = D.k * D.fy( D, t + D.k, x + qx3, y + qy3 );
+
+ x += unSexto * ( qx1 + 2 * qx2 + 2 * qx3 + qx4 );
+ y += unSexto * ( qy1 + 2 * qy2 + 2 * qy3 + qy4 );
+ t += D.k;
+
+ // Imprimo resultados.
+ cout << t << " " << x << " " << y << endl;
+
+ }
+
+}
+
+void nystrom( Datos& D ) {
+
+ Numero gGk2_L = D.g * D.G * D.k * D.k / D.L,
+ xo = D.xo,
+ x1 = ( 1 - gGk2_L * 0.5 ) * D.xo,
+ x2 = 0.0,
+ y = 0.0,
+ fi = 0.0,
+ t = D.to;
+
+ // Imprimo valores iniciales.
+ cout << t << " " << xo << " " << y << endl;
+ y = ( x1 - xo ) / D.k;
+ cout << t << " " << x1 << " " << y << endl;
+
+ while ( t < D.tf ) {
+
+ // Calculo los datos para este punto.
+ fi = D.fi( D, x1, y );
+ x2 = ( ( fi - 1 ) * xo + ( 2 - gGk2_L ) * x1 ) / ( fi + 1 );
+
+ // Prepara para próxima iteración
+ t += D.k;
+ xo = x1;
+ x1 = x2;
+ y = ( x1 - xo ) / D.k;
+
+ // Imprimo resultados.
+ cout << t << " " << x2 << " " << y << endl;
+
+ }
+
+}
--- /dev/null
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+ Version 1.1, March 2000
+
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+ 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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+and independent documents or works, in or on a volume of a storage or
+distribution medium, does not as a whole count as a Modified Version
+of the Document, provided no compilation copyright is claimed for the
+compilation. Such a compilation is called an "aggregate", and this
+License does not apply to the other self-contained works thus compiled
+with the Document, on account of their being thus compiled, if they
+are not themselves derivative works of the Document.
+
+If the Cover Text requirement of section 3 is applicable to these
+copies of the Document, then if the Document is less than one quarter
+of the entire aggregate, the Document's Cover Texts may be placed on
+covers that surround only the Document within the aggregate.
+Otherwise they must appear on covers around the whole aggregate.
+
+
+8. TRANSLATION
+
+Translation is considered a kind of modification, so you may
+distribute translations of the Document under the terms of section 4.
+Replacing Invariant Sections with translations requires special
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+translations of some or all Invariant Sections in addition to the
+original versions of these Invariant Sections. You may include a
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+License, the original English version will prevail.
+
+
+9. TERMINATION
+
+You may not copy, modify, sublicense, or distribute the Document except
+as expressly provided for under this License. Any other attempt to
+copy, modify, sublicense or distribute the Document is void, and will
+automatically terminate your rights under this License. However,
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+License will not have their licenses terminated so long as such
+parties remain in full compliance.
+
+
+10. FUTURE REVISIONS OF THIS LICENSE
+
+The Free Software Foundation may publish new, revised versions
+of the GNU Free Documentation License from time to time. Such new
+versions will be similar in spirit to the present version, but may
+differ in detail to address new problems or concerns. See
+http://www.gnu.org/copyleft/.
+
+Each version of the License is given a distinguishing version number.
+If the Document specifies that a particular numbered version of this
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+of any later version that has been published (not as a draft) by the
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+number of this License, you may choose any version ever published (not
+as a draft) by the Free Software Foundation.
+
+
+ADDENDUM: How to use this License for your documents
+
+To use this License in a document you have written, include a copy of
+the License in the document and put the following copyright and
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+
+ Copyright (c) YEAR YOUR NAME.
+ Permission is granted to copy, distribute and/or modify this document
+ under the terms of the GNU Free Documentation License, Version 1.1
+ or any later version published by the Free Software Foundation;
+ with the Invariant Sections being LIST THEIR TITLES, with the
+ Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.
+ A copy of the license is included in the section entitled "GNU
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+
+If you have no Invariant Sections, write "with no Invariant Sections"
+instead of saying which ones are invariant. If you have no
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+"Front-Cover Texts being LIST"; likewise for Back-Cover Texts.
+
+If your document contains nontrivial examples of program code, we
+recommend releasing these examples in parallel under your choice of
+free software license, such as the GNU General Public License,
+to permit their use in free software.
--- /dev/null
+ GNU GENERAL PUBLIC LICENSE
+ Version 2, June 1991
+
+ Copyright (C) 1989, 1991 Free Software Foundation, Inc.
+ 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ Everyone is permitted to copy and distribute verbatim copies
+ of this license document, but changing it is not allowed.
+
+ Preamble
+
+ The licenses for most software are designed to take away your
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+Foundation's software and to any other program whose authors commit to
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+
+ When we speak of free software, we are referring to freedom, not
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+ For example, if you distribute copies of such a program, whether
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+you have. You must make sure that they, too, receive or can get the
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+
+ We protect your rights with two steps: (1) copyright the software, and
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+
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+that everyone understands that there is no warranty for this free
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+ The precise terms and conditions for copying, distribution and
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+\f
+ GNU GENERAL PUBLIC LICENSE
+ TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
+
+ 0. This License applies to any program or other work which contains
+a notice placed by the copyright holder saying it may be distributed
+under the terms of this General Public License. The "Program", below,
+refers to any such program or work, and a "work based on the Program"
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+Activities other than copying, distribution and modification are not
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+Whether that is true depends on what the Program does.
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+You may charge a fee for the physical act of transferring a copy, and
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+ b) You must cause any work that you distribute or publish, that in
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+ part thereof, to be licensed as a whole at no charge to all third
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+\f
+These requirements apply to the modified work as a whole. If
+identifiable sections of that work are not derived from the Program,
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+under Section 2) in object code or executable form under the terms of
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+\f
+ 4. You may not copy, modify, sublicense, or distribute the Program
+except as expressly provided under this License. Any attempt
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+ 5. You are not required to accept this License, since you have not
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+ 6. Each time you redistribute the Program (or any work based on the
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+This section is intended to make thoroughly clear what is believed to
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+\f
+ 8. If the distribution and/or use of the Program is restricted in
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+ 9. The Free Software Foundation may publish revised and/or new versions
+of the General Public License from time to time. Such new versions will
+be similar in spirit to the present version, but may differ in detail to
+address new problems or concerns.
+
+Each version is given a distinguishing version number. If the Program
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+Software Foundation. If the Program does not specify a version number of
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+Foundation.
+
+ 10. If you wish to incorporate parts of the Program into other free
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+ NO WARRANTY
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+
+ END OF TERMS AND CONDITIONS
+\f
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
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+
+ <one line to give the program's name and a brief idea of what it does.>
+ Copyright (C) <year> <name of author>
+
+ This program is free software; you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation; either version 2 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program; if not, write to the Free Software
+ Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+
+
+Also add information on how to contact you by electronic and paper mail.
+
+If the program is interactive, make it output a short notice like this
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+
+ Gnomovision version 69, Copyright (C) year name of author
+ Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, the commands you use may
+be called something other than `show w' and `show c'; they could even be
+mouse-clicks or menu items--whatever suits your program.
+
+You should also get your employer (if you work as a programmer) or your
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+necessary. Here is a sample; alter the names:
+
+ Yoyodyne, Inc., hereby disclaims all copyright interest in the program
+ `Gnomovision' (which makes passes at compilers) written by James Hacker.
+
+ <signature of Ty Coon>, 1 April 1989
+ Ty Coon, President of Vice
+
+This General Public License does not permit incorporating your program into
+proprietary programs. If your program is a subroutine library, you may
+consider it more useful to permit linking proprietary applications with the
+library. If this is what you want to do, use the GNU Library General
+Public License instead of this License.
--- /dev/null
+# Trabajo Práctico II de Análisis Numérico I
+# Este programa resuelve un sistema de ecuaciones diferenciales
+# resultante de un problema físico de oscilación de líquidos.
+# Copyright (C) 2002 Leandro Lucarella
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/Makefile $
+# $Date: 2002-12-02 18:03:21 -0300 (lun, 02 dic 2002) $
+# $Rev: 35 $
+# $Author: luca $
+#
+
+#### VARIABLES ####
+CPP_OPTS=-O3 -Wall
+CPP_OPTS_DBG=-g3 -Wall
+LIBS=-lm
+
+
+#### GENERAL ####
+all: tp informe
+tp: c d e f
+clean: clean-c clean-d clean-e clean-f clean-informe clean-77891
+
+
+#### PROGRAMA ####
+77891: 77891.cpp
+ c++ $(CPP_OPTS) $(LIBS) -o 77891 77891.cpp
+77891dbg: 77891.cpp
+ c++ $(CPP_OPTS_DBG) $(LIBS) -o 77891dbg 77891.cpp
+clean-77891:
+ rm -f *.o 77891 77891dbg
+
+
+#### PUNTO C ####
+c: c1 c2 c3
+clean-c: clean-c1 clean-c2 clean-c3
+# C.1
+c1e_k0.44.txt: 77891
+ ./77891 e n 2 440 1000 > c1e_k0.44.txt
+c1e_k0.0044.txt: 77891
+ ./77891 e n 2 440 100000 > c1e_k0.0044.txt
+c1e.eps: c1e_k0.44.txt c1e_k0.0044.txt c1e.gnuplot
+ ./c1e.gnuplot
+c1r.txt: 77891
+ ./77891 r n 2 440 1000 > c1r.txt
+c1r.eps: c1r.txt c1r.gnuplot
+ ./c1r.gnuplot
+c1n.txt: 77891
+ ./77891 n n 2 440 1000 > c1n.txt
+c1n.eps: c1n.txt c1n.gnuplot
+ ./c1n.gnuplot
+c1: c1e.eps c1r.eps c1n.eps
+clean-c1:
+ rm -f c1*.txt c1*.eps
+# C.2
+c2e.txt: 77891 calcula_periodo
+ ./77891 e n 2 5000 100000 | ./calcula_periodo /dev/stdin > c2e.txt
+c2r.txt: 77891 calcula_periodo
+ ./77891 r n 2 5000 100000 | ./calcula_periodo /dev/stdin > c2r.txt
+c2n.txt: 77891 calcula_periodo
+ ./77891 n n 2 5000 100000 | ./calcula_periodo /dev/stdin > c2n.txt
+c2.eps: c2e.txt c2r.txt c2n.txt periodo.gnuplot
+ sed "s/<P>/c/g" periodo.gnuplot | sed "s/<PUNTO_DEP>/sin/g" | \
+ sed "s/<PUNTO_FRIC>/sin/g" | sed "s/<PUNTO_FRIC_TIPO>//g" | \
+ gnuplot
+c2: c2.eps
+clean-c2:
+ rm -f c2*.txt c2.eps
+# C.3
+c3c.eps: c3c.gnuplot
+ ./c3c.gnuplot
+c3o.eps: c3o.gnuplot
+ ./c3o.gnuplot
+c3: c3c.eps c3o.eps
+clean-c3:
+ rm -f c3*.eps
+
+
+#### PUNTO D ####
+d: d1 d2 d3 d4
+clean-d: clean-d1 clean-d2 clean-d3 clean-d4
+# D.1
+d1e.txt: 77891
+ ./77891 e l 2 2500 1000000 > d1e.txt
+d1e.eps: d1e.txt d1e.gnuplot
+ ./d1e.gnuplot
+d1r.txt: 77891
+ ./77891 r l 2 2500 1000 > d1r.txt
+d1r.eps: d1r.txt d1r.gnuplot
+ ./d1r.gnuplot
+d1n_k2.0.txt: 77891
+ ./77891 n l 2 2500 1250 > d1n_k2.0.txt
+d1n_k0.0625.txt: 77891
+ ./77891 n l 2 2500 40000 > d1n_k0.0625.txt
+d1n.eps: d1n_k2.0.txt d1n_k0.0625.txt d1n.gnuplot
+ ./d1n.gnuplot
+d1: d1e.eps d1r.eps d1n.eps
+clean-d1:
+ rm -f d1*.txt d1*.eps
+# D.2
+d2e.txt: 77891 calcula_periodo
+ ./77891 e l 2 5000 100000 | ./calcula_periodo /dev/stdin > d2e.txt
+d2r.txt: 77891 calcula_periodo
+ ./77891 r l 2 5000 100000 | ./calcula_periodo /dev/stdin > d2r.txt
+d2n.txt: 77891 calcula_periodo
+ ./77891 n l 2 5000 100000 | ./calcula_periodo /dev/stdin > d2n.txt
+d2.eps: d2e.txt d2r.txt d2n.txt periodo.gnuplot
+ sed "s/<P>/d/g" periodo.gnuplot | sed "s/<PUNTO_DEP>/sin/g" | \
+ sed "s/<PUNTO_FRIC>/con/g" | sed "s/<PUNTO_FRIC_TIPO>/ laminar/g" | \
+ gnuplot
+d2: d2.eps
+clean-d2:
+ rm -f d2*.txt d2.eps
+# D.3
+d3e.txt: 77891
+ ./77891 e l 2 405 100000 > d3e.txt
+d3e.max.txt: d3e.txt calcula_maxmin
+ ./calcula_maxmin max d3e.txt > d3e.max.txt
+d3e.min.txt: d3e.txt calcula_maxmin
+ ./calcula_maxmin min d3e.txt > d3e.min.txt
+d3r.txt: 77891
+ ./77891 r l 2 405 1000 > d3r.txt
+d3r.max.txt: d3r.txt calcula_maxmin
+ ./calcula_maxmin max d3r.txt > d3r.max.txt
+d3r.min.txt: d3r.txt calcula_maxmin
+ ./calcula_maxmin min d3r.txt > d3r.min.txt
+d3n.txt: 77891
+ ./77891 n l 2 405 190 > d3n.txt
+d3n.max.txt: d3n.txt calcula_maxmin
+ ./calcula_maxmin max d3n.txt > d3n.max.txt
+d3n.min.txt: d3n.txt calcula_maxmin
+ ./calcula_maxmin min d3n.txt > d3n.min.txt
+d3: d3e.max.txt d3e.min.txt d3r.max.txt d3r.min.txt d3n.max.txt d3n.min.txt
+clean-d3:
+ rm -f d3*.txt
+# D.4
+d4e.eps: d3e.max.txt d3e.min.txt d4e.gnuplot
+ ./d4e.gnuplot
+d4r.eps: d3r.max.txt d3r.min.txt d4r.gnuplot
+ ./d4r.gnuplot
+d4n.eps: d3n.max.txt d3n.min.txt d4n.gnuplot
+ ./d4n.gnuplot
+d4: d4e.eps d4r.eps d4n.eps
+clean-d4:
+ rm -f d4*.eps fit.log
+
+
+#### PUNTO E ####
+e: e1 e2 e3 e4
+clean-e: clean-e1 clean-e2 clean-e3 clean-e4
+# E.1
+e1e.txt: 77891
+ ./77891 e t 2 1250 100000 > e1e.txt
+e1e.eps: e1e.txt e1e.gnuplot
+ ./e1e.gnuplot
+e1r.txt: 77891
+ ./77891 r t 2 1250 400 > e1r.txt
+e1r.eps: e1r.txt e1r.gnuplot
+ ./e1r.gnuplot
+e1n.txt: 77891
+ ./77891 n t 2 1250 625 > e1n.txt
+e1n.eps: e1n.txt e1n.gnuplot
+ ./e1n.gnuplot
+e1: e1e.eps e1r.eps e1n.eps
+clean-e1:
+ rm -f e1*.txt e1*.eps
+# E.2
+e2e.txt: 77891 calcula_periodo
+ ./77891 e t 2 5000 100000 | ./calcula_periodo /dev/stdin > e2e.txt
+e2r.txt: 77891 calcula_periodo
+ ./77891 r t 2 5000 100000 | ./calcula_periodo /dev/stdin > e2r.txt
+e2n.txt: 77891 calcula_periodo
+ ./77891 n t 2 5000 100000 | ./calcula_periodo /dev/stdin > e2n.txt
+e2.eps: e2e.txt e2r.txt e2n.txt periodo.gnuplot
+ sed "s/<P>/e/g" periodo.gnuplot | sed "s/<PUNTO_DEP>/sin/g" | \
+ sed "s/<PUNTO_FRIC>/con/g" | sed "s/<PUNTO_FRIC_TIPO>/ turbulenta/g" | \
+ gnuplot
+e2: e2.eps
+clean-e2:
+ rm -f e2*.txt e2.eps
+# E.3
+e3e.txt: 77891
+ ./77891 e t 2 405 100000 > e3e.txt
+e3e.max.txt: e3e.txt calcula_maxmin
+ ./calcula_maxmin max e3e.txt > e3e.max.txt
+e3e.min.txt: e3e.txt calcula_maxmin
+ ./calcula_maxmin min e3e.txt > e3e.min.txt
+e3r.txt: 77891
+ ./77891 r t 2 405 1000 > e3r.txt
+e3r.max.txt: e3r.txt calcula_maxmin
+ ./calcula_maxmin max e3r.txt > e3r.max.txt
+e3r.min.txt: e3r.txt calcula_maxmin
+ ./calcula_maxmin min e3r.txt > e3r.min.txt
+e3n.txt: 77891
+ ./77891 n t 2 405 190 > e3n.txt
+e3n.max.txt: e3n.txt calcula_maxmin
+ ./calcula_maxmin max e3n.txt > e3n.max.txt
+e3n.min.txt: e3n.txt calcula_maxmin
+ ./calcula_maxmin min e3n.txt > e3n.min.txt
+e3: e3e.max.txt e3e.min.txt e3r.max.txt e3r.min.txt e3n.max.txt e3n.min.txt
+clean-e3:
+ rm -f e3*.txt
+# E.4
+e4r.txt: 77891
+ ./77891 r t 2 50000 100000 > e4r.txt
+e4r.max.txt: e4r.txt calcula_maxmin
+ ./calcula_maxmin max e4r.txt > e4r.max.txt
+e4r.min.txt: e4r.txt calcula_maxmin
+ ./calcula_maxmin min e4r.txt > e4r.min.txt
+e4r.eps: e4r.max.txt e4r.min.txt e4r.gnuplot
+ ./e4r.gnuplot
+e4n.txt: 77891
+ ./77891 n t 2 50000 25000 > e4n.txt
+e4n.max.txt: e4n.txt calcula_maxmin
+ ./calcula_maxmin max e4n.txt > e4n.max.txt
+e4n.min.txt: e4n.txt calcula_maxmin
+ ./calcula_maxmin min e4n.txt > e4n.min.txt
+e4n.eps: e4n.max.txt e4n.min.txt e4n.gnuplot
+ ./e4n.gnuplot
+e4: e4r.eps e4n.eps
+clean-e4:
+ rm -f e4*.txt e4*.eps
+
+
+#### PUNTO F ####
+f: f1 f2 f3 f4
+clean-f: clean-f1 clean-f2 clean-f3 clean-f4
+# E.1
+f1e.txt: 77891
+ ./77891 e d 0.03 5000 100000 > f1e.txt
+f1e.eps: f1e.txt f1e.gnuplot
+ ./f1e.gnuplot
+f1r.txt: 77891
+ ./77891 r d 0.03 5000 1000 > f1r.txt
+f1r.eps: f1r.txt f1r.gnuplot
+ ./f1r.gnuplot
+f1n.txt: 77891
+ ./77891 n d 0.03 5000 2500 > f1n.txt
+f1n.eps: f1n.txt f1n.gnuplot
+ ./f1n.gnuplot
+f1: f1e.eps f1r.eps f1n.eps
+clean-f1:
+ rm -f f1*.txt f1*.eps
+# E.2
+f2e.txt: 77891 calcula_periodo
+ ./77891 e d 0.03 40000 100000 | ./calcula_periodo /dev/stdin > f2e.txt
+f2r.txt: 77891 calcula_periodo
+ ./77891 r d 0.03 40000 100000 | ./calcula_periodo /dev/stdin > f2r.txt
+f2n.txt: 77891 calcula_periodo
+ ./77891 n d 0.03 40000 100000 | ./calcula_periodo /dev/stdin > f2n.txt
+f2.eps: f2e.txt f2r.txt f2n.txt periodo.gnuplot
+ sed "s/<P>/f/g" periodo.gnuplot | sed "s/<PUNTO_DEP>/con/g" | \
+ sed "s/<PUNTO_FRIC>/con/g" | sed "s/<PUNTO_FRIC_TIPO>/ turbulenta/g" | \
+ gnuplot
+f2: f2.eps
+clean-f2:
+ rm -f f2*.txt f2.eps
+# E.3
+f3e.txt: 77891
+ ./77891 e d 0.03 3400 340000 > f3e.txt
+f3e.max.txt: f3e.txt calcula_maxmin
+ ./calcula_maxmin max f3e.txt > f3e.max.txt
+f3e.min.txt: f3e.txt calcula_maxmin
+ ./calcula_maxmin min f3e.txt > f3e.min.txt
+f3r.txt: 77891
+ ./77891 r d 0.03 3400 3400 > f3r.txt
+f3r.max.txt: f3r.txt calcula_maxmin
+ ./calcula_maxmin max f3r.txt > f3r.max.txt
+f3r.min.txt: f3r.txt calcula_maxmin
+ ./calcula_maxmin min f3r.txt > f3r.min.txt
+f3n.txt: 77891
+ ./77891 n d 0.03 3400 1700 > f3n.txt
+f3n.max.txt: f3n.txt calcula_maxmin
+ ./calcula_maxmin max f3n.txt > f3n.max.txt
+f3n.min.txt: f3n.txt calcula_maxmin
+ ./calcula_maxmin min f3n.txt > f3n.min.txt
+f3: f3e.max.txt f3e.min.txt f3r.max.txt f3r.min.txt f3n.max.txt f3n.min.txt
+clean-f3:
+ rm -f f3*.txt
+# E.4
+f4r.txt: 77891
+ ./77891 r d 0.03 350000 100000 > f4r.txt
+f4r.max.txt: f4r.txt calcula_maxmin
+ ./calcula_maxmin max f4r.txt > f4r.max.txt
+f4r.min.txt: f4r.txt calcula_maxmin
+ ./calcula_maxmin min f4r.txt > f4r.min.txt
+f4r.eps: f4r.max.txt f4r.min.txt f4r.gnuplot
+ ./f4r.gnuplot
+f4n.txt: 77891
+ ./77891 n d 0.03 350000 175000 > f4n.txt
+f4n.max.txt: f4n.txt calcula_maxmin
+ ./calcula_maxmin max f4n.txt > f4n.max.txt
+f4n.min.txt: f4n.txt calcula_maxmin
+ ./calcula_maxmin min f4n.txt > f4n.min.txt
+f4n.eps: f4n.max.txt f4n.min.txt f4n.gnuplot
+ ./f4n.gnuplot
+f4: f4r.eps f4n.eps
+clean-f4:
+ rm -f f4*.txt f4*.eps
+
+
+#### Informe ####
+informe.ps: informe.lyx c1 c2 c3 d1 d2 d4 e1 e2 e4 f1 f2 f4
+ lyx -e ps informe.lyx
+
+informe.pdf: informe.lyx c1 c2 c3 d1 d2 d4 e1 e2 e4 f1 f2 f4
+ lyx -e pdf informe.lyx
+
+informe.tex: informe.lyx c1 c2 c3 d1 d2 d4 e1 e2 e4 f1 f2 f4
+ lyx -e latex informe.lyx
+informe: informe.ps informe.pdf informe.tex
+clean-informe:
+ rm -f informe.ps informe.tex informe.pdf
--- /dev/null
+vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+
+Trabajo Práctico I de Análisis Numérico II
+
+$URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/README $
+$Date: 2002-12-01 03:16:07 -0300 (dom, 01 dic 2002) $
+$Rev: 34 $
+$Author: luca $
+
+Para generar el Informe del trabajo práctico basta con hacer:
+
+ make all
+
+(ver Makefile para otras opciones)
+
+Para que esto funcione debe tener instalados los siguientes programas:
+* GNU make - http://www.gnu.org/software/make/ (o compatible)
+* GNU GCC - http://gcc.gnu.org/ (o compilador de C++ compatible)
+* gnuplot - http://www.gnuplot.info/
+* LyX - http://www.lyx.org/
+* OpenOffice - http://www.openoffice.org/
+* PHP4 - http://www.php.net/ (compilado como cgi)
+
+El gnuplot es necesario para generar los gráficos, el LyX es
+necesario para generar el informe (y es probable que necesite otras
+herramientas para generar el informe en distintos formatos).
+El OpenOffice es opcional ya que se incluye en el paquete el gráfico
+resultante en formato Postscript. El PHP4 es necesario para calcular
+los máximos y mínimos y el período.
+
+Todos los programas de este paquete se encuentran bajo licencia GPL y
+los documentos bajo licencia GFDL. Ambas se incluyen en el paquete.
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/c1e.gnuplot $
+# $Date: 2002-11-24 19:55:46 -0300 (dom, 24 nov 2002) $
+# $Rev: 29 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "c1e.eps"
+
+# Seteos generales.
+set title "Solución por Euler para el caso sin depósitos y sin fricción"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:440]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z) para k = 0.0044"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Altura (z) para k = 0.44"
+set y2range [-25:25]
+set y2tics -25, 5
+set my2tics 5
+
+# Plotea
+plot 'c1e_k0.0044.txt' using 1:2 title "k = 0.0044" with lines linetype 3, \
+ 'c1e_k0.44.txt' using 1:2 axes x1y2 title "k = 0.44" with lines linetype 1
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/c1n.gnuplot $
+# $Date: 2002-11-24 19:55:46 -0300 (dom, 24 nov 2002) $
+# $Rev: 29 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "c1n.eps"
+
+# Seteos generales.
+set title "Solución por Nystrom para el caso sin depósitos y sin fricción"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:440]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'c1n.txt' using 1:2 title "z" with lines linetype 1, \
+ 'c1n.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/c1r.gnuplot $
+# $Date: 2002-11-24 19:55:46 -0300 (dom, 24 nov 2002) $
+# $Rev: 29 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "c1r.eps"
+
+# Seteos generales.
+set title "Solución por RK4 para el caso sin depósitos y sin fricción"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:440]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'c1r.txt' using 1:2 title "z" with lines linetype 1, \
+ 'c1r.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/c3c.gnuplot $
+# $Date: 2002-12-02 18:03:21 -0300 (lun, 02 dic 2002) $
+# $Rev: 35 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "c3c.eps"
+
+# Seteos generales.
+set title "Resolución gráfica del análisis de conservación para RK4."
+set key right top
+
+# Eje X.
+set xlabel "Paso k"
+set xrange [0:10]
+set mxtics 5
+
+# Plotea
+plot x**4 * .02**2 / 4 - x**8 * .02**4 / 2304 - 2 - x**6 * .02**2 / 48 - \
+ x**2 * .02 + x**4 * .02**2 / 24 + abs( x**8 * .02**4 / 2304 + \
+ x**5 * .02**3 / 32 - x**3 * .02**2 / 6 - 3 * x**2 * .02 / 2 ) notitle
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/c3o.gnuplot $
+# $Date: 2002-12-02 18:03:21 -0300 (lun, 02 dic 2002) $
+# $Rev: 35 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "c3o.eps"
+
+# Seteos generales.
+set title "Resolución gráfica del análisis de oscilación para RK4."
+set key right top
+
+# Eje X.
+set xlabel "Paso k"
+set xrange [0:51]
+set mxtics 5
+
+# Plotea
+plot x**4 * .02**3 / 576 + x**3 * .02**2 / 8 - x * .04 / 3 - 6 notitle
--- /dev/null
+#!/usr/bin/php4 -qC
+<?
+// vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+//
+// Trabajo Práctico II de Análisis Numérico I
+// Este programa resuelve un sistema de ecuaciones diferenciales
+// resultante de un problema físico de oscilación de líquidos.
+// Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+//
+// Este programa es Software Libre; usted puede redistribuirlo
+// y/o modificarlo bajo los términos de la "GNU General Public
+// License" como lo publica la "FSF Free Software Foundation",
+// o (a su elección) de cualquier versión posterior.
+//
+// Este programa es distribuido con la esperanza de que le será
+// útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+// implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+// particular. Vea la "GNU General Public License" para más
+// detalles.
+//
+// Usted debe haber recibido una copia de la "GNU General Public
+// License" junto con este programa, si no, escriba a la "FSF
+// Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+// Boston, MA 02111-1307, USA.
+//
+// $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/calcula_maxmin $
+// $Date: 2002-12-01 02:26:18 -0300 (dom, 01 dic 2002) $
+// $Rev: 33 $
+// $Author: luca $
+//
+
+// Muestra ayuda.
+if ( $argc < 3 ) {
+ echo "Modo de uso:\n";
+ echo "$argv[0] modo archivo\n";
+ echo "Donde modo es 'max' para calcular los máximos o 'min' para calcular los mínimos.\n";
+ exit;
+}
+
+// Abre archivos y obtiene datos iniciales.
+$f = fopen( $argv[2], 'r' )
+ or die( "Error al abrir $argv[2] para lectura.\n" );
+$s = fgets( $f, 4096 )
+ or die( "El archivo $argv[2] está vacío.\n" );
+list( $t0, $z0, $dz0 ) = preg_split( '/\s/', $s );
+
+// El primer valor siempre es un máximo.
+if ( $argv[1] == 'max' )
+ echo "$t0 $z0\n";
+
+// Procesa archivo calculando máximos y mínimos.
+while ( ( $s = fgets( $f, 4096 ) ) !== false ) {
+ // Obtiene datos.
+ list( $t, $z, $dz ) = preg_split( '/\s/', $s );
+ // Se fija si la derivada es un "cero decreciente" (si es un máximo).
+ if ( $argv[1] == 'max' and $dz0 > 0 and ( $dz < 0 or $dz == 0 ) )
+ if ( $z0 > $z )
+ echo "$t0 $z0\n";
+ else
+ echo "$t $z\n";
+ // Se fija si la derivada es un "cero creciente" (si es un mínimo).
+ if ( $argv[1] == 'min' and $dz0 < 0 and ( $dz > 0 or $dz == 0 ) )
+ if ( $z0 < $z )
+ echo "$t0 $z0\n";
+ else
+ echo "$t $z\n";
+ // Actualiza valores anteriores.
+ $t0 = $t;
+ $z0 = $z;
+ $dz0 = $dz;
+}
+
+fclose( $f );
+
+?>
--- /dev/null
+#!/usr/bin/php4 -qC
+<?
+// vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+//
+// Trabajo Práctico II de Análisis Numérico I
+// Este programa resuelve un sistema de ecuaciones diferenciales
+// resultante de un problema físico de oscilación de líquidos.
+// Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+//
+// Este programa es Software Libre; usted puede redistribuirlo
+// y/o modificarlo bajo los términos de la "GNU General Public
+// License" como lo publica la "FSF Free Software Foundation",
+// o (a su elección) de cualquier versión posterior.
+//
+// Este programa es distribuido con la esperanza de que le será
+// útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+// implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+// particular. Vea la "GNU General Public License" para más
+// detalles.
+//
+// Usted debe haber recibido una copia de la "GNU General Public
+// License" junto con este programa, si no, escriba a la "FSF
+// Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+// Boston, MA 02111-1307, USA.
+//
+// $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/calcula_periodo $
+// $Date: 2002-12-01 02:26:18 -0300 (dom, 01 dic 2002) $
+// $Rev: 33 $
+// $Author: luca $
+//
+
+// Muestra ayuda.
+if ( $argc < 2 ) {
+ echo "Modo de uso:\n";
+ echo "$argv[0] archivo\n";
+ exit;
+}
+
+// Abre archivo y obtiene datos iniciales.
+$f = fopen( $argv[1], 'r' );
+$s = fgets( $f, 4096 )
+ or die( "El archivo está vacío o no se pudo abrir.\n" );
+list( $t0, $z0 ) = preg_split( '/\s/', $s );
+
+// Procesa archivo calculando período.
+$t_ant = 0;
+$c = 0;
+$t_sum = 0;
+$t_max = 0;
+$tt_max = 0;
+$t_min = 5000;
+$tt_min = 0;
+while ( ( $s = fgets( $f, 4096 ) ) !== false ) {
+ // Obtiene datos.
+ list( $t, $z ) = preg_split( '/\s/', $s );
+ // Se fija si es un "cero decreciente".
+ if ( $z0 > 0 and ( $z < 0 or $z == 0 ) ) {
+ if ( $t_ant ) {
+ $t_actual = $t - $t_ant;
+ if ( $t_actual > $t_max ) {
+ $t_max = $t_actual;
+ $tt_max = $t;
+ }
+ if ( $t_actual < $t_min ) {
+ $t_min = $t_actual;
+ $tt_min = $t;
+ }
+ echo "$t_actual\n";
+ $t_sum += $t_actual;
+ $t_ant = $t;
+ $c++;
+ } else {
+ $t_ant = $t;
+ }
+ }
+ // Actualiza valores anteriores.
+ $t0 = $t;
+ $z0 = $z;
+}
+
+fclose( $f );
+
+// Imprime período.
+echo "Períodos promedio: " . $t_sum / $c . " ($c períodos promediados)\n";
+echo "Período máximo: $t_max (en t = $tt_max)\n";
+echo "Período mínimo: $t_min (en t = $tt_min)\n";
+
+?>
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/d1e.gnuplot $
+# $Date: 2002-11-24 19:55:46 -0300 (dom, 24 nov 2002) $
+# $Rev: 29 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "d1e.eps"
+
+# Seteos generales.
+set title "Solución por Euler para el caso sin depósitos y con fricción laminar"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:2500]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Plotea
+plot 'd1e.txt' using 1:2 title "z" with lines linetype 1
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/d1n.gnuplot $
+# $Date: 2002-11-30 18:54:58 -0300 (sáb, 30 nov 2002) $
+# $Rev: 32 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "d1n.eps"
+
+# Seteos generales.
+set title "Solución por Nystrom para el caso sin depósitos y con fricción laminar"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:2500]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Plotea
+plot 'd1n_k2.0.txt' using 1:2 title "k = 2.0" with lines linetype 1, \
+ 'd1n_k0.0625.txt' using 1:2 title "k = 0.0625" with lines linetype 3
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/d1r.gnuplot $
+# $Date: 2002-11-24 19:55:46 -0300 (dom, 24 nov 2002) $
+# $Rev: 29 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "d1r.eps"
+
+# Seteos generales.
+set title "Solución por RK4 para el caso sin depósitos y con fricción laminar"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:2500]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Plotea
+plot 'd1r.txt' using 1:2 title "z" with lines linetype 1
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/d4e.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "d4e.eps"
+
+# Seteos generales.
+set title "Estimación del tiempo de reposo para Euler en el caso sin depósitos y con fricción laminar"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:300000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set mytics 5
+
+# Primero aproximo por una recta para hallar valores aproximados de los parámetros.
+M(x) = a - b * x
+m(x) = -c - d * x
+fit M(x) 'd3e.max.txt' using 1:2 via a, b
+fit m(x) 'd3e.min.txt' using 1:2 via c, d
+
+# Ahora ahora la verdadera aproximación con una exponencial.
+M(x) = a * exp( -b * x )
+m(x) = -c * exp( -d * x )
+fit M(x) 'd3e.max.txt' using 1:2 via a, b
+fit m(x) 'd3e.min.txt' using 1:2 via c, d
+
+# Plotea
+plot M(x) title "Función aproximante de máximos" with lines linetype 1, \
+ m(x) title "Función aproximante de mínimos" with lines linetype 8
+
+# Se muestran las funciones aproximantes utilizadas.
+print "Funciones aproximantes:"
+print "M(x) = ", a, " * exp( -", b, " * x )"
+print "m(x) = -", c, " * exp( -", d, " * x )"
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/d4n.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "d4n.eps"
+
+# Seteos generales.
+set title "Estimación del tiempo de reposo para Nystrom en el caso sin depósitos y con fricción laminar"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:90000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set mytics 5
+
+# Primero aproximo por una recta para hallar valores aproximados de los parámetros.
+M(x) = a - b * x
+m(x) = -c - d * x
+fit M(x) 'd3n.max.txt' using 1:2 via a, b
+fit m(x) 'd3n.min.txt' using 1:2 via c, d
+
+# Ahora ahora la verdadera aproximación con una exponencial.
+M(x) = a * exp( -b * x )
+m(x) = -c * exp( -d * x )
+fit M(x) 'd3n.max.txt' using 1:2 via a, b
+fit m(x) 'd3n.min.txt' using 1:2 via c, d
+
+# Plotea
+plot M(x) title "Función aproximante de máximos" with lines linetype 1, \
+ m(x) title "Función aproximante de mínimos" with lines linetype 8
+
+# Se muestran las funciones aproximantes utilizadas.
+print "Funciones aproximantes:"
+print "M(x) = ", a, " * exp( -", b, " * x )"
+print "m(x) = -", c, " * exp( -", d, " * x )"
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/d4r.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "d4r.eps"
+
+# Seteos generales.
+set title "Estimación del tiempo de reposo para RK4 en el caso sin depósitos y con fricción laminar"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:90000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set mytics 5
+
+# Primero aproximo por una recta para hallar valores aproximados de los parámetros.
+M(x) = a - b * x
+m(x) = -c - d * x
+fit M(x) 'd3r.max.txt' using 1:2 via a, b
+fit m(x) 'd3r.min.txt' using 1:2 via c, d
+
+# Ahora ahora la verdadera aproximación con una exponencial.
+M(x) = a * exp( -b * x )
+m(x) = -c * exp( -d * x )
+fit M(x) 'd3r.max.txt' using 1:2 via a, b
+fit m(x) 'd3r.min.txt' using 1:2 via c, d
+
+# Plotea
+plot M(x) title "Función aproximante de máximos" with lines linetype 1, \
+ m(x) title "Función aproximante de mínimos" with lines linetype 8
+
+# Se muestran las funciones aproximantes utilizadas.
+print "Funciones aproximantes:"
+print "M(x) = ", a, " * exp( -", b, " * x )"
+print "m(x) = -", c, " * exp( -", d, " * x )"
--- /dev/null
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+%%PageTrailer
+%%Trailer
+%%EOF
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/e1e.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "e1e.eps"
+
+# Seteos generales.
+set title "Solución por Euler para el caso sin depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:1250]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'e1e.txt' using 1:2 title "z" with lines linetype 1, \
+ 'e1e.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/e1n.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "e1n.eps"
+
+# Seteos generales.
+set title "Solución por Nystrom para el caso sin depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:1250]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'e1n.txt' using 1:2 title "z" with lines linetype 1, \
+ 'e1n.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/e1r.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "e1r.eps"
+
+# Seteos generales.
+set title "Solución por RK4 para el caso sin depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:1250]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'e1r.txt' using 1:2 title "z" with lines linetype 1, \
+ 'e1r.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/e4n.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "e4n.eps"
+
+# Seteos generales.
+set title "Estimación del tiempo de reposo para Nystrom en el caso sin depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:50000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set mytics 5
+
+# Plotea
+plot 'e4n.max.txt' title "Máximos" with lines linetype 1, \
+ 'e4n.min.txt' title "Mínimos" with lines linetype 8
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/e4r.gnuplot $
+# $Date: 2002-11-25 03:26:50 -0300 (lun, 25 nov 2002) $
+# $Rev: 30 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "e4r.eps"
+
+# Seteos generales.
+set title "Estimación del tiempo de reposo para RK4 en el caso sin depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:50000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set mytics 5
+
+# Plotea
+plot 'e4r.max.txt' title "Máximos" with lines linetype 1, \
+ 'e4r.min.txt' title "Mínimos" with lines linetype 8
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/f1e.gnuplot $
+# $Date: 2002-11-30 03:48:32 -0300 (sáb, 30 nov 2002) $
+# $Rev: 31 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "f1e.eps"
+
+# Seteos generales.
+set title "Solución por Euler para el caso con depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:5000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'f1e.txt' using 1:2 title "z" with lines linetype 1, \
+ 'f1e.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/f1n.gnuplot $
+# $Date: 2002-11-30 03:48:32 -0300 (sáb, 30 nov 2002) $
+# $Rev: 31 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "f1n.eps"
+
+# Seteos generales.
+set title "Solución por Nystrom para el caso con depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:5000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'f1n.txt' using 1:2 title "z" with lines linetype 1, \
+ 'f1n.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/f1r.gnuplot $
+# $Date: 2002-11-30 03:48:32 -0300 (sáb, 30 nov 2002) $
+# $Rev: 31 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "f1r.eps"
+
+# Seteos generales.
+set title "Solución por RK4 para el caso con depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:5000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set ytics nomirror
+set yrange [-3.5:3.5]
+set ytics -3.5, 0.5
+set mytics 5
+
+# Eje Y secundario.
+set y2label "Velocidad (z')"
+set y2range [-0.5:0.5]
+set y2tics -0.5, 0.1
+set my2tics 5
+
+# Plotea
+plot 'f1r.txt' using 1:2 title "z" with lines linetype 1, \
+ 'f1r.txt' using 1:3 axes x1y2 title "z'" with lines linetype 7
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/f4n.gnuplot $
+# $Date: 2002-11-30 18:54:58 -0300 (sáb, 30 nov 2002) $
+# $Rev: 32 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "f4n.eps"
+
+# Seteos generales.
+set title "Estimación del tiempo de reposo para Nystrom en el caso con depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:350000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set mytics 5
+
+# Plotea
+plot 'f4n.max.txt' title "Máximos" with lines linetype 1, \
+ 'f4n.min.txt' title "Mínimos" with lines linetype 8
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/f4r.gnuplot $
+# $Date: 2002-11-30 18:54:58 -0300 (sáb, 30 nov 2002) $
+# $Rev: 32 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "f4r.eps"
+
+# Seteos generales.
+set title "Estimación del tiempo de reposo para RK4 en el caso con depósitos y con fricción turbulenta"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo (t)"
+set xrange [0:350000]
+set mxtics 5
+
+# Eje Y.
+set ylabel "Altura (z)"
+set mytics 5
+
+# Plotea
+plot 'f4r.max.txt' title "Máximos" with lines linetype 1, \
+ 'f4r.min.txt' title "Mínimos" with lines linetype 8
--- /dev/null
+#LyX 1.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 218
+\textclass book
+\language spanish
+\inputencoding auto
+\fontscheme pslatex
+\graphics default
+\float_placement hbtp
+\paperfontsize default
+\spacing single
+\papersize Default
+\paperpackage widemarginsa4
+\use_geometry 0
+\use_amsmath 0
+\paperorientation portrait
+\secnumdepth 2
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\quotes_times 2
+\papercolumns 1
+\papersides 2
+\paperpagestyle default
+
+\layout Title
+
+Análisis Numérico I
+\newline
+Trabajo Práctico N° 2
+\layout Author
+
+Leandro Lucarella (77.891)
+\layout Date
+
+$Date: 2002-12-05 03:19:47 -0300 (jue, 05 dic 2002) $
+\newline
+$Rev: 36 $
+\layout Standard
+
+Copyright (C) 2002 Leandro Lucarella.
+\layout Standard
+
+Tiene permiso para copiar, distribuir y/o modificar este documento bajo
+ los términos de la GNU Free Documentation License (Licencia de Documentación
+ Libre GNU), Versión 1.1 o cualquier versión posterior publicada por la Free
+ Software Foundation (Fundación de Software Libre); sin Invariant Sections
+ (Secciones Invariantes), sin Front-Cover Texts (Texto de Portada-Delantera),
+ y sin Back-Cover Texts (Texto de Portada-Trasera).
+ Puede obtener una copia de la licencia en inglés en
+\begin_inset LatexCommand \url{http://www.gnu.org/licenses/fdl.txt}
+
+\end_inset
+
+ o en español (sin validez legal) en
+\begin_inset LatexCommand \url{http://www.geocities.com/larteaga/gnu/gfdl.html}
+
+\end_inset
+
+.
+\layout Standard
+
+
+\begin_inset LatexCommand \tableofcontents{}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset LatexCommand \listoffigures{}
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset LatexCommand \listoftables{}
+
+\end_inset
+
+
+\layout Chapter
+
+Introducción.
+\layout Section
+
+Objetivo.
+\layout Standard
+
+Resolver un problema práctico implementando distintas metodologías de discretiza
+ción de ecuaciones diferenciales ordinarias.
+ Analizar el sistema físico bajo estudio y comparar las técnicas de resolución.
+\layout Section
+
+Introducción: Oscilación de líquidos
+\layout Standard
+
+Se estudiará un transitorio hidráulico, consistente en el movimiento de
+ un fluido luego que es apartado de su condición de equilibrio gravitatorio.
+ Como caso particular se presenta la oscilación entre dos depósitos comunicados
+ como se indica en la figura
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 368 283
+file diagrama.eps
+width 1 13
+height 1 10
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:diagrama}
+
+\end_inset
+
+Esquema del modelo físico.
+\end_float
+, donde
+\begin_inset Formula \( z_{1} \)
+\end_inset
+
+ y
+\begin_inset Formula \( z_{2} \)
+\end_inset
+
+ son los apartamientos de las superficies de los depósitos respecto de las
+ posiciones de equilibrio,
+\begin_inset Formula \( A_{1} \)
+\end_inset
+
+ y
+\begin_inset Formula \( A_{2} \)
+\end_inset
+
+ son las áreas horizontales de los depósitos (supuestas constantes),
+\begin_inset Formula \( L \)
+\end_inset
+
+ es la longitud de la tubería de comunicación y
+\begin_inset Formula \( L \)
+\end_inset
+
+ su sección (considerada circular).
+\layout Standard
+
+Considerando un fluido incompresible resulta
+\begin_inset Formula \( z\cdot A=z\cdot A_{1}=z\cdot A_{2} \)
+\end_inset
+
+, con
+\begin_inset Formula \( z \)
+\end_inset
+
+ el desplazamiento de una partícula de fluido dentro de la tubería y respecto
+ de su posición de equilibrio.
+ Así, la ecuación de movimiento del sistema es:
+\begin_inset Formula \begin{equation}
+\label{math:ec_dif_fisica}
+\frac{\textrm{d}^{2}z}{dt^{2}}+\phi _{(z)}\cdot \frac{dz}{dt}+\frac{g\cdot G}{L}\cdot z=0
+\end{equation}
+
+\end_inset
+
+con
+\begin_inset Formula \( t \)
+\end_inset
+
+ el tiempo,
+\begin_inset Formula \( g \)
+\end_inset
+
+ la aceleración de la gravedad,
+\begin_inset Formula \( \phi _{(z)} \)
+\end_inset
+
+ un factor asociado a pérdidas por fricción y
+\begin_inset Formula \( G \)
+\end_inset
+
+ un factor geométrico.
+ Como condiciones iniciales se considerarán:
+\begin_inset Formula \( z_{(0)}=\Delta z \)
+\end_inset
+
+,
+\begin_inset Formula \( \left. \frac{dz}{dt}\right| _{0}=0 \)
+\end_inset
+
+.
+
+\layout Chapter
+
+Discretización.
+\layout Standard
+
+La ecuación
+\begin_inset LatexCommand \vref{math:ec_dif_fisica}
+
+\end_inset
+
+ se deberá discretizar utilizando cada uno de los siguientes esquemas:
+\layout Itemize
+
+Euler explícito (E)
+\layout Itemize
+
+Runge-Kutta de orden 4 (RK4)
+\layout Itemize
+
+Nystrom (N)
+\layout Section
+
+Euler y Runge-Kutta de orden 4.
+\layout Standard
+
+Estos métodos se construyen de forma similar.
+ Primero, al tratarse de una ecuación diferencial de segundo grado, hay
+ que hacer un cambio de variables y plantear un sistema de dos ecuaciones:
+\begin_inset Formula \begin{equation}
+\label{math:cambio_variables}
+\left\{ \begin{array}{rcl}
+x=z & \Rightarrow & \frac{dx}{dt}=\frac{dz}{dt}\\
+y=\frac{dx}{dt}=\frac{dz}{dt} & \Rightarrow & \frac{dy}{dt}=\frac{d^{2}x}{dt^{2}}=\frac{d^{2}z}{dt^{2}}
+\end{array}\right.
+\end{equation}
+
+\end_inset
+
+
+\begin_inset Formula \begin{equation}
+\label{math:sist_ec_dif}
+\Rightarrow \left\{ \begin{array}{lllll}
+\frac{dx}{dt} & = & y & = & f_{x_{(t,x,y)}}\\
+\frac{dy}{dt} & = & -\phi _{\left( x,y_{(x)}\right) }\cdot y-\frac{g\cdot G}{L}\cdot x & = & f_{y_{(t,x,y)}}
+\end{array}\right.
+\end{equation}
+
+\end_inset
+
+ahora discretizamos le sistema de ecuaciones diferenciales de primer grado
+ según el método.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+
+\begin_inset Formula \begin{equation}
+\label{math:euler}
+\left\{ \begin{array}{lllll}
+x_{n+1} & = & x_{n}+k\cdot f_{x_{\left( t_{n},x_{n},y_{n}\right) }} & = & x_{n}+k\cdot y_{n}\\
+y_{n+1} & = & y_{n}+k\cdot f_{y_{\left( t_{n},x_{n},y_{n}\right) }} & = & y_{n}-k\cdot \left( \phi _{\left( x_{n},y_{n}\right) }\cdot y_{n}+\frac{g\cdot G}{L}\cdot x_{n}\right)
+\end{array}\right.
+\end{equation}
+
+\end_inset
+
+
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+
+\begin_inset Formula \begin{equation}
+\label{math:rk4}
+\begin{array}{l}
+\left\{ \begin{array}{lll}
+q_{x_{1}} & = & k\cdot f_{x_{\left( t_{n},x_{n},y_{n}\right) }}\\
+q_{y_{1}} & = & k\cdot f_{y_{\left( t_{n},x_{n},y_{n}\right) }}
+\end{array}\right. \\
+\left\{ \begin{array}{lll}
+q_{x_{2}} & = & k\cdot f_{x_{\left( t_{n}+\frac{k}{2},x_{n}+\frac{q_{x_{1}}}{2},y_{n}+\frac{q_{y_{1}}}{2}\right) }}\\
+q_{y_{2}} & = & k\cdot f_{y_{\left( t_{n}+\frac{k}{2},x_{n}+\frac{q_{x_{1}}}{2},y_{n}+\frac{q_{y_{1}}}{2}\right) }}
+\end{array}\right. \\
+\left\{ \begin{array}{lll}
+q_{x_{3}} & = & k\cdot f_{x_{\left( t_{n}+\frac{k}{2},x_{n}+\frac{q_{x_{2}}}{2},y_{n}+\frac{q_{y_{2}}}{2}\right) }}\\
+q_{y_{3}} & = & k\cdot f_{y_{\left( t_{n}+\frac{k}{2},x_{n}+\frac{q_{x_{2}}}{2},y_{n}+\frac{q_{y_{2}}}{2}\right) }}
+\end{array}\right. \\
+\left\{ \begin{array}{lll}
+q_{x_{4}} & = & k\cdot f_{x_{\left( t_{n}+k,x_{n}+q_{x_{3}},y_{n}+q_{y_{3}}\right) }}\\
+q_{y_{4}} & = & k\cdot f_{y_{\left( t_{n}+k,x_{n}+q_{x_{3}},y_{n}+q_{y_{3}}\right) }}
+\end{array}\right. \\
+\left\{ \begin{array}{lll}
+x_{n+1} & = & x_{n}+\frac{1}{6}\cdot \left( q_{x_{1}}+2\cdot q_{x_{2}}+2\cdot q_{x_{3}}+q_{x_{4}}\right) \\
+y_{n+1} & = & y_{n}+\frac{1}{6}\cdot \left( q_{y_{1}}+2\cdot q_{y_{2}}+2\cdot q_{y_{3}}+q_{y_{4}}\right)
+\end{array}\right.
+\end{array}
+\end{equation}
+
+\end_inset
+
+
+\layout Section
+
+Nystrom.
+\layout Standard
+
+La construcción del método de Nystrom difiere de las anteriores ya que no
+ es necesario hacer un sistema de ecuaciones.
+ Nystrom parte del operador centrado
+\begin_inset Formula \( \frac{d^{2}z}{dt^{2}}=f_{\left( t,z,\frac{dz}{dt}\right) } \)
+\end_inset
+
+, por lo que se obtiene un método de paso múltiple (2 pasos).
+ Resulta entonces
+\begin_inset Formula \begin{equation}
+\label{math:operador_centrado}
+\frac{x_{n+1}-2\cdot x_{n}+x_{n-1}}{k^{2}}\approx \left. \frac{d^{2}x}{dt^{2}}\right| _{n}=f_{\left( t,x,\frac{dx}{dt}\right) }
+\end{equation}
+
+\end_inset
+
+
+\layout Standard
+
+donde
+\begin_inset Formula \( x \)
+\end_inset
+
+ es la discretización de
+\begin_inset Formula \( z \)
+\end_inset
+
+.
+\layout Standard
+
+Desarrollando la ecuación
+\begin_inset LatexCommand \vref{math:ec_dif_fisica}
+
+\end_inset
+
+ obtenemos
+\begin_inset Formula \begin{equation}
+\label{math:nystrom_inicial}
+\frac{x_{n+1}-2\cdot x_{n}+x_{n-1}}{k^{2}}+\phi _{(x)}\frac{x_{n+1}-x_{n-1}}{2\cdot k}+\frac{g\cdot G}{L}\cdot x_{n}=0
+\end{equation}
+
+\end_inset
+
+
+\layout Standard
+
+finalmente, despejando
+\begin_inset Formula \begin{equation}
+\label{math:nystrom}
+x_{n+1}=\frac{\left( \phi _{(x)}-1\right) \cdot x_{n-1}+\left( 2-\frac{g\cdot G\cdot k^{2}}{L}\right) \cdot x_{n}}{\phi _{(x)}+1}
+\end{equation}
+
+\end_inset
+
+
+\layout Subsection
+
+Obtención de valores iniciales.
+\layout Standard
+
+Hay que tener en cuenta que como el método de Nystrom necesita 2 valores
+ iniciales para arrancar, necesitamos obtener el dato correspondiente a
+
+\begin_inset Formula \( x_{1} \)
+\end_inset
+
+.
+ Este valor es obtenido por medio de una aproximación, considerando a la
+ solución una función par.
+ De esta manera suponemos que
+\begin_inset Formula \( x_{-1}=x_{1} \)
+\end_inset
+
+ y evaluamos la función discretizada (ver ecuación
+\begin_inset LatexCommand \ref{math:nystrom_inicial}
+
+\end_inset
+
+)
+\begin_inset Formula \begin{equation}
+\label{math:nystrom_valor_inicial}
+\begin{array}{r}
+\frac{x_{1}-2\cdot x_{0}+x_{-1}}{k^{2}}+\phi _{(x)}\frac{x_{1}-x_{-1}}{2\cdot k}+\frac{g\cdot G}{L}\cdot x_{0}=0\\
+\frac{x_{1}-2\cdot x_{0}+x_{1}}{k^{2}}+\phi _{(x)}\frac{x_{1}-x_{1}}{2\cdot k}+\frac{g\cdot G}{L}\cdot x_{0}=0\\
+\frac{2\cdot x_{1}-2\cdot x_{0}}{k^{2}}+\phi _{(x)}\frac{0}{2\cdot k}+\frac{g\cdot G}{L}\cdot x_{0}=0\\
+\frac{2}{k^{2}}\cdot \left( x_{1}-x_{0}\right) +\frac{g\cdot G}{L}\cdot x_{0}=0\\
+x_{1}-x_{0}+\frac{k^{2}\cdot g\cdot G}{2\cdot L}\cdot x_{0}=0\\
+x_{1}+\left( \frac{k^{2}\cdot g\cdot G}{2\cdot L}-1\right) \cdot x_{0}=0
+\end{array}
+\end{equation}
+
+\end_inset
+
+obteniendo un valor aproximado
+\begin_inset Formula \begin{equation}
+\label{math:nystrom_val_ini}
+x_{1}=\left( 1-\frac{k^{2}\cdot g\cdot G}{2\cdot L}\right) \cdot x_{0}
+\end{equation}
+
+\end_inset
+
+siendo
+\begin_inset Formula \( x_{0} \)
+\end_inset
+
+ dato del problema.
+\layout Subsection
+
+Aproximación de la derivada primera.
+\layout Standard
+
+Para construir la ecuación
+\begin_inset LatexCommand \vref{math:nystrom}
+
+\end_inset
+
+ supusimos un problema lineal aunque el coeficiente
+\begin_inset Formula \( \phi _{(z)} \)
+\end_inset
+
+ no siempre lo es.
+ En los casos donde
+\begin_inset Formula \( \phi _{(z)}\propto \left| \frac{dz}{dt}\right| \)
+\end_inset
+
+ hacemos una aproximación simple
+\begin_inset Formula \begin{equation}
+\label{mat:nystrom_aprox_derivada}
+\left| \frac{dz}{dt}\right| \approx \left| \frac{dx}{dt}\right| \cong \left| \frac{x_{n}-x_{n-1}}{k}\right|
+\end{equation}
+
+\end_inset
+
+
+\layout Standard
+
+No podemos utilizar un operador centrado para hacer la aproximación quedaría
+ en función de
+\begin_inset Formula \( x_{n+1} \)
+\end_inset
+
+ y al tratarse del valor absoluto sería imposible despejarlo para calcular
+ el paso siguiente como en la ecuación
+\begin_inset LatexCommand \vref{math:nystrom}
+
+\end_inset
+
+.
+\layout Section
+
+Valores iniciales.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:valores_iniciales}
+
+\end_inset
+
+Se toman los siguientes valores iniciales:
+\layout Itemize
+
+
+\begin_inset Formula \( D=0.5\, [m] \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( n=10^{-6}\, \left[ \frac{m^{2}}{s}\right] \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( g=9.8\, \left[ \frac{m^{2}}{s}\right] \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( f=0.03\, [-] \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( L=(3\textrm{ últimas cifras del Padrón})+1\, [m]=892\, [m] \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( L_{e}=1.2\, [m] \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( \Delta z=\frac{\sqrt{L}}{10}\, [m]=1070.4\, [m] \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( A_{1}=100A \)
+\end_inset
+
+
+\layout Itemize
+
+
+\begin_inset Formula \( A_{2}=50A \)
+\end_inset
+
+
+\layout Chapter
+
+Caso sin depósitos y sin fricción.
+\layout Standard
+
+Si se desprecian los efectos resistivos y se considera un tubo en U resulta
+
+\begin_inset Formula \( \phi _{(z)}=0 \)
+\end_inset
+
+,
+\begin_inset Formula \( G=2 \)
+\end_inset
+
+.
+\layout Section
+
+Gráfico de la solución en el tiempo.
+\layout Standard
+
+Se grafican aproximadamente 10 períodos de la solución.
+ La solución teórica es conservativa por lo que se espera que la solución
+ numérica también lo sea, aunque como se verá, no siempre es así.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file c1e.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:euler_c1}
+
+\end_inset
+
+Solución por Euler para el caso sin depósitos ni fricción.
+\end_float
+
+\begin_inset LatexCommand \label{sec:euler_diverge}
+
+\end_inset
+
+La solución por este método diverge casi de forma incondicional.
+ En la figura
+\begin_inset LatexCommand \ref{fig:euler_c1}
+
+\end_inset
+
+podemos ver como al graficar variando el paso
+\begin_inset Formula \( k \)
+\end_inset
+
+, el método es mínimamente estable sólo si
+\begin_inset Formula \( k \)
+\end_inset
+
+ es extremadamente pequeño.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file c1r.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:rk4_c1}
+
+\end_inset
+
+Solución por RK4 para el caso sin depósitos ni fricción.
+\end_float
+Luego de probar algunas variantes para el paso, se grafica la solución para
+ un paso de
+\begin_inset Formula \( k=0.44 \)
+\end_inset
+
+.
+ Con este paso la solución parece ser bastante precisa y hasta se podría
+ haber tomado un paso considerablemente mayor.
+ Los resultados pueden verse en la figura
+\begin_inset LatexCommand \ref{fig:rk4_c1}
+
+\end_inset
+
+.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:rk4_estable}
+
+\end_inset
+
+A partir de un paso
+\begin_inset Formula \( k=5 \)
+\end_inset
+
+ la solución parece ser estable.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file c1n.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:nystrom_c1}
+
+\end_inset
+
+Solución por Nystrom para el caso sin depósitos ni fricción.
+\end_float
+
+\begin_inset LatexCommand \label{sec:nystrom_paso_minimo}
+
+\end_inset
+
+Al igual que con el método RK4, se prueban algunos pasos y se utiliza
+\begin_inset Formula \( k=0.44 \)
+\end_inset
+
+, aunque el método es estable y da resultados aparentemente válidos con
+ un paso del orden de
+\begin_inset Formula \( k=10 \)
+\end_inset
+
+.
+ Los resultados pueden verse en la figura
+\begin_inset LatexCommand \ref{fig:nystrom_c1}
+
+\end_inset
+
+.
+\layout Section
+
+Determinación del período.
+\layout Standard
+
+El valor teórico del período es
+\begin_inset Formula \( \tau =\pi \cdot \sqrt{\frac{2\cdot L}{g}}=42.387154 \)
+\end_inset
+
+.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:calculo_periodo}
+
+\end_inset
+
+Para hallar el período numéricamente, se estiman los
+\emph on
+ceros
+\emph default
+
+\emph on
+decrecientes
+\emph default
+ de la solución.
+ Llamamos
+\emph on
+cero decreciente
+\emph default
+ al punto donde la función cruza el eje de las abscisas en dirección decreciente.
+ También puede ser visto como el punto donde la derivada primera es mínima.
+ La diferencia entre un
+\emph on
+cero decreciente
+\emph default
+ y el siguiente es tomado como el período de la función.
+ Para calcularlo resolvió la ecuación númericamente hasta un
+\begin_inset Formula \( t=5000 \)
+\end_inset
+
+, aproximadamente 120 períodos, con un paso
+\begin_inset Formula \( k=0.05 \)
+\end_inset
+
+ para no perder mucha precisión.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file c2.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:periodo_c}
+
+\end_inset
+
+Evolución del Período para el caso sin depósitos y sin fricción
+\end_float
+
+\begin_inset LatexCommand \label{sec:periodo_analisis}
+
+\end_inset
+
+Podemos ver en la figura
+\begin_inset LatexCommand \ref{fig:periodo_c}
+
+\end_inset
+
+ que el período se comporta de la misma forma para todos los métodos analizados.
+ Vemos que sufre una variación leve cerca del período 24 (
+\begin_inset Formula \( t\cong 1000 \)
+\end_inset
+
+) y otra más
+\emph on
+brusca
+\emph default
+ cerca del período 95 (
+\begin_inset Formula \( t\cong 4000 \)
+\end_inset
+
+).
+ Cabe destacar que la variación
+\emph on
+brusca
+\emph default
+ es del orden de
+\begin_inset Formula \( 0.5\% \)
+\end_inset
+
+.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:periodo_euler_c2}
+
+\end_inset
+
+A pesar de los problemas nombrados en el punto
+\begin_inset LatexCommand \vref{sec:euler_diverge}
+
+\end_inset
+
+, el período calculado por Euler da un valor muy cercano al teórico:
+\begin_inset Formula \( \tau =42.382906 \)
+\end_inset
+
+, lo que daría un error porcentual de
+\begin_inset Formula \( e=0.01\% \)
+\end_inset
+
+.
+ Cabe destacar que para el paso
+\begin_inset Formula \( k \)
+\end_inset
+
+ utilizado la solución diverge, pero el período no parece verse afectado.
+\layout Subsection
+
+Runge-Kutta de orden 4 y Nystrom.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:periodo_rk4_nystrom_c2}
+
+\end_inset
+
+Es curioso, ambos métodos promedian un período exactamente igual:
+\begin_inset Formula \( \tau =42.382051 \)
+\end_inset
+
+.
+ Es aún más curioso que este resultado sea más lejano al teórico que el
+ de Euler, aunque por poco.
+ La diferencia es mínima y el error porcentual se mantiene en
+\begin_inset Formula \( e=0.01\% \)
+\end_inset
+
+.
+\layout Section
+
+Análisis de conservación.
+\layout Standard
+
+Para analizar si el método es conservativo se propone igualar la solución
+ numérica con la solución matemática:
+\begin_inset Formula \begin{equation}
+\label{math:conservacion}
+u_{n}\approx c\cdot e^{\alpha \cdot t}\approx c\cdot e^{\alpha \cdot n\cdot k}=c\cdot \left( e^{\alpha \cdot k}\right) ^{n}=c\cdot \gamma ^{n}
+\end{equation}
+
+\end_inset
+
+
+\layout Standard
+
+Para que el método oscile,
+\begin_inset Formula \( \gamma \)
+\end_inset
+
+ debe ser complejo y tener parte imaginaria:
+\begin_inset Formula \begin{equation}
+\label{math:gamma_imaginario}
+\gamma \neq \overline{\gamma }\quad \Leftrightarrow \quad \Im _{(\gamma )}\neq 0
+\end{equation}
+
+\end_inset
+
+y para que el método sea conservativo,
+\begin_inset Formula \( \gamma \)
+\end_inset
+
+ debe tener módulo 1:
+\begin_inset Formula \begin{equation}
+\label{math:gamma_modulo_1}
+\left| \gamma \right| =\sqrt{\Re ^{2}_{(\gamma )}+\Im ^{2}_{(\gamma )}}=1\quad \Leftrightarrow \quad \left| \gamma \right| ^{2}=\Re ^{2}_{(\gamma )}+\Im ^{2}_{(\gamma )}=1
+\end{equation}
+
+\end_inset
+
+
+\layout Subsection
+
+Euler.
+\layout Standard
+
+Como
+\begin_inset Formula \( \phi _{(z)}=0 \)
+\end_inset
+
+, según la ecuación
+\begin_inset LatexCommand \vref{math:euler}
+
+\end_inset
+
+ resulta:
+\begin_inset Formula \[
+\left\{ \begin{array}{rcl}
+x_{n+1} & = & x_{n}+k\cdot y_{n}\\
+y_{n+1} & = & y_{n}-\frac{g\cdot G}{L}\cdot k\cdot x_{n}
+\end{array}\right. \]
+
+\end_inset
+
+reemplazando con la ecuación
+\begin_inset LatexCommand \vref{math:conservacion}
+
+\end_inset
+
+ y expresándolo matricialmente obtenemos:
+\begin_inset Formula \begin{eqnarray*}
+\left( \begin{array}{c}
+C_{x}\cdot \gamma ^{n+1}\\
+C_{y}\cdot \gamma ^{n+1}
+\end{array}\right) & = & \left( \begin{array}{cc}
+1 & k\\
+-k\cdot \frac{g\cdot G}{L} & 1
+\end{array}\right) \cdot \left( \begin{array}{c}
+C_{x}\cdot \gamma ^{n}\\
+C_{y}\cdot \gamma ^{n}
+\end{array}\right) \\
+\left( \begin{array}{c}
+C_{x}\cdot \gamma \\
+C_{y}\cdot \gamma
+\end{array}\right) & = & \left( \begin{array}{cc}
+1 & k\\
+-k\cdot \frac{g\cdot G}{L} & 1
+\end{array}\right) \cdot \left( \begin{array}{c}
+C_{x}\\
+C_{y}
+\end{array}\right) \\
+\left( \begin{array}{c}
+0\\
+0
+\end{array}\right) & = & \left( \begin{array}{cc}
+1-\gamma & k\\
+-k\cdot \frac{g\cdot G}{L} & 1-\gamma
+\end{array}\right) \cdot \left( \begin{array}{c}
+C_{x}\\
+C_{y}
+\end{array}\right)
+\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+Para que la matriz tenga otra solución además de la trivial:
+\begin_inset Formula \[
+\det \left( \begin{array}{cc}
+1-\gamma & k\\
+-k\cdot \frac{g\cdot G}{L} & 1-\gamma
+\end{array}\right) =0\quad \Leftrightarrow \quad \left( 1-\gamma \right) ^{2}+k\cdot \frac{g\cdot G}{L}=0\quad \Leftrightarrow \quad \gamma =1\pm i\cdot k\cdot \sqrt{\frac{g\cdot G}{L}}\]
+
+\end_inset
+
+para que el método resulte oscilatorio, se debe cumplir la ecuación
+\begin_inset LatexCommand \vref{math:gamma_imaginario}
+
+\end_inset
+
+:
+\begin_inset Formula \[
+\Im _{(\gamma )}\neq 0\quad \Leftrightarrow \quad k\neq 0\]
+
+\end_inset
+
+por lo tanto, para que el método de Euler oscile, debe cumplirse
+\begin_inset Formula \( k\neq 0 \)
+\end_inset
+
+ lo que no agrega ninguna limitación.
+ Por otro lado para que el método sea conservativo debe cumplirse la ecuación
+
+\begin_inset LatexCommand \vref{math:gamma_modulo_1}
+
+\end_inset
+
+:
+\begin_inset Formula \[
+\left| \gamma \right| ^{2}=1+k^{2}\cdot \left| \frac{g\cdot G}{L}\right| =1\quad \Leftrightarrow \quad k=0\]
+
+\end_inset
+
+por lo tanto, el método de Euler diverge para cualquier valor de
+\begin_inset Formula \( k \)
+\end_inset
+
+ ya que
+\begin_inset Formula \( \left| \gamma \right| >1\quad \forall k>0 \)
+\end_inset
+
+.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:rk4_conservacion}
+
+\end_inset
+
+Lo primero que hay que hacer es expresar el método de forma más simple,
+ en un sólo par de ecuaciones.
+ Teniendo en cuenta que
+\begin_inset Formula \( \phi _{(z)}=0 \)
+\end_inset
+
+, reemplazando y desarrollando la ecuación
+\begin_inset LatexCommand \vref{math:rk4}
+
+\end_inset
+
+ obtenemos:
+\begin_inset Formula \[
+\left\{ \begin{array}{rcl}
+x_{n+1} & = & \left( 1-\frac{k^{2}\cdot g\cdot G}{2\cdot L}\right) \cdot x_{n}+\frac{k}{2}\cdot \left( 1-\frac{k\cdot g\cdot G}{3\cdot L}\right) \cdot y_{n}\\
+y_{n+1} & = & \left( \frac{k^{3}}{2}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k\cdot g\cdot G}{L}\right) \cdot x_{n}+\left( 1+\frac{k^{4}}{24}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k^{2}\cdot g\cdot G}{2\cdot L}\right) \cdot y_{n}
+\end{array}\right. \]
+
+\end_inset
+
+
+\layout Standard
+
+reemplazando con la ecuación
+\begin_inset LatexCommand \vref{math:conservacion}
+
+\end_inset
+
+ y expresándolo matricialmente obtenemos:
+\begin_inset Formula \begin{eqnarray*}
+\left( \begin{array}{c}
+C_{x}\cdot \gamma ^{n+1}\\
+C_{y}\cdot \gamma ^{n+1}
+\end{array}\right) & = & \left( \begin{array}{cc}
+1-\frac{k^{2}\cdot g\cdot G}{2\cdot L} & \frac{k}{2}\cdot \left( 1-\frac{k\cdot g\cdot G}{3\cdot L}\right) \\
+\frac{k^{3}}{2}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k\cdot g\cdot G}{L} & 1+\frac{k^{4}}{24}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k^{2}\cdot g\cdot G}{2\cdot L}
+\end{array}\right) \cdot \left( \begin{array}{c}
+C_{x}\cdot \gamma ^{n}\\
+C_{y}\cdot \gamma ^{n}
+\end{array}\right) \\
+\left( \begin{array}{c}
+C_{x}\cdot \gamma \\
+C_{y}\cdot \gamma
+\end{array}\right) & = & \left( \begin{array}{cc}
+1-\frac{k^{2}\cdot g\cdot G}{2\cdot L} & \frac{k}{2}\cdot \left( 1-\frac{k\cdot g\cdot G}{3\cdot L}\right) \\
+\frac{k^{3}}{2}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k\cdot g\cdot G}{L} & 1+\frac{k^{4}}{24}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k^{2}\cdot g\cdot G}{2\cdot L}
+\end{array}\right) \cdot \left( \begin{array}{c}
+C_{x}\\
+C_{y}
+\end{array}\right) \\
+\left( \begin{array}{c}
+0\\
+0
+\end{array}\right) & = & \left( \begin{array}{cc}
+1-\frac{k^{2}\cdot g\cdot G}{2\cdot L}-\gamma & \frac{k}{2}\cdot \left( 1-\frac{k\cdot g\cdot G}{3\cdot L}\right) \\
+\frac{k^{3}}{2}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k\cdot g\cdot G}{L} & 1+\frac{k^{4}}{24}\cdot \left( \frac{g\cdot G}{L}\right) ^{2}-\frac{k^{2}\cdot g\cdot G}{2\cdot L}-\gamma
+\end{array}\right) \cdot \left( \begin{array}{c}
+C_{x}\\
+C_{y}
+\end{array}\right)
+\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+Para que la matriz tenga otra solución además de la trivial
+\begin_inset Formula \( \left( a=\frac{g\cdot G}{L}\right) \)
+\end_inset
+
+:
+\begin_inset Formula \begin{eqnarray*}
+\det \left( \begin{array}{cc}
+1-\frac{k^{2}}{2}\cdot a-\gamma & \frac{k}{2}\cdot \left( 1-\frac{k}{3}\cdot a\right) \\
+\frac{k^{3}}{2}\cdot a^{2}-k\cdot a & 1+\frac{k^{4}}{24}\cdot a^{2}-\frac{k^{2}}{2}\cdot a-\gamma
+\end{array}\right) & = & 0\\
+\left( 1-\frac{k^{2}}{2}\cdot a-\frac{k^{3}}{6}\cdot a^{2}-\frac{7\cdot k^{4}}{24}\cdot a^{2}+\frac{k^{5}}{32}\cdot a^{3}+\frac{k^{6}}{48}\cdot a^{3}\right) +\left( k^{2}\cdot a-\frac{k^{4}}{24}\cdot a^{2}-2\right) \cdot \gamma +\gamma ^{2} & = & 0
+\end{eqnarray*}
+
+\end_inset
+
+resultando:
+\begin_inset Formula \[
+\gamma =\frac{-\left( k^{2}\cdot a-\frac{k^{4}}{24}\cdot a^{2}-2\right) \pm \sqrt{\frac{k^{8}\cdot a^{4}}{576}+\frac{k^{5}\cdot a^{3}}{8}-\frac{2\cdot k^{3}\cdot a^{2}}{3}-6\cdot k^{2}\cdot a}}{2}\]
+
+\end_inset
+
+ para que el método resulte oscilatorio, se debe cumplir la ecuación
+\begin_inset LatexCommand \vref{math:gamma_imaginario}
+
+\end_inset
+
+:
+\begin_inset Formula \[
+\Im _{(\gamma )}\neq 0\quad \Leftrightarrow \quad \frac{k^{8}\cdot a^{4}}{576}+\frac{k^{5}\cdot a^{3}}{8}-\frac{2\cdot k^{3}\cdot a^{2}}{3}-6\cdot k^{2}\cdot a<0\quad \Leftrightarrow \quad \frac{k^{4}\cdot a^{3}}{576}+\frac{k^{3}\cdot a^{2}}{8}-\frac{k\cdot 2\cdot a}{3}-6<0\]
+
+\end_inset
+
+
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file c3o.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:oscilacion_rk4}
+
+\end_inset
+
+Análisis de oscilación para RK4.
+\end_float
+Como la inecuación resultante es muy compleja de resolver analíticamente
+ se la resolvió gráficamente y se ve en la figura
+\begin_inset LatexCommand \ref{fig:oscilacion_rk4}
+
+\end_inset
+
+ que para un paso
+\begin_inset Formula \( k<51 \)
+\end_inset
+
+ el método se comporta de forma oscilatoria.
+\layout Standard
+
+Por otro lado para que el método sea conservativo debe cumplirse la ecuación
+
+\begin_inset LatexCommand \vref{math:gamma_modulo_1}
+
+\end_inset
+
+:
+\begin_inset Formula \begin{eqnarray*}
+\left| \gamma \right| ^{2}=\left( \frac{k^{2}\cdot a-\frac{k^{4}}{24}\cdot a^{2}-2}{2}\right) ^{2}+\left( \frac{\sqrt{\frac{k^{8}\cdot a^{4}}{576}+\frac{k^{5}\cdot a^{3}}{8}-\frac{2\cdot k^{3}\cdot a^{2}}{3}-6\cdot k^{2}\cdot a}}{2}\right) ^{2} & = & 1\\
+\frac{k^{4}\cdot a^{2}}{4}-\frac{k^{8}\cdot a^{4}}{2304}-2-\frac{k^{6}\cdot a^{2}}{48}-k^{2}\cdot a+\frac{k^{4}\cdot a^{2}}{24}+\left| \frac{k^{8}\cdot a^{4}}{2304}+\frac{k^{5}\cdot a^{3}}{32}-\frac{k^{3}\cdot a^{2}}{6}-\frac{3\cdot k^{2}\cdot a}{2}\right| & = & 0
+\end{eqnarray*}
+
+\end_inset
+
+
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file c3c.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:convergencia_rk4}
+
+\end_inset
+
+Análisis de convergencia para RK4.
+\end_float
+Nuevamente obtenemos una ecuación muy compleja de resolver analíticamente.
+ En la figura se presenta la resulución gráfica en donde observamos que
+ el método converge para cualquier
+\begin_inset Formula \( k>0 \)
+\end_inset
+
+.
+ Sin embargo para un
+\begin_inset Formula \( k<7 \)
+\end_inset
+
+ aproximadamente vemos que tiende a converger lentamente, mientras que para
+
+\begin_inset Formula \( k>7 \)
+\end_inset
+
+ tiende a converger de forma mucho más brusca (y cada vez más acentuada).
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+Como
+\begin_inset Formula \( \phi _{(z)}=0 \)
+\end_inset
+
+, según la ecuación
+\begin_inset LatexCommand \vref{math:nystrom}
+
+\end_inset
+
+ resulta:
+\begin_inset Formula \[
+x_{n+1}=\left( 2-\frac{g\cdot G}{L}\cdot k^{2}\right) \cdot x_{n}-x_{n-1}\]
+
+\end_inset
+
+ reemplazando con la ecuación
+\begin_inset LatexCommand \vref{math:conservacion}
+
+\end_inset
+
+, obtenemos:
+\begin_inset Formula \[
+\left. \begin{array}{lll}
+\gamma ^{n+1} & = & \left( 2-\frac{g\cdot G}{L}\cdot k^{2}\right) \cdot \gamma ^{n}-\gamma ^{n-1}\\
+\gamma ^{2} & = & \left( 2-\frac{g\cdot G}{L}\cdot k^{2}\right) \cdot \gamma -1\\
+0 & = & \gamma ^{2}-\left( 2-\frac{g\cdot G}{L}\cdot k^{2}\right) \cdot \gamma +1
+\end{array}\right\} \quad \Rightarrow \quad \gamma =\left( 1-\frac{g\cdot G}{2\cdot L}\cdot k^{2}\right) \pm \sqrt{\left( 1-\frac{g\cdot G}{2\cdot L}\cdot k^{2}\right) ^{2}-1}\]
+
+\end_inset
+
+para que el método resulte oscilatorio, se debe cumplir la ecuación
+\begin_inset LatexCommand \vref{math:gamma_imaginario}
+
+\end_inset
+
+:
+\begin_inset Formula \[
+\Im _{(\gamma )}\neq 0\quad \Leftrightarrow \quad \Im _{\left( \sqrt{\left( 1-\frac{g\cdot G}{2\cdot L}\cdot k^{2}\right) ^{2}-1}\right) }\neq 0\quad \Leftrightarrow \quad \left( 1-\frac{g\cdot G}{2\cdot L}\cdot k^{2}\right) ^{2}-1<0\quad \Leftrightarrow \quad 0<k<\sqrt{\frac{4\cdot L}{g\cdot G}}\]
+
+\end_inset
+
+por lo tanto, para que el método de Nystrom oscile, debe cumplirse
+\begin_inset Formula \( k<\sqrt{\frac{4\cdot L}{g\cdot G}}=13.49225 \)
+\end_inset
+
+, que coincide con lo obtenido experimentalmente en el punto
+\begin_inset LatexCommand \vref{sec:nystrom_paso_minimo}
+
+\end_inset
+
+.
+\layout Standard
+
+Para que el método sea conservativo debe cumplirse la ecuación
+\begin_inset LatexCommand \vref{math:gamma_modulo_1}
+
+\end_inset
+
+:
+\begin_inset Formula \[
+\left| \gamma \right| ^{2}=\left( 1-\frac{g\cdot G}{2\cdot L}\cdot k^{2}\right) ^{2}+\left| \left( 1-\frac{g\cdot G}{2\cdot L}\cdot k^{2}\right) ^{2}-1\right| =1\quad \Leftrightarrow \quad k=\sqrt{\frac{4\cdot L}{g\cdot G}}\]
+
+\end_inset
+
+por lo tanto,
+\begin_inset Formula \( k=\sqrt{\frac{4\cdot L}{g\cdot G}}=13.49225 \)
+\end_inset
+
+.
+ Sin embargo en la práctica (ver figura
+\begin_inset LatexCommand \vref{fig:nystrom_c1}
+
+\end_inset
+
+) vemos que, a pesar de que el método debería converger con un paso más
+ pequeño que
+\begin_inset Formula \( k<13.49225 \)
+\end_inset
+
+, no lo hace y da la impresión de comportarse de forma conservativa.
+\layout Chapter
+
+Caso sin depósitos y con fricción laminar.
+\layout Standard
+
+Considerando la fricción laminar y un tubo en U resulta
+\begin_inset Formula \( \phi _{(z)}=\frac{32\cdot n}{D^{2}} \)
+\end_inset
+
+,
+\begin_inset Formula \( G=2 \)
+\end_inset
+
+, siendo
+\begin_inset Formula \( n \)
+\end_inset
+
+ la viscosidad cinemática del líquido y el diámetro del tubo.
+\layout Section
+
+Gráfico de la solución en el tiempo.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:graficos_d}
+
+\end_inset
+
+Se grafican aproximadamente 50 períodos de la solución.
+ Se toman más períodos a pesar de que no se vea muy claro el gráfico porque
+ la solución converge pero de forma lenta y de tomarse pocos períodos casi
+ no se observaría el fenómeno.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file d1e.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:euler_d1}
+
+\end_inset
+
+Solución por Euler para el caso sin depósitos y con fricción laminar.
+\end_float
+El método de Euler sigue teniendo un error muy grande con pasos relativamente
+ pequeños lo que hace que tienda a divergir muy fácilmente (o al menos a
+ contrarrestar la poca convergencia en este caso).
+ Para que esto no suceda, como se ve en la figura
+\begin_inset LatexCommand \ref{fig:euler_d1}
+
+\end_inset
+
+, se tomó un paso
+\begin_inset Formula \( k=0.0025 \)
+\end_inset
+
+, 1000 veces menor que el utilizado para RK4 y Nystrom para obtener los
+ mismos (o incluso mejores) resultados.
+ De tomar un paso más grande, la divergencia del método es muy similar a
+ la presentada en la figura
+\begin_inset LatexCommand \vref{fig:euler_c1}
+
+\end_inset
+
+.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file d1r.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:rk4_d1}
+
+\end_inset
+
+Solución por RK4 para el caso sin depósitos y con fricción laminar.
+\end_float
+Es el método más estable para este caso.
+ El la figura
+\begin_inset LatexCommand \ref{fig:rk4_d1}
+
+\end_inset
+
+ se grafica con un paso de
+\begin_inset Formula \( k=2.5 \)
+\end_inset
+
+, con el cual comienza a comportarse de forma estable.
+ A medida que se disminuye el paso conserva el comportamiento.
+ Si se toma un paso mayor, diverge rápidamente.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file d1n.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:nystrom_d1}
+
+\end_inset
+
+Solución por Nystrom para el caso sin depósitos y con fricción laminar.
+\end_float
+
+\begin_inset LatexCommand \label{sec:nystrom_converge}
+
+\end_inset
+
+Este método se comporta de una forma muy particular.
+ Con un paso de
+\begin_inset Formula \( k=2 \)
+\end_inset
+
+ se mantiene estable y parece dar buenos resultados.
+ Si se toma un paso menor, converge de forma más violenta, dando una solución
+ no válida como se aprecia en la figura
+\begin_inset LatexCommand \ref{fig:nystrom_d1}
+
+\end_inset
+
+.
+ Con un paso mayor se observa el efecto contrario: diverge rápidamente.
+\layout Section
+
+Determinación del período.
+\layout Standard
+
+El valor teórico del período es
+\begin_inset Formula \( \tau =\frac{2\cdot \pi }{\sqrt{\frac{2\cdot g}{L}+\left( \frac{16\cdot n}{D^{2}}\right) ^{2}}}=42.387150 \)
+\end_inset
+
+.
+\layout Standard
+
+Para hallar el período numéricamente se utilizó el mismo método que en el
+ punto
+\begin_inset LatexCommand \vref{sec:calculo_periodo}
+
+\end_inset
+
+, con un paso de
+\begin_inset Formula \( k=0.05 \)
+\end_inset
+
+ y resolviendo hasta
+\begin_inset Formula \( t=5000\cong 120 \)
+\end_inset
+
+ períodos.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file d2.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:periodo_d}
+
+\end_inset
+
+Evolución del Período para el caso sin depósitos y con fricción laminar.
+\end_float
+Podemos ver en la figura
+\begin_inset LatexCommand \ref{fig:periodo_d}
+
+\end_inset
+
+ que el período se comporta de la misma forma que en el punto
+\begin_inset LatexCommand \vref{sec:periodo_analisis}
+
+\end_inset
+
+.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+Al igual que en el punto
+\begin_inset LatexCommand \vref{sec:periodo_euler_c2}
+
+\end_inset
+
+, el período obtenido, a pesar de la divergencia, es
+\begin_inset Formula \( \tau =42.382906 \)
+\end_inset
+
+, un valor muy cercano al teórico.
+ Tiene el mismo error porcentual de
+\begin_inset Formula \( e=0.01\% \)
+\end_inset
+
+.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+También el período hallado por este método coincide con el caso donde no
+ hay fricción (ver punto
+\begin_inset LatexCommand \vref{sec:periodo_rk4_nystrom_c2}
+
+\end_inset
+
+):
+\begin_inset Formula \( \tau =42.382051 \)
+\end_inset
+
+.
+ El error porcentual vuelve a ser del orden de
+\begin_inset Formula \( e=0.01\% \)
+\end_inset
+
+.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+Este es el único método que varía con respecto al caso sin fricción.
+ Con el efecto de la fricción laminar y para un paso tan pequeño el método
+ converge, como se explicó en el punto
+\begin_inset LatexCommand \vref{sec:nystrom_converge}
+
+\end_inset
+
+.
+ Al igual que con Euler, esto no impide calcular el período y también coincidien
+do con Euler, hallamos que el error es menor que para RK4 e incluso que
+ para Euler.
+ El período calculado numéricamente da
+\begin_inset Formula \( \tau =42.387949 \)
+\end_inset
+
+, con lo que observamos que ganamos un dígito de precisión, siendo el error
+ porcentual del orden de
+\begin_inset Formula \( e=0.002\% \)
+\end_inset
+
+.
+\layout Section
+
+Valores de excursión máxima y mínima del menisco.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:max_min_d}
+
+\end_inset
+
+Al calcular estos valores se puso en evidencia la inestabilidad del método
+ de Nystrom para este caso.
+ Hubo que probar valores del paso (sin que sea demasiado grande ni demasiado
+ pequeño por lo explicado en el punto
+\begin_inset LatexCommand \vref{sec:nystrom_converge}
+
+\end_inset
+
+) e ir comparando con RK4 que resulta el más estable.
+ También puede notarse que las crestas no decrecen (en valor absoluto) monótonam
+ente como sucede con Euler y RK4 que tienen un comportamiento similar (aunque
+ Euler necesita un paso extremadamente pequeño para lograr estos resultados).
+\layout Standard
+
+\begin_float tab
+\layout Standard
+\align center
+
+\begin_inset Tabular
+<lyxtabular version="2" rows="12" columns="6">
+<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<row topline="true" bottomline="true" newpage="false">
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Euler (
+\begin_inset Formula \( k=0.00405 \)
+\end_inset
+
+)
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
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+
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+
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+\end_inset
+</cell>
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+
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+
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+\end_inset
+</cell>
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+
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+</cell>
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+
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+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{tab:min_d}
+
+\end_inset
+
+Valores de excursión mínima para el caso sin depósitos y con fricción laminar.
+\end_float
+Pueden verse los resultados en los cuadros
+\begin_inset LatexCommand \ref{tab:max_d}
+
+\end_inset
+
+ y
+\begin_inset LatexCommand \ref{tab:min_d}
+
+\end_inset
+
+.
+\layout Section
+
+Estimación del tiempo de reposo.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:tiempo_reposo}
+
+\end_inset
+
+Para estimar el tiempo de reposo se calculó los máximos de la solución igual
+ que en el punto
+\begin_inset LatexCommand \ref{sec:max_min_d}
+
+\end_inset
+
+ pero para una mayor cantidad de períodos (aproximadamente 1200 en un principio).
+ Luego se ajustaron los datos por mínimos cuadrados a través de una función:
+
+\begin_inset Formula \begin{equation}
+\label{math:aproximacion_minimos_cuadrados}
+z_{(t)}=a\cdot e^{-b\cdot t}
+\end{equation}
+
+\end_inset
+
+
+\layout Standard
+
+Luego de comparar los resultados con cada vez menor cantidad de períodos,
+ se observo que la calidad de la aproximación no empeoraba, incluso al utilizar
+ sólo los primeros 10 períodos obtenidos en el punto
+\begin_inset LatexCommand \ref{sec:max_min_d}
+
+\end_inset
+
+, al menos para el método RK4.
+\layout Standard
+
+Consideramos que se llega al reposo cuando
+\begin_inset Formula \( \left| z\right| \leq 0.01 \)
+\end_inset
+
+, por lo que si despejamos
+\begin_inset Formula \( t \)
+\end_inset
+
+ de la ecuación
+\begin_inset LatexCommand \ref{math:aproximacion_minimos_cuadrados}
+
+\end_inset
+
+, resulta:
+\begin_inset Formula \begin{equation}
+\label{math:reposo}
+t\geq \frac{\ln \left| a\right| -\ln (z)}{b}=\frac{\ln \left| a\right| -\ln (0.01)}{b}=\frac{\ln \left| a\right| +4.60517}{b}
+\end{equation}
+
+\end_inset
+
+
+\layout Subsection
+
+Euler.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file d4e.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:euler_d4}
+
+\end_inset
+
+Estimación del tiempo de reposo para Euler.
+\end_float
+No tiene mucho sentido la estimación por Euler, ya que se pone en evidencia
+ la tendencia a divergir de este método y se obtienen resultados poco precisos
+ cuando se los compara con RK4 o incluso Nystrom, que no es muy estable
+ para este caso.
+ Aún así se presentan los resultados del ajuste de los máximos y mínimos
+ obtenidos en el punto
+\begin_inset LatexCommand \ref{sec:max_min_d}
+
+\end_inset
+
+ en la figura
+\begin_inset LatexCommand \ref{fig:euler_d4}
+
+\end_inset
+
+ porque no tiene sentido aproximar con más períodos porque el paso
+\begin_inset Formula \( k \)
+\end_inset
+
+ debería ser extremadamente pequeño para que no diverja.
+ La funciones aproximantes resultaron:
+\begin_inset Formula \[
+z^{\max }_{(t)}=2.986634\cdot e^{-1.95\cdot 10^{-5}\cdot t}\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+z^{\min }_{(t)}=-2.986629\cdot e^{-1.95\cdot 10^{-5}\cdot t}\]
+
+\end_inset
+
+
+\layout Standard
+
+Para hallar el tiempo de reposo estimado reemplazamos en la ecuación
+\begin_inset LatexCommand \ref{math:reposo}
+
+\end_inset
+
+ obteniendo
+\begin_inset Formula \( t\cong \frac{\ln (2.98663)+4.60517}{1.95\cdot 10^{-5}}\cong 292272\cong 300000 \)
+\end_inset
+
+ como se aprecia en la figura
+\begin_inset LatexCommand \ref{fig:euler_d4}
+
+\end_inset
+
+.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file d4r.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:rk4_d4}
+
+\end_inset
+
+Estimación del tiempo de reposo para RK4.
+\end_float
+Como se explico en el punto
+\begin_inset LatexCommand \ref{sec:tiempo_reposo}
+
+\end_inset
+
+, a pesar de ser posible calcular los máximos y mínimos para muchos más
+ períodos que en el punto
+\begin_inset LatexCommand \vref{sec:max_min_d}
+
+\end_inset
+
+, no es necesario ya que con estos datos se obtiene una precisión que no
+ es mejorada considerablemente aumentando la cantidad de períodos evaluados.
+\layout Standard
+
+La funciones aproximantes resultaron:
+\begin_inset Formula \[
+z^{\max }_{(t)}=2.9862768\cdot e^{-6.4\cdot 10^{-5}\cdot t}\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+z^{\min }_{(t)}=-2.9862509\cdot e^{-6.4\cdot 10^{-5}\cdot t}\]
+
+\end_inset
+
+Para hallar el tiempo de reposo estimado reemplazamos en la ecuación
+\begin_inset LatexCommand \ref{math:reposo}
+
+\end_inset
+
+ obteniendo
+\begin_inset Formula \( t\cong \frac{\ln (2.98626)+4.60517}{6.4\cdot 10^{-5}}\cong 89049\cong 90000 \)
+\end_inset
+
+ como se aprecia en la figura
+\begin_inset LatexCommand \ref{fig:rk4_d4}
+
+\end_inset
+
+.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file d4n.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:nystrom_d4}
+
+\end_inset
+
+Estimación del tiempo de reposo para Nystrom.
+\end_float
+Al igual que con RK4, y a pesar de que para el caso de este método la variación
+ de los máximos y mínimos no es monótona, se obtienen resultados precisos
+ utilizando los datos del punto
+\begin_inset LatexCommand \vref{sec:max_min_d}
+
+\end_inset
+
+.
+\layout Standard
+
+La funciones aproximantes resultaron:
+\begin_inset Formula \[
+z^{\max }_{(t)}=2.977926\cdot e^{-6.68\cdot 10^{-5}\cdot t}\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+z^{\min }_{(t)}=-2.732692\cdot e^{-6.06\cdot 10^{-5}\cdot t}\]
+
+\end_inset
+
+Para hallar el tiempo de reposo estimado reemplazamos en la ecuación
+\begin_inset LatexCommand \ref{math:reposo}
+
+\end_inset
+
+ obteniendo
+\begin_inset Formula \( t\cong \frac{\ln (2.975)+4.60517}{6.37\cdot 10^{-5}}\cong 89409\cong 90000 \)
+\end_inset
+
+ como se aprecia en la figura
+\begin_inset LatexCommand \ref{fig:nystrom_d4}
+
+\end_inset
+
+.
+\layout Chapter
+
+Caso sin depósitos y con fricción turbulenta.
+\layout Standard
+
+Considerando flujo turbulento y un tubo en U resulta
+\begin_inset Formula \( \phi _{(z)}=\frac{f}{2\cdot D}\left| \frac{dz}{dt}\right| \)
+\end_inset
+
+,
+\begin_inset Formula \( G=2 \)
+\end_inset
+
+, siendo
+\begin_inset Formula \( f \)
+\end_inset
+
+ el factor de fricción.
+\layout Section
+
+Gráfico de la solución en el tiempo.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:graficos_e}
+
+\end_inset
+
+Se grafican aproximadamente 25 períodos de la solución.
+ La solución converge de forma más rápida que en el punto
+\begin_inset LatexCommand \ref{sec:graficos_d}
+
+\end_inset
+
+, por lo que se pueden tomar menos períodos lográndose ver el efecto de
+ la convergencia.
+ Se incluyen la derivada primera en los gráficos.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file e1e.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:euler_e1}
+
+\end_inset
+
+Solución por Euler para el caso sin depósitos y con fricción turbulenta.
+\end_float
+Como la convergencia es más violenta en este caso, no es necesario tomar
+ un paso tan pequeño para Euler.
+ Al tomarse un
+\begin_inset Formula \( k=0.0125 \)
+\end_inset
+
+ vemos en la figura
+\begin_inset LatexCommand \ref{fig:euler_e1}
+
+\end_inset
+
+ que el método se comporta de manera razonable.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file e1r.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:rk4_e1}
+
+\end_inset
+
+Solución por RK4 para el caso sin depósitos y con fricción turbulenta.
+\end_float
+Nuevamente este método se comporta de forma estable para un paso bastante
+ grande.
+ El la figura
+\begin_inset LatexCommand \ref{fig:rk4_e1}
+
+\end_inset
+
+ se grafica con un
+\begin_inset Formula \( k=3.125 \)
+\end_inset
+
+, con el cual comienza a comportarse de forma estable.
+ A medida que se disminuye el paso conserva el comportamiento.
+ Si se toma un paso un poco mayor converge rápidamente, y si el paso es
+ mucho mayor, diverge rápidamente.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file e1n.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:nystrom_e1}
+
+\end_inset
+
+Solución por Nystrom para el caso sin depósitos y con fricción turbulenta.
+\end_float
+El método se comporta de la misma forma que en el punto
+\begin_inset LatexCommand \vref{sec:nystrom_converge}
+
+\end_inset
+
+, por lo que solamente se grafica para un paso estable
+\begin_inset Formula \( k=2 \)
+\end_inset
+
+ como se ve en la figura
+\begin_inset LatexCommand \ref{fig:nystrom_e1}
+
+\end_inset
+
+.
+\layout Section
+
+Determinación del período.
+\layout Standard
+
+En este caso no podemos calcular, al menos de forma simple, el valor teórico
+ del período.
+\layout Standard
+
+Para hallar el período numéricamente se utilizó el mismo método que en el
+ punto
+\begin_inset LatexCommand \vref{sec:calculo_periodo}
+
+\end_inset
+
+, con un paso de
+\begin_inset Formula \( k=0.05 \)
+\end_inset
+
+ y resolviendo hasta
+\begin_inset Formula \( t=5000\cong 120 \)
+\end_inset
+
+ períodos.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file e2.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:periodo_e}
+
+\end_inset
+
+Evolución del Período para el caso sin depósitos y con fricción turbulenta.
+\end_float
+Podemos ver en la figura
+\begin_inset LatexCommand \ref{fig:periodo_e}
+
+\end_inset
+
+ que el período se comporta de la misma forma que en el punto
+\begin_inset LatexCommand \vref{sec:periodo_analisis}
+
+\end_inset
+
+.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+Al igual que en el punto
+\begin_inset LatexCommand \vref{sec:periodo_euler_c2}
+
+\end_inset
+
+, el período obtenido, a pesar de la divergencia, es
+\begin_inset Formula \( \tau =42.381624 \)
+\end_inset
+
+, similar al hallado en los casos anteriores.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+El período hallado por este método no sólo es similar a los hallados en
+ casos anteriores, es igual:
+\begin_inset Formula \( \tau =42.382051 \)
+\end_inset
+
+.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+Al igual que con Euler, el período hallado es muy similar al de los casos
+ anteriores:
+\begin_inset Formula \( \tau =42.383419 \)
+\end_inset
+
+.
+\layout Section
+
+Valores de excursión máxima y mínima del menisco.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:max_min_e}
+
+\end_inset
+
+A diferencia del punto
+\begin_inset LatexCommand \ref{sec:max_min_d}
+
+\end_inset
+
+, el método de Nystrom para este caso evoluciona de forma monótona con un
+ paso
+\begin_inset Formula \( k\cong 2 \)
+\end_inset
+
+.
+ Por lo demás es muy similar.
+\layout Standard
+
+\begin_float tab
+\layout Standard
+\align center
+
+\begin_inset Tabular
+<lyxtabular version="2" rows="12" columns="6">
+<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<row topline="true" bottomline="true" newpage="false">
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Euler (
+\begin_inset Formula \( k=0.00405 \)
+\end_inset
+
+)
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+RK4 (
+\begin_inset Formula \( k=0.405 \)
+\end_inset
+
+)
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Nystrom (
+\begin_inset Formula \( k=2.13158 \)
+\end_inset
+
+)
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
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+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true" newpage="false">
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
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+\end_inset
+</cell>
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+
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+\end_inset
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+\end_inset
+</cell>
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+\end_inset
+</cell>
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+\layout Standard
+
+-1.03522
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true" newpage="false">
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+402.98
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+-0.923832
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+402.974
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+-0.912937
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+398.606
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+-0.961757
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{tab:min_e}
+
+\end_inset
+
+Valores de excursión mínima para el caso sin depósitos y con fricción turbulenta.
+\end_float
+Pueden verse los resultados en los cuadros
+\begin_inset LatexCommand \ref{tab:max_e}
+
+\end_inset
+
+ y
+\begin_inset LatexCommand \ref{tab:min_e}
+
+\end_inset
+
+.
+\layout Section
+
+Estimación del tiempo de reposo.
+\layout Standard
+
+Para estimar el tiempo de reposo se probó el método utilizado en el punto
+
+\begin_inset LatexCommand \vref{sec:tiempo_reposo}
+
+\end_inset
+
+, pero no se pudo hallar una buena función aproximante, por lo tanto simplemente
+ se buscó la solución hasta hallar el mismo valor límite de
+\begin_inset Formula \( z=0.01 \)
+\end_inset
+
+ que para el punto
+\begin_inset LatexCommand \ref{sec:tiempo_reposo}
+
+\end_inset
+
+.
+ Para el método de Euler no se justifica calcular la estimación, ya que
+ basándonos en los valores obtenidos por otros métodos hay que llegar a
+ un
+\begin_inset Formula \( t\cong 50000 \)
+\end_inset
+
+ para lo que serían necesario un paso muy pequeño para que no diverja.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file e4r.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:rk4_e4}
+
+\end_inset
+
+Estimación del tiempo de reposo para RK4.
+\end_float
+Para hallar la estimación simplemente se resolvió la ecuación hasta que
+
+\begin_inset Formula \( z\leq 0.01 \)
+\end_inset
+
+.
+ Esto sucedió para
+\begin_inset Formula \( t\cong 50000 \)
+\end_inset
+
+ utilizando un paso de
+\begin_inset Formula \( k=0.5 \)
+\end_inset
+
+ como se observa en la figura
+\begin_inset LatexCommand \ref{fig:rk4_e4}
+
+\end_inset
+
+.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file e4n.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:nystrom_e4}
+
+\end_inset
+
+Estimación del tiempo de reposo para Nystrom.
+\end_float
+Al igual que para RK4, simplemente se resolvió la ecuación hasta que
+\begin_inset Formula \( z\leq 0.01 \)
+\end_inset
+
+.
+ Esto también sucedió para
+\begin_inset Formula \( t\cong 50000 \)
+\end_inset
+
+ pero utilizando un paso mucho mayor
+\begin_inset Formula \( k=2 \)
+\end_inset
+
+ conservando la precisión, como se observa en la figura
+\begin_inset LatexCommand \ref{fig:nystrom_e4}
+
+\end_inset
+
+.
+\layout Chapter
+
+Caso con depósitos y con fricción turbulenta.
+\layout Standard
+
+Considerando flujo turbulento y el sistema hidráulico de la figura
+\begin_inset LatexCommand \vref{fig:diagrama}
+
+\end_inset
+
+, resulta
+\begin_inset Formula \( \phi _{(z)}=\frac{f}{2\cdot D}\cdot \frac{L_{e}}{L}\left| \frac{dz}{dt}\right| \)
+\end_inset
+
+,
+\begin_inset Formula \( G=A\cdot \left( \frac{1}{A_{1}}+\frac{1}{A_{2}}\right) \)
+\end_inset
+
+, siendo
+\begin_inset Formula \( L_{e} \)
+\end_inset
+
+ un factor de pérdidas de carga,
+\begin_inset Formula \( A \)
+\end_inset
+
+ la sección del tubo y
+\begin_inset Formula \( A_{i} \)
+\end_inset
+
+ la sección horizontal del depósito.
+\layout Standard
+
+Considerando los valores iniciales (ver punto
+\begin_inset LatexCommand \vref{sec:valores_iniciales}
+
+\end_inset
+
+), vemos que
+\begin_inset Formula \( G \)
+\end_inset
+
+ es constante:
+\begin_inset Formula \[
+G=A\cdot \left( \frac{1}{A_{1}}+\frac{1}{A_{2}}\right) =A\cdot \left( \frac{1}{100\cdot A}+\frac{1}{50\cdot A}\right) =A\cdot \frac{1+2}{100\cdot A}=\frac{3}{100}=0.03\]
+
+\end_inset
+
+
+\layout Section
+
+Gráfico de la solución en el tiempo.
+\layout Standard
+
+Se grafican aproximadamente 15 períodos de la solución.
+ La solución converge de forma más rápida que en el punto
+\begin_inset LatexCommand \ref{sec:graficos_e}
+
+\end_inset
+
+ y con un período mayor, por lo que se pueden tomar menos períodos lográndose
+ ver el efecto de la convergencia.
+ Se incluyen la derivada primera en los gráficos.
+\layout Standard
+
+Los métodos se comportan igual que en los casos anteriores, al menos para
+ la cantidad de períodos graficados.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file f1e.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:euler_f1}
+
+\end_inset
+
+Solución por Euler para el caso con depósitos y con fricción turbulenta.
+\end_float
+Como la convergencia es más violenta en este caso, no es necesario tomar
+ un paso tan pequeño para Euler.
+ Al tomarse un
+\begin_inset Formula \( k=0.05 \)
+\end_inset
+
+ vemos en la figura
+\begin_inset LatexCommand \ref{fig:euler_f1}
+
+\end_inset
+
+ que el método se comporta de manera razonable.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file f1r.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:rk4_f1}
+
+\end_inset
+
+Solución por RK4 para el caso con depósitos y con fricción turbulenta.
+\end_float
+Nuevamente este método se comporta de forma estable para un paso bastante
+ grande.
+ El la figura
+\begin_inset LatexCommand \ref{fig:rk4_f1}
+
+\end_inset
+
+ se grafica con un
+\begin_inset Formula \( k=5 \)
+\end_inset
+
+.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file f1n.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:nystrom_f1}
+
+\end_inset
+
+Solución por Nystrom para el caso sin depósitos y con fricción turbulenta.
+\end_float
+Volvemos a observar que el método es solamente estable para valores de
+\begin_inset Formula \( k\cong 2 \)
+\end_inset
+
+.
+ En la figura
+\begin_inset LatexCommand \ref{fig:nystrom_f1}
+
+\end_inset
+
+ se grafica para un paso estable
+\begin_inset Formula \( k=2 \)
+\end_inset
+
+.
+\layout Section
+
+Determinación del período.
+\layout Standard
+
+En este caso al igual que en el anterior no podemos calcular, al menos de
+ forma simple, el valor teórico del período.
+\layout Standard
+
+Para hallar el período numéricamente se utilizó el mismo método que en el
+ punto
+\begin_inset LatexCommand \vref{sec:calculo_periodo}
+
+\end_inset
+
+, con un paso de
+\begin_inset Formula \( k=0.04 \)
+\end_inset
+
+ y resolviendo hasta
+\begin_inset Formula \( t=40000\cong 120 \)
+\end_inset
+
+ períodos.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file f2.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:periodo_f}
+
+\end_inset
+
+Evolución del Período para el caso con depósitos y con fricción turbulenta.
+\end_float
+Podemos ver en la figura
+\begin_inset LatexCommand \ref{fig:periodo_f}
+
+\end_inset
+
+ que el período se comporta de la misma forma que en el punto
+\begin_inset LatexCommand \vref{sec:periodo_analisis}
+
+\end_inset
+
+.
+ Cabe destacar que para este caso el período aproximadamente 10 veces superior
+ al de los casos anteriores, por lo que el período 24 se alcanza en
+\begin_inset Formula \( t\cong 8300 \)
+\end_inset
+
+ y el período 95 en
+\begin_inset Formula \( t\cong 33000 \)
+\end_inset
+
+.
+\layout Subsection
+
+Euler.
+\layout Standard
+
+Nuevamente, a pesar de la divergencia, el período
+\begin_inset Formula \( \tau =346.03739 \)
+\end_inset
+
+ parece ser bastante preciso, al menos al compararlo con los resultados
+ obtenidos con los otros métodos.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+El período hallado con este método es:
+\begin_inset Formula \( \tau =346.04435 \)
+\end_inset
+
+.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+El período hallado con este método es:
+\begin_inset Formula \( \tau =346.04695 \)
+\end_inset
+
+.
+\layout Section
+
+Valores de excursión máxima y mínima del menisco.
+\layout Standard
+
+
+\begin_inset LatexCommand \label{sec:max_min_f}
+
+\end_inset
+
+Pueden verse los resultados en los cuadros
+\begin_inset LatexCommand \ref{tab:max_f}
+
+\end_inset
+
+ y
+\begin_inset LatexCommand \ref{tab:min_f}
+
+\end_inset
+
+.
+\layout Standard
+
+\begin_float tab
+\layout Standard
+\align center
+
+\begin_inset Tabular
+<lyxtabular version="2" rows="12" columns="6">
+<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="false" width="" special="">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="" special="">
+<row topline="true" bottomline="true" newpage="false">
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Euler (
+\begin_inset Formula \( k=0.01 \)
+\end_inset
+
+)
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+RK4 (
+\begin_inset Formula \( k=1 \)
+\end_inset
+
+)
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+\end_inset
+</cell>
+<cell multicolumn="1" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Nystrom (
+\begin_inset Formula \( k=2 \)
+\end_inset
+
+)
+\end_inset
+</cell>
+<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true" newpage="false">
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Tiempo (t)
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Altura (z)
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
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+
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+\end_inset
+</cell>
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+
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+\end_inset
+</cell>
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+
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+
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+</cell>
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+\begin_inset Text
+
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+
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+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+3286
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+-0.801451
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{tab:min_f}
+
+\end_inset
+
+Valores de excursión mínima para el caso con depósitos y con fricción turbulenta.
+\end_float
+\layout Section
+
+Estimación del tiempo de reposo.
+\layout Standard
+
+Para estimar el tiempo de reposo se probó el método utilizado en el punto
+
+\begin_inset LatexCommand \vref{sec:tiempo_reposo}
+
+\end_inset
+
+, pero no se pudo hallar una buena función aproximante, por lo tanto se
+ resolvió igual que en el caso anterior, buscando una solución hasta hallar
+ el valor límite de
+\begin_inset Formula \( z=0.01 \)
+\end_inset
+
+ que para el punto
+\begin_inset LatexCommand \ref{sec:tiempo_reposo}
+
+\end_inset
+
+.
+ Para el método de Euler no se justifica calcular la estimación, ya que
+ basándonos en los valores obtenidos por otros métodos hay que llegar a
+ un
+\begin_inset Formula \( t\cong 350000 \)
+\end_inset
+
+ para lo que serían necesario un paso muy pequeño para que se note su tendencia
+ a divergir.
+\layout Subsection
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file f4r.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:rk4_f4}
+
+\end_inset
+
+Estimación del tiempo de reposo para RK4.
+\end_float
+Para hallar la estimación simplemente se resolvió la ecuación hasta que
+
+\begin_inset Formula \( z\leq 0.01 \)
+\end_inset
+
+.
+ Esto sucedió para
+\begin_inset Formula \( t\cong 350000 \)
+\end_inset
+
+ utilizando un paso de
+\begin_inset Formula \( k=3.5 \)
+\end_inset
+
+ como se observa en la figura
+\begin_inset LatexCommand \ref{fig:rk4_f4}
+
+\end_inset
+
+.
+\layout Subsection
+
+Nystrom.
+\layout Standard
+
+\begin_float fig
+\layout Standard
+\align center
+
+\begin_inset Figure size 360 252
+file f4n.eps
+flags 11
+
+\end_inset
+
+
+\layout Caption
+
+
+\begin_inset LatexCommand \label{fig:nystrom_f4}
+
+\end_inset
+
+Estimación del tiempo de reposo para Nystrom.
+\end_float
+Al igual que para RK4, simplemente se resolvió la ecuación hasta que
+\begin_inset Formula \( z\leq 0.01 \)
+\end_inset
+
+.
+ Esto también sucedió para
+\begin_inset Formula \( t\cong 350000 \)
+\end_inset
+
+ pero utilizando un el único paso estable para Nystrom,
+\begin_inset Formula \( k=2 \)
+\end_inset
+
+, como se observa en la figura
+\begin_inset LatexCommand \ref{fig:nystrom_f4}
+
+\end_inset
+
+.
+\layout Chapter
+
+Conclusiones.
+\layout Standard
+
+Este trabajo práctico sirvió para ver las ventajas y desventajas de cada
+ método para distintas situaciones.
+ En términos generales se pueden hacer las siguientes observaciones sobre
+ cada método en particular.
+\layout Section
+
+Euler.
+\layout Standard
+
+El método de Euler es un método que en la práctica sólo sirve para darse
+ una idea de cual es la forma de la solución a grandes rasgos.
+ Al buscar algo de estabilidad y precisión con este método se ven rápidamente
+ sus falencias y para hacer cálculos levemente precisos es necesario un
+ gran número de iteraciones.
+ A pesar de esto tiene como ventaja ser un método muy simple, característica
+ que facilita el análisis teórico para evaluar su comportamiento y su implementa
+ción computacional.
+\layout Section
+
+Runge-Kutta de orden 4.
+\layout Standard
+
+Este método se comporta de forma muy estable en todas las situaciones y
+ aún con pasos relativamente grandes.
+ A pesar de esto es un método lento (evalúa la función
+\begin_inset Formula \( f_{(t,x)} \)
+\end_inset
+
+ de la ecuación diferencial 4 veces por iteración) esto no se convierte
+ en un problema por la posibilidad de usar un paso grande sin perder precisión
+ ni estabilidad.
+ Como desventaja principal se puede nombrar que es un método complejo, no
+ tanto de implementar pero sí de analizar teóricamente para predecir su
+ comportamiento (como se vio en el punto
+\begin_inset LatexCommand \vref{sec:rk4_conservacion}
+
+\end_inset
+
+).
+ También pudimos comprobar experimentalmente que es un método conservativo,
+ al menos para los rangos de
+\begin_inset Formula \( k \)
+\end_inset
+
+ que estuvimos trabajando.
+\layout Section
+
+Nystrom.
+\layout Standard
+
+El método de Nystrom sorprendió por su incondicional estabilidad y bajo
+ error para un paso
+\begin_inset Formula \( k\cong 2 \)
+\end_inset
+
+ y su gran inestabilidad para pasos que no estén en este rango, ya sea convergie
+ndo rápidamente (dejando de ser conservativo) como divergiendo rápidamente
+ (de manera tan brusca que ni pudo ser graficado).
+ El método no es tan complejo de analizar teóricamente, lo que facilita
+ un poco el predecir su comportamiento y es tal vez el más fácil de implementar
+ ya que no tiene que ser convertido en un sistema de ecuaciones diferenciales
+ (aunque necesita un segundo valor inicial para arrancar, que puede ser
+ un problema si no se conoce en absoluto la ecuación que se está discretizando).
+ El hecho de que el paso deba ser del orden de
+\begin_inset Formula \( k\cong 2 \)
+\end_inset
+
+ puede ser visto tanto como una ventaja como una desventaja, ya que no nos
+ permite analizar lo que pasa en la función en intervalos de tiempo pequeño
+ pero nos deja analizar que pasa en intervalos de tiempo relativamente grandes
+ con pocas iteraciones.
+\layout Chapter
+
+Código Fuente.
+\layout Section
+
+Programa Principal (
+\family typewriter
+77891.cpp
+\family default
+).
+\layout LyX-Code
+
+// vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+\layout LyX-Code
+
+//
+\layout LyX-Code
+
+// Trabajo Práctico II de Análisis Numérico I
+\layout LyX-Code
+
+// Este programa resuelve un sistema de ecuaciones diferenciales
+\layout LyX-Code
+
+// resultante de un problema físico de oscilación de líquidos.
+\layout LyX-Code
+
+// Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+\layout LyX-Code
+
+//
+\layout LyX-Code
+
+// Este programa es Software Libre; usted puede redistribuirlo
+\layout LyX-Code
+
+// y/o modificarlo bajo los términos de la "GNU General Public
+\layout LyX-Code
+
+// License" como lo publica la "FSF Free Software Foundation",
+\layout LyX-Code
+
+// o (a su elección) de cualquier versión posterior.
+\layout LyX-Code
+
+//
+\layout LyX-Code
+
+// Este programa es distribuido con la esperanza de que le será
+\layout LyX-Code
+
+// útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+\layout LyX-Code
+
+// implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+\layout LyX-Code
+
+// particular.
+ Vea la "GNU General Public License" para más
+\layout LyX-Code
+
+// detalles.
+\layout LyX-Code
+
+//
+\layout LyX-Code
+
+// Usted debe haber recibido una copia de la "GNU General Public
+\layout LyX-Code
+
+// License" junto con este programa, si no, escriba a la "FSF
+\layout LyX-Code
+
+// Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+\layout LyX-Code
+
+// Boston, MA 02111-1307, USA.
+\layout LyX-Code
+
+//
+\layout LyX-Code
+
+// $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/informe.lyx $
+\layout LyX-Code
+
+// $Date: 2002-12-05 03:19:47 -0300 (jue, 05 dic 2002) $
+\layout LyX-Code
+
+// $Rev: 36 $
+\layout LyX-Code
+
+// $Author: luca $
+\layout LyX-Code
+
+//
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+#include <iostream.h>
+\layout LyX-Code
+
+#include <stdlib.h>
+\layout LyX-Code
+
+#include <math.h>
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Tipos de datos.
+\layout LyX-Code
+
+typedef unsigned int Indice;
+\layout LyX-Code
+
+typedef float Numero;
+\layout LyX-Code
+
+struct Datos;
+\layout LyX-Code
+
+typedef Numero (*Friccion)( Datos&, Numero, Numero );
+\layout LyX-Code
+
+typedef Numero (*Funcion)( Datos&, Numero, Numero, Numero );
+\layout LyX-Code
+
+typedef void (*Metodo)( Datos& );
+\layout LyX-Code
+
+struct Datos {
+\layout LyX-Code
+
+ Numero to; // Tiempo inicial
+\layout LyX-Code
+
+ Numero tf; // Tiempo final
+\layout LyX-Code
+
+ Numero k; // Paso
+\layout LyX-Code
+
+ Numero xo; // X inicial = X(to) = Z(to)
+\layout LyX-Code
+
+ Numero yo; // Y inicial = Y(to) = X'(to) = Z'(to)
+\layout LyX-Code
+
+ Numero D; // Diametro del tubo
+\layout LyX-Code
+
+ Numero n; // Viscosidad cinemática del líquido
+\layout LyX-Code
+
+ Numero g; // Aceleración de la gravedad
+\layout LyX-Code
+
+ Numero f; // Factor de fricción
+\layout LyX-Code
+
+ Numero L; // Longitud del tubo
+\layout LyX-Code
+
+ Numero Le; // Factor de pérdida de carga
+\layout LyX-Code
+
+ Numero G; // Factor geométrico
+\layout LyX-Code
+
+ Funcion fx; // Función asociada a la derivada de x
+\layout LyX-Code
+
+ Funcion fy; // Función asociada a la derivada de y
+\layout LyX-Code
+
+ Friccion fi; // Factor asociado a pérdidas por fricción
+\layout LyX-Code
+
+};
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Factor de fricción nulo.
+\layout LyX-Code
+
+Numero friccionNula( Datos&, Numero, Numero );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Factor de fricción laminar.
+\layout LyX-Code
+
+Numero friccionLaminar( Datos&, Numero, Numero );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Factor de fricción turbulenta.
+\layout LyX-Code
+
+Numero friccionTurbulenta( Datos&, Numero, Numero );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Factor de fricción turbulenta (con depósitos).
+\layout LyX-Code
+
+Numero friccionTurbulentaConDepositos( Datos&, Numero, Numero );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Función asociada a la primera derivada (x'=z').
+\layout LyX-Code
+
+Numero funcionX( Datos&, Numero, Numero, Numero );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Función asociada a la segunda derivada (y'=x''=z'').
+\layout LyX-Code
+
+Numero funcionY( Datos&, Numero, Numero, Numero );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Calcula la ecuación diferencial por el método de Euler.
+\layout LyX-Code
+
+void euler( Datos& );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Calcula la ecuación diferencial por el método de Runge-Kutta de órden
+ 4.
+\layout LyX-Code
+
+void rk4( Datos& );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Calcula la ecuación diferencial por el método de Nystrom.
+\layout LyX-Code
+
+void nystrom( Datos& );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Constantes.
+\layout LyX-Code
+
+const Numero DEFAULT_N = 1000, // Cantidad de pasos por defecto
+\layout LyX-Code
+
+ DEFAULT_to = 0.0,
+\layout LyX-Code
+
+ DEFAULT_tf = 440.0,
+\layout LyX-Code
+
+ DEFAULT_xo = 2.9866369,
+\layout LyX-Code
+
+ DEFAULT_yo = 0.0,
+\layout LyX-Code
+
+ DEFAULT_D = 0.5,
+\layout LyX-Code
+
+ DEFAULT_n = 1e-6,
+\layout LyX-Code
+
+ DEFAULT_g = 9.8,
+\layout LyX-Code
+
+ DEFAULT_f = 0.03,
+\layout LyX-Code
+
+ DEFAULT_L = 892.0,
+\layout LyX-Code
+
+ DEFAULT_Le = 1070.4,
+\layout LyX-Code
+
+ DEFAULT_G = 2.0;
+\layout LyX-Code
+
+const Funcion DEFAULT_fx = &funcionX,
+\layout LyX-Code
+
+ DEFAULT_fy = &funcionY;
+\layout LyX-Code
+
+const Friccion DEFAULT_fi = &friccionNula;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+int main( int argc, char* argv[] ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Se fija que tenga los argumentos necesarios para correr.
+\layout LyX-Code
+
+ if ( argc < 2 ) {
+\layout LyX-Code
+
+ cerr << "Faltan argumentos.
+ Modo de uso:" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+t" << argv[0] << " método fi G tf N to xo yo D n g f L Le" << endl;
+\layout LyX-Code
+
+ cerr << "Desde fi en adelante son opcionales.
+ Los valores por defecto son:" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tfi = n" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tG = " << DEFAULT_G << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+ttf = " << DEFAULT_tf << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tN = " << DEFAULT_N << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tto = " << DEFAULT_to << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+txo = " << DEFAULT_xo << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tyo = " << DEFAULT_yo << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tD = " << DEFAULT_D << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tn = " << DEFAULT_n << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tg = " << DEFAULT_g << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tf = " << DEFAULT_f << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tL = " << DEFAULT_L << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tLe = " << DEFAULT_Le << endl;
+\layout LyX-Code
+
+ return EXIT_FAILURE;
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Selecciona el método deseado.
+\layout LyX-Code
+
+ Metodo metodo = NULL;
+\layout LyX-Code
+
+ switch ( char( argv[1][0] ) ) {
+\layout LyX-Code
+
+ case 'e':
+\layout LyX-Code
+
+ metodo = &euler;
+\layout LyX-Code
+
+ break;
+\layout LyX-Code
+
+ case 'r':
+\layout LyX-Code
+
+ metodo = &rk4;
+\layout LyX-Code
+
+ break;
+\layout LyX-Code
+
+ case 'n':
+\layout LyX-Code
+
+ metodo = &nystrom;
+\layout LyX-Code
+
+ break;
+\layout LyX-Code
+
+ default:
+\layout LyX-Code
+
+ cerr << "Debe especificar un método válido:" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+te: Euler" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tr: Runge-Kutta 4" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tn: Nystrom" << endl;
+\layout LyX-Code
+
+ return EXIT_FAILURE;
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Se inicializan los datos.
+\layout LyX-Code
+
+ Numero N = DEFAULT_N;
+\layout LyX-Code
+
+ Datos D;
+\layout LyX-Code
+
+ D.to = DEFAULT_to;
+\layout LyX-Code
+
+ D.tf = DEFAULT_tf;
+\layout LyX-Code
+
+ D.k = ( D.tf - D.to ) / N;
+\layout LyX-Code
+
+ D.xo = DEFAULT_xo;
+\layout LyX-Code
+
+ D.yo = DEFAULT_yo;
+\layout LyX-Code
+
+ D.D = DEFAULT_D;
+\layout LyX-Code
+
+ D.n = DEFAULT_n;
+\layout LyX-Code
+
+ D.g = DEFAULT_g;
+\layout LyX-Code
+
+ D.f = DEFAULT_f;
+\layout LyX-Code
+
+ D.L = DEFAULT_L;
+\layout LyX-Code
+
+ D.Le = DEFAULT_Le;
+\layout LyX-Code
+
+ D.G = DEFAULT_G;
+\layout LyX-Code
+
+ D.fx = DEFAULT_fx;
+\layout LyX-Code
+
+ D.fy = DEFAULT_fy;
+\layout LyX-Code
+
+ D.fi = DEFAULT_fi;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Si se pasaron datos como argumento, se los va agregando.
+\layout LyX-Code
+
+ switch ( argc ) {
+\layout LyX-Code
+
+ case 15: D.Le = Numero( atof( argv[14] ) );
+\layout LyX-Code
+
+ case 14: D.L = Numero( atof( argv[13] ) );
+\layout LyX-Code
+
+ case 13: D.f = Numero( atof( argv[12] ) );
+\layout LyX-Code
+
+ case 12: D.g = Numero( atof( argv[11] ) );
+\layout LyX-Code
+
+ case 11: D.n = Numero( atof( argv[10] ) );
+\layout LyX-Code
+
+ case 10: D.D = Numero( atof( argv[9] ) );
+\layout LyX-Code
+
+ case 9: D.yo = Numero( atof( argv[8] ) );
+\layout LyX-Code
+
+ case 8: D.xo = Numero( atof( argv[7] ) );
+\layout LyX-Code
+
+ case 7: D.to = Numero( atof( argv[6] ) );
+\layout LyX-Code
+
+ case 6: N = Numero( atof( argv[5] ) );
+\layout LyX-Code
+
+ case 5: D.tf = Numero( atof( argv[4] ) );
+\layout LyX-Code
+
+ // Se recalcula el paso (k) si se cambio to, N o tf.
+\layout LyX-Code
+
+ D.k = ( D.tf - D.to ) / N;
+\layout LyX-Code
+
+ case 4: D.G = Numero( atof( argv[3] ) );
+\layout LyX-Code
+
+ case 3: switch ( char( argv[2][0] ) ) { // Tipo de fricción
+\layout LyX-Code
+
+ case 'n':
+\layout LyX-Code
+
+ D.fi = &friccionNula;
+\layout LyX-Code
+
+ break;
+\layout LyX-Code
+
+ case 'l':
+\layout LyX-Code
+
+ D.fi = &friccionLaminar;
+\layout LyX-Code
+
+ break;
+\layout LyX-Code
+
+ case 't':
+\layout LyX-Code
+
+ D.fi = &friccionTurbulenta;
+\layout LyX-Code
+
+ break;
+\layout LyX-Code
+
+ case 'd':
+\layout LyX-Code
+
+ D.fi = &friccionTurbulentaConDepositos;
+\layout LyX-Code
+
+ break;
+\layout LyX-Code
+
+ default:
+\layout LyX-Code
+
+ cerr << "Debe especificar un tipo de fricción válido:"
+ << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tn: Nula" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tl: Laminar" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+tt: Turbulenta" << endl;
+\layout LyX-Code
+
+ cerr << "
+\backslash
+td: Turbulenta (con depósitos)" << endl;
+\layout LyX-Code
+
+ return EXIT_FAILURE;
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Imprime el paso utilizado.
+\layout LyX-Code
+
+ cerr << "Paso k = " << D.k << endl;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Ejecuta el método correspondiente con los datos correspondientes.
+\layout LyX-Code
+
+ metodo( D );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ return EXIT_SUCCESS;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+Numero friccionNula( Datos& D, Numero x, Numero y ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ return 0.0;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+Numero friccionLaminar( Datos& D, Numero x, Numero y ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ return 32 * D.n / ( D.D * D.D );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+Numero friccionTurbulenta( Datos& D, Numero x, Numero y ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ return D.f * fabs( y ) / ( 2 * D.D );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+Numero friccionTurbulentaConDepositos( Datos& D, Numero x, Numero y ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ return D.f * fabs( y ) * D.Le / ( 2 * D.D * D.L);
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+Numero funcionX( Datos& D, Numero t, Numero x, Numero y ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ return y;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+Numero funcionY( Datos& D, Numero t, Numero x, Numero y ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ return - D.fi( D, x, y ) * y - D.g * D.G * x / D.L;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+void euler( Datos& D ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ Numero xo = D.xo,
+\layout LyX-Code
+
+ yo = D.yo,
+\layout LyX-Code
+
+ x = 0.0,
+\layout LyX-Code
+
+ y = 0.0,
+\layout LyX-Code
+
+ t = D.to;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ while ( t < D.tf ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Calculo los datos para este punto.
+\layout LyX-Code
+
+ x = xo + D.k * D.fx( D, t, xo, yo );
+\layout LyX-Code
+
+ y = yo + D.k * D.fy( D, t, xo, yo );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Imprimo resultados.
+\layout LyX-Code
+
+ cout << t << " " << x << " " << y << endl;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Reemplazo valores iniciales.
+\layout LyX-Code
+
+ xo = x;
+\layout LyX-Code
+
+ yo = y;
+\layout LyX-Code
+
+ t += D.k;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+void rk4( Datos& D ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ Numero x = D.xo,
+\layout LyX-Code
+
+ y = D.yo,
+\layout LyX-Code
+
+ t = D.to,
+\layout LyX-Code
+
+ qx1 = 0.0,
+\layout LyX-Code
+
+ qx2 = 0.0,
+\layout LyX-Code
+
+ qx3 = 0.0,
+\layout LyX-Code
+
+ qx4 = 0.0,
+\layout LyX-Code
+
+ qy1 = 0.0,
+\layout LyX-Code
+
+ qy2 = 0.0,
+\layout LyX-Code
+
+ qy3 = 0.0,
+\layout LyX-Code
+
+ qy4 = 0.0,
+\layout LyX-Code
+
+ unSexto = 1.0 / 6.0;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Imprimo datos iniciales.
+\layout LyX-Code
+
+ cout << t << " " << x << " " << y << endl;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ while ( t < D.tf ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Calculo los datos para este punto.
+\layout LyX-Code
+
+ qx1 = D.k * D.fx( D, t, x, y );
+\layout LyX-Code
+
+ qy1 = D.k * D.fy( D, t, x, y );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ qx2 = D.k * D.fx( D, t + D.k / 2.0, x + qx1 / 2.0, y + qy1 / 2.0 );
+\layout LyX-Code
+
+ qy2 = D.k * D.fy( D, t + D.k / 2.0, x + qx1 / 2.0, y + qy1 / 2.0 );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ qx3 = D.k * D.fx( D, t + D.k / 2.0, x + qx2 / 2.0, y + qy2 / 2.0 );
+\layout LyX-Code
+
+ qy3 = D.k * D.fy( D, t + D.k / 2.0, x + qx2 / 2.0, y + qy2 / 2.0 );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ qx4 = D.k * D.fx( D, t + D.k, x + qx3, y + qy3 );
+\layout LyX-Code
+
+ qy4 = D.k * D.fy( D, t + D.k, x + qx3, y + qy3 );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ x += unSexto * ( qx1 + 2 * qx2 + 2 * qx3 + qx4 );
+\layout LyX-Code
+
+ y += unSexto * ( qy1 + 2 * qy2 + 2 * qy3 + qy4 );
+\layout LyX-Code
+
+ t += D.k;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Imprimo resultados.
+\layout LyX-Code
+
+ cout << t << " " << x << " " << y << endl;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+void nystrom( Datos& D ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ Numero gGk2_L = D.g * D.G * D.k * D.k / D.L,
+\layout LyX-Code
+
+ xo = D.xo,
+\layout LyX-Code
+
+ x1 = ( 1 - gGk2_L * 0.5 ) * D.xo,
+\layout LyX-Code
+
+ x2 = 0.0,
+\layout LyX-Code
+
+ y = 0.0,
+\layout LyX-Code
+
+ fi = 0.0,
+\layout LyX-Code
+
+ t = D.to;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Imprimo valores iniciales.
+\layout LyX-Code
+
+ cout << t << " " << xo << " " << y << endl;
+\layout LyX-Code
+
+ y = ( x1 - xo ) / D.k;
+\layout LyX-Code
+
+ cout << t << " " << x1 << " " << y << endl;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ while ( t < D.tf ) {
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Calculo los datos para este punto.
+\layout LyX-Code
+
+ fi = D.fi( D, x1, y );
+\layout LyX-Code
+
+ x2 = ( ( fi - 1 ) * xo + ( 2 - gGk2_L ) * x1 ) / ( fi + 1 );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Prepara para próxima iteración
+\layout LyX-Code
+
+ t += D.k;
+\layout LyX-Code
+
+ xo = x1;
+\layout LyX-Code
+
+ x1 = x2;
+\layout LyX-Code
+
+ y = ( x1 - xo ) / D.k;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ // Imprimo resultados.
+\layout LyX-Code
+
+ cout << t << " " << x2 << " " << y << endl;
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+}
+\layout Section
+
+Utilidades.
+\layout Standard
+
+Todas las utilidades listadas aquí son Software Libre; puede redistribuirlas
+ y/o modificarlas bajo los términos de la "GNU General Public License" como
+ lo publica la "FSF Free Software Foundation", o (a su elección) de cualquier
+ versión posterior.
+\layout Subsection
+
+
+\family typewriter
+calcula_periodo
+\layout LyX-Code
+
+#!/usr/bin/php4 -qC
+\layout LyX-Code
+
+<?
+\layout LyX-Code
+
+// Muestra ayuda.
+\layout LyX-Code
+
+if ( $argc < 2 ) {
+\layout LyX-Code
+
+ echo "Modo de uso:
+\backslash
+n";
+\layout LyX-Code
+
+ echo "$argv[0] archivo
+\backslash
+n";
+\layout LyX-Code
+
+ exit;
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Abre archivo y obtiene datos iniciales.
+\layout LyX-Code
+
+$f = fopen( $argv[1], 'r' );
+\layout LyX-Code
+
+$s = fgets( $f, 4096 )
+\layout LyX-Code
+
+ or die( "El archivo está vacío o no se pudo abrir.
+\backslash
+n" );
+\layout LyX-Code
+
+list( $t0, $z0 ) = preg_split( '/
+\backslash
+s/', $s );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Procesa archivo calculando período.
+\layout LyX-Code
+
+$t_ant = 0;
+\layout LyX-Code
+
+$c = 0;
+\layout LyX-Code
+
+$t_sum = 0;
+\layout LyX-Code
+
+$t_max = 0;
+\layout LyX-Code
+
+$tt_max = 0;
+\layout LyX-Code
+
+$t_min = 5000;
+\layout LyX-Code
+
+$tt_min = 0;
+\layout LyX-Code
+
+while ( ( $s = fgets( $f, 4096 ) ) !== false ) {
+\layout LyX-Code
+
+ // Obtiene datos.
+\layout LyX-Code
+
+ list( $t, $z ) = preg_split( '/
+\backslash
+s/', $s );
+\layout LyX-Code
+
+ // Se fija si es un "cero decreciente".
+\layout LyX-Code
+
+ if ( $z0 > 0 and ( $z < 0 or $z == 0 ) ) {
+\layout LyX-Code
+
+ if ( $t_ant ) {
+\layout LyX-Code
+
+ $t_actual = $t - $t_ant;
+\layout LyX-Code
+
+ if ( $t_actual > $t_max ) {
+\layout LyX-Code
+
+ $t_max = $t_actual;
+\layout LyX-Code
+
+ $tt_max = $t;
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+ if ( $t_actual < $t_min ) {
+\layout LyX-Code
+
+ $t_min = $t_actual;
+\layout LyX-Code
+
+ $tt_min = $t;
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+ echo "$t_actual
+\backslash
+n";
+\layout LyX-Code
+
+ $t_sum += $t_actual;
+\layout LyX-Code
+
+ $t_ant = $t;
+\layout LyX-Code
+
+ $c++;
+\layout LyX-Code
+
+ } else {
+\layout LyX-Code
+
+ $t_ant = $t;
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+ }
+\layout LyX-Code
+
+ // Actualiza valores anteriores.
+\layout LyX-Code
+
+ $t0 = $t;
+\layout LyX-Code
+
+ $z0 = $z;
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+fclose( $f );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Imprime período.
+\layout LyX-Code
+
+echo "Períodos promedio: " .
+ $t_sum / $c .
+ " ($c períodos promediados)
+\backslash
+n";
+\layout LyX-Code
+
+echo "Período máximo: $t_max (en t = $tt_max)
+\backslash
+n";
+\layout LyX-Code
+
+echo "Período mínimo: $t_min (en t = $tt_min)
+\backslash
+n";
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+?>
+\layout Subsection
+
+
+\family typewriter
+calcula_maxmin
+\layout LyX-Code
+
+#!/usr/bin/php4 -qC
+\layout LyX-Code
+
+<?
+\layout LyX-Code
+
+// Muestra ayuda.
+\layout LyX-Code
+
+if ( $argc < 3 ) {
+\layout LyX-Code
+
+ echo "Modo de uso:
+\backslash
+n";
+\layout LyX-Code
+
+ echo "$argv[0] modo archivo
+\backslash
+n";
+\layout LyX-Code
+
+ echo "Donde modo es 'max' para calcular los máximos o 'min' para calcular
+ los mínimos.
+\backslash
+n";
+\layout LyX-Code
+
+ exit;
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Abre archivos y obtiene datos iniciales.
+\layout LyX-Code
+
+$f = fopen( $argv[2], 'r' )
+\layout LyX-Code
+
+ or die( "Error al abrir $argv[2] para lectura.
+\backslash
+n" );
+\layout LyX-Code
+
+$s = fgets( $f, 4096 )
+\layout LyX-Code
+
+ or die( "El archivo $argv[2] está vacío.
+\backslash
+n" );
+\layout LyX-Code
+
+list( $t0, $z0, $dz0 ) = preg_split( '/
+\backslash
+s/', $s );
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// El primer valor siempre es un máximo.
+\layout LyX-Code
+
+if ( $argv[1] == 'max' )
+\layout LyX-Code
+
+ echo "$t0 $z0
+\backslash
+n";
+\layout LyX-Code
+
+
+\layout LyX-Code
+
+// Procesa archivo calculando máximos y mínimos.
+\layout LyX-Code
+
+while ( ( $s = fgets( $f, 4096 ) ) !== false ) {
+\layout LyX-Code
+
+ // Obtiene datos.
+\layout LyX-Code
+
+ list( $t, $z, $dz ) = preg_split( '/
+\backslash
+s/', $s );
+\layout LyX-Code
+
+ // Se fija si la derivada es un "cero decreciente" (si es un máximo).
+\layout LyX-Code
+
+ if ( $argv[1] == 'max' and $dz0 > 0 and ( $dz < 0 or $dz == 0 ) )
+\layout LyX-Code
+
+ if ( $z0 > $z )
+\layout LyX-Code
+
+ echo "$t0 $z0
+\backslash
+n";
+\layout LyX-Code
+
+ else
+\layout LyX-Code
+
+ echo "$t $z
+\backslash
+n";
+\layout LyX-Code
+
+ // Se fija si la derivada es un "cero creciente" (si es un mínimo).
+\layout LyX-Code
+
+ if ( $argv[1] == 'min' and $dz0 < 0 and ( $dz > 0 or $dz == 0 ) )
+\layout LyX-Code
+
+ if ( $z0 < $z )
+\layout LyX-Code
+
+ echo "$t0 $z0
+\backslash
+n";
+\layout LyX-Code
+
+ else
+\layout LyX-Code
+
+ echo "$t $z
+\backslash
+n";
+\layout LyX-Code
+
+ // Actualiza valores anteriores.
+\layout LyX-Code
+
+ $t0 = $t;
+\layout LyX-Code
+
+ $z0 = $z;
+\layout LyX-Code
+
+ $dz0 = $dz;
+\layout LyX-Code
+
+}
+\layout LyX-Code
+
+fclose( $f );
+\layout LyX-Code
+
+?>
+\the_end
--- /dev/null
+#!/usr/bin/gnuplot
+# vim: set tabstop=4 softtabstop=4 shiftwidth=4 expandtab:
+#
+# Trabajo Práctico II de Análisis Numérico I
+# Genera gráficos con los resultados de las corridas utilizando
+# GNU Plot.
+# Copyright (C) 2002 Leandro Lucarella <leandro@lucarella.com.ar>
+#
+# Este programa es Software Libre; usted puede redistribuirlo
+# y/o modificarlo bajo los términos de la "GNU General Public
+# License" como lo publica la "FSF Free Software Foundation",
+# o (a su elección) de cualquier versión posterior.
+#
+# Este programa es distribuido con la esperanza de que le será
+# útil, pero SIN NINGUNA GARANTIA; incluso sin la garantía
+# implícita por el MERCADEO o EJERCICIO DE ALGUN PROPOSITO en
+# particular. Vea la "GNU General Public License" para más
+# detalles.
+#
+# Usted debe haber recibido una copia de la "GNU General Public
+# License" junto con este programa, si no, escriba a la "FSF
+# Free Software Foundation, Inc.", 59 Temple Place - Suite 330,
+# Boston, MA 02111-1307, USA.
+#
+# $URL: http://www.llucax.hn.org:81/svn/facultad/75.12/tp2/periodo.gnuplot $
+# $Date: 2002-11-30 03:48:32 -0300 (sáb, 30 nov 2002) $
+# $Rev: 31 $
+# $Author: luca $
+#
+
+# Seteo terminal para que "dibuje" en un PS.
+set term postscript eps enhanced color
+set encoding iso_8859_1
+set output "<P>2.eps"
+
+# Seteos generales.
+set title "Evolución del período para el caso <PUNTO_DEP> depósitos y <PUNTO_FRIC> fricción<PUNTO_FRIC_TIPO>"
+set key right top
+
+# Eje X.
+set xlabel "Tiempo"
+set mxtics 5
+
+# Eje Y.
+set ylabel "Período (s)"
+set ytics nomirror
+set mytics 5
+
+# Plotea
+plot '<P>2e.txt' every ::::110 title "Período con Euler" with lines linetype 1, \
+ '<P>2r.txt' every ::::110 title "Período con RK4" with lines linetype 3, \
+ '<P>2n.txt' every ::::110 title "Período con Nystrom" with lines linetype 7
projects / z.facultad / 75.12 / tp2.git / commitdiff