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12.010 Computational Methods of Scientific Programming. Lecturers Thomas A Herring, Room 54-618, tah@mit.edu Chris Hill, Room 54-1511, cnh@gulf.mit.edu Web page http://www-gpsg.mit.edu/~tah/12.010. Overview Today. Review some aspects of Homework 4 (Mathematica).
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12.010 Computational Methods of Scientific Programming Lecturers Thomas A Herring, Room 54-618, tah@mit.edu Chris Hill, Room 54-1511, cnh@gulf.mit.edu Web page http://www-gpsg.mit.edu/~tah/12.010
Overview Today • Review some aspects of Homework 4 (Mathematica). • Look at formatting of question 1 • Look at FindRoot and NDSolve for Question 3. • Final Project: Examine Matlab N-body project to show: • methods Animation • Solution using odeXX differential equation solver in Matlab • End by looking at some graphics packages 12.010 Lec 20
Mathematica Homework • Question 1: Formatting output was tricky:The notebook 12.010_Lec20_Mathematica.nb gives examples for some solutions • Question 2: OK, features from 5.2 Mathematica used • Question 3: NDSolve was the way to solve this problem along with FindRoot 12.010 Lec 20
Final Project: • Example case for N-body problem: • N-body planetary problem that use Runge-Kutta numerical integration with variable step size. • Demonstrate basic use of this program (tar file on web site) • Graphics usage in gui-form: Basic methods can be used in your projects. 12.010 Lec 20
Graphics real-time output in Matlab • In Matlab: we can consider “real time” out put the results • When a plot is made you can set an “EraseMode” • Options are: • none — leaves all points on the screen • background — paints old points with background color. Erases old points but also erases any other information such as grid lines and text • xor — Exclusive or. Erases just the previously plotted points, leaves the background intact 12.010 Lec 20
Generating animated sequence with Matlab • Basic mode of use: • Plot first point keeping the graphics handle • p = plot3(x,y,z,’.’, ‘EraseMode’,’xor’); • Set the axis limits: (Need to think of values to use here. • axis([-100 100 -100 100 -100 100]); • hold on • Now generate the sequence of points • Loop over time steps, compute new x y z • set(p,’Xdata’,x,’Ydata’,y,’Zdata’,z) • Drawnow • These ideas were demonstrated in Matlab M-file 12.010 Lec 20
ODE Solvers • Use of ODE Solvers in Matlab (demonstrated in class) • Vector y is 2-d position and velocity (1:4). y0 = [0.0; 0.0; vx; vz]; [t,y,te,ye,ie] = ode23(@bacc,[0:1:tmax],y0,options); • The bacc routine computes accelerations. dy/dt is returned so that dy[1]=d(pos)/dt=y[3]; dy[2]=y[4]; and dy[3] and dy[4] are new accelerations function dy = bacc(t, y) % acc: Computes accelerations • Options sets ability to detect event such as hitting ground options = odeset('AbsTol',[terr 1 1 1],'Events','hit'); function [value,isterminal,direction] = hit(t,y) Value returns the height. • Look through Matlab help and use demo program 12.010 Lec 20
Graphics Programs • Basic data plotting programs: • There a number of programs which fall into this basic category: • KaleidaGraph — commercial program runs on PC and Mac (~$150) (web http://www.synergy.com) • Tecplot — Available on Athena (Unix and PC) ($1300) (web http://www.amtec.com/) • Grace -- Similar to KaleidaGraph available free for Unix systems. The old version is called xmgr. (http://plasma-gate.weizmann.ac.il/Grace/ • Demo of basic features of these types of programs 12.010 Lec 20
Summary • Looked at Mathematica homework solution • Example of 3-D Numerical integration Project • Graphics packages • Matlab homework due Thursday Dec 1. 12.010 Lec 20
