Jason Yalim. Here we perform a few tasks to ensure a clean working environment for our script.

Transcription

1 Table of Contents Initialization... 1 Part One -- Computation... 1 Part Two -- Plotting... 5 Part Three -- Plotting Incorrectly... 7 Part Four -- Showing Output Adjacent to Code... 8 Part Five -- Creating a Table in Output... 9 Part Six -- Calling and Publishing External Functions MAT 275 Lab with Lopez and Kim Fridays, 10:30-3:50 Publishing Example Initialization Here we perform a few tasks to ensure a clean working environment for our script. clear all format compact close all Part One -- Computation This clears previous variable definitions from the workspace, preventing the use of old variables. This makes our output, if we have any, not take up as much space. This closes any old figure windows that might have been open, ensuring that the only plots/figures visible after running the script are freshly made. We will modeling the Mandelbrot fractal. The governing linear map is where is a complex initial condition. Note that. N = 60; Number of Iterations M = 3000; Size of 1-D Domain numsnp = 6; Number of Snap Shots x = linspace(-2,1,m); y = linspace(-1.5,1.5,m); [X,Y] = meshgrid( x, y ); Z0 = complex( X, Y ); count = ones( size( Z0 ) ); Z = Z0; for k = 1:N 1

2 Z = Z.^2 + Z0; c = abs(z) <= 2.; count = count + c; if ( mod(k,floor(n/numsnp)) == 0 ) Takes numsnp snapshots. colormap( [ jet() ;flipud( jet() ); ] ) imagesc(x,y,log(count)); title(strcat('iteration ', num2str(k))) snapnow; Note outer contours. count = log(count); 2

3 3

4 4

5 Part Two -- Plotting Here we plot the Mandelbrot fractal with three different methods. Note that pcolor and surf have edgecolors that are black by default, which must be changed to see details. Notice that we have specified figures for each plot. This is done so each plot actually shows up with publish. Without it, only the last plot will show. This will further be demonstrated in part 3. figure colormap( [ jet() ;flipud( jet() ); ] ) imagesc(x,y,count); title('imagesc Plot') xlabel 'x', ylabel 'y' figure colormap( [ jet() ;flipud( jet() ); ] ) h2 = pcolor(x,y,count); set( h2, 'edgecolor', 'none' ) title('pseudo Color Plot') xlabel 'x', ylabel 'y' figure colormap( [ jet() ;flipud( jet() ); ] ) h3 = surf(x,y,count); set( h3, 'edgecolor', 'none' ) title('surface Plot') xlabel 'x', ylabel 'y', zlabel 'Convergence Value' 5

6 6

7 Note how the size of the figures in the html publish are depent on the size of the figure window when publishing occurs. Their sizes can also be affected by the publishing options. Part Three -- Plotting Incorrectly Here we try to plot three graphs, but only up with one of them. This demonstrates that we need separate figure windows for plots within sections to have them successfully appear in our published output. We will attempt to plot identically as the last section, but without the figure specifications. colormap( [ jet() ;flipud( jet() ); ] ); h = imagesc(x,y,count); title('imagesc Plot') xlabel 'x', ylabel 'y' h2 = pcolor(x,y,count); set( h2, 'edgecolor', 'none' ) title('psuedo Color Plot') xlabel 'x', ylabel 'y' h3 = surf(x,y,count); set( h3, 'edgecolor', 'none' ) title('surface Plot') xlabel 'x', ylabel 'y', zlabel 'Convergence Value' 7

8 Thus MATLAB publish will only include one plot per section unless the figure command it used to create a new figure for each new plot. Part Four -- Showing Output Adjacent to Code Here we demonstrate how to include output next to its respective calling operation: A = rand(3); B = magic(3); display(a) display(b) C = A*B D = B*A A = e e e e e e e e e-01 B = C = e e e e e e e e e+00 8

9 E = A*B-B*A D = e e e e e e e e e+00 E = e e e e e e e e e+00 Part Five -- Creating a Table in Output Here we follow the style of exercises 1 and 3 from lab 3 using Runge Kutta 4 instead of Runge Kutta 1 or 2 (Euler's and Improved Euler's). Thus we will be approximating the differential equation, which has the exact solution, We will approximating with 2, 8, 32, 128, and 512 points, computing the absolute error of our approximation at $ t =.5 $, and we will compute a ratio between our errors ( which we expect to be around 4^4 = 256 ). N = (2.^(1:2:10))'; Note that this is a Column Vector M = length(n); tspan = [ 0,.5 ]; x_sol = 3 * exp( 2 * tspan() ); f 2*x; abs_err = zeros(m,1); approxx = zeros(m,1); err_rto = zeros(m,1); cell_array = { 'N', 'x(.5)', 'Error', 'Ratio' }; table_cell_array = cell( M+1,4 ); for k = 1:length(cell_array) table_cell_array{1,k} = cell_array{k}; Sets up Table Header for k = 1:M [ tt, xx ] = RK4( f, tspan, 3, N(k) ); approxx(k) = xx(); abs_err(k) = abs( xx() - x_sol ); table_cell_array{k+1,1} = N(k); Sets up First Column table_cell_array{k+1,2} = approxx(k); et c. table_cell_array{k+1,3} = abs_err(k); et c. Now we must compute the error ratio. Since comparing e5 to e5 is nonsensical, we first set its value to N/A, for not applicable. 9

10 table_cell_array{2,4} = 'N/A'; Then we compute the others as the absolute error of the previous. Since we are working with RK4, we expect these ratios to be of the order 4^4 = 256, as each level has a timestep that is a factor of four smaller, and the RK4 method is fourth order accurate, O(h^4). for k = 2:M err_rto(k) = abs_err(k-1)/abs_err(k); table_cell_array{k+1,4} = err_rto(k); This is a sufficient example of a table using a matrix format shorte table_matrix = [ N, approxx, abs_err, err_rto ]; fprintf('s\n','results for Runge Kutta 4 Displayed with Matrix') fprintf('12s 12s 12s 12s\n','N','x(.5)','Error', 'Ratio') disp(table_matrix) Results for Runge Kutta 4 Displayed with Matrix N x(.5) Error Ratio e e e e e e e e e e e e e e e e e e e+02 This is a sufficient example of a table using a cell array format shorte Results_for_Runge_Kutta_4_Table=table_cell_array Results_for_Runge_Kutta_4_Table = 'N' 'x(.5)' 'Error' 'Ratio' [ 2] [8.1250e+00] [2.9845e-02] 'N/A' [ 8] [8.1548e+00] [2.5132e-05] [1.1875e+03] [ 32] [8.1548e+00] [7.1634e-08] [3.5084e+02] [128] [8.1548e+00] [2.5952e-10] [2.7603e+02] [512] [8.1548e+00] [1.0019e-12] [2.5903e+02] This is an overly sufficient example of a table fprintf('\n') Space out example output tm = size(table_cell_array,1); tn = size(table_cell_array,2); for j = 1:tM if ( j == 1 ) fprintf('\n') for i = 1:(tN * 14) fprintf('-') fprintf('\n38s\n','results for Runge Kutta 4') for i = 1:(tN * 14) fprintf('-') 10

11 fprintf('\n') for k = 1:tN token = table_cell_array{j,k}; if ( isstr(token) ) fprintf('11s',token) else fprintf('11g',token) if ( k < tn ) fprintf(' ') if ( j == 1 ) fprintf('\n') for i = 1:(tN * 14) fprintf('-') fprintf('\n') for i = 1:(tN * 14) fprintf('-') fprintf('\n') Results for Runge Kutta N x(.5) Error Ratio N/A e e e e And thus we see that our method is indeed fourth order accurate, as when we decrease the timestep by a factor of four, our error ratio is of the order of 256. Now we include the external function file that we used, RK4.m The next section, Part Six, focuses on calling and publishing external functions. See RK4.m below: type('rk4.m') RK4.m Computes a RK4 on f over time domain tspan with initial condition y0 with N timesteps. Note that f = f(t,y). 11

12 INPUTS: f -- f = f(t,y) = y'(t) tspan -- two vector with initial time and final time as elements. y0 -- Initial condtion vector. N -- Number of points in time domain OUTPUTS: t -- time, returns time domain y -- solution vector/matrix over time domain. Is a matrix if y0 was a vector. function [t,y] = RK4( f, tspan, y0, N ) t = linspace(tspan(1), tspan(), N); h = t(2) - t(1); y = zeros( length(y0), N ); y(:,1) = y0; for k = 1:N-1 tk = t(k); yt = y(:,k); K1 = h * f( tk, yt ); K2 = h * f( tk +.5 * h, yt +.5 * K1 ); K3 = h * f( tk +.5 * h, yt +.5 * K2 ); K4 = h * f( tk + 1. * h, yt + K3 ); y(:,k+1) = y(:,k) + 1/6 * ( K1 + 2 * ( K2 + K3 ) + K4 ); Returning column vector and corresponding column solns. t = t'; y = y'; Part Six -- Calling and Publishing External Functions Here we demonstrate how to call an external function and how to include its respective code within the published output. Notice how the command type is under the function call. This produces the output of the function before printing its respective code. Finally, notice how the command type does not style the code. This means that any LaTeX or comments will not be processed by MATLAB's publishing engine. t = linspace(0,20,5000); function_example_stable_focus( t ); 12

13 function_example_stable_focus.m type( 'function_example_stable_focus.m' ) function_example_stable_focus.m INPUTS: t -- time lin space vector that specifies the existence of the dynamical solution. OUTPUTS: A figure over the time range for the parametric equation: $$ e^{-t} \langle \cos(t), \sin(t), e^t (t_{max} - t) \rangle $$ function [ z ] = function_example_stable_focus( t ) d = exp(-t); x = d.* cos(10*pi*t); y = d.* sin(10*pi*t); z = t(:-1:1); figure(1) plot3( x, y, z, 'b-' ) xlabel('x') ylabel('y') zlabel('-t') title('stable Focus') grid on 13

14 Published with MATLAB R2013b 14