Plotting Error

The following is my code, everything works how I want it to but I need to plot the error on another plot, how do I do this?

%%Analytical
simplify(dsolve('Dy=-x/y','y(0)=5','x'));
%%Numerical
f=@(x) (-x^2+25)^(1/2); 
dydx=@(x,y) -(x/y); 
[x1,y1]=eulode(dydx, [0 5],5,.5);
[x2,y2]=eulode(dydx,[0 5],5,.1); 
[x3,y3]=eulode(dydx,[0 5],5,.01); 
disp([x1,y1])
disp([x2,y2]) 
disp([x3,y3])
%%Percent Error 
x1=0:.5:5; 
x2=0:.1:5;
x3=0:.01:5; 
analytical_step1= (-x1.^2+25).^(1/2);
analytical_step2=(-x2.^2+25).^(1/2);
analytical_step3=(-x3.^2+25).^(1/2);
numerical_1=[y1]'; 
numerical_2=[y2]';
numerical_3=[y3]';
Percent_Error1=abs((analytical_step1-numerical_1)/analytical_step1)*100%answer displayed in percent 
Percent_Error2=abs((analytical_step2-numerical_2)/analytical_step2)*100%answer displayed in percent 
Percent_Error3=abs((analytical_step3-numerical_3)/analytical_step3)*100%answer displayed in percent
%%Plot 
plot(x1,y1,'k',x2,y2,'b',x3,y3,'g','linewidth',2)
hold on 
fplot(f,[0 5],'Linewidth',2,'r') 
legend('STEP 0.5','STEP 0.1','STEP 0.01','Analytical','Location','W') 
grid
title('Plot of Analytical Function and Numerical Approximations')

Answer 1

[Matt Tearle]

Use figure to create another plot window

Answer 2

[John]

I have tried using figure(2)

however, I do not know which array to call my error and how to plot it I am a little confused.

[Matt Tearle]

Sorry, didn't understand your question, I guess. So what are you trying to plot? Do you want to see how the error evolves as a function of x? (Or y?) Or are you trying to see the order of the numerical method? In that case you want to plot a single value for the error as a function of stepsize? That's pretty common, but in your case you only have three stepsizes, so the resulting graph will be less than visually riveting.

[Matt Tearle]

Ahhhh.... just noticed something: you're using / instead of ./ in the calculation of the error. That returns a scalar from the vectors analytical_step and numerical, but it's probably not what you want.

[John]

I am wanting to plot the error as a function of the step size, in other words a plot of the error at each point as the function evolves. I think I can just put a ./ in my percent error calculation and plot those values correct?

Answer 3

[John]

my new code looks like this (I needed a scaler value for error so I left it in my code.

   %%Analytical
      simplify(dsolve('Dy=-x/y','y(0)=5','x'))
      %%Numerical
      f=@(x) (-x^2+25)^(1/2)
      dydx=@(x,y) -(x/y);
      [x1,y1]=eulode(dydx, [0 5],5,.5);
      [x2,y2]=eulode(dydx,[0 5],5,.1);
      [x3,y3]=eulode(dydx,[0 5],5,.01);
      disp([x1,y1])
      disp([x2,y2])
      disp([x3,y3])
      %%Percent Error
      x1=0:.5:5;
      x2=0:.1:5;
      x3=0:.01:5;
      analytical_step1= (-x1.^2+25).^(1/2);
      analytical_step2=(-x2.^2+25).^(1/2);
      analytical_step3=(-x3.^2+25).^(1/2);
      numerical_1=[y1]';
      numerical_2=[y2]';    
      numerical_3=[y3]';
      Percent_Error1=abs((analytical_step1-numerical_1)/analytical_step1)*100%answer displayed in percent
      PE1=abs((analytical_step1-numerical_1)./analytical_step1)*100;
      Percent_Error2=abs((analytical_step2-numerical_2)/analytical_step2)*100%answer displayed in percent
      PE2=abs((analytical_step2-numerical_2)./analytical_step2)*100;
      Percent_Error3=abs((analytical_step3-numerical_3)/analytical_step3)*100%answer displayed in percent
      PE3=abs((analytical_step3-numerical_3)./analytical_step3)*100;
      %%Plot
      plot(x1,y1,'k',x2,y2,'b',x3,y3,'g','linewidth',2)
      hold on
      fplot(f,[0 5],'Linewidth',2,'r')
      legend('STEP 0.5','STEP 0.1','STEP 0.01','Analytical','Location','W')
      grid
      figure
      plot(x1,PE1,x2,PE2,x3,PE3,'linewidth',2)
      grid
      legend('STEP 0.5','STEP 0.1','STEP 0.01','NW')
    Title('Percent Error')
    ylabel('Percent')

[Walter Roberson]

If you are expecting a scalar for the PE* variables then plotting versus step size is just going to be three single points, such as would be plotted as

scatter([.01 .1 .5], [PE3, PE2, PE1], 'c*');

[John]

No I am not wanting the scaler values of the percent error I want to plot it as the function progresses, is the correct the way I have it coded?

[Matt Tearle]

In that case, yes, your code is working as you want, as far as that goes. Some minor things: "title", not "Title"; you're missing a 'location' keyword in your legend command; and you might want to consider using semilogy rather than plot.

The bigger issue is still your calculation of Percent_ErrorX. Are you sure this is doing what you want? In MATLAB, x = A/B is interpreted as "solve the system of equations x*B = A (for x)". Your A & B are the row vectors analytical_step1 and (analytical_step1-numerical_1). xB = A is equivalent to B'x = A'. This is the standard setup for a regression, with A' as the response variable and B' as the matrix of predictors. Hence, you're essentially doing a regression of the form y = kx where x is the analytic solution, and y is the error. However, if you plot these vectors against each other, you'll see that they are not related by that equation. (You could possibly get a linear regression, but you'd need a constant term.)

Bottom line: I don't think (true-numeric)/true is what you want. Generally, numerical error is reduced to a single value by taking some simple vector-norm aggregate. ie Percent_Error1 = norm(PE1,1)

or norm(PE1,2) or norm(PE1,Inf).

But....! This will cause problems because your last error is Inf, due to the analytic solution being 0. So you'll need to figure out how you want to deal with that. norm(PE1(isfinite(PE1)),Inf) would be one way.