Taylor's method of order 2 programming

So I am writing a program to solve an IVP problem, I'm trying to write it with not using the function script because I want to plot the error later.

This is my coding:

clear;

h=0.1; t=1:h:2; n=length(t);

%initial condition y=zeros(n,1); y(1)=-0.5;

for i = 1:n-1 y(i+1)=y(i)+h*RHS(t(i),y(i))+(h.^2/2)*RHS3(t(i),y(i)); end

truesol= (sqrt(3)/(2*t))*tan((sqrt(3)/2)*log(t))-1/2*t;

plot(t,y,'.',t,truesol,'*')

[t' y truesol']

And I wrote my RHS(which is my original function) and RHS3(which is my df/dt) separately:

function result = RHS(t,y)

result= y.^2+(1/(t.^2));

end

and

function result= RHS3(t,y)

result = 2*y.^3+((2*y)/(t.^2))-(2/(t.^3));

end

Everything seems to be okay, I even tried a simple "true solution" and it works. It only doesn't work when I tried the correct true solution, which is

truesol= (sqrt(3)/(2*t))*tan((sqrt(3)/2)*log(t))-1/2*t;

it gives me an error of "??? Error using ==> mldivide Matrix dimensions must agree."

Thanks for any input.

Answer 1

[Star Strider]

See if vectorising this line (replacing (/) with (./) and (*) with (.*)) solves the problem:

truesol= (sqrt(3)./(2*t)).*tan((sqrt(3)/2)*log(t))-1/2*t;

Consider vectorising other lines in your code as well. See Array vs. Matrix Operations for details.

[ivordes greenleaf]

Thank you so much, that really does help.

I also want to plot the logarithm of error log|yexact-yapprox| versus the logarithm of step size h, and then find the slope of that straight line through the graph. I wrote a line of code stated

 err(i) = (abs(truesol(i)-y(i)));

with

truesol= (sqrt(3)*tan((sqrt(3)/2)*log(t))-1)./(2*t);

and then can I do a plot(loglog(h,err)), it does give me something, and it's a graph with a line with slope 1. I think this is a little too simple to be right, so what should I do here?

Thank you so much for your help.

[Star Strider]

My pleasure. I’m glad it worked.

I did not run your code, but it would seem to me that a smaller step size would result in a smaller error. You are plotting log(err) against log(h), so I would expect a linear relationship. I believe your plot is likely correct.