Polyfit error warning for SDE

I'm getting this Warning everytime I use polyfit, and I can't seem to find the error behind my code. Here's the warning:

Warning: Polynomial is not unique; degree >= number of data points.

I'm trying to find the slope of 4 points from using the Euler-Maruyama method applied to my SDE. Here's the code:

% Euler-Maruyama method on linear SDE
% SDE is dX = -cos^3(X)*sin(X)*dt + cos^2(X)*dW
% X(0) = Xzero, Xzero = 0
% For 0 <= t <= 1
% Time steps: 1/50, 1/100, 1/150, 1/200
% Exact Solution: X(t) = tan^-1(W(t))
% K = 50, 100, 150, 200; % number of steps
% T = 1; % end of interval
% tau = T/K; % time step

% A for loop for the 4 timesteps (1/50, 1/100, 1/150, 1/200)
for f = 1:4
    K = 50*f; % So we can get 50, 100, 150, 200
    T = 1; 
    tau = T/K;
    N_s = 2000; % number of trajectories
    W = zeros(N_s,K+1);  % setting up discrete Brownian path
    dW = zeros(N_s,K);  % setting up Brownian increments
    X_app = zeros(N_s,K+1); % setting up Approx solution
    X_exact = zeros(N_s,K+1); % setting up Exact solution
    Xerr = zeros(1,K+1); % setting up Global error
% Compute 2000 independent Brownian Motion trajectories
for q = 1:N_s 
    for p = 1:K 
        W(q,p+1) = W(q,p) + sqrt(tau)*normrnd(0,1);
    end
end
% Use these Brownian Motion trajectories to define the increments
for c = 1:N_s
    for n = 1:K
        dW(c,n) = W(c,n+1) - W(c,n);
    end
end
% Generate 2000 solution trajectories using the EM update & these increments
for j = 1:N_s
    for k = 1:K
        Xzero = 0;
        X_app(j,1) = Xzero;
        X_app(j,k+1) = X_app(j,k) + (-cos(X_app(j,k))^3)*(sin(X_app(j,k)))*tau + (cos(X_app(j,k))^2)*dW(j,k);
    end
end
% Estimate the global error of tau for the timestep using the exact solution
for tk = 1:K+1
    for g = 1:N_s
        X_exact(g,tk) = atan(W(g,tk));
        Xerr(g,tk) = abs(X_exact(g,tk) - X_app(g,tk));
    end
    Err(tk) = (1/N_s)*sum(Xerr(:,tk));
end
GErr = max(Err)
% Make a log-log plot of the data through a scatter plot
scatter(log(tau),log(GErr),'filled'), hold on
end
hold off
% Finding the slop of the line is the strong order of convergence
di = polyfit(log(tau),log(GErr),1);
slope = di(1)

I don't understand why I'm getting the error. It should be a 1 term polynomial (y = mx+b), and I only need the slope for my strong order of convergence. I get an answer with the warning, but I don't want to feel contempt with my answer when the warning is happening.

Answer 1

[Jan]

tau and GErr are scalars, so you have one single point for the polynomial only. Then you cannot determine a slope.

I guess you want to create vectors instead. Then replace all tau and GErr to tau(f) and GErr(f):

...
    tau(f) = T/K;
...
     W(q,p+1) = W(q,p) + sqrt(tau(f))*normrnd(0,1);
 ...
     GErr(f) = max(Err);
 ...

Then this will be working:

     di = polyfit(log(tau),log(GErr),1);

By the way, this is not useful:

    for k = 1:K
        Xzero = 0;           % ??? Set this once outside the loops
        X_app(j,1) = Xzero;  % ??? Why setting this in each iteration again
        X_app(j,k+1) = X_app(j,k) + (-cos(X_app(j,k))^3)*(sin(X_app(j,k)))*tau + (cos(X_app(j,k))^2)*dW(j,k);
    end

This:

for c = 1:N_s
    for n = 1:K
        dW(c,n) = W(c,n+1) - W(c,n);
    end
end

can be simpified to:

dW = diff(W, 1, 2)

and

for tk = 1:K+1
    for g = 1:N_s
        X_exact(g,tk) = atan(W(g,tk));
        Xerr(g,tk) = abs(X_exact(g,tk) - X_app(g,tk));
    end
    Err(tk) = (1/N_s)*sum(Xerr(:,tk));
end

to

 X_exact = atan(W);
 Xerr    = abs(X_exact - X_app);
 Err     = sum(Xerr, 1) / N_s;

[Mel Hernandez]

Thank you so much! I completely forgot about tau(f) and GErr(f). It fixed it! And you didn't have to clean up my code. I do appreciate it though. Realized I did have a few too many lines going on. Thank you again!