Clear Filters
Clear Filters

mldivide versus least squares: X\(eye(m)) versus ( (X'X)\eye(m))*X'

3 views (last 30 days)
Dear all,
I am fitting a polynomial to data. I construct a polynomial basis X. I use some algorithm to update Y. I could use the mldivide to obtain coefficients theta or use . But i don't know which one is more robust/accurate for my applicaiton. The system is normally overdetermined, but it might be exactly determined.
To obtain the coefficients of the polynomials I would normally do:
theta = X\Y;
However, since I have to do this repeatedly and X does not change I want to use:
X_inv = X\eye(m);
%In each iteration:
theta = X_inv*Y;
where m is size(X,1).This should save computation time of the mldivide.
Now my questions is, for the Minimization of Squared Errors sometimes people also use . In that case I should define:
X_regr = ( (X'*X)\eye(m) )*X';
%In each iteration:
theta = X_regr*Y;
Should one method be preferred to the other (when overdetermined or exactly determined)? Or is that another method that is even better?

Accepted Answer

Matt J
Matt J on 6 Nov 2021
Edited: Matt J on 6 Nov 2021
However, since I have to do this repeatedly and X does not change
If that's the case you should organize the different Y into the columns of a single matrix and do
Doing (X'*X)\ is not as numerically well-conditioned as X\, because the operation X'*X basically squares the condition number of X. Nevertheless, if X has many rows and few columns, X'*X\ will often run faster, and sometimes people will give priority to speed, especially if the cond(X) is known to be good.
Matt J
Matt J on 7 Nov 2021
It depends in part on how many Y_i you have. If n is less then m, then the computation of X\eye(m) alone requires more inversions than X\[Y1,Y2,Y3,...,Yn].
Elapsed time is 0.596263 seconds.
Elapsed time is 0.175316 seconds.
Sargondjani on 9 Nov 2021
Wow, this helps a lot! I didn't expect this result, because I didn't completely understand. But it makes sense now, and it speeds up my code quite a bit!
Again, many thanks!

Sign in to comment.

More Answers (0)


Find more on Operating on Diagonal Matrices in Help Center and File Exchange

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!