[x,s] = wsolve(A,y,dy) Solve a potentially over-determined system with uncertainty in the values. A x = y +/- dy Use QR decomposition for increased accuracy. Estimate the uncertainty for the solution from the scatter in the data. The returned structure s contains normr = sqrt( A x - y ), weighted by dy R such that R'R = A'A df = n-p, n = rows of A, p = columns of A See polyconf for details on how to use s to compute dy. The covariance matrix is inv(R'*R). If you know that the parameters are independent, then uncertainty is given by the diagonal of the covariance matrix, or dx = sqrt(N*sumsq(inv(s.R'))') where N = normr^2/df, or N = 1 if df = 0. Example 1: weighted system A=[1,2,3;2,1,3;1,1,1]; xin=[1;2;3]; dy=[0.2;0.01;0.1]; y=A*xin+randn(size(dy)).*dy; [x,s] = wsolve(A,y,dy); dx = sqrt(sumsq(inv(s.R'))'); res = [xin, x, dx] Example 2: weighted overdetermined system y = x1 + 2*x2 + 3*x3 + e A = fullfact([3,3,3]); xin=[1;2;3]; y = A*xin; dy = rand(size(y))/50; y+=dy.*randn(size(y)); [x,s] = wsolve(A,y,dy); dx = s.normr*sqrt(sumsq(inv(s.R'))'/s.df); res = [xin, x, dx] Note there is a counter-intuitive result that scaling the uncertainty in the data does not affect the uncertainty in the fit. Indeed, if you perform a monte carlo simulation with x,y datasets selected from a normal distribution centered on y with width 10*dy instead of dy you will see that the variance in the parameters indeed increases by a factor of 100. However, if the error bars really do increase by a factor of 10 you should expect a corresponding increase in the scatter of the data, which will increase the variance computed by the fit.
Package: optim