2.13 Function wsolve, another linear solver

Helptext:

[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.