Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a round-off error from doing it manually. Thanks.
I'm learning multivariable root finding, starting with some simple sets of nonlinear equations in several variables -- solving for F = 0. I'm making up easy equations in which I know some obvious multivariable roots, and I am choosing starting points just a bit away from the roots, in order to get solutions from Matlab's fsolve algorithm. At the moment, I'm trying to learn how to interpret the algorithm's feedback, since my results differ from Matlab's, which is here:
The sum of squared function values, r = 6.701089e-19, is less than sqrt(options.FunctionTolerance) = 1.000000e-03. The relative norm of the gradient of r, 7.127651e-09, is less than options.OptimalityTolerance = 1.000000e-06.
relative norm(grad r) = 7.13e-09
r = 6.70e-19
OptimalityTolerance = 1e-06 (default)
sqrt(FunctionTolerance) = 1.0e-03 (default)
When I manually compute the sum of squared function values in the Command Window, I am getting something that's roughly 1e-7, which is nothing close to their r-value of 6.7e-19. What is this value r? Matlab calls it the "sum of squared function values", too. They also give a "relative norm" of grad r. I know of the usual Euclidean norm, p-norms, the infinity norm, and the Frobenius norm, but I've never heard of a relative norm. So, I suspect that Matlab's definitions are proprietary and not standard -- or, manual computations in the Command Window might have round-off errors.