relax SVN archive

Implemented back end for estimating r2eff and errors by exponential curve fitting with scipy.optimize.leastsq.

task #7822(https://gna.org/task/index.php?7822): Implement user function to estimate R2eff and associated errors for exponential curve fitting.

Modified check for model, to accept model as input, for error printing.

task #7822(https://gna.org/task/index.php?7822): Implement user function to estimate R2eff and associated errors for exponential curve fitting.

Added front end user function relax_disp.r2eff_estimate() to estimate R2eff and errors by exponential curve fitting in scipy.optimize.leastsq.

task #7822(https://gna.org/task/index.php?7822): Implement user function to estimate R2eff and associated errors for exponential curve fitting.

Implemented initial systemtest Relax_disp.test_estimate_r2eff for setting up the new user function to estimate R2eff and errors by scipy.

task #7822(https://gna.org/task/index.php?7822): Implement user function to estimate R2eff and associated errors for exponential curve fitting.

Moved the target function for minimisation of exponential fit into the target functions folder.

task #7822(https://gna.org/task/index.php?7822): Implement user function to estimate R2eff and associated errors for exponential curve fitting.

Added verification script, that shows that using scipy.optimize.leastsq reaches the exact same parameters as minfx for exponential curve fitting.

The profiling shows that scipy.optimize.leastsq is 10X as fast as using minfx (with no linear constraints.)
scipy.optimize.leastsq is a wrapper around wrapper around MINPACK's lmdif and lmder algorithms.

MINPACK is a FORTRAN90 library which solves systems of nonlinear equations, or carries out the least squares minimization of the residual of a set of linear or nonlinear equations.

The verification script also shows, that a very heavy and time consuming monte carlo simulation of 2000 steps, reaches the same
errors as the errors reported by scipy.optimize.leastsq.

The return from scipy.optimize.leastsq, gives the estimated co-variance.
Taking the square root of the co-variance corresponds with 2X error reported by minfx.

This could be an extremely time saving step, when performing model fitting in R1rho, where
the errors of the R2eff values, are estimited by Monte-Carlo simulations.

The following setup illustrates the problem.
This was analysed on a: MacBook Pro, 13-inch, Late 2011.
Witn no multi-core setup.

Script running is:
test_suite/shared_data/dispersion/Kjaergaard_et_al_2013/2_pre_run_r2eff.py

This script analyses just the R2eff values for 15 residues.
It estimates the errors of R2eff based on 2000 Monte Carlo simulations.
For each residues, there is 14 exponential graphs.

The script was broken after 35 simulations.
This was measured to 20 minutes.
So 500 simulations would take about 4.8 Hours.
The R2eff values and errors can by scipy.optimize.leastsq can instead be calculated in: 15 residues * 0.02 seconds = 0.3 seconds.

Further improved the profiling of relax curve fit.

This profiling shows, that Python code is about twice as slow as the C code implemented.

But it also shows that optimising with scipy.optimize.leastsq is 20 X faster.
It also gives reasonable error values.

Combining a function for a linear fit to guess the initial values, together
with scipy optimise, will be an extreme time win for estimating R2eff values fast.

A further test would be to use relax Monte-Carlo simulations for say 1000-2000 iterations,
and compare to the errors extracted from estimated covariance.

Further extended the profiling script for curve fitting.

Now profiling is in place for the implemented C code method in relax.

A similar code should now be devised for numpy array for comparing.

But this profiling shows that when contraints=True, is slowing down this procedure by a factor 10 X !

Added Nikolai's original Matlab code to the lib.dispersion.ns_r1rho_2site module docstring.

This is the code taken directly form the original funNumrho.m file, which was the origin of the code
in this module.

Fix for looping over data indices, where tilt_angles has the si index.

bug #22461(https://gna.org/bugs/?22461): NS R1rho 2-site_fit_r1 has extremely high chi2 value in systemtest Relax_disp.test_r1rho_kjaergaard_missing_r1.