Levenberg Marquardt curve-fitting

minimize sum of weighted squared residuals. Javascript version of matlab library from Henri Gavin.

See example for usage

--------- INPUT VARIABLES -----------

func = function of n independent variables, 't', and m parameters, 'p',

returning the simulated model: y_hat = func(t,p,c)

p = n-vector of initial guess of parameter values

t = m-vectors or matrix of independent variables (used as arg to func)

y_dat = m-vectors or matrix of data to be fit by func(t,p)

weight = weighting vector for least squares fit ( weight >= 0 ) ...

inverse of the standard measurement errors

Default: sqrt(d.o.f. / ( y_dat' * y_dat ))

dp = fractional increment of 'p' for numerical derivatives

dp(j)>0 central differences calculated

dp(j)<0 one sided 'backwards' differences calculated

dp(j)=0 sets corresponding partials to zero; i.e. holds p(j) fixed

Default: 0.001;

p_min = n-vector of lower bounds for parameter values

p_max = n-vector of upper bounds for parameter values

c = an optional matrix of values passed to func(t,p,c)

opts = vector of algorithmic parameters

parameter defaults meaning

opts(1) = prnt 3 >1 intermediate results; >2 plots

opts(2) = MaxIter 10*Npar maximum number of iterations

opts(3) = epsilon_1 1e-3 convergence tolerance for gradient

opts(4) = epsilon_2 1e-3 convergence tolerance for parameters

opts(5) = epsilon_3 1e-3 convergence tolerance for Chi-square

opts(6) = epsilon_4 1e-2 determines acceptance of a L-M step

opts(7) = lambda_0 1e-2 initial value of L-M paramter

opts(8) = lambda_UP_fac 11 factor for increasing lambda

opts(9) = lambda_DN_fac 9 factor for decreasing lambda

opts(10) = Update_Type 1 1: Levenberg-Marquardt lambda update

2: Quadratic update

3: Nielsen's lambda update equations

##---------- OUTPUT VARIABLES -----------

p = least-squares optimal estimate of the parameter values

X2 = Chi squared criteria

Henri Gavin, Dept. Civil & Environ. Engineering, Duke Univ. 22 Sep 2013 modified from: [http://octave.sourceforge.net/optim/function/leasqr.html] using references by

Press, et al., Numerical Recipes, Cambridge Univ. Press, 1992, Chapter 15.

Sam Roweis [http://www.cs.toronto.edu/~roweis/notes/lm.pdf]

Manolis Lourakis [http://www.ics.forth.gr/~lourakis/levmar/levmar.pdf]

Hans Nielson [http://www2.imm.dtu.dk/~hbn/publ/TR9905.ps]

Mathworks optimization toolbox reference manual

K. Madsen, H.B., Nielsen, and O. Tingleff

[http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3215/pdf/imm3215.pdf]