Direct Force Field User Manual

Fit Methods

Levenberg-Marquardt Method

The Levenberg-Marquardt method 1 2 with constrained minimization technique is implemented. The least squares problem is written as:

The first term is the normal least squares expression of input data and parameter-dependent calculated data . is the weighting factor, which can be related to the Boltzmann factor for energy. The second term is a penalty term in which the parameter is restrained to preferred value . The penalty factor is adjustable.

Linear SVD Method

In addition, the SVD method 3 is used to fit the linear equations,

where is adjustable parameter and is the variable. The least-squares equation is then reduced to a linear equation:

This method is especially useful when matrix A is singular or very close to singular.

References

Footnotes

  1. Levenberg, Kenneth (1944). "A Method for the Solution of Certain Non-Linear Problems in Least Squares". Quarterly of Applied Mathematics. 2 (2): 164–168. doi:10.1090/qam/10666. ↩

  2. Marquardt, Donald (1963). "An Algorithm for Least-Squares Estimation of Nonlinear Parameters". SIAM Journal on Applied Mathematics. 11 (2): 431–441. doi:10.1137/0111030. hdl:10338.dmlcz/104299. ↩

  3. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical recipes 3rd edition: The art of scientific computing; Cambridge University Press, 2007. ↩