LevenbergMinimizer

class triqs_maxent.minimizers.levenberg_minimizer.LevenbergMinimizer(convergence=None, maxiter=1000, miniter=0, J_squared=False, marquardt=False, mu0=1e-18, nu=1.3, max_mu=1e+20, verbose_callback=None)

Bases: Minimizer

The Levenberg minimization algorithm.

The task of this algorithm is to minimize a function \(Q\) with respect to a quantity \(H\). That is equivalent to searching a solution to \(f := \partial Q/\partial H = 0\). We assume that the equation is parametrized by a solution vector \(v\) (i.e., we are looking for a solution in \(H(v)\)). We then calculate the matrix \(J = \frac{\partial}{\partial v} \frac{\partial Q}{\partial H}\).

Depending on the settings of the parameters, the solution is searched for in the following way:

• J_squared=False, marquardt = False: \((J + \mu 1)\delta v = f\)

• J_squared=False, marquardt = True: \((J + \mathrm{diag} J)\delta v = f\)

• J_squared=True, marquardt = False: \((J^TJ + \mu 1)\delta v = f\)

• J_squared=True, marquardt = True: \((J^TJ + \mathrm{diag} J^T J)\delta v = f\)

Then, \(v\) is updated by subtracting \(\delta v\).

The step length is determined by the damping parameter \(\mu\), which is chosen so that \(Q\) is minimal.

Parameters:

convergenceConvergenceMethod

method to check convergence; the default is MaxDerivativeConvergenceMethod(1.e-4) | RelativeFunctionChangeConvergenceMethod(1.e-16)

maxiterint

the maximum number of iterations

miniterint

the minimum number of iterations

J_squaredbool

if True, the algorithm solves \((J^TJ +\mu 1) \delta v = J^T f\), else \((J +\mu 1) \delta v = f\).

marquardtbool

if True, the algorithm uses \(\mathrm{diag}(J)\) or \(\mathrm{diag}(J^T J)\) (depending on the parameter J_squared), instead of the identity matrix.

mu0float

the initial value of the Levenberg damping parameter

nufloat

the relative increase/decrease of mu when necessary

max_mufloat

the maximum number of mu, to prevent infinite loops

verbose_callbackfunction

a function used to print verbosity information e.g., the print function

Attributes:

n_iterint

the total number of iterations performed since the creation of the class

n_iter_lastint

the number of iterations performed when minimize was last called

convergedbool

whether the algorithm converged in fewer than maxiter loops when minimze was last called

minimize(function, v0)

Perform the minimization.

Parameters:

functionDoublyDerivableFunction

the function to be minimized

v0array

the initial function argument \(v\)

Returns:

varray

the vector \(v\) at the minimum