Package numdifftools :: Module core :: Class Hessian
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Class Hessian

     object --+        
              |        
Common_diff_par --+    
                  |    
         Derivative --+
                      |
                     Hessian


Estimate Hessian matrix, with error estimate


Input arguments
===============
fun = function to differentiate.

**kwds
------
derOrder : Derivative order is always 2
metOrder : Integer from 1 to 4 defining order of basic method used.
             (For 'central' methods, it must be from the set [2,4].
             (Default 2)
method   : Method of estimation.  Valid options are:
              'central', 'forward' or 'backwards'.     (Default 'central')
numTerms : Number of Romberg terms used in the extrapolation.
             Must be an integer from 0 to 3.  (Default 2)
             Note: 0 disables the Romberg step completely.
stepFix  : If not None, it will define the maximum excursion from x0
             that is used and prevent the adaptive logic from working.
             This will be considerably faster, but not necessarily
             as accurate as allowing the adaptive logic to run.
            (Default: None)
stepMax  : Maximum allowed excursion from x0 as a multiple of x0. (Default 100)
stepRatio: Ratio used between sequential steps in the estimation
             of the derivative (Default 2)
vectorized : True  - if your function is vectorized.
               False - loop over the successive function calls (default).

Uses a semi-adaptive scheme to provide the best estimate of the
derivative by its automatic choice of a differencing interval. It uses
finite difference approximations of various orders, coupled with a
generalized (multiple term) Romberg extrapolation. This also yields the
error estimate provided. See the document DERIVEST.pdf for more explanation
of the algorithms behind the parameters.

 Note on metOrder: higher order methods will generally be more accurate,
         but may also suffer more from numerical problems. First order
         methods would usually not be recommended.
 Note on method: Central difference methods are usually the most accurate,
        but sometimes one can only allow evaluation in forward or backward
        direction.



HESSIAN estimate the matrix of 2nd order partial derivatives of a real
valued function FUN evaluated at X0. HESSIAN is NOT a tool for frequent
use on an expensive to evaluate objective function, especially in a large
number of dimensions. Its computation will use roughly  O(6*n^2) function
evaluations for n parameters.

Assumptions
-----------
fun : SCALAR analytical function
    to differentiate. fun must be a function of the vector or array x0,
    but it needs not to be vectorized.

x0 : vector location
    at which to differentiate fun
    If x0 is an N x M array, then fun is assumed to be a function
    of N*M variables.

Examples
--------

#Rosenbrock function, minimized at [1,1]
>>> rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> Hfun = Hessian(rosen)
>>> h = Hfun([1, 1]) #  h =[ 842 -420; -420, 210];
>>> Hfun.error_estimate
array([[  2.86982123e-12,   1.92513461e-12],
       [  1.92513461e-12,   9.62567303e-13]])

#cos(x-y), at (0,0)
>>> cos = np.cos
>>> fun = lambda xy : cos(xy[0]-xy[1])
>>> Hfun2 = Hessian(fun)
>>> h2 = Hfun2([0, 0]) # h2 = [-1 1; 1 -1];
>>> Hfun2.error_estimate
array([[  4.34170696e-15,   4.34170696e-15],
       [  4.34170696e-15,   4.34170696e-15]])

>>> Hfun2.numTerms = 3
>>> h3 = Hfun2([0,0])
>>> Hfun2.error_estimate
array([[  1.70965039e-14,   1.29284572e-12],
       [  1.29284572e-12,   1.70965039e-14]])


See also
--------
Gradient,
Derivative,
Hessdiag,
Jacobian

Method Summary
  __call__(self, x00)
  hessian(self, x00)
Hessian matrix i.e., array of 2nd order partial derivatives
    Inherited from Derivative
  __init__(self, fun, **kwds)
  derivative(self, x00)
Return estimate of n'th derivative of fun at x0 using romberg extrapolation
  _derivative(self, x00)
  _fder(self, f_x0i, x0i, h)
Return derivative estimates of f at x0 for a sequence of stepsizes h...
  _fun(self, xi)
  _gradient(self, x00)
  _hessdiag(self, x00)
  _hessian(self, x00)
  _partial_der(self, x00)
Return partial derivatives
    Inherited from Common_diff_par
  _check_params(self)
check the parameters for acceptability
  _fdamat(self, parity, nterms)
Return matrix for fda derivation.
  _fdiff_b(self, f_x0i, x0i, h)
Return backward differences...
  _fdiff_c(self, f_x0i, x0i, h)
Return central differences...
  _fdiff_f(self, f_x0i, x0i, h)
Return forward differences...
  _rombextrap(self, der_init)
Return Romberg extrapolated derivatives and error estimates based on the initial derivative estimates...
  _set_all_der_par(self)
Set derivative parameters: stepsize, differention rule and romberg extrapolation
  _set_delta(self)
Set the steps to use in derivation.
  _set_fdarule(self)
Generate finite differencing rule in advance.
  _set_fdiff(self)
Set _fdiff fun according to method
  _set_rombexpon(self)
Member variables used...
    Inherited from object
  __delattr__(...)
x.__delattr__('name') <==> del x.name
  __getattribute__(...)
x.__getattribute__('name') <==> x.name
  __hash__(x)
x.__hash__() <==> hash(x)
  __reduce__(...)
helper for pickle
  __reduce_ex__(...)
helper for pickle
  __repr__(x)
x.__repr__() <==> repr(x)
  __setattr__(...)
x.__setattr__('name', value) <==> x.name = value
  __str__(x)
x.__str__() <==> str(x)
    Inherited from type
  __new__(T, S, ...)
T.__new__(S, ...) -> a new object with type S, a subtype of T

Class Variable Summary
str _hessian_txt = "\n\n    Input arguments\n    ===========...

Method Details

hessian(self, x00)

Hessian matrix i.e., array of 2nd order partial derivatives

See also derivative, gradient, hessdiag, jacobian

Class Variable Details

_hessian_txt

Type:
str
Value:
"""

    Input arguments
    ===============
    fun = function to differentiate.

    **kwds
    ------
...                                                                    

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