Variance Minimization

            See also

1. ..\integration\MonteCarlo\VarianceFromZeroMCError.doc .htm Original derivation of error in the expectation value of the energy.
2. The Monte Carlo method for selecting values begins in ..\integration\MonteCarlo\Welcome.htm
3. Tests of the selection method being in ..\integration\MonteCarlo\FMonte\Fmonte.doc .htm
4. ..\integration\MonteCarlo\ExpectationValues.doc .htm – A derivation of the standard deviation in the expectation value of any operator. Equation (1.8) with a Y factored out of the parenthesis and x_j and c explicitly introduced into the wave function and w becomes
5. ..\diffeqns\shoot.doc .htm – exact method for integrating a 1-d differential equation. – Surprising infinities for an incorrect <A>

          (1.1)

Define

                                  (1.2)

                        So that

          (1.3)

            Or

            (1.4)

Then define

                                       (1.5)

And

                                                   (1.6)

Then (1.1) has the form

                               (1.7)

The wave function and weight function are slowly varying functions of the constants while near the minimum the difference (A Y)/Y - <A> oscillates rapidly about zero.  Thus for minimization purposes

      Then with f_j = <A>(c) = input constant (1.7) becomes

                                     (1.8)

This is the form in ..\Fittery\FitData.doc .htm (1). 

Infinite Y

In Fitting with Poisson random variables.doc .htm it is shown that if the dependence of e_j on f_A is ignored in the sense that its derivatives are not used in finding the best f_A that a “better” fit results.  This is due to the fact that the minimum c² that occurs by making e large is not desired. 

            If the set {e_j} is not varied, A à H, and f_i becomes E, minimizing (1.8) is a method for solving 

                      (1.9)

In one dimension, a good way to solve (1.9) is discussed in ..\diffeqns\shoot.doc#OverFlow .htm. The c’s in this case become the values of the wave function at each of the points.  The figure shows that at r @ 40 a_B, the numerical solution for the Hydrogen wave function switches from + infinity to – infinity when the 16^th digit E_in is changed by 1. This would not prevent (1.8) with fixed {e_j} from giving a very small value of χ².  The wave function, however, will be difficult to calculate numerically, making it hard to find the value of f_A in (1.6).  An only slightly worse minimum will not have this problem making it frustratingly difficult to minimize (1.8).

Further derivation and suggested code are in Variance Minimization2.doc .htm.