Improving the initial guess

While the initial value of x is the measured vector in contrast to the linear case we need also an initial guess (say) y of the unmeasured y. Clearly, this y will be chosen somewhere in the admissible region of variables. The better is the guess, the better is our expectation that the procedure will converge. Given such y% we can compute the residual 

We have again the partition (10.2.4) of the Jacobi matrix Dg. Considering the residual as function g (y), thus at fixed x the minimum condition for [g reads 2g TB = 0 thus 

If B is of full column rank J (y observable), B TB is invertible and the solution u is unique. Else we subjectu to an additional condition. First,if rankB = L Jthen some J-L rows of B are linearly dependent on the remaining ones. By elimination 

Then u will be subjected to a condition requiring that y = y + u is not too distant from the initial guess y, to avoid y escaping from the admissible region. Formally, let us minimize the square norm 

Here and in the sequel, the choice of S (as well as that of T above) modifies only the strategy, but (according to the uniqueness hypothesis) not the final reconciled value x of the measured vector. We then compute vector u by (10.4.5b). Observe that if rankB = 7 then M is invertible and u is independent of the choice of S in fact it equals the unique solution of (10.4.3) (to within possible numerical errors, of course). 

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There are three common schemes that can be used to improve the initial guess and the successive solutions. These are... 

The parameter Y depends on the unknown ammonium nitrate concentration, making an iterative solution necessary. For this approach, one assumes a value of the ammonium nitrate concentration, calculates the corresponding K and finds from Figure 10.21 the corresponding Kp. Then one can follow the same approach as that for solid ammonium nitrate and calculate the actual ammonium nitrate concentration. If the assumed value and the calculated one arc the same, then this value is the solution of the problem. If not, these values arc used to improve the initial guess and the process is repeated. 

One of the main problems with this TDHF procedure, though, is its poor convergence behavior, which becomes worse as a pole is approached. This simple approach is particularly hard to converge above the first pole, and conventional random phase approximation methods (to be discussed later) work better in this region. More work needs to be done to improve the initial guess and to develop better convergence accelerators. 

Improving the initial guess according to Subsection 10.4.1 not obligatory]... 

To improve an initial guess (x ° y ), we reach above this point and project tangent planes from the surfaces of Ri and R2. The improved guess is the point... 

From the initial guess x°, we calculate the m values of f(x°) and their derivatives relative to each Xj. Solving the least-square system, we get an improved estimate of x, that we use as the initial value for the next iteration until the values cease to change significantly. Indicating the fcth estimate by the superscript k, we can write... 

In the second loop, the matrix of species concentrations C is computed rowwise by the Newton-Raphson function. Each solution is analysed individually. To expedite the computations, the initial guesses for the component concentrations are the result of the previous solution (apart from the first one). If the Newton-Raphson function returns an error these initial guesses need to be improved. 

The method uses the fact that any response F (v) always has a broader distribution than the input distribution W(y). Hence, if a distribution Fj(v) is broader than F (v), the assumed W (y) must be sharpened to give a response closer to F (v). Using F ( v) as the initial guess for W(y), subsequent improved estimates are calculated by ... 

In a self-consistent calculation of defects, the initial guess of the potential 1/ in the form (5.4) has to be checked or improved toward self-consistency this requires a calculation of the density matrix of the imperfect crystal. This can be done as follows. 

An improved method was developed by Chirlian and Francl and called CHELP (CHarges from ELectrostatic Potentials). Their method, which uses a Lagrangian multiplier method for fitting the atomic charges, is fast and noniterative and avoids the initial guess required in the standard least-squares methods. In this approach, the best least-squares fit is obtained by minimizing y ... 

To improve an initial guess (x(o), y(o)), we reach above this point and project tangent planes from the surfaces of R and R2. The improved guess is the point (x(1), y(1)) where these tangent planes intersect each other and the table top. We repeat this process until each residual function is less than a negligible value. 

This justifies the initial guess ylf ) = o, and it also suggests that if convergence of the Newton iteration is poor, this might be improved by taking a smaller timestep, since this effectively places yl(o) closer to the solution of the equation. 

It is often difficult for a nonlinear solver to locate a feasible solution (one that satisfies the constraints), especially when the initial guessed values are poor. When the user provides a feasible starting point, the likelihood of successful convergence to an optimal solution is greatly improved. 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

The next estimate of y is unique if rankB (= rankB ) = J thus (L = J thus) B"[7, J] is invertible. In that case, we can apply an arbitrary linear reconciliation method to (10.4.50), for instance also (9.2.24) or (9.2.30). Else we can use again the solution minimizing the difference y-y where y is the initial guess, either y or improved according to Subsection 10.4.1. The linearized equation (10.4.51a) with x = reads... 

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