Numerical Implementation

On considering the dual formulation, the solution for the Lagrange multipliers can be written as [16,17]

$${\lambda}^* = \mathop{\rm argmin}\limits F({\lambda}), \quad F({\lambda}) := \ln Z({\lambda}),$$ (5)

where ${\lambda}^*$ is the optimal solution that is desired. Since $F$ is strictly convex in the interior of $D$, convex optimization algorithms (for example, Newton's method and families of gradient descent) are a natural choice. The steps in these algorithm are:

1. Start with iteration counter $k = 0$. The initial guess ${\lambda}^0 = {0}$ and let $\epsilon$ be the desired convergence tolerance. For the convergence tolerance, $\epsilon = 10^{-14}$-$10^{-10}$ is suitable (see sample input data files in the tests sub-directory);
2. Compute ${g}^k := {\nabla}_{{\lambda}}F({\lambda}^k)$ (gradient of $F$) and ${H}^k := {\nabla}_{{\lambda}} {\nabla}_{{\lambda}} F({\lambda}^k)$ (Hessian of $F$);
3. Determine a suitable search direction, $\Delta {\lambda}^k$. For steepest descent, $\Delta {\lambda}^k = -{g}^k$ and for Newton's method, $\Delta {\lambda}^k = - \left( {H}^k \right)^{-1} {g}^k$ (matrix-vector notation) are used;
4. Update: ${\lambda}^{k+1} = {\lambda}^{k} + \alpha \Delta {\lambda}^k$, where $\alpha$ is the step size. For steepest descent, a variable step size (line search) algorithm, which is presented in Reference [18], is used to determine $\alpha$, and for Newton's method (damped or guarded), the line search is used to set the step size if the error is greater than $10^{-4}$ and otherwise, $\alpha = 1$ is used;
5. Check convergence: if ${\lvert{g}^{k+1}\rvert} > \epsilon$, increment the iteration counter, $k \leftarrow k+1$, and goto 2, else continue;
6. Set ${\lambda}^* = {\lambda}^{k+1}$ and compute the max-ent basis functions using Eq. (4).