Numerical Implementation

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

- Start with iteration counter . The initial guess
and let be the desired
convergence tolerance. For the convergence tolerance,
- is
suitable (see sample input data files in the
`tests`sub-directory); - Compute (gradient of ) and (Hessian of );
- Determine a suitable search direction, . For steepest descent, and for Newton's method, (matrix-vector notation) are used;
- Update:
, where is the step size.
For steepest descent, a variable step size (line
search) algorithm, which is presented in Reference [18],
is used to determine
, and for Newton's method (
*damped*or*guarded*), the line search is used to set the step size if the error is greater than and otherwise, is used; - Check convergence: if , increment the iteration counter, , and goto 2, else continue;
- Set and compute the max-ent basis functions using Eq. (4).

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