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Method 1 (requiring starting point)

The second derivatives

$\displaystyle H_p^{u_k, u_l} = \sum_{i=1}^N \frac{-B_{ik}B_{il}}{\lambda_i^2},$

can be used in the Newton-Raphson iteration to maximize $H_p$. We have found that the optimization has very nice convergence properties and converges usually in about 5 to 10 iterations. The algorithm is implemented in software/me_solve.m.