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Implementation of minimization of tpd by using a second order newton#167

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fedebenelli wants to merge 24 commits intomainfrom
method_tpd_newton
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Implementation of minimization of tpd by using a second order newton#167
fedebenelli wants to merge 24 commits intomainfrom
method_tpd_newton

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This PR changes the method to obtain the minima of the $tm$ function.

It uses the second order newton

$\min tm(\mathbf{W}): 1 + \sum_i W_i \left( \ln W_i + \ln \hat{\phi}_i(\mathbf{W}) - d_i -1\right)$

With its gradient and Hessian:

  • $g_i = \ln W_i + \ln \hat{\phi}_i(\mathbf{W}) - d_i$
  • $H_{ij} = \frac{\delta_{ij}}{W_i} + \frac{\partial{\ln\hat{\phi}_i}}{\partial W_j}$

Michelsen and Mollerup recommend to use a trust-region method, by solving the step with:

  • $(\mathbf{H} + \eta S)\Delta \mathbf{W} + g_i = 0$

More details on page 236 of their book.

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