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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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6.7. Stability and convergence

Let us now study the stability and the convergence of the algorithm of the deterministic gradient.

By taking the recursive expressions of the coefficient vector and by translation:

images

The following expressions

images

enable us to write: αK+1 = (Id − 2μR)αK Id: identity matrix.

By writing R in the form

images

and by premultiplying αK+1 by Q−1, we obtain:

images

Thus:

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or: image

and: image

Thus, the algorithm is stable and convergent if

images

Thus, if and only if image

Thus, if and only if: image

In addition, we ...

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