Avoid computing the inverse - Extended Least Square

Question

The extended least square estimates this polynomial equation:

$$A(q)y(t) = B(q)y(t) + C(q)e(t)$$

By using:

$$\epsilon(t) = y(t) - \phi^T(t-1)\hat \theta(t-1)$$ $$\hat \theta(t) = \theta(t-1) + P(t)\theta(t-1)\epsilon$$ $$P^{-1}(t) = P^{-1}(t-1) + \phi(t-1)\phi^T(t-1)$$

Is there any way for me to avoid computing the inverse of $P(t)$?

I have created a question for a long time ago about Recursive Least Sqares and Extended Least Squares is not the same as Recursive Least Squares. Here is an example how to update the $P$ matrix.

>> P = rand(5,5);
>> phi = rand(5,1);
>> A = inv(P) + phi*phi' % Extended Least Squares P update
A =

   -2.02448    0.59870    0.95827    0.95451   -0.83298
    1.58485    3.63862   -0.11856    1.79290   -3.02662
    5.15738   -6.45723    1.96129    0.62033    3.42288
   12.20361   -7.25674    2.19398    7.91516   -2.78538
  -13.93087   11.70728   -3.85160   -7.97718    4.38994

>> B = P - (P*phi*phi'*P)/(1+phi'*P*phi) % Recursive Least Squares P update
B =

  -0.1947838   0.2169201   0.1270665  -0.0609621  -0.0251604
   0.0503370   0.2747009   0.1513839   0.0038340   0.0833390
   0.4232505   0.4797054   0.4333511  -0.3326665  -0.1379224
   0.2636843  -0.2302613  -0.1209715   0.3929447   0.2349241
   0.0981407  -0.0417592   0.1598960   0.2184892   0.2315819

I assume this is recursive least squares, in which case you already asked a question about this. — Kwin van der Veen, Jun 27 '19 at 04:57
@Kwinvanderveen RLS and ELS are not the same, but do you think i can borrow some equations from RLS so I don't need to compute the inverse? — euraad, Jun 27 '19 at 09:00
@KwinvanderVeen Nope. That does not work. I have updated my question. — euraad, Jun 27 '19 at 11:21
But A is the inverse of the update P, namely inv(A) should be equal to B. — Kwin van der Veen, Jun 27 '19 at 15:55

Avoid computing the inverse - Extended Least Square

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