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I am a green-hand in calculating the time complexity. Given a calculation as follows:

\begin{equation} \mathbf{x}=\mathbf{A^T}(\mathbf{AA^T}+\lambda\mathbf{I}_n)^{-1}\mathbf{b} \end{equation} where $\mathbf{A}\in R^{n\times p}$, $\mathbf{I}_n\in R^{n\times n}$, $\mathbf{b}\in R^{n}$ and $\lambda\in R$.

Could some one kindly give me the complexity of the above formula in details? Thanks a lot.

Raphael
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aaronyxt
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  • What have you tried? Where did you get stuck? We do not want to just do your (home-)work for you; we want you to gain understanding. However, as it is we do not know what your underlying problem is, so we can not begin to help. See here for a relevant discussion. If you are uncertain how to improve your question, why not ask around in [chat]? You may also want to check out our reference questions. – Raphael Aug 04 '15 at 06:19
  • Formulae don't have "time complexities" (what you really mean is "evaluation cost"), algorithms do. So how do you transpose, add, multiply and invert matrices? Take the thing apart, and then put the cost together. – Raphael Aug 04 '15 at 06:20
  • @Raphael Thanks for your reply and reminder. In fact, this problem confused me when I was reading a publication. But the author just gave the direct result of the complexity, i.e., $O(n^3+n^2p)$. Here the complexity refers to the operation in calculation the formula denoted by big O. – aaronyxt Aug 07 '15 at 09:35
  • In fact, I have been stuck in understanding the multiplication of matrices. I know that the multiplication of $AA^T$ is $O(n^2p)$. Then the inverse of the matrix cost $O(n^3)$. But then at least, the complexity is $O(n^2p+n^3)$. But the formula has not been completely calculated. It should also consider the matrice multiplication of $A^T\times inv(result) \times b$. Then the complexity should be larger than $O(n^2p+n^3)$. It is wired. If you could give some hints, I would be highly appreciated. – aaronyxt Aug 07 '15 at 09:35
  • As far as I can tell, the other operation have the same $O$-bounds so the terms fall together. You should read up on what $O$ really means; our reference questions on asymptotics and algorithm analysis should get you started. – Raphael Aug 07 '15 at 09:49

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