Circulant matrix-vector product procedure

A circulant matrix $$C$$ can be represented as

$$C = F^{-1} \mbox{diag}(Fc) \, F$$

When $$C$$ is multiplied by vector $$b$$

$$C b = F^{-1} \mbox{diag}(Fc) \, (F b)$$

My question only about procedure. As I understood it, to compute $$Cb$$:

1. Find FFT of $$b$$.
2. Find FFT of $$c$$
3. Compute $$\mbox{diag}(Fc)*(Fb)$$
4. Find IFFT of matrix from the third step.

Am I right?

• @RodrigodeAzevedo i am sorry,my bad,mixed up. I am already fixed it. So, what are you think about my question? – Аделина Артуровна May 4 at 10:12
• This is confusing. What are the inputs and outputs? – Rodrigo de Azevedo May 4 at 10:50
• @RodrigodeAzevedo this is process of computing circulant matrix - vector multiplication O(nlogn) . input- circulant matrix and vector , output- vector. whats wrong? sorry for my english :) – Аделина Артуровна May 4 at 13:19
• If $F$ is a Fourier matrix, it does not need to be inverted. That is the whole point. – Rodrigo de Azevedo May 4 at 16:50
• @RodrigodeAzevedo . b-input vector. c-first row of C – Аделина Артуровна May 4 at 20:49