# Estimating rate of decay of residual norms in gradient descent

I am using gradient descent to solve the linear system $$Ax=b$$, where matrix $$A$$ is symmetric and positive definite. More precisely, I am attempting to solve the following quadratic program

$$\text{minimize} \quad \frac{1}{2}x^TAx-b^Tx$$

A logarithmic plot of the residual norms ($$\rho_k$$) after $$K$$ steps forms a straight line sloping down to the right. I was asked to produce a least-squares approximation to $$\log(\rho_k)$$ where $$k \in \{K, \dots, N\}$$ in order to determine $$\beta$$ and $$\gamma$$ where

$$\log(\rho_k) \approx \beta + \gamma(k-K)$$

and am not sure what to do. Any help would be appreciated.

• Is $K$ the dimension of $A$? – LutzL Apr 26 at 19:14
• Using uppercase for both matrices and integers is a huge turnoff. – Rodrigo de Azevedo Apr 27 at 16:15
• Take the part of the curve that looks linear and use least-squares. – Rodrigo de Azevedo Apr 27 at 16:19