Derivation of the conjugate gradient method: Difference between revisions

Revision as of 23:22, 5 August 2024

Revision as of 04:52, 17 February 2025

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\end{cases}</math>.

'''Proof.''' By construction, we have <math>\mathbf{r}_{i+1} = \mathbf{r}_i - \alpha_kalpha_i \mathbf{A p}_i</math>, thus<math display="block">\boldsymbol r_{k+1}^T \boldsymbol A \boldsymbol p_i = \boldsymbol r_{k+1}^T \frac{\boldsymbol r_{i} -\boldsymbol r_{i+1}}{\alpha_i}</math>Now apply lemma 1.