Optimization_Problem_6

Problem

Using $\beta_k = \frac{r_k^TAp_{k-1}}{p_{k-1}^TAp_{k-1}}$ verify that $ span[r_0, ..., r_k] = span[Ar_0, ..., A^kr_0] $ and $span[p_0, ..., p_k] = span[Ar_0, ..., A^kr_0] $ hold for k = 1

Solution

  • Show that $span[r_0, r_1] = span[r_0, Ar_0]$

    • By definition $r_0 \in span[r_0 , Ar_0]$

    • Show $r_k \in span[r_0 , Ar_0]$

      • let $r_1 = Ax_1 - b$ where $x_1 = x_0 + \alpha p_0$ and $p_0 = -r_0$

        $$r_1 = A(x_0 + \alpha p_0) - b$$

        $$r_1 = A(x_0 + \alpha -r_0) - b$$

        $$r_1 = A x_0 + -\alpha A r_0) - b$$

    • Thus we have $r_1 \in span[r_0 , Ar_0]$

2) To show the other direction we subsiture $r_0$ with $- p_0$ and solve in the same fashion

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