Fill in the details in a direct derivation of the Conjugate Gradient algorithm following the ideas outlined in the text: starting with the initial guess x₀ and corresponding residual vector w₁ = r₀ = b, at the kth step in the algorithm, given the approximation x_k and residual r_k = b − Ax_k, the k^th conjugate direction is chosen so that w_k+1 = r_k + s_kw_k satisfies the conjugacy conditions (9.113). The next approximation x_k+1 = x_k + t_k+1w_k+1 is chosen so that its residual r_k+1 = b − Ax_k+1 is as small as possible.

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((W₂, W;)) = w/ Aw, = 0, W i ‡ j. (9.113)