By Schewchuk.

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Hence, 1 ♥ is the point on 0 ♥ ✮ å 1 that minimizes ♣ 1♥ . 1 ➵ that -conjugacy of the search direction and ✾ ➵ ➺ ❀ the error term is equivalent to minimizing (and therefore➵ ➵✾♣ ➺ ) along the search direction. However, ♠ after Conjugate Directions takes a second step, minimizing ♣ along a second search direction Ô 1 ♥ , why ➵ ✾ ➵ ➺ ♠ should we expect that ♣ still be minimized in the direction of Ô 0 ♥ ? After taking ➀ steps, why will ♠ ✆ ❀ ✳ ✆ ♠ ✧ ✶ be minimized over allwill ✧ of 0 ♥ ✮ å ? ♥ Jonathan Richard Shewchuk 28 ✆✥♠ ✆♠ 0 ♥✮ å ✆✭♠ 1 0 ✆♠ ♥ ✆♠ 1 r ♠ 1♥ ♠ ♠Ô ♣ 1♥ 1♥ ♥ Ô ♠ 0♥ ✆✥♠ ✆ 1 0 ♥✮ å ✆ ♠ ✆♠ Ô ♠ 0♥ 0 ♥✮ å r ♠ 1♥ ♥ Ô ♠ 1♥ Ô ♠ 0♥ 0 ♥ ✆♠ ✆✭♠ 2 ♣ ♠ 1♥ ✆ (b) 1 1♥ r ♠ 1♥ Ô ♠ 1♥ ✆✥♠ ♥ 1 (a) ✆ ♠ 0 0 ♥✮ å ✆✥♠ 2 Ô ♠ 0♥ ♥✆ 1 ♥ 0 ♥✮ å 1 ♠ r 1♥ Ô ♠ 1♥ ✭✆ ♠ ✆ 2♥ ✆✭♠ 0 ♥ ✆✭♠ 0 ♥✮ å 2 (d) (c) Figure 27: Optimality of the Method of Conjugate Directions.

Even when it is too expensive to compute the full Hessian ❀ ❪ ❪ , it is often reasonable to compute its diagonal Jonathan✆ Richard Shewchuk 48 2 6 4 2 -4 -2 ✆♠ 0 ♥ ✆ 2 4 6 1 -2 Figure 41: The preconditioned nonlinear Conjugate Gradient Method, using the Polak-Ribi`ere formula and a diagonal preconditioner. The space has been “stretched” to show the improvement in circularity of the contour lines around the minimum. ✆ for use as a preconditioner. However, be forewarned that if is sufficiently far from a local minimum, the diagonal elements of the Hessian may not all be positive.

To show this fact mathematically, premultiply Equation 35 by Ô ✧ ♥ : ✯ Ô ✢♠ ✧ ☎ ♥ Ô ✢♠ ✧ ✽ ✯ ✗➲ 1 ♠ ↔ Ô ↔ ★❫✧ Ù ♥ 0 ➀♦❿ ❊ ♣ ♠↔ ♥ ✞ ♠ ✞ ♥r ↔♥ ✢♠ ✧ ☎ Ô ♠ ♥ ↔♥ (38) ☎ (by -orthogonality of Ô -vectors). ➭ (39) We could have derived this identity by another tack. Recall that once we take a step in a search direction, ☎ we need never step in that direction again; the error term is evermore -orthogonal to all the old search ♠ ♠ ✉ ✞ ✝ ✯ ☎ directions. Because r ✧ ♥ ♣ ✧ ♥ , the residual is evermore orthogonal to all the old search directions.