By Jonathan R Shewchuk
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Additional resources for An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
In linear CG, there are several equivalent expressions for the value of . In nonlinear CG, these different Ü best choice. Two choices are the expressions are no longer equivalent; researchers are still investigating the Fletcher-Reeves formula, which we used in linear CG for its ease of computation, and the Polak-Ribi`ere formula: ✢♠ ♠ ✧✩❹ ✢♠ ♠ ✧✩❹ ✯ ♠ ✧ ✶ ♠ ✧✩❹ ✸✷✶✹ Ü ✞ r ✧❸❹ ✢ 1♥ r ♠ 1♥ 1♥ r ♠✧♥ r ✧♥ ❊ ✠ Ü ♠ ✧✩❹ ✹ ✞ r ✩✧ ❹ 1♥ ✳➶r ✢ 1♠ ♥ r ♥ 1♥ r ♠✧♥ r ✧♥ The Fletcher-Reeves method converges if the starting point is sufficiently close to the desired minimum, whereas the Polak-Ribi`ere method can, in rare cases, cycle infinitely without converging.
In this example, CG is not nearly as effective as in the linear case; this function is deceptively difficult to minimize. Figure 37(c) shows a cross-section of the surface, corresponding to the first line search in Figure 37(b). Notice that there are several minima; the line search finds a value of ② corresponding to a nearby minimum. Figure 37(d) shows the superior convergence of Polak-Ribi`ere CG. Because CG can only generate ☛ conjugate vectors in an ☛ -dimensional space, it makes sense to restart CG every ☛ iterations, especially if ☛ is small.
However, a three-dimensional example is more revealing. Figures 27(c) and 27(d) each show two concentric ellipsoids. ✆♠ ✆♠ The point 1 ♥ lies on the outer ellipsoid, and 2♥ lies on the inner ellipsoid. Look carefully at these figures: ✆♠ ✆ ♠ the plane 0 ♥ ✮ å 2 slices through the larger ellipsoid, and is tangent to the smaller ellipsoid at 2 ♥ . The ✆ ♠ In Figure 27(b), Ô 0 ♥ point is at the center of the ellipsoids, underneath the plane. ✆✭♠ Looking at Figure 27(c), we can rephrase ➵ ➵ our ✆✥♠ question.