By Jonathan R Shewchuk
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Extra info for An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Canned Algorithms B5.
Comparing Equations 52 and 28, it is clear that the convergence of CG is much quicker than that of Steepest Descent (see Figure 35). However, it is not necessarily true that every iteration of CG enjoys faster convergence; for example, the first iteration of CG is an iteration of Steepest Descent. The factor of 2 in Equation 52 allows CG a little slack for these poor iterations. 10. Complexity The dominating operations during an iteration of either Steepest Descent or CG are matrix-vector products.
A conservative solution is to not precondition (set when the Hessian cannot be guaranteed to be positive-definite. Figure 41 demonstrates the convergence of diagonally preconditioned nonlinear CG, with the Polak-Ribi`ere formula, on the same function illustrated ❀ ❪ ❪ at the solution point ✆ to precondition every in Figure 37. Here, I have cheated by using the diagonal of iteration. ✬ A Notes Conjugate Direction methods were probably first presented by Schmidt  in 1908, and were independently reinvented by Fox, Huskey, and Wilkinson  in 1948.