By Jonathan R Shewchuk
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Extra resources for An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Figures 27(a) and 27(c) demonstrate the behavior of Conjugate Directions in ➌ 2 and ➌ 3 ; lines that appear perpendicular in these illustrations are orthogonal. On the other hand, Figures 27(b) and 27(d) show the same drawings in spaces that are stretched (along the eigenvector axes) so that the ellipsoidal ☎ contour lines become spherical. Lines that appear perpendicular in these illustrations are -orthogonal. ✆♠ ♠ In Figure 27(a), the Method of Conjugate Directions begins at 0 ♥ , takes a step in the direction of Ô 0 ♥ , ✆✭♠ ♠ ☎ ♠ and stops at the point 1 ♥ , where the error vector ♣ 1 ♥ is -orthogonal to Ô 0 ♥ .
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. In the early fifties, the method of Conjugate Gradients was discovered independently by Hestenes  and Stiefel ; shortly thereafter, they jointly published what is considered the seminal reference on CG . Convergence bounds for CG in terms of Chebyshev polynomials were developed by Kaniel .
Of course, the number 50 is arbitrary; for large ☛ , ➞ ☛ might be appropriate. If the tolerance is large, the residual need not be corrected at all (in practice, this correction is rarely used). If the tolerance is close to the limits of the floating point precision of the machine, a test should be added after is evaluated to check if ❰ 2 0 , and if this test holds true, the exact Ù reevaluated. This prevents Ù Ù the procedure from terminating early residual should also be recomputed and Ù due to floating point roundoff error.
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain by Jonathan R Shewchuk