By Jonathan R Shewchuk
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Additional resources for An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
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.
To find their values, use the same trick used to find ✧✽ ➲ 1 Û ✢✧ ☎ Ô ♠ ↔ ♥ ✮ ★ ✧ ➧ØÔ ✢Û ✧ ☎ Ô ♠ ➧ ✧ ↔➥0Ô Ü ✢ ♠ ☎ ↔♥ ✮ ↔♥ ☎ ✢ ♠ Ü ✧ ✯ Û ✢ ☎ Ô ♠↔ ♥ Ô ♠↔ ♥ Ô ↔ ♥ ✢♠ ☎ Ô ♠ ➧♥ ↔♥ Ô ♠↔ ♥ ❊ ➀ ✱ ➭ Ù ↔: ☎ (by -orthogonality of Ô vectors) (37) The difficulty with using Gram-Schmidt conjugation in the method of Conjugate Directions is that all ✶ the old search vectors must be kept in memory to construct each new one, and furthermore Þ✍✳➶☛ 3 operations ßdá u à0 ßdá (0) (0) u+ u1 u* ßdá (1) Figure 24: Gram-Schmidt conjugation of two vectors.
The vector â 1 is composed of two components: â✣ã , which is ❚ -orthogonal (or conjugate) ×➙➤ ➦ â❙ä , which is parallel to ×➛➤ 0➦ . After conjugation, only the ❚ -orthogonal portion remains, and ×➙➤ ➦ ❱❦â➐ã 1 . Set 0 to 0 , and . 1 Jonathan ✆ Richard Shewchuk 26 2 4 2 ✆ -4 -2 2 4 6 1 -2 -4 -6 Figure 25: The method of Conjugate Directions using the axial unit vectors, also known as Gaussian elimination. are required to generate the full set. In fact, if the search vectors are constructed by conjugation of the axial unit vectors, Conjugate Directions becomes equivalent to performing Gaussian elimination (see Figure 25).
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain by Jonathan R Shewchuk