An Introduction to the Conjugate Gradient Method Without the by Jonathan R Shewchuk

By Jonathan R Shewchuk

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Extra resources for An Introduction to the Conjugate Gradient Method Without the Agonizing Pain

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There is one other possibility for early termination: 0♥ may already be -orthogonal to ♠ ☎ ✆ some of the eigenvectors of . If eigenvectors are missing from the expansion of 0♥ , their eigenvalues may be omitted from consideration in Equation 50. 5 1 Figure 32: Chebyshev polynomials of degree 2, 5, 10, and 49. We also find that CG converges more quickly when eigenvalues are clustered together (Figure 31(d)) ✧ than when they are irregularly distributed between ❻✣Ï ✗ and ❻✣Ï♦Ð ❵ , because it is easier for CG to choose a polynomial that makes Equation 50 small.

These evaluations may be inexpensive if Ô ✢❾❀ ❪ ❪ Ô can be ❀ ❪ ❪ must be evaluated repeatedly, the algorithm is prohibitively analytically simplified, but if the full matrix slow. For some applications, it is possible to circumvent this problem by performing an approximate line ❀ ❪❪ search that uses only the diagonal elements of . Of course, there are functions for which it is not possible ❀ ❪ ❪ to evaluate at all. ❀ ❪❪ ❄❃ ❃ To perform an exact line search without computing , the Secant method approximates the second ✶ ✞ 0 and ✞ , where is ❀ ✆ derivative of ✳ ✮❨② Ô by evaluating the first derivative at the distinct points ② ② an arbitrary small nonzero number: ❅✻ ❇❆ ❃ ❏ ③ ❀ ✳✆ ✶ ▲ ★ ✯Ý❏ ③ ❀ ✳ ✆ ✶▲ ★ Ô Ô ✮ ❆ ② ✮ ❆ ② ④ ④ ③⑤④ ③⑤④ ✞ ❏ ❀ ❪ ✳ ✆ ✮ Ô ✶ ▲ ✢ Ô ✯Ý❏ ❀ ❪ ✳ ✆▼✶ ▲ ✢ Ô ❊ which becomes a better approximation to the second derivative as ② and ✶ Ô 2 ❀ ✳✆ Ô → ✮ ② 2 Ô② ❃ ❃ ❃ 0 ✞❧ 0 (58) ❃ approach zero.

This curve is a scaled version of the Chebyshev polynomial of degree 2. 183 times its initial value. Compare with Figure 31(c), where it is known that there are only two eigenvalues. 2 ✌ ✧ ✜✡ ✛✣✜✡ ✢✚✛✣✤ ✢✚❹ ✤ ✡ ✽ ✛ ✡ ø ✛ ❜ ✽ ❜ ✡ ➾ ✌ ➽ ✧ ✡✡ ✣✛✣✛ ✢✚✢✚✤✤ ❹✽ ✡✡ ✛✛ øø ❜❜ ➾ ø ➽ ✧ ✳ ✶❝✞ ✞ ❻✣Ï♦Ð ❵ It is shown in Appendix C3 that Equation 50 is minimized by choosing ✞ ✧ ✳➜❻ ✶❝✞ 2 This polynomial has the oscillating properties of Chebyshev polynomials on the domain ❻✣Ï ✧ ✗ ❰ ❻ ❰ ❻✣Ï♦Ð ❵ (see Figure 33). The denominator enforces our requirement that 0 1.

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