1. Example: Introduction to Krylov Subspace Methods DEF: Krylov sequence 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 1 11 118 1239 12717 1 12 141 1651 19446 1 10 100 989 9546 1 10 106 1171 13332

2. Example: Introduction to Krylov Subspace Methods DEF: Krylov subspace 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Krylov subspace DEF: Krylov matrix

3. Example: Introduction to Krylov Subspace Methods Remark: DEF: Krylov matrix

4. Conjugate Gradient Method We want to solve the following linear system

5. 0K=1K=2K=3K=4 x1 x2 x3 X4 Conjugate Gradient Method Example: Solve: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Conjugate Gradient Method 0 0.4716 0.9964 1.0015 1.0000 0 1.9651 1.9766 1.9833 2.0000 0 -0.8646 -0.9098 -1.0099 -1.0000 0 1.1791 1.0976 1.0197 1.0000 31.7 5.1503 1.0433 0.1929 0.0000

6. constants vectors Conjugate Gradient Method

7. We want to solve the following linear system Define: quadratic function Example:

8. Conjugate Gradient Method Example: Remark: Why not max?

9. Conjugate Gradient Method Remark: IDEA: Search for the minimum Problem (1)

10. Conjugate Gradient Method Example: minimum

11. Conjugate Gradient Method Method: “search direction” “step length” Method:

12. Conjugate Gradient Method Method:

13. Conjugate Gradient Method Method: Conjugate Gradient Method

14. INNER PRODUCT

15. Inner Product DEF: We say that Is an inner product if Example:

16. Inner Product DEF: We say that Is an inner product if Example: where H is SPD We define the norm

17. Inner Product DEF: We say that Is symmetric bilinear form if Example: where H is Symmetric

18. Inner Product DEF: Example: DEF: where H is SPD

19. Conjugate Gradient

20. Conjugate Gradient Method Method: Conjugate Gradient Method

21. Method:

22. Conjugate Gradient Method Method: Conjugate Gradient Method

23. Lemma :[Elman,Silvester,Wathen Book]

24. Conjugate Gradient Method 6.0000 5.1362 0.0427 -0.0108 25.0000 0.1121 0.0562 0.1185 -11.0000 -0.4424 -0.8384 0.0698 15.0000 -0.7974 -0.6530 -0.1395 6.0000 4.9781 -0.1681 -0.0123 0.0000 25.0000 -0.5464 0.0516 0.1166 -0.0000 -11.0000 -0.1526 -0.8202 0.0985 0.0000 15.0000 -1.1925 -0.6203 -0.1172 -0.0000 0.0000 0.4716 0.9964 1.0015 1.0000 0.0000 1.9651 1.9766 1.9833 2.0000 0.0000 -0.8646 -0.9098 -1.0099 -1.0000 0.0000 1.1791 1.0976 1.0197 1.0000 0.0786 0.1022 0.1193 0.1411 0.0713 0.0263 0.0410 0.0342