Example:

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

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

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

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

5. Conjugate Gradient Method Conjugate Gradient Method Example: Solve: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 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. Conjugate Gradient Method Conjugate Gradient Method vectors constants

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

10. Conjugate Gradient Method Example: minimum

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

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: 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: DEF: where H is SPD Example:

19. Conjugate Gradient

20. Conjugate Gradient Method Method: Conjugate Gradient Method

21. Conjugate Gradient Method Method:

22. Conjugate Gradient Method Method: Conjugate Gradient Method

23. Conjugate Gradient Method Lemma:[Elman,Silvester,Wathen Book]

24. Conjugate Gradient Method 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 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 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 0.0786 0.1022 0.1193 0.1411 0.0713 0.0263 0.0410 0.0342