1. Stochastic Differential Equations and Random Matrices
Alan Edelman
MIT: Dept of Mathematics,
Computer Science Artificial Intelligence Laboratory
SIAM Applied Linear Algebra
July 18, 2003
11/8/2018

2. Subject + Random = Wow! Quantum Mechanics Statistical Mechanics
Randomized Algorithms
Random Variation & Natural Selection
Option Pricing Model
Why not applied linear algebra????
11/8/2018

3. Everyone’s Favorite Tridiagonal-2 1 1 n2 … … … … … d2
dx2
11/8/2018

4. Everyone’s Favorite Tridiagonal-2 1 G 1
(βn)1/2 1 n2 … … … + … … dW
β1/2 d2
dx2 +
11/8/2018

5. G
11/8/2018

6. G
11/8/2018

7. 7 G O
11/8/2018

8. 7 G O
11/8/2018

9. 7 G O
11/8/2018

10. 7 G O
11/8/2018

11. 7 G O
11/8/2018

12. 7 G O
11/8/2018

13. 7 G O
11/8/2018

14. 7 G O
11/8/2018

15. 7 G O
11/8/2018

16. 7 G O
11/8/2018

17. 7 G O 6
11/8/2018

18. 7 G O 6
11/8/2018

19. 7 G O 6
11/8/2018

20. 7 G O 6
11/8/2018

21. 7 G O 6
11/8/2018

22. 7 G O 6
11/8/2018

23. 7 G O 6
11/8/2018

24. 7 G O 6
11/8/2018

25. 7 G O 6
11/8/2018

26. 7 G O 6 5
11/8/2018

27. 7 G O 6 5
11/8/2018

28. 7 G O 6 5
11/8/2018

29. 7 G O 6 5
11/8/2018

30. 7 G O 6 5
11/8/2018

31. 7 G O 6 5
11/8/2018

32. 7 G O 6 5
11/8/2018

33. 7 G O 6 5
11/8/2018

34. 7 G O 6 5 4
11/8/2018

35. 7 G O 6 5 4
11/8/2018

36. 7 G O 6 5 4
11/8/2018

37. 7 G O 6 5 4
11/8/2018

38. 7 G O 6 5 4
11/8/2018

39. 7 G O 6 5 4
11/8/2018

40. 7 G O 6 5 4
11/8/2018

41. 7 G O 6 5 4 3
11/8/2018

42. 7 G O 6 5 4 3
11/8/2018

43. 7 G O 6 5 4 3
11/8/2018

44. 7 G O 6 5 4 3
11/8/2018

45. 7 G O 6 5 4 3
11/8/2018

46. 7 G O 6 5 4 3
11/8/2018

47. 7 G O 6 5 4 3 2
11/8/2018

48. 7 G O 6 5 4 3 2
11/8/2018

49. 7 G O 6 5 4 3 2
11/8/2018

50. 7 G O 6 5 4 3 2
11/8/2018

51. 7 G O 6 5 4 3 2
11/8/2018

52. 7 G O 6 5 4 3 2 1
11/8/2018

53. 7 G O 6 5 4 3 2 1
11/8/2018

54. G 6 5 4 3 2 1 Same idea: sym matrix to tridiagonal form
11/8/2018

55. G 6 5 4 3 2  Same idea: General beta beta:
1: reals 2: complexes 4: quaternions
11/8/2018

56. Stochastic Operator Limit, dW β 2 x dx d + - ,
N(0,2) χ n β 2 1 ~ H 2) (n 1) ÷ ø ö ç è æ - … … … , G β 2 H n +
11/8/2018

57. Largest Eigenvalue Plots
11/8/2018

58. MATLAB beta=1; n=1e9; opts.disp=0;opts.issym=1;
alpha=10; k=round(alpha*n^(1/3)); % cutoff parameters
d=sqrt(chi2rnd( beta*(n:-1:(n-k-1))))';
H=spdiags( d,1,k,k)+spdiags(randn(k,1),0,k,k);
H=(H+H')/sqrt(4*n*beta);
eigs(H,1,1,opts)
11/8/2018

59. Tricks to get O(n9) speedup
Sparse matrix storage (Only O(n) storage is used)
Tridiagonal Ensemble Formulas (Any beta is available due to the tridiagonal ensemble)
The Lanczos Algorithm for Eigenvalue Computation ( This allows the computation of the extreme eigenvalue faster than typical general purpose eigensolvers.)
The shift-and-invert accelerator to Lanczos and Arnoldi (Since we know the eigenvalues are near 1, we can accelerate the convergence of the largest eigenvalue)
The ARPACK software package as made available seamlessly in MATLAB (The Arnoldi package contains state of the art data structures and numerical choices.)
The observation that if k = 10n1/3 , then the largest eigenvalue is determined numerically by the top k × k segment of n. (This is an interesting mathematical statement related to the decay of the Airy function.)
11/8/2018

60. Open Problems The distribution for general beta
Seems to be governed by a convection-diffusion equation
11/8/2018