Stochastic Differential Equations and Random Matrices Alan Edelman

Subject + Random = Wow! Quantum Mechanics Statistical Mechanics Randomized Algorithms Random Variation &

Everyone’s Favorite Tridiagonal 1 n 2 -2 1 1 -2 1 … … …

Same idea: sym matrix to tridiagonal form 11/28/2020 G 6 6 G 5 5

Same idea: General beta G 6 beta: 1: reals 2: complexes 4: quaternions 6

MATLAB beta=1; n=1 e 9; opts. disp=0; opts. issym=1; alpha=10; k=round(alpha*n^(1/3)); % cutoff parameters

Tricks to get O(n 9) speedup • Sparse matrix storage (Only O(n) storage is

Open Problems The distribution for general beta Seems to be governed by a convection-diffusion

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Stochastic Differential Equations and Random Matrices Alan Edelman MIT: Dept of Mathematics, Computer Science Artificial Intelligence Laboratory SIAM Applied Linear Algebra July 18, 2003 11/28/2020 1

Subject + Random = Wow! Quantum Mechanics Statistical Mechanics Randomized Algorithms Random Variation & Natural Selection Option Pricing Model Why not applied linear algebra? ? 11/28/2020 2

Everyone’s Favorite Tridiagonal 1 n 2 -2 1 1 -2 1 … … … 1 1 -2 d 2 dx 2 11/28/2020 3

Everyone’s Favorite Tridiagonal 1 n 2 -2 1 1 -2 1 … … … 1 1 -2 G d 2 dx 2 11/28/2020 G 1 +(βn)1/2 G + d. W β 1/2 4

11/28/2020 G G G G G G G G G G G G G 5

11/28/2020 G G G G G G G G G G G G G 6

11/28/2020 7 O O O G G G G G G G G G G G 7

11/28/2020 7 O O O G G G G G G G G G G G 8

11/28/2020 7 O O O G G G G G G G G G G G 9

11/28/2020 7 O O O G G G G G G G G G G G 10

11/28/2020 7 O O O G G G G G G G G G G G 11

11/28/2020 7 O O O G G G G G G G G G G G 12

11/28/2020 7 O O O G G G G G G G G G G G 13

11/28/2020 7 O O O G G G G G G G G G G G 14

11/28/2020 7 O O O G G G G G G G G G G G 15

11/28/2020 7 O O O G G G G G G G G G G G 16

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 17

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 18

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 19

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 20

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 21

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 22

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 23

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 24

11/28/2020 7 O O O G 6 O O O G G G G G G G G G 25

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 26

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 27

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 28

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 29

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 30

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 31

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 32

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G G G G G 33

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G G G G G 34

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G G G G G 35

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G G G G G 36

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G G G G G 37

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G G G G G 38

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G G G G G 39

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G G G G G 40

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G G 41

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G G 42

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G G 43

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G G 44

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G G 45

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G G 46

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G 2 O G G G G 47

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G 2 O G G G G 48

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G 2 O G G G G 49

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G 2 O G G G G 50

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G 2 O G G G G 51

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G 2 O G G G 1 52

11/28/2020 7 O O O G 6 O O O G G 5 O O G G G 4 O O O G G 3 O O G G G 2 O G G G 1 53

Same idea: sym matrix to tridiagonal form 11/28/2020 G 6 6 G 5 5 G 4 4 G 3 3 G 2 2 G 1 1 G 54

Same idea: General beta G 6 beta: 1: reals 2: complexes 4: quaternions 6 G 5 5 G 4 4 G 3 3 G 2 2 G G 11/28/2020 55

Largest Eigenvalue Plots 11/28/2020 57

MATLAB beta=1; n=1 e 9; opts. disp=0; opts. issym=1; alpha=10; k=round(alpha*n^(1/3)); % cutoff parameters d=sqrt(chi 2 rnd( 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) 11/28/2020 58

Tricks to get O(n 9) 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 = 10 n 1/3 , 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. ) 11/28/2020 59

Open Problems The distribution for general beta Seems to be governed by a convection-diffusion equation 11/28/2020 61