1. Circuit Simulation via Matrix Exponential Operators CK Cheng UC San Diego 1

2. Outline General Matrix Exponential Krylov Space and Arnoldi Orthogonalization Matrix Exponential Method – Krylov Subspace Approximation – Invert Krylov Subspace Approximation – Rational Krylov Subspace Approximation 2

3. General Matrix Exponential 3

4. Krylov Space and Arnoldi Orthonormalization Input A and v 1 =x 0 /|x 0 | Output AV=VH+h m+1 v m+1 e m T For i=1, …, m T i+1 =Av i For j=1, …, I – h ji = – T i+1 =T i+1 -h ji v j End For h i+1,i =|T i+1 | v i+1 =1/h i+1 T i+1 End For 4

5. Standard Krylov Space Generate: AV=VH+h m+1 v m+1 e m T Thus, we have e Ah v 1 ≈ Ve Hh e 1 Residual r=Cdx/dt-Gx=-h m+1 Cv m+1 e m T e Hh e 1 Derivation: Cdx/dt-Gx=CVHe Hh e 1 -GVe Hh e 1 =(CVH-GV)e Hh e 1 = C(VH-C -1 GV)e Hh e 1 =C(VH-VH-h m+1 v m+1 e m T )e Hh e 1 =-h m+1 Cv m+1 e m T e Hh e 1 5

6. Standard Mexp Error trend 6 sweep m and h

7. Invert Krylov Space Generate: A -1 V=VH+h m+1 v m+1 e m T Let H=H -1, we have e Ah v 1 ≈ Ve Hh e 1 Residual r=Cdx/dt-Gx=h m+1 Gv m+1 e m T He Hh e 1 Derivation: Cdx/dt-Gx=CVHe Hh e 1 -GVe Hh e 1 =(CVH-GV)e Hh e 1 = G(G -1 CVH-V)e Hh e 1 =G(A -1 VH-V)e Hh e 1 =h m+1 Gv m+1 e m T He Hh e 1 7

8. large step size with less dimension Invert Matrix Exponential 8 sweep m and h

9. Rational Krylov Space Generate: (1-rA) -1 V=VH+h m+1 v m+1 e m T Let H=1/r (I-H -1 ) we have e Ah v 1 ≈ Ve Hh e 1 Residual r=Cdx/dt-Gx=-h m+1 (C/r-G)v m+1 e m T H -1 e Hh e 1 Derivation: Cdx/dt-Gx=CVHe Hh e 1 -GVe Hh e 1 =(CVH-GV)e Hh e 1 = (1/r CV(I-H -1 )-GV)e Hh e 1 =(1/rCV(H-1)-GVH)H -1 e Hh e 1 =((1/rC-G)VH-1/rCV)H -1 e Hh e 1 =-h m+1 (C/r-G)v m+1 e m T H -1 e Hh e 1 9

10. large step size with less dimension Rational Matrix Exponential 10 fix , sweep m and h

11. Different  11 needs large m

12. Different  12

13. Spectral Transformation –  = 10f Small RC mesh, 100 by 100 Different h for Krylov subspace Different  for rational Krylov subspace 13

14. Spectral Transformation–  = 1p Small RC mesh, 100 by 100 Different h for Krylov subspace Different  for rational Krylov subspace 14

15. Spectral Transformation–  = 100p Small RC mesh, 100 by 100 Different h for Krylov subspace Different  for rational Krylov subspace 15

16. Sweep  for Large Range 16

17. Sweep  for Large Range 17

18. Difference Between Inverted and Rational 18

19. Fixed  = 1p, sweep time step h 19

20. Fixed  = 1n, sweep time step h 20

21. Fixed  = 1u, sweep time step h 21

22. Fixed  = 1m, sweep time step h 22

23. Fixed  = 1, sweep time step h 23

24. Fixed  = 1k, sweep time step h 24

25. Fixed  = 1M, sweep time step h 25

26. Krylov Space Residual Generate: AV=VH+h m+1 v m+1 e m T Thus, we have e Ah v 1 ≈ Ve Hh e 1 Residual r=Cdx/dt-Gx=-h m+1 Cv m+1 e m T e Hh e 1 Derivation: 1. Set Y=[e 1 He 1 H 2 e 1 … H m-1 e 1 ] 2. We have YC=HY where C= z m +c m-1 z m-1 +…+c 1 z+c 0 =0 has roots 1, 2,… m 26 0-c 0 10-c 1 10-c 2 ……… 10-c m-2 1-c m-1

27. Krylov Space Residual 27 1 1. 1 m-1 1 2. 2 m-1........... 1 m. m m-1

28. Invert Krylov Space Residual 28