A Technique to Accelerate Vector Fitting Algorithm for Interconnect Simulations

1. A Technique to Accelerate Vector Fitting Algorithm for Interconnect Simulations Institute of Design Problems in Microelectronics (IPPM RAS) M. Gourary, S. Rusakov, S. Ulyanov, M. Zharov

2. Introduction. Rational approximation by Vector Fitting (VF) method Structure of VF linear system QR factorization of VF matrix Solving under simple orthogonalization Algorithm with reorthogonalization Testing results Conclusions Outline

3. Introduction. Rational approximation by Vector Fitting method The problem: approximation of experimental/simulated data of linear device by vector transfer functions (TF) by rational functions Classical least squares (LS) method: fn(sk) – the value of n-th TF at k-th frequency Overdetermined linear system to define amn, bm: (LS problem): Shortcomings of LS method: the matrix is poorly conditioned

4. Introduction. Rational approximation by Vector Fitting method B. Gustavsen and A. Semlyen, IEEE Trans. Power Delivery, vol. 14, no. 3, 1999 VF approximates rational TF defined by partial fractions m = 1…Np – pole index rmn, dn, hn, pm – approximation paramerters The problem is linear with respect to residues rmn, dn, hn nonlinear with respect to poles pm

5. Introduction. Rational approximation by Vector Fitting method VF iterations by the usage of -function: 1) Solve with respect to 2) Determine new poles as zeroes of -function: 3) Repeat 1-2 with p = pNEWuntil convergence

6. Structure of VF linear system for full set of measured/simulated data Block structure of VF system Sizes of blocks

7. QR factorization of VF matrix Step 1. QR factorization of block Blocks q, r provide QR factorization of submatrix AF : Sizes of blocks

8. QR factorization of VF matrix Factorized matrix after Step 1: (identity matrix) Step 2. Orthogonalize matrix AF with respect to Qσ

9. QR factorization of VF matrix Factorized matrix after Step 2: Step 3. QR factorization of matrix Bσ : QR factorized matrix:

10. Solving of VF system Solution:

11. Solving of VF system Insufficient accuracy of the orthogonalization in step 2. Maximal entry of matrix The reorthogonalization essentially improves accuracy of the orthogonalization.

12. Algorithm with reorthogonalization The structure of Q is the same as the structure of A

13. Testing results (Transmission line) SizeCPU Time sec.Residual norm

14. Testing results (TF with known poles) SizeCPU Time sec.Residual norm

15. new algorithm exploiting the special form of Vector Fitting (VF) matrix has been developed the effectiveness of new algorithm is achieved by the replacement of full matrix QR decomposition by the sequence of block operations it is shown that to provide sufficient accuracy of the solver the reorthogonalization procedure is needed new solver essentially reduces CPU time and provides less memory requirements and better accuracy (less residual norm) than the standard solver the advantages of new solver increase with increasing the size of the system Conclusions