Fang Gong gongfang@ucla.edu Homework 3 Fang Gong gongfang@ucla.edu.

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Fang Gong gongfang@ucla.edu Homework 3 Fang Gong gongfang@ucla.edu

(G+ sC)x(s) = Bu(s) (Laplace(s) domain) PRIMA review Any RLC circuit can be represented by a first order differential equation G x(t) + C = B u(t) (G+ sC)x(s) = Bu(s) (Laplace(s) domain) Can we reduce the equation size? Reduce the number of variables (column # of G and C) Reduce the number of equations (row # of G and C) dx(t) dt

Projection Framework qxn G qxq nxn nxq

Intuitive view of Krylov subspace choice for Uq Taylor series expansion: A=-G-1C, R=G-1B change base and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point

Orthonormalization of Uq:The Arnoldi Algorithm For i = 1 to q-1 Generates k+1 vectors! Orthogonalize new vector: Remove the projection on other normalized vectors For j = 1 to i end Normalize new vector end 5

Combine point and moment matching: multipoint moment matching Multiple expansion points give larger band Moment (derivates) matching gives more accurate behavior in between expansion points 6

We know how to select Uq now… but how about Vq?

PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98) Select Vq=Uq with Arnoldi Krylov Projection Framework: Use Arnoldi: Numerically very stable

Moment Matching Theorem PRIMA preserves the moments of the transfer function up to the q-th order, i.e., Original System Reduced System 9

Homework 3 [due Feb 6 ] [3] Modify the PRIMA code with single frequency expansion to multiple points expansion. You should use a vector fspan to pass the frequency expansion points. Compare the waveforms of the reduced model between the following two cases: 1. Single point expansion at s=1e4. 2. Four-point expansion at s=1e3, 1e5, 1e7, 1e9. 10

Matlab Files We provide two matlab files: prima.m PRIMA on single point expansion demo2_11.m perform single-point MOR, calculate and compare corresponding time and frequency domain response between original matrix and MATLAB reduced matrix. prima function is called.

Format of the input matrices for test 1 3 -0.000542882 0.0 1 4 0.000152585 -7.5288e-15 1 5 0.000464074 -2.9702e-15 1 6 -0.000542801 0.0 1 68 -19.4595 0.0 2 1 0.0 -2.9702e-15 2 2 3.66672 2.44291e-13 2 3 0.0 -2.3594e-13 2 4 0.0 -5.3806e-15 2 72 -1.425 0.0 2 329 -2.06075 0.0 2 341 -0.091255 0.0 2 343 -0.0897199 0.0 3 1 -2.44188e-06 0.0 3 2 -0.000464141 -2.3594e-13 3 3 40.8898 2.42089e-13 ….. The input files GC8 and GC9 each has 4 columns. They are: row number m, column number n, (m,n) entry in G matrix - G(m,n), (m,n) entry in C matrix - C(m,n) . If both G(m,n) and C(m,n) are zero, that entry is omitted in input file. 12

Frequency and time domain response for single point expansion

Impulse response for single point expansion