P. Feldmann R. W. Freund. of a linear circuit via the Lanczos process [5]. This. it produces. The computational cost per order of approximation

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1 Ecient Linear Circuit Analysis by Pade Approximation via the Lanczos Process P Feldmann R W Freund AT&T Bell Laboratories AT&T Bell Laboratories Murray Hill, NJ 09{03 Murray Hill, NJ 09{03 Abstract This paper describes a highly ecient algorithm for the iterative computation of dominant poles and zeros of large linear networks The algorithm is based on a new implementation of the Pade approximation via the Lanczos process This implementation has superior numerical properties, maintains the same computational eciency as its predecessors, and provides a bound on the approximation error 1 Introduction Circuit simulation tasks, such as the accurate prediction of interconnect eects at the board and chip level, or analog circuit analysis with full accounting of parasitic elements, reuire the solution of very large linear networks The use of SPICE-like simulators, would be inecient oreven prohibitive for such large problems In the last few years, the Asymptotic Waveform Evaluation algorithm (AWE) [1, 2] based on Pade approximation [3] has emerged as the method of choice for the ecient analysis of large linear circuits AWE is based on approximating the Laplace-domain transfer function of a linear network by a reduced-order model, containing only a relatively small number of dominant poles and zeros Such reduced-order models can be used to predict the timedomain or freuency-domain response of the linear network over a predetermined range of excitation freuencies AWE, however, suers from a number of fundamental numerical limitations In particular, each run of AWE produces only a fairly small number of accurate poles The proposed remedial techniues are sometimes heuristic, hard to apply automatically, and may be computationally expensive Another shortcoming of AWE is its inability to estimate the accuracy of the approximating reduced-order model [2, ] In this paper, we introduce a new, numerically stable algorithm that computes the Pade approximation of a linear circuit via the Lanczos process [5] This algorithm, called PVL (Pade Via Lanczos), can be used to generate an arbitrary number of poles and zeros with little numerical degradation Moreover, PVL computes a uality measure for the poles and zeros it produces The computational cost per order of approximation is practically the same as for AWE The paper is organized as follows In Section 2, we review system-order reduction by Pade approximation In Section 3, we demonstrate the numerical limitations of existing algorithms In Section, we derive the new PVL algorithm and discuss some of its properties In Section 5, we present results of numerical experiments with PVL for a variety of examples 2 System Reduction by Pade Approx Using any standard circuit-euation formulation method such as modied nodal analysis, sparse tableau, etc [], a lumped, linear, time-invariant circuit can be described by the following system of rst-order dierential euations: C _x = y = l T x: Gx + bu; (1) Here, the vector x represents the circuit variables, the matrix G represents memoryless elements, such as resistors, C represents memory elements, such as capacitors and inductors, y is the output of interest, and bu represents excitations from independent sources We are interested in determining the impulse-response of the linear circuit with zero initial-conditions, which, in turn, can be used to determine the response to any excitation We apply the Laplace transform to the system (1) and obtain scx = Y = l T X; GX + bu; (2) where X, U, and Y denote the Laplace transform of x, u, and y, respectively It follows from (2) that the Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery To copy otherwise, or to republish, reuires a fee and/or specic permission (c) 199 ACM /9/

2 Laplace-domain impulse response, or transfer function, dened as H(s) =Y(s)=U(s), is given by H(s) =l T (G+sC) 1 b: (3) Let s 0 2 C be an arbitrary, but xed expansion point such that the matrix G + s 0 C is nonsingular Using the change of variables s = s 0 + and setting A = (G + s 0 C) 1 C; r =(G+s 0 C) 1 b; () we can rewrite (3) as follows: H(s 0 + ) =l T (I A) 1 r: (5) When matrix A is diagonalizable, we obtain H(s 0 + ) = {z} l T S (I ) = f T {z } = g 1 S 1 r ; () where A = SS 1, = diag( 1 ; 2 ;:::; N )isadiagonal matrix whose diagonals elements are the eigenvalues of A, and the matrix S contains the corresponding eigenvectors as columns From (), we get H(s 0 + ) = NX f j g j 1 j ; () where f j and g j are components of f and g The numerical computation of all eigenvectors and eigenvalues of the matrix A becomes prohibitive as soon as its size reaches a few hundreds, and therefore, the only practical way to obtain an expression for the impulse response is through an approximation For each pair of integers p; 0, the Pade approximation (of type (p=)) to the network impulse response H(s 0 + ) is dened as the rational function H p; (s 0 + ) = b p p ++b 1 +b 0 a ++a 1 +1 (8) whose Taylor series about = 0 agrees with the Taylor series of H(s 0 + ) in the rst p terms, ie, H p; (s 0 + ) =H(s 0 +)+O(s p++1 ): The coecients a 1 ;:::;a ;b 0 ;b 1 ;:::;b p of the Pade approximation (8) are uniuely determined by the rst p Taylor coecients of the impulse response The roots of the denominator and numerator polynomials in (8) represent the dominant poles and zeros of the system, respectively In the context of impulse-response approximations, it is very natural to choose p = 1 in (8), so that the Pade approximation is of the same form as the original impulse response In the following, we refer to H := H 1; as the th Pade approximant to the impulse response H By using a partial-fraction decomposition, we can write H in the form H (s 0 + ) = X k j p j : (9) The Taylor coecients necessary for the Pade approximant H result from the following expansion of H(s) about s 0 : H(s 0 + ) =l T where X 1 I+A+ 2 A 2 + r= m k k ; k=0 m k = l T A k r; k =0;1::: : (10) Note that in the case when the expansion point is chosen as the origin, ie, s 0 = 0, the coecients m k, are, up to a constant factor, the time-domain moments of the circuit response Because of this analogy, we will always refer to the Taylor coecients (10) as the moments of the impulse-response function H(s 0 + ) 3 Limitations of Current Algorithms In AWE, the Pade approximant H is obtained via explicit computation of the leading 2 moments, m k, of H To this end, one rst generates the vectors u 0 = r; u 1 = Ar; u 2 = A 2 r;:::;u 2 1 = A 2 1 r by recursive solution of the linear systems (G + s 0 C) u k = Cu k 1 ; k =1;2;:::;2 1; (11) with the initial vector u 0 =(G+s 0 C) 1 b Observe that the recursive computation of the vectors u k can be performed very eciently because the matrix G + s 0 C is LU-factored exactly once The moments are then computed as m k = l T u k, k =0;1;2;:::;2 1 As the next step, AWE computes the coecients of the denominator polynomial of the representation (8) of H via solution of the linear system 2 a 3 2 m 3 a M 1 m +1 a 1 5 = m ; (12) where M =[m j+k 2 ] j;k=1;2;:::; is the so-called moment matrix The poles p j of H in (9) are then obtained as the roots of the denominator polynomial Finally, the residues k j in (9) are computed by solving another linear system of order, see, eg, []

3 Voltage gain (db) Exact AWE, 2 iter AWE, 5 iter AWE, 8 iter (M ) (M (1) ) (M (2) ) (M (3) ) 2 959e+05 22e+00 22e+00 10e e+15 51e e+03 21e e+2 113e e e e+35 23e e e+10 38e+ 22e+1 238e+1 599e e e+18 18e+18 2e e+2 15e+18 50e+1 538e+1 Table 1: Condition numbers of moment matrices for simulation of voltage gain with AWE Freuency (Hz) Figure 1: Results for simulation of voltage gain with AWE As is increased, one would expect more and more accurate approximations H of the exact impulse response H Unfortunately, this is not the case when the Pade approximant H is generated with AWE Indeed, typically H improves only for values of up to about 10, and after that the process stagnates Figure 1 illustrates this behavior Here, we tried to simulate the voltage gain of a lter with our own implementation of AWE written in MATLAB TM [] We show the exact voltage gain and the approximations generated by AWE for =2;5;8 The results for >8 showed no further improvement The severe numerical problems with the AWE approach of obtaining Pade approximants via direct computation of the moments can be explained as follows The generation of the vectors u k = A k r in (11) corresponds to vector iteration with the matrix A, and this process converges rapidly to an eigenvector corresponding to an eigenvalue of A with largest absolute value As a result, the moments m k = l T u k contain only information corresponding to one eigenvalue of A, even for fairly small values of k As a result, the moment matrix M becomes rapidly ill-conditioned The condition number, (M ), of the matrix M is a measure for how round-o error aects the accuracy of the numerically computed solution of (12) Each increase of (M )by a factor of 10 signals the loss of one decimal digit of accuracy in the computed solution In the rst column of Table 1, we list (M ) for the moment matrices corresponding to the simulation of the voltage gain of a lter Clearly, the moment matrices are extremely ill-conditioned Scaling of moments, proposed as remedy to this problem, in [2] and [], provides only a modest improvement in the matrix condition number In Table 1, we also list the condition numbers of the scaled moment matrices using three proposed scaling strategies The PVL Algorithm We now describe the PVL algorithm that exploits the intimate connection [8] between Pade approximation and the Lanczos process to elude the direct computation of the moments First, we recall the classical Lanczos process [5] and some of its key properties Algorithm 1 (Lanczos algorithm [5]) 0) Set v = r, w = l, v 0 = w 0 = 0, and 0 =1 For n =0;1;:::; do : 1) Compute n+1 = kvk 2 and n+1 = kwk 2 If n+1 =0or n+1 = 0, then stop 2) Set v n+1 = v n+1 ; w n+1 = w n+1 ; (13) n = w T nv n ; n = wt nav n n ; (1) n = n n n 1 ; n = n n n 1 ; (15) v = Av n v n n v n 1 n ; (1) w = A T w n w n n w n 1 n : (1) We remark that in Algorithm 1 a breakdown will occur if one encounters n =0oreven n 0 in (1) Therefore, our implementation of the PVL algorithm employs the look-ahead Lanczos algorithm described in [9] that remedies the breakdown problem The uantities generated by Algorithm 1 have the following properties:

4 1 The vectors fv n g +1 and fw n=1 ng +1 n=1 are biorthogonal: w T j ; if j = k, j v k = (18) 0; if j = k, for all j; k =1;2;:::;+1 Thus, setting V =[v 1 v 2 v ]; W =[w 1 w 2 w ]; wehaved =W T V = diag( 1 ; 2 ;:::; ) 2 The tridiagonal matrices ~T and T dened by T = ; ~T = satisfy ~ T T = D T D 1 3 The matrices ~T and T have the following relation to the matrix A: AV = V T +[00 v +1 ] +1 ; A T W = W ~T +[0 0w +1 ] +1 : (19) Next, we demonstrate the connection of the Lanczos process to Pade approximation Using the relations in (19) and the fact that T is tridiagonal, one can show that, for all j =0;1;:::; 1, A j r = 1 A j v 1 = 1 A j V e 1 = 1 V T j e 1 ; (20) l T A j = 1 1 e T 1 T j D 1 W T ; where e 1 =[1 0 0] T is the rst unit vector On the other hand, by (10), each moment m k can be written as follows: m k = l T A k r = lt A k0 A k00 r ; (21) where k = k 0 + k 00 If k 2 2, we can nd 0 k 0 ;k 00 1, and from (20){(21) it follows that m k = e T 1 T k0 D 1 W T V {z T } k 00 e 1 : = D Note that = l T r, and thus, from (22), we get (22) m k =(l T r) e T 1T k e 1 ; k=0;1;:::;2 2: (23) Furthermore, it can be shown that the relation (23) also holds for k =2 1 Thus, by (23), we have l T r 1X k=0 e T 1 T k e 1 k = 2 1 X k=0 m k k + O( 2 ); (2) and conseuently, H (s 0 + ) =l T re T 1 (I T ) 1 e 1 (25) is just the th Pade approximant ofh In analogy to the representation () of the exact impulse response H, we can rewrite the expression (25) of H in terms of the eigendecomposition T = S S 1 of the Lanczos matrix T : H (s 0 + ) =l T r e T 1S = X {z } = T l T r j j 1 j : (I ) 1 S 1 e 1 {z } = (2) Here, = diag( 1 ; 1 ;:::; ) contains the eigenvalues of T, and j and j are the components of the vectors and Finally, from (2), we obtain the pole/residue representation of the Pade approximant: H (s 0 + ) = X j =0 l T r j j = j 1= j : (2) The previous derivation shows that the Pade approximant H can be obtained by running the Lanczos algorithm and by computing an eigendecomposition of the Lanczos matrix T The resulting computational procedure is the PVL algorithm Algorithm 2 (Sketch of the PVL algorithm) 1) Run steps of the Lanczos process (Algorithm 1) to obtain the tridiagonal matrix T 2) Compute an eigendecomposition T = S diag( 1 ; 1 ;:::; )S 1 (28) of T, and set = S T e 1 and = S 1 e 1 3) Compute the poles and residues of H by setting p j =1= j and k j = lt r j j j j =1;2;:::: for all (29) We remark that the PVL algorithm and AWE reuire roughly the same amount of computational work Similar to AWE, the dominating cost is the computation of the LU factorization of the matrix G + s 0 C which needs to be computed only once The vectors Av n and A T w n reuired in step 2) of Algorithm 1

5 Voltage gain (db) Exact PVL, 2 iter PVL, 8 iter PVL, 28 iter Freuency (Hz) Figure 2: Results for simulation of voltage gain with PVL are then obtained using forward-backward substitutions PVL reuires 2 substitutions to generate the th Pade approximant, exactly as AWE As a rst example, we reran the simulation of the voltage gain of the lter, now with the PVL algorithm instead of AWE The results of three runs with = 2;8;28 are shown in Figure 2 In contrast to the simulation with AWE (cf Figure 1), the PVLgenerated Pade approximant for = 28 gives a perfect match of the exact voltage gain The zeros of the reduced-order model H can also be computed easily from the Lanczos matrix T In fact, it can be shown that zeros of H are just the inverses of the eigenvalues of T 0, the matrix obtained from T by deleting the rst row and column Finally, we briey sketch how the PVL algorithm can be used to obtain bounds for the pole approximation error Recall that, by (29), the poles of the th Pade approximant H are just the inverses of the eigenvalues of the Lanczos matrix T, On the other hand, in view of (), the poles of the exact impulse response H are the inverses of the eigenvalues of A Therefore, a uality measure for the poles of H can be obtained by checking how well the eigenvalues of T approximate the eigenvalues of A It can be shown that Az j j z j = v s j, where s j is the last component of the vector s j By taking norms, and since kv +1 k 2 = 1, it follows that kaz j j z j k 2 kak 2 ks j k 2 +1js j j n(a) ks j k 2 =Q j : (30) where n(a) is an estimate of kak 2, easily obtained from the algorithm Thus the number Q j represents a measure of how well the pair ( j ; z j ) approximates an eigenpair of the matrix A, and conseuently a uality measure of the approximate pole 1= j produced by the PVL algorithm Note that Q j can easily be computed from the uantities generated by PVL 5 Discussion and Examples The Pade approximation generates poles that correspond to the dominant poles of the original system and a few poles that do not correspond to poles in the original system, but account for the eects of the remaining original poles The true poles can be identied using the bound presented in Section and by comparing the poles obtained at consecutive iterations of the algorithm The true poles that have converged should not change signicantly between iterations The rst example is a lumped-element euivalent circuit for three-dimensional electromagnetic problem modeled via PEEC [10] (partial element euivalent circuit) The circuit consists of 2100 capacitors, 12 inductors, and 990 inductive couplings, resulting in a 3030 fairly dense MNA matrix The Pade approximation generated by AWE, reproduces the transferfunction of the euivalent circuit accurately up to 1GHz [11] In [12], Chiprout et al, through the use of multi-point moment matching, obtain a sucient number of accurate poles and zeros to extend the validity of the approximation up to 5GHz However, their method involves several complex circuit matrix factorizations and, therefore, is signicantly more expensive computationally We applied our PVL algorithm to the same circuit and, after 0 iterations, obtained a reduced-order system with a better match up to 5GHz than the one obtained from multi-point moment matching (Figure 3) Moreover, since PVL reuires only one real circuit matrix factorization, the cost of the computation is similar to AWE The second example models a complete power grid for a standard cell mixed signal ASIC, including some of the substrate contacts and substrate coupling/decoupling, as described in [13] The model contains 10 power bus segments, 3 models for cells, and a coarse, substrate grid The resulting MNA matrix has a size of 1 1 We are interested to determine the eects of the switching current in cells on the VDD and GND rails Figure shows the magnitude of the corresponding transfer function produced by the PVL algorithm in 50 iterations compared to the same transfer function produced by an ACsweep The agreement is perfect

6 Current (Amps) Exact PVL, 0 iter Current gain (db) AC analysis PVL, 50 iter Freuency (GHz) Freuency (Hz) Figure 3: Results for the PEEC circuit, 0 PVL iterations Conclusions This paper introduces PVL, a novel algorithm for linear circuit analysis based on the connection between the Lanczos process and the Pade approximation Its superior numerical stability allows the computation of more accurate and higher order Pade approximations with no sacrice in eciency As a conseuence the usefulness of Pade approximation based linear circuit analysis techniues is considerably extended Acknowledgments We would like to thank R Melville, R Rutenbar, H Heeb, and B Stanisic for providing us with interesting circuit examples We also acknowledge the original AWE developers, L Pillage, X Huang, and R Rohrer, who demonstrated the usefulness of the Pade approximation in the analysis of large linear circuits References [1] LT Pillage and RA Rohrer, \Asymptotic waveform evaluation for timing analysis," IEEE Trans Computer-Aided Design, vol 9, pp 352{3, Apr 1990 [2] X Huang, \Pade approximation of linear(ized) circuit responses," PhD diss, Dept of Electrical and Computer Engineering, Carnegie Mellon Univ, Pittsburgh, PA, Nov 1990 [3] GA Baker, Jr and P Graves-Morris, Pade Approximants, Part I: Basic Theory Reading, MA: Addison- Wesley, 1981 [] E Chiprout and MS Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Figure : Results for the mixed-signal circuit magnitude, 50 PVL iterations Analysis Norwell, MA: Kluwer Academic Publishers, 199 [5] C Lanczos, \An iteration method for the solution of the eigenvalue problem of linear dierential and integral operators," J Res Nat Bur Standards, vol 5, pp 255{282, 1950 [] J Vlach and K Singhal, Computer Methods for Circuit Analysis and Design New York, NY: Van Nostrand Reinhold, 1983 [] MATLAB User's Guide The MathWorks, Inc, Natick, MA, 1992 [8] WB Gragg, \Matrix interpretations and applications of the continued fraction algorithm," Rocky Mountain J Math, vol, pp 213{225, 19 [9] RW Freund, MH Gutknecht, and NM Nachtigal, \An implementation of the look-ahead Lanczos algorithm for non-hermitian matrices," SIAM J Sci Comput, vol 1, pp 13{158, Jan 1993 [10] AE Ruehli, \Euivalent circuit models for threedimensional multiconductor systems," IEEE Trans Microwave Theory and Tech, vol 22, pp 21{221, Mar 19 [11] H Heeb, AE Ruehli, JE Bracken, and RA Rohrer, \Three dimensional circuit oriented electromagnetic modeling for VLSI interconnects," in Proc Int Conf Computer Design: VLSI in Computers & Processors, Oct 1992 [12] E Chiprout, H Heeb, MS Nakhla, and AE Ruehli, \Simulating 3-D retarded interconnect models using complex freuency hopping (CFH)," in Tech Dig IEEE/ACM Int Conf Computer-Aided Design,Nov 1993 [13] BR Stanisic, RA Rutenbar, and LR Carley, \Mixed-signal noise-decoupling via simultaneous power distribution design and cell customization in rail," in Proc IEEE Custom Integrated Circuits Conf, 199