Method of determining high-speed VLSI reduced-order interconnect by non-symmetric lanczos algorithm

Abstract

Two-sided projection-based model reductions has become a necessity for efficient interconnect modeling and simulations in VLSI design. In order to choose the order of the reduced system that can really reflect the essential dynamics of the original interconnect, the element of reduced model of the transfer function can be considered as a stopping criteria to terminate the non-symmetric Lanczos iteration process. Furthermore, it can be found that the approximate transfer function can also be expressed as the original interconnect model with some additive perturbations. The perturbation matrix only involves at most rank-2 modification at the previous step of the non-symmetric algorithm. The information of stopping criteria will provide a guideline for the order selection scheme used in the Lanczos model-order reduction algorithm.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to a reduced-order circuit model, and more particularly to a rapid and accurate reduced-order interconnect circuit model which can be used for signal analysis of high-speed and very-large IC interconnect.

2. Description of Related Art

With rapid development of semiconductor techniques, the parasitic effect has no longer been ignored during design of high-speed and very-large IC interconnect. This technology was proposed in 2002 by M. Celik, L. T. and A. Odabasioglu “IC Interconnect Analysis,” Kluwer Academic Publisher.

Given the fact of more complex circuit, the corresponding order of mathematical model will be increased in order to simulate accurately the characteristics of interconnect circuits. Therefore, an efficient model reduction method has become a necessary know-how for interconnect modeling and simulation. The well-proven technologies, such as U.S. Pat. Nos. 6,789,237, 6,687,658, 6,460,165, 6,135,649, 601,170, 6,023,573, are proposed in 2000 by R. W. Freund, “Krylov-Subspace Methods for Reduced-Order Modeling in Circuit Simulation,” Journal of Computational and Applied Mathematics, Vol. 123, pp. 395-421; in 2002 by J. M. Wang, C. C. Chu, Q. Yu and E. S. Khu, “On Projection Based Algorithms for Model Order Reduction of Interconnects,” IEEE Trans. on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 49, No. 1, pp. 1563-1585.

In recent years, the common methods for circuit model reduction include:

Asymptotic Waveform Evaluation (AWE)(L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, Vol. 9, No. 4, pp. 352-366, 1990);

PVL (Pade via Lanczos)(P. Feldmann and R. W. Freund, “Efficient linear circuit analysis by Pad'e approximation via the Lanczos process,” IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, Vol. 14 pp. 639-649, 1995);

SyMPVL (Symmetric Matrix Pade via Lanczos)(P. Feldmann and R. W. Freund, “The SyMPVL algorithm and its applications to interconnect simulation,” Proc. 1997 Int. Conf. on Simulation of Semiconductor Processes and Devices, pp. 113-116, 1997);

Arnoldi Algorithm (e.g. U.S. Pat. No. 6,810,506);

PRIMA (Passive Reduced-order Interconnect Macromodeling Algorithm)(A. Odabasioglu, M. Celik and L. T. Pileggi, “PRIMA: passive reduced-order interconnect macromodeling algorithm,” IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, Vol. 17 pp. 645-653, 1998).

All of the aforementioned model reduction techniques employ Krylov Subspace Projection Method, which utilizes projection operator to obtain the state variables of reduced circuit system after projecting the state variable of original circuit system. The projection operator is established by Krylov Algorithm iteration process, of which the order of reduced circuit is the number of iteration. For model reduction algorithm of applied projection method, another important job is to determine the order of reduced circuit, since it is required to find out an appropriate order such that the reduced circuit can reflect accurately important dynamic behavior of original circuit.

SUMMARY OF THE INVENTION

The present invention provides an improved non-symmetric Lanczos Algorithm. Based on error estimation of linear circuit is reduced model and original model, it should thus be possible for improvement of submicron IC interconnect model.

The present invention will present a detailed description of the relationship between original circuit system and reduced circuit system, of which the reduced circuit is to obtain project-based matrix and then reduced-order circuit model by employing non-symmetric Lanzcos Algorithm. Based on δ_q+1 and δ_q+1 calculated by the algorithm, it is required to set a termination iteration condition in order to obtain a balance point between complexity of computation and accuracy of reduced model.

In addition, the present invention will prove that, after transfer function of original circuit is added with some additive perturbations, the moment of transfer function fully matches that of reduced model by employing non-symmetric Lanczos Algorithm in various orders. Since well-proven technology has demonstrated that q-th moments of reduced system are equivalent to those of original system, so q-th order moments of original system plus perturbed system are equivalent to those of original system. Of which, perturbation matrix is related to component generated by non-symmetric Lanczos Algorithm, with the cyclomatic number up to 2, so no additional computational resources are required. The algorithm of the present invention will provide an efficient guideline of selecting reduced-order circuit by Krylov Subspace Model Reduction Algorithm.

In a certain embodiment, the present invention has simplified circuit model by employing non-symmetric Lanczos Algorithm, which includes the following steps: (1) input a mesh circuit; (2) input an expand frequency point; (3) set up a state space matrix of circuit; (4) reduce submicron IC interconnect model by employing an improved non-symmetric Lanczos Algorithm.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the pseudo code of the traditional non-symmetric Lanczos Algorithm.

FIG. 2 shows the flow process diagram of reduced circuit by employing non-Symmetric Lanczos Algorithm.

FIG. 3 shows the pseudo code of improved non-Symmetric Lanczos Algorithm

FIG. 4 shows the simplified embodiment of the present invention.

FIG. 5 shows the curve diagram of termination conditions of the present invention.

FIG. 6 shows the frequency response diagram of simplified embodiment.

FIG. 7 shows the error analysis of reduced-order models of simplified embodiment.

FIG. 8 shows the analysis pattern of order of simplified embodiment and moment value of system.

DESCRIPTION OF THE MAIN COMPONENTS

• • (102) Original system
  • (104) Reduced order q=1
  • (106) Bi-orthogonal v_q and w_q of non-symmetric Lanczos Algorithm
  • (108) Solution of δ_q+1, δ_q+1$\begin{array}{cc}{\lambda }_q=\frac{{\delta }_{q+1}}{{\mathrm{AV}}_q}\le \varepsilon \mathrm{and}\frac{{\beta }_{q+1}}{A^{\prime }{W}_q}\le \varepsilon & \left(110\right)\end{array}$
  • (112) q++
  • (114) Calculate the model of reduced system

DETAILED DESCRIPTION OF THE INVENTION

The conventional methods, such as Modified Nodal Analysis (MNA), Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL), are used for analyzing the characteristics of very-large IC interconnects. The circuits can be expressed as the following state space matrixes: $\begin{array}{cc}M\frac{dx\left(t\right)}{dt}=-\mathrm{Nx}\left(t\right)+\mathrm{bu}\left(t\right),y\left(t\right)=c^{T}x\left(t\right),& \left(1\right)\end{array}$ Where, M,NεR^n×n, x,b,cεRⁿ and y(t)εRⁿ. Matrix M comprises capacitance c and inductance L, matrix N comprises conductance G and resistance R, state matrix x(t) comprises node voltage and branch current. And, u(t) is input signal, and y(t) is output signal. Let A =−N⁻¹M and r=N⁻¹b, formula (1) can be expressed as: $\begin{array}{cc}A\frac{dx\left(t\right)}{dt}=x\left(t\right)-\mathrm{ru}\left(t\right),y\left(t\right)=c^{T}x\left(t\right),& \left(2\right)\end{array}$

Model order reduction aims to reduce the order of circuit system, and reflect efficiently the reduced circuit system of original circuit system. The state space matrix of reduced circuit can be expressed as: $\begin{array}{cc}\hat{A}\frac{d\hat{x}\left(t\right)}{dt}=\hat{x}\left(t\right)-\hat{r}u\left(t\right),\hat{y}\left(t\right)={\hat{c}}^T\hat{x}\left(t\right),& \left(3\right)\end{array}$ Where, {circumflex over (x)}(t)εR^q, ÂεR^q×q, {circumflex over (r)}, ĉεR^q and q<<n.

Let X(s)=L[x(t)] and {circumflex over (X)}(s)=L[{circumflex over (x)}(t)] are pulse responses of original system and reduced system in Laplace Domain, X(s) and {circumflex over (X)}(s) can be expressed as follows: X(s)=(I_n−sA)⁻¹,{circumflex over (X)}(s)=(I_q−sÂ)⁻¹{circumflex over (r)}  (4) Where, I_n is unit matrix of n×n, and I_m is unit matrix of q×q. The transfer function H(s) of original system and transfer function Ĥ(s) of reduced system can be expressed separately as: H(s)=c^TX(s)=c^T(I_n−sA)⁻¹r  (5) And Ĥ(s)=ĉ^T{circumflex over (X)}(s)=ĉ^T(I_q−sÂ)⁻¹{circumflex over (r)}  (6) Modeling Reduction Technique

To calculate the reduced model of very-large IC interconnects, well-known non-symmetric Lanczos Algorithm (P. Feldmann and R. W. Freund, “Efficient Linear Circuit Analysis by Pade Approximation via Lanczos Process”, IEEE Trans. on CAD of ICS, Vol. 14, No. 5, 1995) is employed to set up two projection matrixes V_q and W_q, and generate reduced models by two-sided projections, with the pseudo code of the algorithm as shown in FIG. 1. The algorithm is required to provide an order q of reduced model. To keep the characteristic consistency of reduced model and original system, it is required to increase the order q. However, in order to minimize the computational complexity in system simulation, it is required to reduce the order q. To address the aforesaid tradeoff, the present invention attempts to improve original non-symmetric PVL, and judge the iteration termination conditions during computation. It aims to realize a maximum accuracy nearby expand frequency point under the lowest level of computational complexity, with the improved flow process as shown in FIG. 2. With input parameters of various passive components in original circuit in Step (102), it is possible to establish the corresponding Modified Nodal Analysis Equation for comparison of reduced circuit model. In Step (104), projection technique of original circuit is applied to generate a reduced-order system by firstly setting the order of reduced model q=1. In Step (106), non-symmetric Lanczos Algorithm is used to input matrix A and its transpose matrix as well as two original vectors b and c′, thereby obtaining bi-orthogonal matrix V_q=└v₁,v₂,∀,v_q┘ and W_q=└w₁,w₂,∀,w_q┘, namely: W_q′V_q=I, of which IεR^q×q. Moreover, V_q exists in Krylov Subspace K_q(A,b)=span{└b Ab A²b ∀ A^q−1b┘}, which can be developed from the Basis of K_q. Meanwhile, W_q exists in Krylov Subspace L_q(A′,c′)=span{└c′ (A′)c′ (A′)²c′ ∀ (A′)^q−1c′┘}, which can be developed from the Basis of L_q. Furthermore, every iteration process will yield new bi-orthogonal vectors v_q and w_q. In addition, original system's matrix A can be reduced to a tri-diagonal matrix according to non-symmetric Lanczos Algorithm: $T_q=\left[\begin{array}{cccccc}{\alpha }_1& {\beta }_2& & & & \\ {\delta }_2& {\alpha }_2& {\beta }_3& & & \\ & {\delta }_3& {\alpha }_3& {\beta }_4& & \\ & & {\delta }_4& {\rm O}& {\rm O}& \\ & & & {\rm O}& {\rm O}& {\beta }_q\\ & & & & {\delta }_q& {\alpha }_q\end{array}\right]$

The calculation of the aforesaid matrixes will satisfy: AV_q=V_qT_q+δ_q+1v_q+1e_q′  (7) A′W_q=W_qT_q′+β_q+1w_q+1e_q′  (8) and W_q′AV_q=T_q, where e_q is q-th row vector in unit matrix IεR^q×q.

Owing to the error of moment between reduced model and original model, formula (7) and (8) can be expressed as the following three-term recursive equations in order to reduce efficiently the computation of system analysis and minimize the error: Av_q=β_qv_q−1+α_qv_q+δ_q+1v_q+1  (9) A′w_q=δ_qw_q−1+α_qw_q+β_q+1v_q+1  (10)

Since δ_q+1 can be treated as the component of new vector v_q+1 in AV_q, and β_q+1 treated as the component of new vector w_q+1 in A′W_q, Step (108) selects δ_q+1 and β_q+1 as a reference indicator. Similarly, Step (110) takes ${\lambda }_q=\frac{{\delta }_{q+1}}{{\mathrm{AV}}_q},{\mu }_q=\frac{{\beta }_{q+1}}{A^{\prime }{W}_q}$ as an indicator for terminating the iteration process. Assuming that λ_q and μ_q are less than tolerances for termination conditions, the reduced system will be very similar to original system. If above-specified conditions are not met, the order of reduced model will be gradually increased in Step (112). Every iteration will generate new bi-orthogonal vectors v_q and w_q as well as new δ_q+1 and β_q+1. When both λ_q and μ_q meets the conditions as specified in Step (110), non-symmetric Lanczos Algorithm iteration process will be stopped, in such case q is an optimal order of reduced model. In Step (114), order q is used for reduction of system model. Addition of Perturbed System

In original interconnect circuit system, Additive Perturbation Matrix can be added to demonstrate the reliability of this method. Suppose that the circuit is Modified Nodal Analysis Equation is: $\begin{array}{cc}\left(A-\Delta \right)\frac{d{x}_{\Delta }\left(t\right)}{dt}=x_{\Delta }\left(t\right)-\mathrm{ru}\left(t\right),y_{\Delta }\left(t\right)=c^T{x}_{\Delta }\left(t\right)& \left(11\right)\end{array}$ Where, Δ represents additive perturbations of system: Δ=Δ₁+Δ₂  (12) Where, Δ₁=v_q+1δ_q+1w_q′, Δ₂=v_qβ_q+1w_q+1′, and q is the order of reduced model.

The transfer function of original circuit is reduced model can be expressed as Ĥ(s), as shown in formula (6). Under the condition of formula (12), the transfer function H_Δ(S) of original system plus perturbed system will be equal to transfer function Ĥ(s) of reduced system. Let expend frequency point s=s₀+σ, and l′r=(β₁w₁)′δ₁v₁=β₁δ₁(w₁′v₁)=β₁δ₁, the transfer function of reduced model can be simplified as: $\begin{array}{cc}\begin{array}{c}\hat{H}\left(s_0+\sigma \right)={{\hat{l}}^{\prime }\left(I_q-\sigma \hat{A}\right)}^{-1}\hat{r}\\ =l^{\prime }{V_m\left(I_q-\sigma T_q\right)}^{-1}{W}_m^{\prime }r\\ ={\beta }_1{w}_1^{\prime }{V_q\left(I_q-\sigma T_q\right)}^{-1}{W}_q^{\prime }{v}_1{\delta }_1\\ =\left(l^{\prime }r\right){e_1^{\prime }\left(I_q-\sigma T_q\right)}^{-1}{e}_1,\end{array}& \left(13\right)\end{array}$ And the transfer function of perturbed system can be simplified as: $\begin{array}{cc}\begin{array}{c}{H}_{\Delta }\left(s_0+\sigma \right)={l^{\prime }\left(I_n-\sigma \left(A-\Delta \right)\right)}^{-1}r\\ =w_1^{\prime }{{\beta }_1\left(I_n-\sigma \left(A-\Delta \right)\right)}^{-1}{\delta }_1{v}_1\\ =\left(l^{\prime }r\right){w_1^{\prime }\left(I_n-\sigma \left(A-\Delta \right)\right)}^{-1}{v}_1.\end{array}& \left(14\right)\end{array}$ By using formula (7), it can be shown that various moments of reduced model via PVL method are equivalent to those of original model with additive perturbed system Δ. Firstly, subtract ΔV_q at both sides of the equation, right side of the equation can be reduced as: V_qT_q+δ_q+1v_q+1e_q′−ΔV_q=V_qT_q+δ_q+1v_q+1e_q′−(v_q+1δ_q+1w_q′+v_qβ_q+1w_q+1′)V_q=V_qT_q

Formula (7) can be rewritten as: AV_q−ΔV_q=V_qT_q+δ_q+1v_q+1e_q′−ΔV_q(A−Δ)V_q=V_qT_q

If we multiply −σ and add V_q at both sides of the aforesaid equation, it can be rewritten as: V_q−(A−Δ)V_q=V_q−σV_qT_q  (15)

Thus (I_n−σ(A−Δ))V_q=V_q(I_q−σT_q)  (16)

If we multiply (I_n−σ(A−Δ))⁻¹ at matrix left of the equation, and multiply (I_q−σT_q)⁻¹ at matrix right of the equation, then formula (16) can be rewritten as: V_q(I_q−σT_q)⁻¹=(I_n−σ(A−Δ))⁻¹V_q  (17)

Finally, multiply w₁′ at matrix left of the equation, e₁ at matrix right, and multiply constant l′r=β₁δ₁ at both sides, then: w₁′V_q(I_m−σT_q)⁻¹e₁=w₁′(I_n−σ(A−Δ))⁻¹V_qe₁ (l′r)e₁′(I_q−σT_q)⁻¹e₁=(l′r)w₁′(I_n−σ(A−Δ))⁻¹v₁  (18)

By comparing formula (19) and (13)/(14): Ĥ(s₀+σ)=H_Δ(s₀+σ)  (19)

The reduced model derived from the aforementioned equations demonstrates non-symmetric PVL Algorithm, where various moments of transfer function are equal to those of original system with additive perturbed system Δ.

The model reduction method of present invention for high-speed very-large IC employs an improved non-symmetric Lanczos Algorithm, with its pseudo code shown in FIG. 3.

SIMPLE EMBODIMENT OF THE INVENTION

The present invention tests a simple embodiment in order to verify the validity of proposed algorithm. FIG. 4 depicts a circuit model with 12 lines. The line parameters are: resistance: 1.0 Ω/cm; capacitance: 5.0 pF/cm; inductance: 1.5 nH/cm; driver resistance: 3Ω, and load capacitance: 1.0 pF. Each line is 30 mm long and divided into 10 sections. Thus, the dimension of MNA matrix is: n=238. Under a frequency from 0 to 15 GHZ, it should be possible to observe the frequency response of V_out node voltage of the embodiment, and set the expand frequency point of reduced model s₀=0H_Z. If non-symmetric Lanczos Algorithm is performed, record the values of β_i+1 and δ_i+1 in tri-diagonal matrix. If you continue to enable non-symmetric Lanczos Algorithm iteration process, and set the tolerance error of termination conditions ε=10⁻³, i.e. u_q<10⁻³, it is should be possible to obtain the optimal solution of accuracy and reduce computational complexity when reduced model is q=14. FIG. 5 depicts the curve diagram of termination conditions ${\lambda }_q=\frac{{\delta }_{q+1}}{{\mathrm{AV}}_q},{\mu }_q=\frac{{\beta }_{q+1}}{A^{\prime }{W}_q}$ during computational process of algorithm. FIG. 6 shows the frequency response diagram of the reduced model, where H(s), Ĥ(s) and H_Δ⁻(s) represent respectively the transfer function of original circuit, the transfer function of system after performing non-symmetric Lanczos order-reduction method, and the transfer function of original circuit with additive perturbed system. As shown in FIG. 6, the analysis of three perturbations involves Δ₁, Δ₂ and Δ perturbed systems. FIG. 7 analyzes the error between three perturbed systems and applied non-symmetric Lanczos Algorithm. It can thus be found that, perturbed system Δ will vary from the reduced-order system, and maintain a high-level consistency with non-symmetric Lanczos Algorithm. As illustrated in FIG. 8, the moment values of system are observed. With the increase of order, it can be found that the moment values are far less than the floating accuracy of operating system (EPS, about 2.22e-16). So, the error arising from inaccuracy of operational factors may be ignored.

In brief, the present invention has derived very-large RLC interconnect, and implemented the model reduction method by employing non-symmetric Lanczos Algorithm, thereby helping to judge automatically the order of reduced model while maintaining the accuracy and reducing computation complexity. At the same time, the present invention also derived that, transfer function of original circuit with additive perturbations can represent approximation transfer function. Of which, perturbation matrix is related to the component generated by non-symmetric Lanczos Algorithm, so the computational quantity is very small.

The above-specified, however, are only used to describe the operating principle of the present invention, but not limited to its application range. However, it should be appreciated that a variety of embodiments and various modifications are embraced within the scope of the following claims, and should be deemed as a further development of the present invention.

Claims

1. A method of determining high-speed VLSI reduced-order interconnect by non-symmetric Lanczos Algorithm, which includes the following steps: a) input a mesh circuit; b) input an expand frequency point; c) set a state space matrix of circuit; and d) judge the order of reduced model for model reduction as per iteration termination conditions.

2. A high-speed VLSI reduced-order interconnect determined by non-symmetric Lanczos Algorithm defined in claim 1, wherein including iteration termination conditions; and since δq+1 can be treated as component of new vector vq+1 in AVq, and βq+1 treated as component of new vector wq+1 in A′Wq, it is required to satisfy the order q of:

3. A high-speed VLSI reduced-order interconnect determined by non-symmetric Lanczos Algorithm defined in claim 1, wherein the model reduction method means that the transfer function of perturbed system is equal to transfer function H(s) of original system with additive perturbations; the transfer function HΔ(s) of modified nodal analysis can be expressed as follows: