Skip to main content

Predictor/Corrector Newton-Raphson (PCNR): A Simple, Flexible, Scalable, Modular, and Consistent Replacement for Limiting in Circuit Simulation

  • Conference paper
  • First Online:
Scientific Computing in Electrical Engineering (SCEE 2018)

Part of the book series: Mathematics in Industry ((TECMI,volume 32))

  • 474 Accesses


Modern circuit simulators predominantly use Newton-Raphson (NR) iteration to solve circuit equations. To improve NR convergence, circuit simulators use a practice called “limiting”. This ensures that sensitive circuit quantities (such as diode voltages) do not change too much between successive NR iterations. However, in most simulators, the implementation of limiting tends to be inflexible, non-modular, inconsistent, and confusing. We therefore propose Predictor/Corrector Newton-Raphson (PCNR), a replacement for limiting that overcomes these disadvantages while incurring modest computational overhead. The key ideas behind PCNR are, (1) to add each limited circuit quantity as an extra unknown to the circuit’s Modified Nodal Analysis (MNA) system of equations, (2) to split each NR iteration into a “prediction” phase followed by a “correction” phase, and (3) to mitigate the computational cost of the extra unknowns by eliminating them from all Ax = b solves using a Schur complement based technique.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. 1.

    PCNR works for differential-algebraic equations as well, but for simplicity, we only consider algebraic equations in this write-up.

  2. 2.

    Equivalently, one can also compute the Jacobian dgdx, and an “RHS” function given by \(\mathrm {RHS}(\mathbf {x}) = \left ( \frac {d\mathbf {g}}{d\mathbf {x}} \right ) .\, \mathbf {x} - \mathbf {g}(\mathbf {x})\), at each iteration x i, which is the approach traditionally followed by SPICE simulators.

  3. 3.

    In practice, each diode also returns a Boolean flag to the simulator telling it whether limiting was or was not applied, which the simulator uses to determine whether NR has truly converged or not.

  4. 4.


  1. Nagel, L.W.: SPICE2: a computer program to simulate semiconductor circuits. Ph.D. thesis, The University of California at Berkeley (1975)

    Google Scholar 

  2. Ho, C.W., Ruehli, A., Brennan, P.: The modified nodal approach to network analysis. IEEE Trans. Circuits Syst. 22(6), 504–509 (1975)

    Article  Google Scholar 

  3. Roychowdhury, J.: Numerical simulation and modelling of electronic and biochemical systems. Found. Trends Electron. Des. Autom. 3(2–3), 97–303 (2009)

    MATH  Google Scholar 

  4. Sangiovanni-Vincentelli, A.L.: Computer design aids for VLSI circuits. In: Circuit Simulation, pp. 19–112. Springer, Dordrecht (1984)

    Google Scholar 

  5. Keiter, E.R., Hutchinson, S.A., Hoekstra, R.J., Russo, T.V., Waters, L.J.: Xyce® parallel electronic simulator design: mathematical formulation. Tech. Rep. SAND2004-2283, Sandia National Laboratories, Albuquerque, NM (2004)

    Google Scholar 

  6. Neamen, D.A.: Semiconductor Physics and Devices: Basic Principles, 4th edn. McGraw-Hill, New York (2011)

    Google Scholar 

  7. Kao, W.H.: Comparison of quasi-Newton methods for the DC analysis of electronic circuits. Master’s thesis, The University of Illinois at Urbana-Champaign (1972)

    Google Scholar 

  8. Wang, T., Roychowdhury, J.: Well-posed models of memristive devices (2016). ArXiv. e-prints

    Google Scholar 

Download references


This work was sponsored by the Laboratory Directed Research and Development (LDRD) Program at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Author information

Authors and Affiliations


Corresponding authors

Correspondence to Karthik V. Aadithya , Eric R. Keiter or Ting Mei .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Aadithya, K.V., Keiter, E.R., Mei, T. (2020). Predictor/Corrector Newton-Raphson (PCNR): A Simple, Flexible, Scalable, Modular, and Consistent Replacement for Limiting in Circuit Simulation. In: Nicosia, G., Romano, V. (eds) Scientific Computing in Electrical Engineering. SCEE 2018. Mathematics in Industry(), vol 32. Springer, Cham.

Download citation

Publish with us

Policies and ethics