JosephsonCircuits.jl

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Numerical references ​

These references describe the numerical methods used or adapted in the package. The implementation notes and tests define the package-specific choices; a reference to an algorithm does not imply that every detail of its published implementation is used unchanged.

Harmonic balance and circuit models ​

The home-page bibliography lists the circuit and harmonic-balance texts and the published device comparisons.

Nonlinear and linear solvers ​

  • S. C. Eisenstat and H. F. Walker, “Choosing the Forcing Terms in an Inexact Newton Method,” SIAM Journal on Scientific Computing 17(1), 16–32 (1996). DOI, authors' technical report.

  • Y. Saad and M. H. Schultz, “GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems,” SIAM Journal on Scientific and Statistical Computing 7(3), 856–869 (1986). DOI.

  • D. G. Anderson, “Iterative Procedures for Nonlinear Integral Equations,” Journal of the ACM 12(4), 547–560 (1965). DOI.

  • H. F. Walker and P. Ni, “Anderson Acceleration for Fixed-Point Iterations,” SIAM Journal on Numerical Analysis 49(4), 1715–1735 (2011). DOI.

  • J. M. Tang, R. Nabben, C. Vuik, and Y. A. Erlangga, “Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods,” Journal of Scientific Computing 39(3), 340–370 (2009). DOI.

These cover the forcing terms by which Newton–Krylov sets the accuracy of each linear solve, the GMRES that solves it, the Anderson acceleration of QuasiNewton, and the A-DEF1 form of the optional Floquet deflation.

Stability ​

W.-J. Beyn, “An Integral Method for Solving Nonlinear Eigenvalue Problems,” Linear Algebra and its Applications 436(10), 3839–3863 (2012). DOI.

ContourIntegral reduces the moments of the inverse of the perturbation operator around its circle to the poles inside, as in this paper; see the stability guide.

Quantum noise ​

The noise conventions, symmetrized noise in quanta with the vacuum at half a photon, and the quantum limit of a phase-preserving amplifier, half a photon of added noise referred to its input, follow

  • C. M. Caves, “Quantum Limits on Noise in Linear Amplifiers,” Physical Review D 26(8), 1817–1839 (1982). DOI.

  • A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to Quantum Noise, Measurement, and Amplification,” Reviews of Modern Physics 82(2), 1155–1208 (2010). DOI, arXiv.

The noise a passive network emits in equilibrium, (nbar + 1/2)(I - S*S'), the noise of a passive block and of calcCnoise, is Bosma's theorem: H. Bosma, On the Theory of Linear Noisy Systems, Ph.D. thesis, Technische Hogeschool Eindhoven (1967).

Gauss collocation ​

E. Hairer, Geometric Numerical Integration, Lecture 2: Symplectic integrators (2010), discusses Gauss collocation and preservation of quadratic invariants. Lecture notes. These results concern the underlying integration method; algebraic projection and history interpolation require their own analysis.

Rational fitting ​

  • B. Gustavsen and A. Semlyen, “Rational Approximation of Frequency Domain Responses by Vector Fitting,” IEEE Transactions on Power Delivery 14(3), 1052–1061 (1999). Paper.

  • B. Gustavsen, “Improving the Pole Relocating Properties of Vector Fitting,” IEEE Transactions on Power Delivery 21(3), 1587–1592 (2006). Paper.

  • D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of Multiport Systems Using a Fast Implementation of the Vector Fitting Method,” IEEE Microwave and Wireless Components Letters 18(6), 383–385 (2008). Paper.

  • B. Gustavsen, “An Efficient Residue Perturbation Scheme for Passivity Enforcement of S-Parameter Rational Models,” IEEE Transactions on Electromagnetic Compatibility 67(3), 913–920 (2025). Paper.

  • C. Eckart and G. Young, “The Approximation of One Matrix by Another of Lower Rank,” Psychometrika 1(3), 211–218 (1936). Paper.

  • J. R. Rice and K. H. Usow, “The Lawson Algorithm and Extensions,” Mathematics of Computation 22(101), 118–127 (1968). Paper.

These cover the relocation of common poles, the relaxed normalization, the per-entry reduction that accelerates a multiport fit, and the perturbation of the residues that enforces passivity. The last two serve the order search: a fit of N poles is no closer to the samples than their best approximation of rank N + 1 (Eckart and Young), so an order whose bound exceeds the tolerance is not fitted, and Lawson's reweighted least squares bounds from below the error of every fit with given poles (Rice and Usow), so an order whose poles cannot meet the tolerance is not made passive.

Transfer-matrix norms ​

S. Boyd and V. Balakrishnan, “A Regularity Result for the Singular Values of a Transfer Matrix and a Quadratically Convergent Algorithm for Computing its L-infinity-norm,” Systems & Control Letters 15(1), 1–7 (1990). Author's page and paper.

The package finds the frequencies where a singular value of a rational realization crosses a level from the Hamiltonian matrix or a pencil, as in this paper, to validate a realization supplied directly at its tolerance. See passivityassessment for interpreting its verdicts.

Traveling-wave line models ​

H. W. Dommel, “Digital Computer Solution of Electromagnetic Transients in Single- and Multiphase Networks,” IEEE Transactions on Power Apparatus and Systems PAS-88(4), 388–399 (1969). DOI.

This is a historical circuit-transient reference for transmission-line models based on traveling-wave delay relations. The package's centered history interpolation, Gauss stages, and discrete adjoint are described separately in the transient theory.