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circuit-sim

circuit simulation

SPICE

SystemTechniques · 5Primitives · 10 circuit-sim Continuation methodsnumerics · medium: Homotopy and source or gmin stepping to make Newton converge from a far startFill-reducing orderingnumerics · medium: Permute a sparse symmetric matrix before Cholesky/LU so the factor has few new nonzeros; minimum degree, nested dissection and RCM trade quality, speed and parallelism.›Modified nodal analysisnumerics · medium: Build circuit equations from netlists by element stamps (MNA), including voltage sourcesSparse direct solvernumerics · medium: Solve sparse Ax = b exactly by reordering, symbolic analysis, numeric LLᵀ/LU and triangular solves; fill-in, not flops per entry, decides the cost.›Graph partitioninggraphs · medium: Split a graph into k balanced parts with few edges between them; multilevel coarsen-partition-refine (METIS, KaHIP) for quality, streaming heuristics (LDG, Fennel) for one-pass scale.›LU and Cholesky factorizationnumerics · low: Factor A = PLU (any square matrix, partial pivoting) or A = LLᵀ (SPD) once in O(n³), then solve each right-hand side in O(n²).›Newton root findingnumerics · low: Linearize at the guess and jump to the tangent root; quadratic convergence near a simple root, made robust by brackets (1D) or line search.›Sparse matrices (CSR)numerics · low: Compressed sparse row stores only nonzeros as row pointers, column indices and values; SpMV streams them once, so it is memory-bandwidth bound.›Stiff solvers (implicit, BDF)numerics · low: Implicit steps solve a small nonlinear system so fast decaying modes stay stable at any step size; BDF methods are the stiff-ODE workhorse.›Fast Fourier transformnumerics · low: Computes the DFT in O(n log n) by recursively splitting into even and odd halves; the basis of fast convolution and spectral methods.›Floating-point errornumerics · low: How IEEE 754 rounding error accumulates and how to contain it: compensated (Kahan/Neumaier) and pairwise summation, error-free transformations with FMA, ULP comparisons.›Adaptive Runge-Kutta (Dormand-Princ…numerics · low: An embedded 5(4) Runge-Kutta pair estimates each step's error for free and resizes the step to meet a tolerance; the default non-stiff solver.›d-ary heapdata-structures · low: Array heap with d children per node (d = 4 typical): half the binary heap's depth, siblings in one cache line, faster priority queues.›PRNGs (PCG, xoshiro)numerics · low: Small, fast, statistically strong non-cryptographic generators: PCG32 and xoshiro256**, with seeding, parallel streams and unbiased ranges.›Runge-Kutta 4numerics · low: The classic four-stage, fourth-order ODE step; error drops 16× when the step halves, with no derivatives or past steps needed.›

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