[lispandfound]

Closes #249

Motivated by an issue in the pSA calculation that introduces an error proportional to xi * dt / T. this PR swaps out the Newmark solver for an exact Nigam-Jennings solver. Unlike Newmark, Nigam-Jennings is a solver specifically for SDOF oscillators that model pSA, and it solves it exactly under the assumption that acceleration is linear between sampled points (which is the same assumption that Newmark makes in its linear configuration). The algorithm is simple,

1. Initialise a state vector [u, u', p, p'] = [0, 0, -a(0), -a'(0)] where a is the acceleration timeseries,
2. At each timestep i, calculate the exact solution for u(i + 1) according to standard ODE theory. The original 1969 paper calculates the solution explicitly and lists the coefficients, but its cheaper and more accurate to use the matrix exponential form in the linked article. The matrix exponentiation is achieved using scipy.
3. Update the state vector [u, u', p, p'] = [u(i + 1), u'(i + 1), -a(i + 1), a'(i + 1)].

The algorithm is not only theoretically simpler (relying only on an introductory course to ODEs), but faster and considerably more accurate than Newmark at short periods.

I had Claude survey the solvers used by various toolkits. The exact Nigam-Jennings solver is used overwhelmingly through the codebases of peer groups (including notably NGA-West). So this change addresses the issue, aligns us internationally, is faster, easier to interpret and more accurate than a correctly implement Newmark method.

I've adopted Nigam-Jennings with the 10 PPW rule that NGA-West3 appears to apply. This resamples the waveform if there are fewer than 10 points per period T.

Solver benchmark

Who	Solver	Short-period handling	Confidence
NGA-West3	Nigam–Jennings (via gmprocess)	Sinc upsampling to ≥10 samples/period	Medium (processing not fully published)
NGA-West2	Nigam–Jennings (Boore's programs)	Not stated	Program verified, method inferred
NGA-East	Nigam–Jennings (Boore; Boore & Goulet 2014)	Sinc upsampling	Partial
USGS gmprocess / ShakeMap	Nigam–Jennings (C port of Boore's icmpmx)	Sinc upsampling to ≥10 samples/period	Verified (code)
Boore TSPP / SMSIM	Nigam–Jennings (rdrvaa, icmpmx)	Linear subdivision; sinc upsampling since 2018	Verified (code)
RCTC (UCLA/PEER 2017/09)	Exact recurrence (Chopra §5.2)	Sinc upsampling ×8	Verified (code)
USGS PRISM	Nigam–Jennings per report (code appears to use constant-input hold, inferred)	FFT resample to 200 samples/s	Verified (report)
K-NET/KiK-net flatfile (GFZ, 2026)	Nigam–Jennings	Sinc upsampling to 400 Hz	Verified
GEM SMT (OpenQuake toolkit)	Nigam–Jennings (default)	None	Verified (code)
eqsig	Nigam–Jennings	Linear interpolation in one method only; SA = PGA for T < 6·dt	Verified (code)
PySeismoSoil	Exact recurrence (Chopra §5.2)	None	Verified (code)
Chile flatfile (Bastías & Montalva 2016)	Nigam–Jennings	Unknown	Low (search snippet)
CSMIP / CESMD	Likely Nigam–Jennings (Caltech heritage)	Unknown	Not verified
pyrotd (Kottke)	Frequency domain	Spectral zero-padding to ≥10 samples/period	Verified (code)
Strata (Kottke)	Frequency domain	Spectral zero-padding to 5× oscillator frequency	Verified (code)
pystrata (Kottke)	Frequency domain	None	Verified (code)
SeisComP scwfparam	Newmark, average acceleration (β = ¼)	None	Verified (code)
OpenSees sdfResponse	Newmark β = ¼ (nonlinear SDOF)	User-chosen step	Verified (code)
GEM SMT "NewmarkBeta" (non-default)	Central difference (β = 0)	None	Verified (code)
NZGMDB (IM_calculation v2025.12.1)	Newmark with c2 bug	None	Verified
NZ Strong Motion Database (Van Houtte 2017)	"Duhamel's integral", scheme not stated	Not stated	Ambiguous
GeoNet operational products	"Linear SDOF oscillators", algorithm not stated	Not stated	Not verified
ObsPy	Basic SDOF via seismometer simulation (FFT), not a response spectrum	None	Verified (code)
NGA-Sub, ESM/ITACA, JMA, Taiwan, China, Turkey, Mexico	Unknown	Unknown	Not found