IMRPhenomXHM intermediate amplitude from supplied targets
Scope and status
Component xhm.amplitude.intermediate.inverse-polynomial.v1 implements the
six-constraint algebra in García-Quirós et al., arXiv:2001.10914v2,
Sec. V.1.1, Eq. (37). It accepts
calibrated amplitudes and endpoint derivatives as inputs; it does not supply
the fit surfaces, interval boundaries, veto policy, or a release-122022
provider. This component is not connected to the production XHM waveform.
Inputs, conventions, and options
For positive frequency \(f\in[f_s,f_e]\), set \(v(f)=A(f)/A_0\), where the same constant \(A_0\) applies across the join. The paper's ansatz is
Mathematics and components
The six constraints are \(v(f_s)\), \(dv/df(f_s)\), two interior values at \(f_s+(f_e-f_s)/3\) and \(f_s+2(f_e-f_s)/3\), \(v(f_e)\), and \(dv/df(f_e)\). The native kernels solve in \(u=(f-f_s)/(f_e-f_s)\) to avoid poor conditioning at small physical frequencies. Their endpoint inverse-polynomial derivative is \(d(1/v)/du=-(f_e-f_s)(dv/df)/v^2\). The independent 70-digit mpmath reference solves directly for the paper's physical-frequency coefficients.
Implementation mapping
The Torch helper checks finite positive inputs, ordered endpoints, in-interval evaluation, and conservatively certifies a positive denominator over the whole interval by Bernstein subdivision. The JAX helper is a JIT-compatible algebra kernel for caller-validated inputs; it does not certify denominator positivity. Both preserve gradients through the solve.
Evidence and limitations
Two supplied-target cases with different interval widths and amplitude
slopes are checked at eleven frequencies and nine-point batched grids.
The gate compares scalar endpoint and interior collocations, both endpoint
first derivatives, full native values, frequency derivatives, and
sensitivity to an interior amplitude target. The native value tolerance is
2e-11 relative/absolute; frequency derivatives use 1e-8 relative and
2e-9 absolute, and target sensitivity uses 2e-7 relative and 2e-8
absolute. These are algebra tests, not empirical XHM accuracy claims.
The published Eq. (37) also has exceptional lower-degree and extra pre-intermediate constructions; the present component covers the ordinary two-interior-node degree-five branch only. Releasing a full mode additionally requires a current-release amplitude provider, region cuts and vetoes, the calibrated spheroidal (3,2) amplitude and mixing, then strict binary and NR holdouts. The pinned XHM/XPHM qualification receipt remains failed independently of this component.