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Exact shape derivatives of the incomplete gamma's logs (#797) - #848
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The derivatives of log P and log Q in the shape were five-point differences of scipy's values: 1e-10 off mpmath's typically, up to 4e-9 (at a = 0.5, x = 1), wherever the Gamma's hazard is taken from them (a standardised time below 30; above it the hazard has its own fraction, #786). They are now carried along the series of P (below x = a + 1) and Legendre's continued fraction of Q (at or above it), in real forward mode, each to rounding: within 1e-14 of mpmath, median 3e-16. Where neither settles (a shape of millions with x near it), and at x of 0 or inf, they are differenced as before; the shape curvature a Hessian takes is the difference of the exact first derivative. Decided with the user: the cost is accepted. A GammaPH fit on 400 rows takes 0.32 s where it took 0.21 s (GammaAFT 0.17 to 0.24 s, a censored Gamma 0.04 to 0.05 s); the estimates are the same. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01TRhKL2fJBiAAfNo9rihwts
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Fixes #797.
The derivatives of log P and log Q in the shape were five-point
differences of scipy's values: 1e-10 off mpmath's typically, up to 4e-9
(at a = 0.5, x = 1), wherever the Gamma's hazard is taken from them (a
standardised time below 30; above it the hazard has its own fraction,
#786). They are now carried along the series of P (below x = a + 1)
and Legendre's continued fraction of Q (at or above it), in real
forward mode, each to rounding: within 1e-14 of mpmath, median 3e-16.
Where neither settles (a shape of millions with x near it), and at x of
0 or inf, they are differenced as before; the shape curvature a Hessian
takes is the difference of the exact first derivative.
Decided with the user: the cost is accepted. A GammaPH fit on 400 rows
takes 0.32 s where it took 0.21 s (GammaAFT 0.17 to 0.24 s, a censored
Gamma 0.04 to 0.05 s); the estimates are the same.
🤖 Generated with Claude Code
https://claude.ai/code/session_01TRhKL2fJBiAAfNo9rihwts
Generated by Claude Code