PyREC scaffolding (from YREC development, 7be4133)/ boole.fGlossary

boole.f

82 lines · 1 definitions · on GitHub (7be4133)

Python version: boole.py, 73 lines, on the right (switch to Python above the code).

Purpose

This file integrates tabulated data with Boole's rule (5-point Newton–Cotes), after interpolating the data onto an even grid with a cubic spline. Its only use is the acoustic depth in CALCAD, $\int dr / c_s$.

How it fits

  • Inputs: the arguments X(N) and Y(N), and NN, the number of points of the even grid (it should be $1 + 4p$).
  • Who calls it: CALCAD, twice: over the whole star and from the base of the convection zone to the surface.
  • What it calls: SPLINJ (the spline's second derivatives) and SPLINT (spline evaluation).
  • Output: the integral in S.

The method

  1. Rescale: $x' = 10^{-11} x$, $y' = 10^{7} y$ (SCALEX, SCALEY; the comment says this brings the radius and the sound-speed vector to values near 1).

  2. Fit a natural cubic spline to $(x', y')$ and evaluate it at NN evenly spaced points $x'_k = x'_1 + (k-1)h$, $h = (x'_N - x'_1)/(NN-1)$.

  3. Apply Boole's rule to each group of 5 points: $$\int_{x_0}^{x_4} f\,dx \approx \frac{2h}{45}\left(7f_0 + 32f_1 + 12f_2 + 32f_3 + 7f_4\right),$$ over $p = \lfloor (NN-1)/4 \rfloor$ groups. If $NN - 1$ is not a multiple of 4, the last points would be left out without a message; CALCAD, the only caller, rounds $NN - 1$ up to a multiple of 4 in both calls (calcad.f, lines 310–316↗, calcad.f, lines 375–381↗), except for a profile of at most 3 points (calcad.f, suspected bug 7) (settled from the code, Nov 9, 2026).

  4. Undo the scaling: $S = S'/(10^{-11}\cdot 10^{7})$.

The local arrays XI, YI, XD, YD, D2X have sizes set by the arguments N and NN (automatic arrays, standard in Fortran 90), and H and S are arrays of length 1 used as scalars (lines 36–37↗).

Definitions

subroutineBOOLE

lines 28–79 · GitHub

Spline-resamples $(X, Y)$ onto NN even points and integrates with Boole's rule; returns $\int Y\,dX$ in S.

Called by CALCAD ×2

Flowchart of its calls, in order (2)
SPLINJ48
for I=1,NN (DO 20), lines 49–55
SPLINT51

Change log

The changes since rp_1

Changed since rp_1. Declarations only: IMPLICIT NONE with explicit declarations; SCALEX is now the double-precision 1D-11 (it was the single-precision 1e-11, about $4\times10^{-9}$ off in relative terms), which changes only rounding, since the result is scaled back; the header comment now names SPLINJ and SPLINT (question 4 of rp_1, the header naming "numerical recipes SPLINE," is settled and removed).