Scalar growth sensitivity convergence¶
Original D/Dprime NumPy/SciPy DOP853 oracle, restarted at every Akima knot; complex-step mass derivatives.
Environment: jax 0.6.2, jaxlib 0.6.2, diffrax 0.7.0, quadax 0.2.13, scipy 1.18.0. CPU float64.
Both schemes use the frozen scalar thermal prescription; the public JAX polynomial table is rebuilt at import.
| source | mass | Ocb | z | reltol | dD/dm | abs error D | relative error D | df/dm | abs error f | relative error f |
|---|---|---|---|---|---|---|---|---|---|---|
| cb | 0.06 | 0.3 | 1 | 1e-08 | -7.7935471950e-02 | 1.037e-06 | 1.331e-05 | -2.9306273553e-02 | 9.619e-10 | 3.282e-08 |
| cb | 0.06 | 0.3 | 1 | 1e-09 | -7.7936658483e-02 | 1.494e-07 | 1.916e-06 | -2.9306272792e-02 | 2.003e-10 | 6.834e-09 |
| cb | 0.06 | 0.3 | 1 | 1e-10 | -7.7936592864e-02 | 8.373e-08 | 1.074e-06 | -2.9306272571e-02 | 2.007e-11 | 6.848e-10 |
| cb | 0.06 | 0.3 | 1 | 1e-11 | -7.7936564827e-02 | 5.569e-08 | 7.146e-07 | -2.9306272548e-02 | 4.318e-11 | 1.473e-09 |
| cb | 0.06 | 0.3 | 1 | 1e-12 | -7.7936491012e-02 | 1.812e-08 | 2.325e-07 | -2.9306272619e-02 | 2.757e-11 | 9.409e-10 |
| cb | 0.06 | 0.3 | 1 | 1e-13 | -7.7936522131e-02 | 1.300e-08 | 1.668e-07 | -2.9306272621e-02 | 2.928e-11 | 9.993e-10 |
| m | 0.06 | 0.3 | 1 | 1e-08 | 2.4434490608e-04 | 2.215e-08 | 9.066e-05 | 1.2175195945e-02 | 1.124e-08 | 9.233e-07 |
| m | 0.06 | 0.3 | 1 | 1e-09 | 2.4436976882e-04 | 2.709e-09 | 1.108e-05 | 1.2175205554e-02 | 1.632e-09 | 1.340e-07 |
| m | 0.06 | 0.3 | 1 | 1e-10 | 2.4436825502e-04 | 1.195e-09 | 4.889e-06 | 1.2175207739e-02 | 5.530e-10 | 4.542e-08 |
| m | 0.06 | 0.3 | 1 | 1e-11 | 2.4436754014e-04 | 4.799e-10 | 1.964e-06 | 1.2175207448e-02 | 2.624e-10 | 2.155e-08 |
| m | 0.06 | 0.3 | 1 | 1e-12 | 2.4436743359e-04 | 3.733e-10 | 1.528e-06 | 1.2175207426e-02 | 2.398e-10 | 1.970e-08 |
| m | 0.06 | 0.3 | 1 | 1e-13 | 2.4436706779e-04 | 7.557e-12 | 3.092e-08 | 1.2175207453e-02 | 2.664e-10 | 2.188e-08 |
| cb | 0.3 | 0.3 | 1 | 1e-08 | -7.5494475544e-02 | 2.445e-07 | 3.238e-06 | -2.8742843908e-02 | 2.741e-10 | 9.535e-09 |
| cb | 0.3 | 0.3 | 1 | 1e-09 | -7.5494772727e-02 | 5.273e-08 | 6.984e-07 | -2.8742843564e-02 | 6.976e-11 | 2.427e-09 |
| cb | 0.3 | 0.3 | 1 | 1e-10 | -7.5494683749e-02 | 3.625e-08 | 4.802e-07 | -2.8742843638e-02 | 3.918e-12 | 1.363e-10 |
| cb | 0.3 | 0.3 | 1 | 1e-11 | -7.5494718835e-02 | 1.164e-09 | 1.542e-08 | -2.8742843631e-02 | 2.735e-12 | 9.514e-11 |
| cb | 0.3 | 0.3 | 1 | 1e-12 | -7.5494715377e-02 | 4.623e-09 | 6.123e-08 | -2.8742843634e-02 | 2.177e-13 | 7.574e-12 |
| cb | 0.3 | 0.3 | 1 | 1e-13 | -7.5494720538e-02 | 5.388e-10 | 7.138e-09 | -2.8742843635e-02 | 8.891e-13 | 3.093e-11 |
| m | 0.3 | 0.3 | 1 | 1e-08 | 2.4451078231e-03 | 6.030e-11 | 2.466e-08 | 1.1857048789e-02 | 6.579e-10 | 5.548e-08 |
| m | 0.3 | 0.3 | 1 | 1e-09 | 2.4451086288e-03 | 7.455e-10 | 3.049e-07 | 1.1857049358e-02 | 8.937e-11 | 7.537e-09 |
| m | 0.3 | 0.3 | 1 | 1e-10 | 2.4451068877e-03 | 9.957e-10 | 4.072e-07 | 1.1857049483e-02 | 3.554e-11 | 2.998e-09 |
| m | 0.3 | 0.3 | 1 | 1e-11 | 2.4451078690e-03 | 1.441e-11 | 5.893e-09 | 1.1857049459e-02 | 1.133e-11 | 9.559e-10 |
| m | 0.3 | 0.3 | 1 | 1e-12 | 2.4451078439e-03 | 3.943e-11 | 1.613e-08 | 1.1857049453e-02 | 5.330e-12 | 4.495e-10 |
| m | 0.3 | 0.3 | 1 | 1e-13 | 2.4451079035e-03 | 2.017e-11 | 8.247e-09 | 1.1857049444e-02 | 3.124e-12 | 2.635e-10 |
| cb | 0.75 | 0.3 | 1 | 1e-08 | -6.6568287925e-02 | 1.685e-08 | 2.531e-07 | -2.7715740306e-02 | 1.395e-09 | 5.033e-08 |
| cb | 0.75 | 0.3 | 1 | 1e-09 | -6.6568283303e-02 | 1.223e-08 | 1.837e-07 | -2.7715741621e-02 | 7.948e-11 | 2.868e-09 |
| cb | 0.75 | 0.3 | 1 | 1e-10 | -6.6568274920e-02 | 3.845e-09 | 5.776e-08 | -2.7715741696e-02 | 4.854e-12 | 1.751e-10 |
| cb | 0.75 | 0.3 | 1 | 1e-11 | -6.6568272695e-02 | 1.620e-09 | 2.434e-08 | -2.7715741702e-02 | 1.704e-12 | 6.148e-11 |
| cb | 0.75 | 0.3 | 1 | 1e-12 | -6.6568271360e-02 | 2.857e-10 | 4.292e-09 | -2.7715741701e-02 | 1.429e-13 | 5.155e-12 |
| cb | 0.75 | 0.3 | 1 | 1e-13 | -6.6568270618e-02 | 4.567e-10 | 6.861e-09 | -2.7715741701e-02 | 1.951e-14 | 7.039e-13 |
| m | 0.75 | 0.3 | 1 | 1e-08 | 2.5079990962e-03 | 1.845e-09 | 7.357e-07 | 1.1252750342e-02 | 3.442e-10 | 3.059e-08 |
| m | 0.75 | 0.3 | 1 | 1e-09 | 2.5080010675e-03 | 1.262e-10 | 5.033e-08 | 1.1252750632e-02 | 5.507e-11 | 4.894e-09 |
| m | 0.75 | 0.3 | 1 | 1e-10 | 2.5080009389e-03 | 2.374e-12 | 9.466e-10 | 1.1252750687e-02 | 6.606e-13 | 5.870e-11 |
| m | 0.75 | 0.3 | 1 | 1e-11 | 2.5080009567e-03 | 1.550e-11 | 6.181e-09 | 1.1252750686e-02 | 3.276e-13 | 2.911e-11 |
| m | 0.75 | 0.3 | 1 | 1e-12 | 2.5080009354e-03 | 5.785e-12 | 2.307e-09 | 1.1252750687e-02 | 5.265e-14 | 4.679e-12 |
| m | 0.75 | 0.3 | 1 | 1e-13 | 2.5080009419e-03 | 6.559e-13 | 2.615e-10 | 1.1252750687e-02 | 4.177e-14 | 3.712e-12 |
| cb | 0.1 | 0.5 | 2 | 1e-08 | -3.8732794481e-02 | 1.500e-07 | 3.873e-06 | -3.4681346045e-02 | 1.469e-11 | 4.236e-10 |
| cb | 0.1 | 0.5 | 2 | 1e-09 | -3.8732793234e-02 | 1.488e-07 | 3.841e-06 | -3.4681345989e-02 | 7.068e-11 | 2.038e-09 |
| cb | 0.1 | 0.5 | 2 | 1e-10 | -3.8732640935e-02 | 3.525e-09 | 9.102e-08 | -3.4681345596e-02 | 4.637e-10 | 1.337e-08 |
| cb | 0.1 | 0.5 | 2 | 1e-11 | -3.8732648375e-02 | 3.914e-09 | 1.011e-07 | -3.4681346102e-02 | 4.207e-11 | 1.213e-09 |
| cb | 0.1 | 0.5 | 2 | 1e-12 | -3.8732651876e-02 | 7.416e-09 | 1.915e-07 | -3.4681346067e-02 | 6.672e-12 | 1.924e-10 |
| cb | 0.1 | 0.5 | 2 | 1e-13 | -3.8732648079e-02 | 3.619e-09 | 9.342e-08 | -3.4681346061e-02 | 1.327e-12 | 3.826e-11 |
| m | 0.1 | 0.5 | 2 | 1e-08 | -5.2267678043e-04 | 1.564e-09 | 2.992e-06 | 5.8310708292e-04 | 4.626e-09 | 7.933e-06 |
| m | 0.1 | 0.5 | 2 | 1e-09 | -5.2267560925e-04 | 2.735e-09 | 5.233e-06 | 5.8310096356e-04 | 1.493e-09 | 2.561e-06 |
| m | 0.1 | 0.5 | 2 | 1e-10 | -5.2267813711e-04 | 2.072e-10 | 3.964e-07 | 5.8310194312e-04 | 5.139e-10 | 8.813e-07 |
| m | 0.1 | 0.5 | 2 | 1e-11 | -5.2267845398e-04 | 1.097e-10 | 2.098e-07 | 5.8310254254e-04 | 8.556e-11 | 1.467e-07 |
| m | 0.1 | 0.5 | 2 | 1e-12 | -5.2267845517e-04 | 1.109e-10 | 2.121e-07 | 5.8310277366e-04 | 3.167e-10 | 5.431e-07 |
| m | 0.1 | 0.5 | 2 | 1e-13 | -5.2267834504e-04 | 7.180e-13 | 1.374e-09 | 5.8310240788e-04 | 4.911e-11 | 8.421e-08 |
Tighter primal tolerances need not produce monotonically smaller sensitivity errors. Absolute errors matter for small/cancelled derivatives; the table is evidence, not a promise of monotonic convergence.