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Fix readme (#73)
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* Update README.md

* Update README.md
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MikaelSlevinsky authored Jul 2, 2021
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Expand Up @@ -52,17 +52,17 @@ The library is compiled by Apple LLVM version 10.0.1 as above with the compiler
| ----------------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: |
| **Forward** | 0.000101 | 0.000446 | 0.001952 | 0.006277 | 0.026590 | 0.141962 | 0.775724 | 4.870621 |
| **Backward** | 0.000101 | 0.000448 | 0.001918 | 0.006357 | 0.027123 | 0.145149 | 0.815881 | 5.040577 |
| **ϵ<sub>2</sub>** | 5.42e-16 | 7.79e-16 | 9.23e-16 | 1.27e-15 | 1.80e-15 | 2.52e-15 | 3.54e-15 | 4.98e-15 |
| **ϵ<sub>∞</sub>** | 1.33e-15 | 2.55e-15 | 4.55e-15 | 5.22e-15 | 9.33e-15 | 1.11e-14 | 1.81e-14 | 3.80e-14 |
| <b>ϵ<sub>2</sub></b> | 5.42e-16 | 7.79e-16 | 9.23e-16 | 1.27e-15 | 1.80e-15 | 2.52e-15 | 3.54e-15 | 4.98e-15 |
| <b>ϵ<sub>∞</sub></b> | 1.33e-15 | 2.55e-15 | 4.55e-15 | 5.22e-15 | 9.33e-15 | 1.11e-14 | 1.81e-14 | 3.80e-14 |

### Proriol series to bivariate Chebyshev series (`tri2cheb`) with (α, β, γ) = (0, 0, 0)

| Degree | 63 | 127 | 255 | 511 | 1023 | 2047 | 4095 | 8191 |
| ----------------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: |
| **Forward** | 0.000063 | 0.000300 | 0.001048 | 0.003886 | 0.019123 | 0.113299 | 0.699465 | 4.555086 |
| **Backward** | 0.000065 | 0.000289 | 0.000987 | 0.003837 | 0.019529 | 0.114718 | 0.699051 | 4.697958 |
| **ϵ<sub>2</sub>** | 2.30e-15 | 3.94e-15 | 8.14e-15 | 1.68e-14 | 3.55e-14 | 7.30e-14 | 1.44e-13 | 2.97e-13 |
| **ϵ<sub>∞</sub>** | 1.09e-14 | 1.84e-14 | 5.49e-14 | 1.49e-13 | 9.06e-13 | 1.95e-12 | 3.73e-12 | 1.05e-11 |
| <b>ϵ<sub>2</sub></b> | 2.30e-15 | 3.94e-15 | 8.14e-15 | 1.68e-14 | 3.55e-14 | 7.30e-14 | 1.44e-13 | 2.97e-13 |
| <b>ϵ<sub>∞</sub></b> | 1.09e-14 | 1.84e-14 | 5.49e-14 | 1.49e-13 | 9.06e-13 | 1.95e-12 | 3.73e-12 | 1.05e-11 |

The error growth rate is significantly reduced if (α, β, γ) = (-0.5, -0.5, -0.5).

Expand All @@ -72,16 +72,16 @@ The error growth rate is significantly reduced if (α, β, γ) = (-0.5, -0.5, -0
| ----------------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: |
| **Forward** | 0.000209 | 0.001077 | 0.003618 | 0.013243 | 0.064321 | 0.369272 | 2.182001 | 16.15872 |
| **Backward** | 0.000208 | 0.001027 | 0.003487 | 0.012903 | 0.064544 | 0.372493 | 2.191747 | 16.43340 |
| **ϵ<sub>2</sub>** | 7.60e-16 | 9.35e-16 | 1.31e-15 | 1.84e-15 | 2.56e-15 | 3.59e-15 | 5.05e-15 | 7.15e-15 |
| **ϵ<sub>∞</sub>** | 1.78e-15 | 2.66e-15 | 5.33e-15 | 7.11e-15 | 1.27e-14 | 1.79e-14 | 2.91e-14 | 1.66e-13 |
| <b>ϵ<sub>2</sub></b> | 7.60e-16 | 9.35e-16 | 1.31e-15 | 1.84e-15 | 2.56e-15 | 3.59e-15 | 5.05e-15 | 7.15e-15 |
| <b>ϵ<sub>∞</sub></b> | 1.78e-15 | 2.66e-15 | 5.33e-15 | 7.11e-15 | 1.27e-14 | 1.79e-14 | 2.91e-14 | 1.66e-13 |

### Rotation of a spherical harmonic series (`sph_rotation`) with ZYZ Euler angles (α, β, γ) = (0.1, 0.2, 0.3)

| Degree | 63 | 127 | 255 | 511 | 1023 | 2047 | 4095 | 8191 |
| ----------------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: | -------: |
| **Time** | 0.000141 | 0.000453 | 0.001598 | 0.006742 | 0.031256 | 0.170632 | 1.082894 | 10.35029 |
| **ϵ<sub>2</sub>** | 7.75e-15 | 8.29e-15 | 1.08e-14 | 1.77e-14 | 2.93e-14 | 5.12e-14 | 1.09e-13 | 1.87e-13 |
| **ϵ<sub>∞</sub>** | 1.47e-14 | 2.55e-14 | 8.01e-14 | 2.59e-13 | 7.36e-13 | 2.45e-12 | 6.74e-12 | 1.55e-11 |
| <b>ϵ<sub>2</sub></b> | 7.75e-15 | 8.29e-15 | 1.08e-14 | 1.77e-14 | 2.93e-14 | 5.12e-14 | 1.09e-13 | 1.87e-13 |
| <b>ϵ<sub>∞</sub></b> | 1.47e-14 | 2.55e-14 | 8.01e-14 | 2.59e-13 | 7.36e-13 | 2.45e-12 | 6.74e-12 | 1.55e-11 |

## References

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