uxarray.utils.computing.diff_of_products

uxarray.utils.computing.diff_of_products#

uxarray.utils.computing.diff_of_products(a, b, c, d)#

Kahan’s accurate a*b - c*d using two_prod and two_sum.

Naive evaluation of a*b - c*d loses all significant bits when the two products are nearly equal (catastrophic cancellation). This routine computes each product exactly via two_prod, subtracts the rounded high parts, then folds the residual low parts back in. The result has rounding error bounded by one ulp of the true value regardless of cancellation.

This is the core operation that makes cross products accurate: every component of a x b is a difference of two products of exactly this form.

Parameters:

  • a (float) – Input scalars; computes a*b - c*d.

  • b (float) – Input scalars; computes a*b - c*d.

  • c (float) – Input scalars; computes a*b - c*d.

  • d (float) – Input scalars; computes a*b - c*d.

Returns:

  • hi (float) – High-order part of the accurate result.

  • lo (float) – Low-order correction term; hi + lo equals the accurate value.

References

Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9780898718027

Jeannerod, C.-P., Louvet, N., and Muller, J.-M. (2013). Further analysis of Kahan’s algorithm for the accurate computation of 2 × 2 determinants. Mathematics of Computation, 82, 2245-2264. https://doi.org/10.1090/S0025-5718-2013-02679-8