Towards a universal data type for scientific computing

Modern scientific computing uses an abundance of data types. Besides floating point numbers, we routinely use intervals, univariate Taylor series, Taylor series with interval coefficients, and more recently multivariate Taylor series. Newer are Taylor models, which allow verified calculations like intervals, but largely avoid many of their limitations, including the cancellation effect, dimensionality curse, and low-order scaling of resulting width to domain width. Another more recent structure is the Levi-Civita numbers, which allow viewing many aspects of scientific computation as an application of arithmetic and analysis with infinitely small numbers, and which axe useful for a variety of purposes including the assessment of differentiability at branch points. We propose new methods based on partially ordered Levi-Civita algebras that allow for a unification of all these various approaches into one single data type.