Introduce 'nullCombination' and 'affineCombination'

[henrikt-ma]

Driven by #3748, this PR introduces two new built-in functions which we are quite far from being possible to express as user-defined functions. By having these in the language, the discussion about how to infer absoluteValue can avoid getting stuck on the issues that these two functions solve cleanly.

The concepts captured by these functions are well established in mathematics, and represented in several other software environments where the distinction between absolute and relative quantities is of importance. Adding them to the language as a preparation for #3748 is similar to how we have recently added language features to prepare for good unit checking rules, such as semantics for raising to an integer power, rational exponents of units, the nthRoot, and unitful literals.

The examples demonstrate how the functions can be used with (non-coherent) absolute Celsius temperatures, but in principle they are equally important when working with absolute quantities of any unit, coherent or not.

When combined with future unit semantics, these functions will provide a powerful combination of features I didn't see in any of the other softwares in my survey:

• Static (translation-time) analysis of absolute vs relative.
• Possibility to consider weight variability when checking validity of the sum of the weights.
• Interplay with units in two different but meaningful ways for the two functions.

Possible extension: homothety

The PR currently does not include a related operation in the same family, namely homothety, which could be defined as a syntactic sugar:

homothety(k, p, center = c) := affineCombination({1 - k, k}, {c, p})

The form of an operator where the center of the transformation is required to be given as a named argument would clarify intent when expressing things like "today the temperature is twice as warm as yesterday":

tToday = homothety(2, tYesterday, center = T_ice)

(I refuse to use the ridiculously named Constants.T_zero; the T_ice is just an alternative name which makes the expression above look more meaningful.)

In case the name homothety would be rejected because it sounds too similar to homotopy, alternative name candidates include dilation and scale. With these less specific names, it could be argued that the operator should also allow scaling a relative quantity, and then the presence of center would mark the difference between operating on relative or absolute quantities.

[HansOlsson]

We need to discuss this at the next meeting.

An alternative, which I believe MapleSoft had implemented at least for affine combination, would be to support this without any new functions. E.g., stating that sum(w.*x)/sum(w) and/or sum(w*x)/sum(w) is an affine combination of the vector x provided the same variable/parameter/constant w is used in both places.

Thus, one question is how users view having a new special function affineCombination (both in terms of reading code and writing it) compared to being required to write such calls in special ways (assuming we don't handle possible all cases).

Another is how complicated it is to detect such special cases for tools. Note that finding those special cases is important even if we introduce affineCombination in order to find cases where users should have used it.

In many (almost all?) cases the affine combination will be a convex combination; should that be noted?

For media there are a number convex combinations where we for species concentrations have two different formulations, either sum(X)=1 or cat(1,X,{1-sum(X)) . I don't know if we need to consider something special for either of these cases.

Looking at Modelica.Media I expected to find uses, but the first somewhat related case I found was Modelica.Media.Interfaces.PartialMixtureMedium.moleToMassFractions which isn't an affine combination - but the weights for it computed as: moleFractions.*MMX/sum(moleFractions.*MMX), which seems like a variant to also consider.

However, Media also have more normal cases like:

• Modelica.Media.IdealGases.Common.MixtureGasNasa.h_TX
• Modelica.Media.IdealGases.Common.MixtureGasNasa.specificHeatCapacityCp

Extra:
I focus on affine combinations as I don't know how common null combinations are in practice, except the trivial ones. Clearly it should work to write a-b instead of nullCombination({1,-1},{a,b}).

Note that we be should be that using affineCombination (or similarly) doesn't require/make that the physics correspond to an affine (sub-)space.