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Only a few object-oriented languages actually allow this (for example, Python when typechecked with mypy). C++, Java and most other languages that support overloading and/or shadowing would interpret this as a method with an overloaded or shadowed name. However, Sather supported both covariance and
The type coercion for function types may be given by f'(t) = coerce S 2 → T 2 (f(coerce T 1 → S 1 (t))), reflecting the contravariance of parameter values and covariance of return values. The coercion function is uniquely determined given the subtype and supertype. Thus, when multiple subtyping relationships are defined, one must be careful ...
In object-oriented programming, a covariant return type of a method is one that can be replaced by a "narrower" (derived) type when the method is overridden in a subclass. A notable language in which this is a fairly common paradigm is C++.
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z ( x ) on a domain D , a covariance function C ( x , y ) gives the covariance of the values of the random field at the two ...
The explicit form of a covariant transformation is best introduced with the transformation properties of the derivative of a function. Consider a scalar function f (like the temperature at a location in a space) defined on a set of points p , identifiable in a given coordinate system x i , i = 0 , 1 , … {\displaystyle x^{i},\;i=0,1,\dots ...
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern differential geometry uses more sophisticated index-free methods to represent tensors. In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector ...