Dear Ulrich,
besides of real numbers we compute with vectors and matrices of real
numbers. The dot product is a basic operation in the vector- and matrix
spaces.
It can be used for guaranteed evaluation of polynomials and other
arithmetic expressions, and it is simple and fast, not more complicated
than an adder tree for fast multiplication.
I wonder how you can use the dot product for *guaranteed* evaluation of
polynomials, since the dot product can store only products of precision
twice as large as the inputs, whereas for a polynomial with leading term
an*x^n you need accumulators of precision n+1 times that of the inputs.
Moreover there are more general models of computation, for example the Real
RAM model, which can be very efficiently implemented, see for example the
iRRAM package from Norbert Müller in Trier: http://irram.uni-trier.de/.
I wonder why we focus on "complete arithmetic" and "exact dot product"
on this list, and don't consider those more general models...
Best regards,
Paul