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Am 15.09.2013 18:25, schrieb Richard
Fateman:
You are right, CA requires an extended accumulator. The EDP, however, does not require it. In our first Pascal-XSC (1980) we provided a function scalp(a,b,n) where a and b are real vectors and n is an integer parameter which specifies the roudning mode (to nearest, downwards, or upwards) and the precision of the result after rounding (single, double, quadruple, ..., or full precision). The method with which scalp is implemented remains hidden allowing different ways of realization. Experience showed that the method which uses the long accumulator is superior and it allows realization by very fast hardware. Please see my mail of May 15, 2013 together with its attachments! You really should not be shocked by the size of the long accumulator. Including a very fast carry resolution technique it requires less hardware resources than an adder tree for fast multiplication which now is standard technology in every modern processor. The EDP is the additive equivalent to the adder tree for fast multiplication. It brings a similar speed up and eliminates many unnecessary errors (cancellations) in floating-point arithmetic. If you consider the EDP as an isolated object it may look gratuitous in the context of interval arithmetic. But the reality is different. In earlier mails I mentioned many applications of the EDP in interval analysis. I repeat a few of them: -- guaranteed evaluation of polynomials or arithmetic expressions. To see what you get without and with the EDP just have a look at the explicit examples given in my book. -- Rump's method for inverting arbitrarily ill conditions matrices. The method computes an enclosure of the solution. -- the implmentation and applications of multiprecision interval arithmetic, like accurate evaluation of polynomials in several variables, or -- computer assisted proof of the existence of a solution of a partial differential equation. In these and many other problems the EDP is an essntial engredient. Let me just discuss an explicit example more closely, computing the dot product of two vectors with interval components. What you would like to have is the least enclosure of the set of all dot products of real vectors out of the two interval vectors. Computing the interval dot product in conventional interval arithmetic (what we are going to standardize in P1788) for each interval product of two vector components you round the minimum of all products of the interval bounds downwards and the maximum upwards. During the following addition of the product to the intermediate sum you again round the lower bound downwards and the upper bound upwards. What you get is always different and frequently a large superset of what you would like to have. With the EDP you get the desired answer. The interval multiplications deliver the minima and the maxima of the products and these are now exactly accumulated by the EDP. The sum of the minima would then be rounded downwards and the sum of the maxima rounded upwards only once at the very end of the accumulation. This process deliver the desired answer. Interval analysis has developed a wide variety of applications. The standard P1788 should clearly state what the basic engredients are. It is out of question that the EDP is one of them. We shall never get it if we don't require it. On the other side I am convinced that we shall get it if we require it. Restricting P1788 to the four basic operations for intervals would be a big mistake. Summary: Motion 50 just requires an EDP. Its implementation is left open. By not requiring its implementation via CA any risk that it might damage the success of P1788 disappears. So please vote YES on Motion 50. With best regards Ulrich -- Karlsruher Institut für Technologie (KIT) Institut für Angewandte und Numerische Mathematikg D-76128 Karlsruhe, Germany Prof. Ulrich Kulisch Telefon: +49 721 608-42680 Fax: +49 721 608-46679 E-Mail: ulrich.kulisch@xxxxxxx www.kit.edu www.math.kit.edu/ianm2/~kulisch/ KIT - Universität des Landes Baden-Württemberg und nationales Großforschungszentrum in der Helmholtz-Gesellschaft |