CA and EDP

[Ulrich Kulisch <ulrich.kulisch@xxxxxxx>]

It certainly makes a big difference if you compute a sum of matrix or matrix-vector products
s := a×b+c×d+e×f in the conventional way in floating-point arithmetic or if you compute each component of s by the exact dot product and round this only once to a desired precision at the very end of the summation . This kind of apllication appears again and again in scientific computing.

A conventional numerical computation does not tell you anything about the accuracy of the result. If you don't trust your result a frequent remedy is extending the precision. But again the result does not tell you anything about the accuracy. Here long interval atihmetic is the appropriate tool for a guaranteed answer.

The position paper of Motion 9 is entitled:
The exact dot product as basic tool for long interval arithmetic.
I attach section 9.7.2 of my book which shows how long interval arithmetic or multiple precision interval arithmetic is defined. It shows that the EDP indeed is necessary for long interval arithmetic. It is needed for vectors the components of which are floating-point numbers. A correctly or otherwise rounded dot product is not sufficient. The four reduction operations sum, dot, sumSquare, and sumAbs with different roundings are simple consequences of it.

Long interval arithmetic or multiple precision interval arithmetic is a general tool for highly accurate guaranteed evaluation of arithmetic expressions. By not requiring an EDP in IEEE P1788 we give up a fundamental instrument for success of interval arithmetic. 

A very impressive application is considered in [5] in the references, an iteration with the logistic equation (dynamical system)
x_n+1 := 3.75 · x_n · (1 − x_n), n >= 0.
For the initial value x₀ = 0.5 the system shows chaotic behavior. Double precision floating-point or interval arithmetic totally fail (no correct digit) after 30 iterations while long interval arithmetic still computes correct digits of a guaranteed enclosure after 2790 iterations.

Experience shows that for numerical computing manufacturers only implement what the standard requires. I am absolutely convinced that we shall get the EDP if we require it. (The reaction on my poster which states that IEEE P1788 requires an EDP (Motion 9) was very positive). On the other hand I have severe doubts whether we shall get it if we just recommend it. So I am asking everybody to vote YES on M0047:Motion45Amendment-1. International societies required it repeatedly. See [15], [16], [1], [2] in the references.

I shall comment on the history of the EDP in another mail.

With best regards
Ulrich

-- 
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch

Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
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www.math.kit.edu/ianm2/~kulisch/

KIT - Universität des Landes Baden-Württemberg 
und nationales Großforschungszentrum in der 
Helmholtz-Gesellschaft

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