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From: g-rh@cca.UUCP
Newsgroups: comp.lang.c
Subject: Re: C and Floating Point
Message-ID: <14681@cca.CCA.COM>
Date: Sat, 4-Apr-87 16:25:18 EST
Article-I.D.: cca.14681
Posted: Sat Apr  4 16:25:18 1987
Date-Received: Mon, 6-Apr-87 23:41:14 EST
References: <15958@sun.uucp> <5716@brl-smoke.ARPA>
Reply-To: g-rh@CCA.CCA.COM.UUCP (Richard Harter)
Organization: Computer Corp. of America, Cambridge, MA
Lines: 55
Keywords: C Fortran Floating Point

One class of usage for floating point equality has been suggested,
detection of exact bit patterns.  I suggest that this is not advisable,
both in principle and in practice.  I say 'in principle' because you
are using an operation with one set of semantics to simulate an
operation with a different set of semantics.  I say 'in practice'
because you can get unexpected results if the bit patterns are
equivalent to unnormalized floating point numbers.

However let us consider the following rather common situation:
We have a iterative process which generates a floating point number,
x, with the following iteration equation

	x <- x + delta

where delta is a function of x and where it is expected that the
iteration converges to some limit point x0.  We have the generic
problem of determining when the iterative sequence has converged,
taking into account the nature of floating point arithmetic.

Typically we will find the the calculation of the delta in question
becomes numerically unstable when x is sufficiently close to x0.
Let us suppose that we have addressed this problem, so that we
are simply left with the convergence issue.

The iteration will have converged if the iteration equation does
not change the value of x; i.e. we will have found the limit point
of the sequence, given that we are using floating point arithmetic
of the machine in question.  The question then is, how do we test
for this condition?  Is the following pseudo code acceptable?

	if ((x+delta)==x) then terminate_iteration
	else continue_iteration

If not, why not?  Alternatively should one use

	if (delta>0.) then
		if ((x+delta)<=x) then terminate_iteration
		else continue_iteration
	else if (delta<0.) then
		if ((x+delta)>=x) then terminate_iteration
		else continue_iteration
	else terminate_iteration

or is this also unacceptable?  If so, why?  Is there a best method?
What is it, and what are the issues involved?  If it is not
guaranteed the the calculation of delta is stable, what further
issues are involved?

You will be given a test on this at the end of the week.  No grade
will be given, but, if your answers are incorrect, World War III
will happen.

-- 

Richard Harter, SMDS Inc. [Disclaimers not permitted by company policy.]