Re: Rational approximation link

From: helmut.zeisel_at_[hidden]
Date: 2001-10-22 04:23:49

--- In boost_at_y..., brianjparker_at_h... wrote:
> --- In boost_at_y..., Daryle Walker <darylew_at_m...> wrote:

I uploaded a first version of the rational approximation:

http://groups.yahoo.com/group/boost/files/big_int/farey_20011022.zip 

The prototype is 

template<class Int, class Float> 
boost::rational<Int> farey(const Float& v, const Int& lim);

> > 
> > Maybe we should have both.
> 
> Yes, that is my conclusion as well. 

This would be no problem for me - I would appreciate
to have different options.

> It may be possible to overload 
> the function based on the error term type- double or float for 
> relative error and integer for max. denominator, although this may 
be 
> too fragile and separate function names may be better.

Do not try to be too clever.
Since there are different options for rational approximation,
it will be difficult to find a single interface
for all of them.

I chose the name "farey" to point out
that it is just one algorithm
and not the only solution for rational approximation.

> An advantage of the relative error approach is that a default small 
> value close to the epsilon of the floating point representation 
could 
> be specified, so that the error argument could be ignored in many 
> cases.

If you just want to control the relative error,
there is a simple rule how to specify the max. denominator.
You can find it in my preliminary documentation.

Observe that the return type is rational<Int>,
so we need one way to secify the integer type to be used.
Specifying it by the type of the second parameter
is IMHO the most convenient solution.
 
> Also, as was pointed out in a previous posting, the Netlib code 
needs 
> to have two defines updated for IEEE754 floating point- I will look 
> into this in the future.
> 

IMHO updating for IEEE754 is not enough -
the algorithm should also work for some future
high precision floating point type.

A better solution would be to choose these
values from std::numeric_limits.
I fear, however, that this will not be possible
without deeper understanding of the algorithm used there,
and this (thanks to the gotos) will not be possible
without a look into the book referenced there.

Helmut