Fractions Toolbox

create and manipulate fractions (K+N/D) using exact arithmetic

Description

The fractions toolbox allows users to create and manipulate fractions and fraction arrays of the form K+N/D, e.g.

  fr(1,3) % returns 1 / 3
  fr(pi) % returns 3 + 4703 / 33215

All the standard arithmetic and comparison operations are valid:
   fr(1,3)+fr(1,2) % returns 5 / 6
   fr(1,3)>0.3 % returns 1

Linear equations:
   A = fr(ones(2),[2,3;5,7]);
   B = fr(ones(2,1),[11;13]);
   A\B % returns [-3+49/143; 4+37/143]
   lsq(fr([1;1]),[0;1]) % returns 1/2

The treatment of singular and non-square systems is different from that of the built-in "\" so please read the documentation, e.g. for reasons of personal preference "\" does not do least-squares by default - use lsq instead.

Partial fractions and arbitrary-base digits can be computed:
  [d,r]=digits(fr(1,7),4,3) % 4 digits of base-3 expansion of 1/7 plus remainder
  % returns d=[0,1,0,2] and r= 4/567

Continued fractions expansions of fractions and square roots:
  [cf,rep] = cfracsqrt(fr(13,5)) % continued fraction of sqrt(13/5)
  [r1,r2] = bestrat(cf,rep,1000) % best rational approximations with denominator limit 1000

A powerful feature of the toolbox is that the numerator and denominator can theoretically be any data types that accept the standard arithmetic and comparison operations as well as gcd and mod. For example, if you have John D'Errico's Variable Precision Integer Toolbox (20 July 2009 release or later; see link below):

   prod(fr(1,vpi(2:7)).^10)

ans =
      1 / 10575608481180064985917685760000000000

If there exists a suitably defined polynomial object, this toolbox could be used to perform partial fraction and series expansions of rational functions.

See the demo and help files for a full list of features.

The functions have been tested with doubles and vpi integers, but message me if you encounter any problems, and let me know how it goes with other data types.

Acknowledgements

The author wishes to acknowledge the following in the creation of this submission:
Variable Precision Integer Arithmetic

• MATLAB release: MATLAB 7.9 (2009b)

[John D'Errico]

Nifty

[Khaled Hamed]

It seems to require vpi to be in the Matlab path, or a previous instance of a vpi.

>> fr(1,7)
??? Error using ==> superiorto
Unknown class 'vpi' listed in 'SUPERIORTO'.

Error in ==> fr.fr at 201
superiorto('vpi','double','single','int8','uint8','int16', ...

[Ben Petschel]

Hi Khaled, thanks for pointing that out (I forgot to run "rehash path" when testing without the vpi toolbox). I've submitted an update which should become available soon. Let me know if you have any further problems.

[Khaled Hamed]

Very useful toolbox, especially with the added precision of the vpi toolbox.

I think the demo_fr.m file needs an "echo on" at the beginning to see the header of each of the demo problems, rather than answers only (followed by echo off at the end).

[Derek O'Connor]

This is a very useful toolbox, especially when used with
John D'Errico's Variable Precision Integer Toolbox.

Here are two tests I ran :

   function z = RumpFrac(x,y)

   % Testing John D'Errico's Variable Precision Integer Toolbox
   % and Ben Petschel's Fractions Toolbox using
   % Rump's polynomial. Derek O'Connor Aug 01 2009

     x = fr(vpi(x));
     y = fr(vpi(y));

    R1 = (33375/100)*y^6+ x^2*(11*x^2*y^2-y^6- 121*y^4- 2)
    R2 = (55/10)*y^8
    R3 = x/(2*y)
    z1 = R1+R2
    z = z1 + R3;

%
% R1 =
% -7917111340668961361101134701524942850
% R2 =
% 7917111340668961361101134701524942848
% R3 =
% 1 + 11425 / 66192
% z1 =
% -2
% z =
% -1 + 11425 / 66192 = -54767/66192 --- Correct.

% z = -1.180591620717411e+021 without first two statements

and

function z = JuddFrac(x,y)
   % Testing John D'Errico's Variable Precision Integer Toolbox
   % and Ben Petschel's Fractions Toolbox using
   % Judd's polynomial. Derek O'Connor Aug 01 2009

     x = fr(vpi(x));
     y = fr(vpi(y));

 J1 = 1682*x*y^4
 J2 = 3*x^3
 J3 = 29*x*y^2
 J4 = - 2*x^5
 z1 = J1+J2
 z2 = z1+J3
 z3 = z2+J4
 z4 = z3+832
 z = z4/107751

% z = JuddFrac(192119201,35675640);
% J1 =
% 523460426438903533308340192814390277120000
% J2 =
% 21273236588999014470832803
% J3 =
% 7091078862999671298158400
% J4 =
% -523460426438903561672655644813075853992002
% z1 =
% 523460426438903554581576781813404747952803
% z2 =
% 523460426438903561672655644813076046111203
% z3 =
% 192119201
% z4 =
% 192120033
% z =
% 1783 --- Correct.

% z = 7.721506064908910e-003 without first two statements

[Matt Fig]

Great work!

[Erdal Bizkevelci]

 

[Bill McKeeman]

I used this toolbox in the computation of pi (see FX 29504).

[Christophe Lauwerys]

Great stuff, but I wonder how your two statements quoted below can be unified.

In other words: how can you define for instance SIGN and ABS for objects that represent polynomials? Not to mention GCD for multivariate polynomials ... Not an expert but do you need Groebner bases for this?

Thanks

Christophe

A)

% Non-standard objects must include 0, 1, -1 and require the following
% operations to be defined in order to create a fraction object:
% gcd
% rem
% sign
% abs
% +, - , .*, ./
% ==, <, <=, >, >=, ~=
%
% The following additional operation definitions are recommended:
% *, .^
% sort
% floor
% factor
% gcd (3-output form)
% rat (if floor(x) or mod(x,1) is not always equal to x)

B) If there exists a suitably defined polynomial object, this toolbox could be used to perform partial fractions.

[Ben Petschel]

Ok, to do that you'd need to define a total ordering on the polynomials by partitioning R[x] into P, -P and {0}, so p(x)>0 if p(x) is in P. See Lang's Algebra chapter 11 (real fields) for examples and details on the theory - e.g. the monomial orderings used for Groebner basis calculation are valid. This way you have sign(p)=1 if p is in P, etc, and similarly abs(p)=sign(p)*p.

Also I forgot to mention that MOD is required, but once this is defined then it's easy to write a GCD function. To define MOD you just need to define the representative elements of the cosets R[x]/p(x), such that MOD(q,p)>=0.

If you're willing to put in the effort implementing this, I'd be keen to see the results, but otherwise you're probably better off with a professional package such as Mathematica (which has an affordable home-use version) or the Symbolic Toolbox.