Re: Continued Fractions

• To: mathgroup at smc.vnet.net
• Subject: [mg75098] Re: Continued Fractions
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Mon, 16 Apr 2007 20:21:42 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <evvbni\$b5u\$1@smc.vnet.net>

```Wu Weidong wrote:
> Hi,
> I'm working on a function which takes 2 integer values as input, a and b an=
> d outputs a list of integers corresponding to the continued fraction repres=
> entation of the rational number a/b.
>
> My function is as follows:
>
> build[a_Integer,b_Integer, l_List] :== Module[{r==1,q,temp},
>  While[r!==0,
>   If[a>==b,
>    q==Quotient[a,b];
>    l==Append[l,q];
>    r == a-b*q;
>    a == r,
>
>    temp == a;
>    a == b;
>    b == temp;
>   ];
>  ];
>  l
> ];
>
> Essentially, if the fraction given is less than 1, the first number in the =
> list is 0. The function then proceeds to find the quotient of the numerator=
>  and denominator (and add that to the list) and the remainder. If the remai=
> nder is less than one, it takes the reciprocal and splits it into a whole-n=
> umber part plus another fraction which will be less than 1 and repeat. It s=
> tops when the remainder is 0.
>
> When I tried to run the program, I get the following error messages:
>
> Set::shape: Lists {} and {2} are not the same shape.
>
> Set::setraw: Cannot assign to raw object 45.
>
> Set::shape: Lists {} and {2} are not the same shape.
>
> Set::setraw: Cannot assign to raw object 45.
>
> Set::shape: Lists {} and {2} are not the same shape.
>
> General::stop: Further output of Set::shape
>      will be suppressed during this calculation.
>
> Set::setraw: Cannot assign to raw object 45.
>
> General::stop: Further output of Set::setraw
>      will be suppressed during this calculation.
>
>
> However, if I just run the While part of the function with a==45, b==16, l=
> =={}, r==1 (so I can enter the while loop), I can find the right result (2,=
> 1,4,3), so I believe my algorithm should be correct. The errors only appear=
>  when I put the while loop under the Module structure.
>
> What is wrong with my program?
>
> Thank you.
>
> Regards,
> Rayne
>

Hi Rayne,

Contrary to conventional programming language, the list of arguments to
a function (a, b, and l, in your case) does declare neither some local
variables nor some references to global variables. The list of arguments
is a list of patterns, indeed; patterns that are going to be replaced in
the right-hand side of the expression before evaluation.

Say you ask Mathematica to evaluate build[45, 16, {}]. First,
Mathematica replaces any occurrence of the pattern a by 45, b by 16, and
l by {}. Only then it enters the Module and evaluates (executes) the
While loop. So everything is fine until Mathematica reaches the line "l
= Append[l, q]" in your code, which is now {} = Append[{}, 2]. This
means that you have lost the name l and it has been replaced by its
value (the empty list). (It is like to attempt to evaluate 1 = 2.)

You can see what is going on by using, among many others, the *Trace*
command, as in

build[45, 16, {}]//Trace

Now, you will find below a fixed version of the code you posted. Note
that I have renamed the original arguments m and n and created two local
variables a and b within the Module. Also, it seems that it is not
necessary to give the list l as argument: you can just defined and
initialize it inside the Module.

In[1]:=
build[m_Integer, n_Integer] :=
Module[{a = m, b = n, r = 1, q, temp, lst = {}},
While[r != 0,
If[a >= b,
q = Quotient[a, b];
lst = Append[lst, q];
r = a - b*q;
a = r,
temp = a;
a = b;
b = temp;
];
];
lst
];

In[2]:=
build[45,16]

Out[2]=
{2,1,4,3}

From here, of course, you have to test your code to check if the
results are consistent.

Regards,
Jean-Marc

```

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