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\title{CISC 822 Algebraic Algorithms \hfill Fall 1998\\
First Exercise Set \hfill Due Wed, Oct 21}
\author{Collins \& Saunders}
\maketitle

\begin{enumerate}
\item
Prove the first 4 statements of Theorem 1 of Collins, {\em The computing
time of the Euclidean algorithm}, JACM 3, 1974.  This paper was handed
out in class.  
Let $f, f', g, g'$ be non-negative real-valued functions, and let $c$ be 
a positive real number. The four statements to prove are:
\begin{enumerate}
\item[(a)] $f \sim cf$.
\item[(b)] if $f \preceq g$ and $f' \preceq g'$,
then $f + f' \preceq g + g'$ and $f f' \preceq g g'$.
\item[(c)] $f \preceq g$ and $f' \preceq g$ implies $f + f' \preceq g$.
\item[(d)] max$(f,g) \sim f + g $.
\end{enumerate}

\item This question relates to Knuth section 4.3.3 exercise 13.
The exercise is about multiplying an m bit number A by an n bit number B,
when n is substantially larger than m.
An approach is given in the answers section in the back of the book 
along with an analysis of it's run time.  The basic idea of that
method is to split B into m bit parts and multiply
each part by A. 
The method implemented
in the \verb|KARATSUBA| procedure in saclib is a bit different.
It splits B into two n/2 bit numbers and recursively calls itself
on the two parts (see \verb|Step2| of \verb|KARATSUBA|).
\begin{enumerate}
\item
Show this method has running time codominant with that given in 
Knuth's book.
\item
Which method do you think is better in practice? why?
\end{enumerate}

\item
Section 4.5.1 Exercise 6.

\item 
Section 4.5.2 Exercise 17.

\item
Saclib has representations for a number of mathematical domains
including integers, rational numbers, and integer coefficient polynomials
in one or more variables.  All of these domains contain a zero element,
and in all of the representations it is represented by the zero Word 
(all bits zero).  All of these domains also contain the integers 
in a natural way.  For instance rational numbers with denominator 1 are
integers, constant polynomials are integers.  Yet the  
representation of an integer differs in each of these saclib representations.
For example, consider 2.
In the rationals it is (2, 1), in the univariate integer polynomials
it is (0, 2), in the bivariate polynomials it is (0, (0, 2)), etc.

Explain why those rational numbers
which are integers are not simply represented as they are in the integers.
(Show how ambiguities would arise, which would confuse the rational arithmetic
  procedures.)  Sketch a design of a system supporting arbitrary integers, 
rationals, polynomials, in which the representation of all integers is 
like the representation of zero in saclib:  the same, regardless of the 
domain in which it is embedded.  What are some pros and cons of your
design relative to saclib's?
This is an essay question; the expected answer length is a page or so .  

\item
Write and test a program to meet this specification:
\begin{verbatim}
/*===========================================================================
			 n <-- RLOG2(R,k; n,f)

Rational number logarithm, base 2.

Inputs
  R : a positive rational number,
  k : a positive BETA-digit.

Output
  n : a bit BETA-digit, the integer part of L, where L is log_2(R).
  f : a BETA-digit such that f/2^k <= L < (f+1)/2^k.
===========================================================================*/
#include "saclib.h"

void RLOG2(R,k, n_, f_)
       Word R,k, *n_,*f_;
{
   ///
}
\end{verbatim}

\item
The first \verb|for| loop in IPEMV could be replaced by
\begin{verbatim}
 /* assuming "Word ap;" is declared above in the code  */
 ap = IEXP(a, e1 - e2) 
 if (rp == 0)
   B = IPROD(ap,B);
 else
   B = IPIP(rp,ap,B);
\end{verbatim}

And the second \verb|for| loop could be similarly replaced.

The idea of the change is to compute $(a^k)B$ rather than $(a (a \ldots (a B)\ldots))$, for $k = e1 - e2$, a conjectured advantage if B is an extensive polynomial.

\begin{enumerate}
\item
Do an analysis to produce a good dominating function, preferably codominant
function, for the cost of the algorithm with and without this change.
Compare. 
Measure the cost in terms of the degree of the polynomial in the main variable 
and the total number of terms.
\item
implement the change and test the performance with and without the 
change on a variety of inputs to IPEMV.
\end{enumerate}

\end{enumerate}


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