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\section{The Pythagorean sum}
In this section, we analyse the computation of the Pythagorean sum,
which is used in two different computations, that is the norm of a complex
number and the 2-norm of a vector of real values.
In the first part, we briefly present the mathematical formulas for these
two computations.
We then present the naïve algorithm based on these mathematical formulas.
In the second part, we make some experiments in Scilab and compare our
naïve algorithm with the \emph{abs} and \emph{norm} Scilab primitives.
In the third part, we analyse
why and how floating point numbers must be taken into account when the
Pythagorean sum is to compute.
\subsection{Theory}
\subsection{Experiments}
% TODO : compare both abs and norm.
\lstset{language=Scilab}
\lstset{numbers=left}
\lstset{basicstyle=\footnotesize}
\lstset{keywordstyle=\bfseries}
\begin{lstlisting}
// Straitforward implementation
function mn2 = mynorm2(a,b)
mn2 = sqrt(a^2+b^2)
endfunction
// With scaling
function mn2 = mypythag1(a,b)
if (a==0.0) then
mn2 = abs(b);
elseif (b==0.0) then
mn2 = abs(a);
else
if (abs(b)>abs(a)) then
r = a/b;
t = abs(b);
else
r = b/a;
t = abs(a);
end
mn2 = t * sqrt(1 + r^2);
end
endfunction
// With Moler & Morrison's
// At most 7 iterations are required.
function mn2 = mypythag2(a,b)
p = max(abs(a),abs(b))
q = min(abs(a),abs(b))
//index = 0
while (q<>0.0)
//index = index + 1
//mprintf("index = %d, p = %e, q = %e\n",index,p,q)
r = (q/p)^2
s = r/(4+r)
p = p + 2*s*p
q = s * q
end
mn2 = p
endfunction
function compare(x)
mprintf("Re(x)=%e, Im(x)=%e\n",real(x),imag(x));
p = abs(x);
mprintf("%20s : %e\n","Scilab",p);
p = mynorm2(real(x),imag(x));
mprintf("%20s : %e\n","Naive",p);
p = mypythag1(real(x),imag(x));
mprintf("%20s : %e\n","Scaling",p);
p = mypythag2(real(x),imag(x));
mprintf("%20s : %e\n","Moler & Morrison",p);
endfunction
// Test #1 : all is fine
x = 1 + 1 * %i;
compare(x);
// Test #2 : more difficult when x is large
x = 1.e200 + 1 * %i;
compare(x);
// Test #3 : more difficult when x is small
x = 1.e-200 + 1.e-200 * %i;
compare(x);
\end{lstlisting}
\begin{verbatim}
***************************************
Example #1 : simple computation with Scilab 5.1
x(1)=1.000000e+000, x(2)=1.000000e+000
Scilab : 1.414214e+000
Naive : 1.414214e+000
Scaling : 1.414214e+000
Moler & Morrison : 1.414214e+000
***************************************
Example #2 : with large numbers ?
Scilab : Inf
Naive : Inf
Scaling : 1.000000e+200
Moler & Morrison : 1.000000e+200
***************************************
Example #3 : with small numbers ?
x(1)=1.000000e-200, x(2)=1.000000e-200
Scilab : 0.000000e+000
Naive : 0.000000e+000
Scaling : 1.414214e-200
Moler & Morrison : 1.414214e-200
***************************************
> Conclusion : Scilab is naive !
Octave 3.0.3
***************************************
octave-3.0.3.exe:29> compare(x);
***************************************
x(1)=1.000000e+000, x(2)=1.000000e+000
Octave : 1.414214e+000
Naive : 1.414214e+000
Scaling : 1.414214e+000
Moler & Morrison : 1.414214e+000
***************************************
x(1)=1.000000e+200, x(2)=1.000000e+000
Octave : 1.000000e+200
Naive : Inf
Scaling : 1.000000e+200
Moler & Morrison : 1.000000e+200
***************************************
octave-3.0.3.exe:33> compare(x)
x(1)=1.000000e-200, x(2)=1.000000e-200
Octave : 1.414214e-200
Naive : 0.000000e+000
Scaling : 1.414214e-200
Moler & Morrison : 1.414214e-200
***************************************
> Conclusion : Octave is not naive !
With complex numbers.
***************************************
Re(x)=1.000000e+000, Im(x)=1.000000e+000
Scilab : 1.414214e+000
Naive : 1.414214e+000
Scaling : 1.414214e+000
Moler & Morrison : 1.414214e+000
***************************************
Re(x)=1.000000e+200, Im(x)=1.000000e+000
Scilab : 1.000000e+200
Naive : Inf
Scaling : 1.000000e+200
Moler & Morrison : 1.000000e+200
***************************************
Re(x)=1.000000e-200, Im(x)=1.000000e-200
Scilab : 1.414214e-200
Naive : 0.000000e+000
Scaling : 1.414214e-200
Moler & Morrison : 1.414214e-200
***************************************
> Conclusion : Scilab is not naive !
\end{verbatim}
\subsection{Explanations}
\subsection{References}
The paper by Moler and Morrisson 1983 \cite{journals/ibmrd/MolerM83} gives an
algorithm to compute the Pythagorean sum $a\oplus b = \sqrt{a^2 + b^2}$
without computing their squares or their square roots. Their algorithm is based on a cubically
convergent sequence.
The BLAS linear algebra suite of routines \cite{900236} includes the SNRM2, DNRM2
and SCNRM2 routines which conpute the euclidian norm of a vector.
These routines are based on Blue \cite{355771} and Cody \cite{Cody:1971:SEF}.
In his 1978 paper \cite{355771}, James Blue gives an algorithm to compute the
Euclidian norm of a n-vector $\|x\| = \sqrt{\sum_{i=1,n}x_i^2}$.
The exceptionnal values of the \emph{hypot} operator are defined as the
Pythagorean sum in the IEEE 754 standard \cite{P754:2008:ISF,ieee754-1985}.
The \emph{\_\_ieee754\_hypot(x,y)} C function is implemented in the
Fdlibm software library \cite{fdlibm} developed by Sun Microsystems and
available at netlib. This library is used by Matlab \cite{matlab-hypot}
and its \emph{hypot} command.
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