reduce-historical/r37/doc/manual/map.tex

\section{MAP Operator}
The operator MAP applies a uniform evaluation pattern
to all members of a composite structure: a matrix, a list or the arguments
of an operator expression. The evaluation pattern can be a
unary procedure, an operator, or an algebraic expression with
one free variable. MAP is used with the syntax:
\begin{verbatim}
  MAP(EXPRN1:algebraic,EXPRN2:expression)
\end{verbatim}
{\tt EXPRN2} is a list, a matrix or an operator expression.

{\tt EXPRN1} is
\begin{itemize}
\item the name of an operator for a single argument: the operator
 is evaluated once with each element of {\tt EXPRN1} as its single argument,
\item an algebraic expression with exactly one free variable, that is
a variable preceded by the tilde symbol: the expression
 is evaluated for each element of {\tt EXPRN1} where the element is
 substituted for the free variable,
\item a replacement rule of the form
\begin{verbatim}
  VAR => EXPRN3
\end{verbatim} 
  where {\tt VAR} is a variable and {\tt EXPRN3} is an expression
  which contains {\tt VAR}.
  Here {\tt EXPRN3} is evaluated for each element of {\tt EXPRN1} where 
  the element is substituted for  {\tt VAR}. {\tt VAR} may be 
  optionally preceded by a tilde.
\end{itemize}
The rule form for {\tt EXPRN2} is needed when more than
one free variable occurs.

{\it Examples:}
\begin{verbatim}
     % collect absolute values
   map(abs,{1,-2,a,-a});
               
                      ->

   {1,2,abs(a),abs(a)}

     % integrate a matrix
   map(int(~w,x), mat((x^2,x^5),(x^4,x^5)));

                      ->
 
        [  3     6 ]
        [ x     x  ]
        [----  ----]
        [ 3     6  ]
        [          ]
        [  5     6 ]
        [ x     x  ]
        [----  ----]
        [ 5     6  ]

     % multiply an equation
   map(~w*6 , x^2/3 = y^3/2 -1);
   
                      ->

         2     3
      2*x =3*(y  - 2)

    
\end{verbatim}
{\tt MAP} can be applied in nested mode:
\begin{verbatim}
   map(sub(x=y,~q)-sub(x=0,~q),map(mat(int(~w,x),(x^2-y,x^3))));
                      ->

       [  2             4 ]
       [ y *(y - 3)    y  ]
       [------------  ----]
       [     3         4  ]

    % The following example needs the rule form because there are two
    % free variables in the rightmost expression.
 
   map(~w=>map(int(~r,x),w),{mat((x^2,x^4)),mat((x^3,x^5))});

                      ->
       {

        [  3     5 ]
        [ x     x  ] 
        [----  ----]
        [ 3     5  ]

        ,

        [  4     6 ]
        [ x     x  ]
        [----  ----]
        [ 4     6  ]

        }
\end{verbatim}



\section{SELECT Operator}
The operator SELECT extracts from a list 
or from the arguments of an n--ary operator elements corresponding
to a boolean predicate. The predicate pattern can be a
unary procedure, an operator or an algebraic expression with
one free variable. SELECT is used with the syntax:
\begin{verbatim}
  SELECT(EXPRN1:expression,EXPRN2:list)
\end{verbatim}

{\tt EXPRN1} is
\begin{itemize}
\item the name of an operator for a single argument: the operator
 is evaluated once with each element of {\tt EXPRN2} as its single argument,
\item an algebraic expression with exactly one free variable, that is
a variable preceded by the tilde symbol at least once: the expression
 is evaluated for each element of {\tt EXPRN2} where the element is
 substituted for the free variable,
\item a replacement rule of the form
\begin{verbatim}
  VAR => EXPRN3
\end{verbatim} 
  where {\tt VAR} is a variable and {\tt EXPRN3} is an expression
  which contains {\tt VAR}.
  Here {\tt EXPRN3} is evaluated for each element of {\tt EXPRN2} where 
  the element is substituted for  {\tt VAR}. {\tt VAR} may be 
  optionally preceded by a tilde.
\end{itemize}
The rule form for {\tt EXPRN1} is needed when more than
one free variable occurs. The result of the operation is built
from those elements of {\tt EXPRN2} which let {\tt EXPRN1} evaluate
to a value different from {\tt 0} or {\tt nil}, using the
leading operator of  {\tt EXPRN2}.


Examples:

\begin{verbatim}
  select( ~w>0, {1,-1,2,-3,3})
                      ->
                    {1,2,3}

   % select the tersm with even powers of y 
  select(evenp deg(~w,y), part((x+y)^5,0):=list);

                      -> 
          5     3  2      4
        {x ,10*x *y ,5*x*y }

   % select elements of a sum directly
  select(evenp deg(~w,x), 2x^2+3x^3+4x^4);

                      ->
          2    4
        2x  +4x    
\end{verbatim}