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-version 1.1.1.1, 1999/12/08 05:47:44
+version 1.10, 2003/11/27 12:23:58
 Line 1
 Line 1
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+ @comment $OpenXM: OpenXM/src/asir-doc/parts/asir.texi,v 1.9 2003/11/27 12:08:58 ohara Exp $
+ \BJP
  @node $B%f!<%68@8l(B Asir,,, Top
  @chapter $B%f!<%68@8l(B Asir
+ \E
+ \BEG
+ @node User language Asir,,, Top
+ @chapter User language @b{Asir}
+ \E
  @noindent
+ \BJP
  @b{Asir} $B$NAH$_9~$_H!?t$O(B, $B0x?tJ,2r(B, GCD $B$J$I$N7W;;$r9T$&$b$N(B, $B%U%!(B
  $B%$%kF~=PNO$r9T$&$b$N(B, $B$"$k$$$O?t<0$N0lIt$r<h$j=P$9$b$N$J$I$5$^$6$^$J$b$N(B
  $B$,MQ0U$5$l$F$$$k$,(B, $B%f!<%6$,<B:]$K9T$$$?$$$3$H$r<B9T$5$;$k$?$a$K$O0l(B
  $BHL$K$O%f!<%68@8l$K$h$k%W%m%0%i%`$r=q$/I,MW$,$"$k(B. $B%f!<%68@8l$b(B
  @b{Asir} $B$H8F$P$l$k(B. $B0J2<$G$O(B, $B%f!<%68@8l$NJ8K!5,B'$*$h$S<B:]$N%f!<(B
  $B%68@8l%W%m%0%i%`$rNc$H$7$?%W%m%0%i%`$N=q$-J}$K$D$$$F=R$Y$k(B.
+ \E
+ \BEG
+ @b{Asir} provides many built-in functions, which perform algebraic
+ computations, e.g., factorization and GCD computation, file I/O,
+ extract a part of an algebraic expression, etc.
+ In practice, you will often encounter a specific problem for which
+ @b{Asir} does not provide a direct solution.  For such cases, you have
+ to write a program in a certain user language.  The user language for
+ @b{Asir} is also called @b{Asir}.  In the following, we describe the
+ Syntax and then show how to write a user program by several examples.
+ \E
  @menu
+ \BJP
  * $BJ8K!(B (C $B8@8l$H$N0c$$(B)::
  * $B%f!<%6Dj5AH!?t$N=q$-J}(B::
+ \E
+ \BEG
+ * Syntax (Difference from C language)::
+ * Writing user defined functions::
+ \E
  @end menu
+ \BJP
  @node $BJ8K!(B (C $B8@8l$H$N0c$$(B),,, $B%f!<%68@8l(B Asir
  @section $BJ8K!(B (C $B8@8l$H$N0c$$(B)
+ \E
+ \BEG
+ @node Syntax (Difference from C language),,, User language Asir
+ @section Syntax --- Difference from C language
+ \E
  @noindent
+ \BJP
  @b{Asir} $B$NJ8K!$O(B C $B8@8l$K=`5r$7$F$$$k(B.
  $B$*$b$JAj0cE@$O<!$NDL$j$G$"$k(B. $B0J2<$G(B, $BJQ?t$H$O(B @b{Asir} $B$K$*$1$k(B
  $B%W%m%0%i%`MQ$NJQ?t(B, $B$9$J$o$ABgJ8;z$G;O$^$kJ8;zNs$r0UL#$9$k$3$H$H$9$k(B.
+ \E
+ \BEG
+ The syntax of @b{Asir} is based on C language.
+ Main differences are as follows.
+ In this section, a variable does not mean an indeterminate, but
+ a program variable which is written by a string which begins with a
+ capital alphabetical letter in @b{Asir}.
+ \E
  @itemize @bullet
  @item
- $BJQ?t$N7?$,$J$$(B.
+ \JP $BJQ?t$N7?$,$J$$(B.
+ \EG No types for variables.
+ @*
+ \BJP
  $B4{$K@bL@$7$?$H$*$j(B, @b{Asir} $B$G07$o$l$kBP>]<+?H$OA4$F2?$i$+$N7?(B
  $B$r;}$C$F$$$k(B. $B$7$+$7(B, $B%W%m%0%i%`JQ?t<+BN$O(B, $B$I$N$h$&$JBP>]$G$b(B
  $BBeF~$G$-$k$H$$$&0UL#$G7?$,$J$$$N$G$"$k(B.
+ \E
+ \BEG
+ As is already mentioned, any object in @b{Asir} has their respective
+ types.  A program variable, however, is type-less, that is, any typed
+ object can be assigned to it.
+ \E
  @example
  [0] A = 1;
-Line 43
+Line 91
 Line 43
 Line 91
  @end example
  @item
+ \BJP
  $BH!?tFb$NJQ?t$O(B, $B%G%U%)%k%H$G$O2>0z?t$r$3$a$F$9$Y$F6I=jJQ?t(B.
+ @*
  $B$?$@$7(B, @code{extern} $B@k8@$5$l$?JQ?t$O(B, $B%H%C%W%l%Y%k$K$*$1$kBg0hJQ?t$H$J$k(B.
  $B$9$J$o$A(B, $BJQ?t$N%9%3!<%W$OBg0hJQ?t$H6I=jJQ?t$N(B 2 $B<oN`$KC1=c2=$5$l$F$$$k(B.
  $B%H%C%W%l%Y%k(B, $B$9$J$o$A%W%m%s%W%H$KBP$7$FF~NO$5$l$?JQ?t$OA4$FBg0hJQ?t(B
  $B$H$7$FEPO?$5$l$k(B. $B$^$?H!?tFb$G$O<!$N$$$:$l$+$H$J$k(B.
+ \E
+ \BEG
+ Variables, together with formal parameters, in a function (procedure)
+ are all local to the function by default.
+ @*
+ Variables can be global at the top level,
+ if they are declared with the key word @code{extern}.
+ Thus, the scope rule of @b{Asir} is very simple.
+ There are only two types of variables: global variables and local
+ variables.
+ A name that is input to the @b{Asir}'s prompt at the top level
+ is denotes a global variable commonly accessed at the top level.
+ In a function (procedure) the following rules are applied.
+ \E
  @enumerate
  @item
+ \BJP
  $BH!?t$,Dj5A$5$l$k%U%!%$%k$K$*$$$F(B, $B$=$NH!?tDj5A0JA0$K(B, $B$"$k(B
  $BJQ?t$,(B @code{extern} $B@k8@$5$l$F$$$k>l9g(B, $BH!?tFb$N$=$NJQ?t$bBg0hJQ?t(B
  $B$H$7$F07$o$l$k(B.
+ \E
+ \BEG
+ If a variable is declared as global by an @code{extern} statement in
+ a function, the variable used in that function denotes a global variable
+ at the top level.
+ Furthermore, if a variable in a function is preceded by an @code{extern}
+ declaration outside the function but in a file where the function is
+ defined, all the appearance of that variable in the same file denote
+ commonly a global variable at the top level.
+ \E
  @item
- @code{extern} $B@k8@$5$l$F$$$J$$JQ?t$O$=$NH!?t$K6I=jE*$H$J$k(B.
+ \JP @code{extern} $B@k8@$5$l$F$$$J$$JQ?t$O$=$NH!?t$K6I=jE*$H$J$k(B.
+ \BEG
+ A variable in a function is local to that function, if it is not declared
+ as global by an @code{extern} declaration.
+ \E
  @end enumerate
  @example
-Line 77  end$
+Line 155  end$
 Line 77  end$
 Line 155  end$
  @end example
  @item
- $B%W%m%0%i%`JQ?t$OBgJ8;z$G;O$^$j(B, $BITDj85(B, $BH!?t$O>.J8;z$G;O$^$k(B.
+ \JP $B%W%m%0%i%`JQ?t$OBgJ8;z$G;O$^$j(B, $BITDj85(B, $BH!?t$O>.J8;z$G;O$^$k(B.
+ \EG Program variables and algebraic indeterminates are distinguished in @b{Asir}.
+ @*
+ \BJP
  $B$3$NE@$O(B, $B4{B8$N?t<0=hM}%7%9%F%`$N$[$H$s$I$H0[$J$kE@$G$"$k(B. @b{Asir}
  $B$,$3$N;EMM$r:NMQ$7$?$N$O(B, $B%f!<%6$,ITDj85$N$D$b$j$G;HMQ$7$?JQ?t$K(B
  $B$J$s$i$+$NCM$,BeF~$5$l$F$$$?>l9g$K:.Mp$r>7$/(B, $B$H$$$&(B, $B4{B8$N(B
  $B%7%9%F%`$K$"$j$,$A$J>u67$rHr$1$k$?$a$G$"$k(B.
+ \E
+ \BEG
+ The names of program variables must begin with a capital letter;
+ while the names of indeterminates and functions must begin with
+ a small letter.
+ This is an unique point that differs from almost all other existing
+ computer algebra systems.  The distinction between program variables
+ and indeterminates is adopted to avoid the possible and usual confusion
+ that may arise in a situation where a name is used as an indeterminate
+ but, as it was, the name has been already assigned some value.
+ To use different type of letters, capital and small, was a matter of
+ syntactical convention like Prolog, but it is convenient to distinguish
+ variables and indeterminates in a program.
+ \E
  @item
- @code{switch} $BJ8(B, @code{goto} $B$,$J$$(B.
+ \JP @code{switch} $BJ8(B, @code{goto} $B$,$J$$(B.
+ \EG No @code{switch} statements, and @code{goto} statements.
+ @*
+ \JP @code{goto} $B$,$J$$$?$a(B, $BB?=E%k!<%W$r0lEY$KH4$1$k$N$,$d$dJ#;($K$J$k>l9g$,$"$k(B.
+ \EG Lack of @code{goto} statement makes it rather bothering to exit from within multiple loops.
- @code{goto} $B$,$J$$$?$a(B, $BB?=E%k!<%W$r0lEY$KH4$1$k$N$,$d$dJ#;($K$J$k>l9g$,$"$k(B.
  @item
- $B%3%s%^<0$O(B, @code{for (A;B;C)} $B$^$?$O(B, @code{while(A)} $B$N(B @code{A}, @code{B}, @code{C} $B$K$N$_;H$&$3$H$,$G$-$k(B.
+ \BJP
+ $B%3%s%^<0$O(B, @code{for (A;B;C)} $B$^$?$O(B, @code{while(A)} $B$N(B @code{A},
+ @code{B}, @code{C} $B$K$N$_;H$&$3$H$,$G$-$k(B.
+ \E
+ \BEG
+ Comma expressions are allowed only in @code{A}, @code{B} and @code{C}
+ of the constructs @code{for (A;B;C)} or @code{while(A)}.
+ \E
+ @*
+ \JP $B$3$l$O(B, $B%j%9%H$r@5<0$J%*%V%8%'%/%H$H$7$F2C$($?$3$H$K$h$k(B.
+ \EG This limitation came from adopting lists as legal data objects for @b{Asir}.
- $B$3$l$O(B, $B%j%9%H$r@5<0$J%*%V%8%'%/%H$H$7$F2C$($?$3$H$K$h$k(B.
  @end itemize
  @noindent
- $B0J>e$O@)8B$G$"$k$,(B, $B3HD%$H$7$F$O<!$NE@$,5s$2$i$l$k(B.
+ \JP $B0J>e$O@)8B$G$"$k$,(B, $B3HD%$H$7$F$O<!$NE@$,5s$2$i$l$k(B.
+ \EG The above are limitations; extensions are listed as follows.
  @itemize @bullet
  @item
- $BM-M}<0$KBP$9$k7W;;$r(B, $BDL>o$N(B C $B$K$*$1$k7W;;$HF1MM$K$G$-$k(B.
+ \JP $BM-M}<0$KBP$9$k7W;;$r(B, $BDL>o$N(B C $B$K$*$1$k7W;;$HF1MM$K$G$-$k(B.
+ \BEG
+ Arithmetic for rational expressions can be done in the
+ same manner as is done for numbers in C language.
+ \E
  @item
- $B%j%9%H$,07$($k(B.
+ \JP $B%j%9%H$,07$($k(B.
+ \EG Lists are available for data objects.
+ \BJP
  $B9=B$BN$rMQ$$$k$^$G$b$J$$MWAG$N=89gBN$r(B, $B%j%9%H$GI=$9$3$H$,$G$-(B,
  C $B$GD>@\=q$/>l9g$KHf3S$7$F%W%m%0%i%`$,C;$/(B, $BFI$_$d$9$/=q$1$k(B.
+ \E
+ \BEG
+ Lists are conveniently used to represent a certain collection of objects.
+ Use of lists enables to write programs more easily, shorter and more
+ comprehensible than use of structure like C programs.
+ \E
+ @item
+ \JP $B%f!<%6Dj5AH!?t$K$*$1$k%*%W%7%g%s;XDj(B.
+ \EG Options can be specified in calling user defined functions.
+ \JP $B$3$l$K4X$7$F$O(B, @xref{$B%*%W%7%g%s;XDj(B}.
+ \EG @xref{option}.
  @end itemize
+ \BJP
  @node $B%f!<%6Dj5AH!?t$N=q$-J}(B,,, $B%f!<%68@8l(B Asir
  @section $B%f!<%6Dj5AH!?t$N=q$-J}(B
+ \E
+ \BEG
+ @node Writing user defined functions,,, User language Asir
+ @section Writing user defined functions
+ \E
  @menu
+ \BJP
  * $B%f!<%6Dj5AH!?t(B::
  * $BJQ?t$*$h$SITDj85(B::
  * $B0z?t(B::
-Line 122  C $B$GD>@\=q$/>l9g$KHf3S$7$F%W%m%0%i%`$,C;$/(B, $BF
+Line 254  C $B$GD>@\=q$/>l9g$KHf3S$7$F%W%m%0%i%`$,C;$/(B, $BF
 Line 122  C $B$GD>@\=q$/>l9g$KHf3S$7$F%W%m%0%i%`$,C;$/(B, $BF
 Line 254  C $B$GD>@\=q$/>l9g$KHf3S$7$F%W%m%0%i%`$,C;$/(B, $BF
  * return $BJ8(B::
  * if $BJ8(B::
  * $B%k!<%W(B break return continue::
+ * $B9=B$BNDj5A(B::
  * $B$5$^$6$^$J<0(B::
  * $B%W%j%W%m%;%C%5(B::
+ * $B%*%W%7%g%s;XDj(B::
+ * $B%b%8%e!<%k(B::
+ \E
+ \BEG
+ * User defined functions::
+ * variables and indeterminates::
+ * parameters and arguments::
+ * comments::
+ * statements::
+ * return statement::
+ * if statement::
+ * loop break return continue::
+ * structure definition::
+ * various expressions::
+ * preprocessor::
+ * option::
+ * module::
+ \E
  @end menu
+ \BJP
  @node $B%f!<%6Dj5AH!?t(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $B%f!<%6Dj5AH!?t(B
+ \E
+ \BEG
+ @node User defined functions,,, Writing user defined functions
+ @subsection User defined functions
+ \E
  @noindent
+ \BJP
  $B%f!<%6$K$h$kH!?t$NDj5A$O(B @samp{def} $BJ8$G9T$&(B. $BJ8K!%(%i!<$OFI$_9~$_;~$K(B
  $B$"$kDxEY%A%'%C%/$5$l(B, $B$*$*$h$=$N>l=j$,I=<($5$l$k(B.
  $B4{$K(B($B0z?t$N8D?t$K4X78$J$/(B)$BF1L>$NH!?t$,Dj5A$5$l$F$$$k>l9g$K$O(B,
  $B$=$NH!?t$O:FDj5A$5$l$k(B. @code{ctrl()} $BH!?t$K$h$j(B @code{verbose} $B%U%i%0(B
  $B$,(B on $B$K$J$C$F$$$k>l9g(B,
+ \E
+ \BEG
+ To define functions by an user himself, @samp{def} statement must be used.
+ Syntactical errors are detected in the parsing phase
+ of @b{Asir}, and notified with an indication of where @b{Asir} found the error.
+ If a function with the same name is already defined (regardless to
+ its arity,) the new definition will override the old one, and the user
+ will be told by a message,
+ \E
  @example
  afo() redefined.
  @end example
  @noindent
+ \BJP
  $B$H$$$&%a%C%;!<%8$,I=<($5$l$k(B. $B$"$kH!?t$NDj5A$K$*$$$F(B, $B$^$@L$Dj5A$NH!?t(B
  $B$r8F$S=P$7$F$$$F$b(B, $BDj5A;~$K$O%(%i!<$K$J$i$J$$(B. $B<B9T;~$KL$Dj5A$NH!?t(B
  $B$r8F$S=P$=$&$H$7$?>l9g$K%(%i!<$H$J$k(B.
+ \E
+ \BEG
+ on the screen when a flag @code{verbose} is set to a non-zero value by
+ @code{ctrl()}.
+ Recursive definition, and of course, recursive use of functions are
+ available.
+ A call for an yet undefined function in a function definition is not
+ detected as an error.  An error will be detected at execution of the
+ call of that yet undefined function.
+ \E
  @example
  @tex
  /* $X!$ */
-Line 158  def f(X) @{
+Line 335  def f(X) @{
 Line 158  def f(X) @{
 Line 335  def f(X) @{
  @}
  @tex
- /* ${_i}C_j ( 0 \le i \le N, 0 \le j \le i )$ */
+ \JP /* ${_i}C_j ( 0 \le i \le N, 0 \le j \le i )$ */
+ \EG /* ${_i}C_j ( 0 \le i \le N, 0 \le j \le i )$ */
  @end tex
  def c(N)
-Line 171  def c(N)
+Line 349  def c(N)
 Line 171  def c(N)
 Line 349  def c(N)
          @}
      return A;
  @}
+ @tex
+ /* $A+B$ */
+ @end tex
+ def add(A,B)
+ "add two numbers."
+ @{
+     return A+B;
+ @}
  @end example
  @noindent
+ \BJP
 $B$DL\$NNc$G$O(B, $BD9$5(B @code{N+1} $B$N%Y%/%H%k(B (@code{A}$B$H$9$k(B) $B$,JV$5$l$k(B.
  @code{A[I]} $B$OD9$5(B @code{I+1} $B$NG[Ns$G$"$j(B, $B$=$N$=$l$>$l$NMWAG$,(B
+ \E
+ \BEG
+ In the second example, @code{c(N)} returns a vector, say @code{A}, of length
+ @code{N+1}.  @code{A[I]} is a vector of length @code{I+1}, and
+ each element is again a vector which contains
+ \E
  @iftex
  @tex
  ${_I}C_J$
-Line 184  ${_I}C_J$
+Line 379  ${_I}C_J$
 Line 184  ${_I}C_J$
 Line 379  ${_I}C_J$
  @ifinfo
  ICJ
  @end ifinfo
- $B$rMWAG$H$9$kG[Ns$G$"$k(B.
+ \JP $B$rMWAG$H$9$kG[Ns$G$"$k(B.
+ \EG as its elements.
  @noindent
+ \BJP
+$B$DL\$NNc$G$O(B, $B0z?tJB$S$N$"$H$KJ8;zNs$,CV$+$l$F$$$k$,!"$3$l$O(B
+ Emacs-Lisp $B$N4X?tDj5A$KN`;w$N5!G=$G!"%X%k%WMQ$NJ8;zNs$G$"$k!#(B
+ $B$3$NNc$N>l9g!"(B@code{help(add)} $B$K$h$C$F$3$NJ8;zNs$,=PNO$5$l$k!#(B
+ \E
+ @table @t
+ \JP @item $B;2>H(B
+ \EG @item References
+ @fref{help}.
+ @end table
+ @noindent
+ \BJP
  $B0J2<$G$O(B, C $B$K$h$k%W%m%0%i%_%s%0$N7P83$,$J$$?M$N$?$a$K(B, @b{Asir} $B8@8l(B
  $B$K$h$k%W%m%0%i%`$N=q$-J}$r2r@b$9$k(B.
+ \E
+ \BEG
+ In the following, the manner of writing @b{Asir} programs is exhibited
+ for those who have no experience in writing C programs.
+ \E
+ \BJP
  @node $BJQ?t$*$h$SITDj85(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $BJQ?t$*$h$SITDj85(B
+ \E
  @noindent
+ \BJP
  $B4{$K=R$Y$?DL$j(B, @b{Asir} $B$K$*$$$F$O%W%m%0%i%`JQ?t$HITDj85$rL@3N$K(B
  $B6hJL$7$F$$$k(B.
+ \E
+ \BEG
+ @node variables and indeterminates,,, Writing user defined functions
+ @subsection variables and indeterminates
+ \E
  @table @b
+ \BJP
  @item $BJQ?t(B
  $BBgJ8;z$G;O$^$j(B, $B%"%k%U%!%Y%C%H(B, $B?t;z(B, @samp{_} $B$+$i$J$kJ8;zNs(B
+ \E
+ \BEG
+ @item variables (program variables)
+ A program variable is a string that begins with a capital
+ alphabetical letter followed by any numbers of alphabetical letters,
+ digits and @samp{_}.
+ \E
+ \BJP
  $BJQ?t$"$k$$$O%W%m%0%i%`JQ?t$H$O(B, @b{Asir} $B$N$5$^$6$^$J7?$NFbIt7A<0$r(B
  $B3JG<$9$k$?$a$NH"$G$"$j(B, $B3JG<$5$l$?FbIt7A<0$,(B, $B$3$NJQ?t$NCM$G$"$k(B.  $BJQ(B
  $B?t$,<0$NMWAG$H$7$FI>2A$5$l$k;~$O(B, $B$=$3$K<}$a$i$l$?CM$KCV$-49$($i$l$k(B.
  $B$9$J$o$A(B, $BFbIt7A<0$NCf$K$O%W%m%0%i%`JQ?t$O8=$l$J$$(B. $BJQ?t$OA4$F(B 0 $B$G(B
  $B=i4|2=$5$l$F$$$k(B.
+ \E
+ \BEG
+ A program variable is thought of a box (a carrier) which can contain
+ @b{Asir} objects of various types.  The content is called the `value'
+ of that variable.  When an expression in a program is to be evaluated,
+ the variable appearing in the expression is first replaced by its value
+ and then the expression is evaluated to some value and stored in
+ the memory.  Thus, no program variable appears in objects in the
+ internal form.
+ All the program variables are initialized to the value 0.
+ \E
  @example
  [0] X^2+X+1;
-Line 216  ICJ
+Line 458  ICJ
 Line 216  ICJ
 Line 458  ICJ
  @end example
+ \BJP
  @item $BITDj85(B
  $B>.J8;z$G;O$^$j(B, $B%"%k%U%!%Y%C%H(B, $B?t;z(B, @samp{_} $B$+$i$J$kJ8;zNs(B
  $BITDj85$H$O(B, $BB?9`<04D$r9=@.$9$k:]$KE:2C$5$l$kJQ?t$r$$$&(B. @b{Asir} $B$K(B
  $B$*$$$F$O(B, $BITDj85$OCM$r$b$?$J$$D61[E*$J85$G$"$j(B, $BITDj85$X$NCM$NBeF~$O(B
  $B5v$5$l$J$$(B.
+ \E
+ \BEG
+ @item indeterminates
+ An indeterminate is a string that begins with a small alphabetical letter
+ followed by any numbers of alphabetical letters, digits and @samp{_}.
+ An indeterminate is a transcendental element, so-called variable,
+ which is used to construct polynomial rings.
+ An indeterminate cannot have any value.  No assignment is allowed to it.
+ \E
  @example
  [3] X=x;
-Line 231  x^2+x+1
+Line 484  x^2+x+1
 Line 231  x^2+x+1
 Line 484  x^2+x+1
  @end example
  @end table
+ \BJP
  @node $B0z?t(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $B0z?t(B
+ \E
+ \BEG
+ @node parameters and arguments,,, Writing user defined functions
+ @subsection parameters and arguments
+ \E
  @example
  def sum(N) @{
-Line 243  def sum(N) @{
+Line 502  def sum(N) @{
 Line 243  def sum(N) @{
 Line 502  def sum(N) @{
  @end example
  @noindent
+ \BJP
  $B$3$l$O(B, 1 $B$+$i(B @code{N} $B$^$G$N<+A3?t$NOB$r5a$a$kH!?t(B @code{sum()} $B$N(B
  $BDj5A$G$"$k(B. $B$3$NNc$K$*$1$k(B @code{sum(N)} $B$N(B @code{N} $B$,0z?t$G$"$k(B.
  $B$3$NNc$O(B, 1 $B0z?tH!?t$NNc$G$"$k$,(B, $B0lHL$K0z?t$N8D?t$OG$0U$G$"$j(B,
-Line 252  def sum(N) @{
+Line 512  def sum(N) @{
 Line 252  def sum(N) @{
 Line 512  def sum(N) @{
  $B9TNs$r0z?t$KEO$7$?>l9g$G$"$k(B. $B$3$N>l9g$b(B, $BEO$5$l$?JQ?t$=$N$b$N$r=q$-(B
  $BBX$($k$3$H$O(B, $B$=$NH!?t$K6I=jE*$JA`:n$G$"$k$,(B, $BMWAG$r=q$-49$($?>l9g(B,
  $B$=$l$O(B, $B8F$S=P$7B&$N%Y%/%H%k(B, $B9TNs$NMWAG$r=q$-49$($k$3$H$K$J$k(B.
+ \E
+ \BEG
+ This is an example definition of a function that sums up integers
+ from 1 to @code{N}.  The @code{N} in @code{sum(N)} is called the
+ (formal) parameter of @code{sum(N)}.
+ The example shows a function of the single argument.
+ In general, any number of parameters can be specified by separating
+ by commas (@samp{,}).
+ A (formal) parameter accepts a value given as an argument (or an actual
+ parameter) at a function call of the function.
+ Since the value of the argument is given to the formal parameter,
+ any modification to the parameter does not usually affect the argument
+ (or actual parameter).  However, there are a few exceptions: vector
+ arguments and matrix arguments.
+ Let @code{A} be a program variable and assigned to a vector value
+ @code{[ a, b ]}.
+ If A is given as an actual parameter to a formal parameter, say @code{V},
+ of a function, then an assignment in the function to the vector element
+ designator @code{V[1]}, say @code{V[1]=c;}, causes modification of the
+ actual parameter @code{A} resulting @code{A} to have an altered value
+ @code{[ a c ]}.  Thus, if a vector is given to a formal parameter of
+ a function, then its element (and subsequently the vector itself) in
+ the calling side is modified through modification of the formal parameter
+ by a vector element designator in the called function.
+ The same applies to a matrix argument.
+ Note that, even in such case where a vector (or a matrix) is given to
+ a formal parameter, the assignment to the whole parameter itself has
+ only a local effect within the function.
+ \E
  @example
  def clear_vector(M) @{
-Line 263  def clear_vector(M) @{
+Line 553  def clear_vector(M) @{
 Line 263  def clear_vector(M) @{
 Line 553  def clear_vector(M) @{
  @end example
  @noindent
+ \BJP
  $B$3$NH!?t$O(B, $B0z?t$N%Y%/%H%k$r(B 0 $B%Y%/%H%k$K=i4|2=$9$k$?$a$NH!?t$G$"$k(B.
  $B$^$?(B, $B%Y%/%H%k$r0z?t$KEO$9$3$H$K$h$j(B, $BJ#?t$N7k2L$r0z?t$N%Y%/%H%k$K(B
  $B<}G<$7$FJV$9$3$H$,$G$-$k(B. $B<B:]$K$O(B, $B$3$N$h$&$J>l9g$K$O(B, $B7k2L$r%j%9%H(B
  $B$K$7$FJV$9$3$H$b$G$-$k(B. $B>u67$K1~$8$F;H$$$o$1$9$k$3$H$,K>$^$7$$(B.
+ \E
+ \BEG
+ This function will clear off the vector given as its argument to the
+ formal parameter @code{M} and return a 0 vector.
+ Passing a vector as an argument to a function enables returning
+ multiple results by packing each result in a vector element.
+ Another alternative to return multiple results is to use a list.
+ Which to use depends on cases.
+ \E
+ \BJP
  @node $B%3%a%s%H(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $B%3%a%s%H(B
+ \E
+ \BEG
+ @node comments,,, Writing user defined functions
+ @subsection comments
+ \E
  @noindent
- C $B$HF1MM(B @samp{/*} $B$H(B @samp{*/} $B$G0O$^$l$?ItJ,$O%3%a%s%H$H$7$F07$o$l$k(B.
+ \JP C $B$HF1MM(B @samp{/*} $B$H(B @samp{*/} $B$G0O$^$l$?ItJ,$O%3%a%s%H$H$7$F07$o$l$k(B.
+ \BEG
+ The text enclosed by @samp{/*} and @samp{*/} (containing @samp{/*} and
+ @samp{*/}) is treated as a comment and has no effect to the program
+ execution as in C programs.
+ \E
  @example
  /*
-Line 283  def afo(X) @{
+Line 595  def afo(X) @{
 Line 283  def afo(X) @{
 Line 595  def afo(X) @{
  @end example
  @noindent
+ \BJP
  $B%3%a%s%H$OJ#?t9T$KEO$C$F$b9=$o$J$$$,(B, $BF~$l;R$K$9$k$3$H$O$G$-$J$$(B.
  @samp{/*} $B$,$$$/$D$"$C$F$b:G=i$N$b$N$N$_$,M-8z$H$J$j(B, $B:G=i$K8=$l$?(B
  @samp{*/} $B$G%3%a%s%H$O=*N;$7$?$H8+$J$5$l$k(B. $B%W%m%0%i%`$J$I$G(B, $B%3%a%s%H(B
  $B$r4^$`2DG=@-$,$"$kItJ,$r%3%a%s%H%"%&%H$7$?>l9g$K$O(B, @code{#if 0},
- @code{#endif}$B$r;H$($P$h$$(B. (@xref{$B%W%j%W%m%;%C%5(B})
+ @code{#endif}$B$r;H$($P$h$$(B. (@xref{$B%W%j%W%m%;%C%5(B}.)
+ \E
+ \BEG
+ A comment can span to several lines, but it cannot be nested.
+ Only the first @samp{/*} is effective no matter how many @samp{/*}'s
+ in the subsequent text exist, and the comment terminates at the first
+ @samp{*/}.
+ In order to comment out a program part that may contain comments in it,
+ use the pair, @code{#if 0} and @code{#endif}. (@xref{preprocessor}.)
+ \E
  @example
  #if 0
-Line 297  def bfo(X) @{
+Line 620  def bfo(X) @{
 Line 297  def bfo(X) @{
 Line 620  def bfo(X) @{
  #endif
  @end example
+ \BJP
  @node $BJ8(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $BJ8(B
+ \E
+ \BEG
+ @node statements,,, Writing user defined functions
+ @subsection statements
+ \E
  @noindent
+ \BJP
  @b{Asir} $B$N%f!<%6H!?t$O(B,
  @example
-Line 310  def $BL>A0(B($B0z?t(B,$B0z?t(B,...,$B0z?t(B) @
+Line 640  def $BL>A0(B($B0z?t(B,$B0z?t(B,...,$B0z?t(B) @
 Line 310  def $BL>A0(B($B0z?t(B,$B0z?t(B,...,$B0z?t(B) @
 Line 640  def $BL>A0(B($B0z?t(B,$B0z?t(B,...,$B0z?t(B) @
      ...
      $BJ8(B
  @}
+ \E
+ \BEG
+ An user function of @b{Asir} is defined in the following form.
+ @example
+ def name(parameter, parameter,...,parameter) @{
+     statement
+     statement
+     ...
+     statement
+ @}
+ \E
  @end example
  @noindent
+ \BJP
  $B$H$$$&7A$GDj5A$5$l$k(B. $B$3$N$h$&$K(B, $BJ8$OH!?t$N4pK\E*9=@.MWAG$G$"$j(B, $B%W%m(B
  $B%0%i%`$r=q$/$?$a$K$O(B, $BJ8$,$I$N$h$&$J$b$N$G$"$k$+CN$i$J$1$l$P$J$i$J$$(B.
  $B:G$bC1=c$JJ8$H$7$F(B, $BC1J8$,$"$k(B. $B$3$l$O(B,
+ \E
+ \BEG
+ As you can see, the statement is a fundamental element of the
+ function.
+ Therefore, in order to write a program, you have to learn what
+ the statement is.  The simplest statement is the simple statement.
+ One example is an expression with a terminator (@samp{;} or @samp{$}.)
+ \E
  @example
  S = sum(N);
  @end example
  @noindent
+ \BJP
  $B$N$h$&$K(B, $B<0$K=*C<5-9f(B (@samp{;} $B$^$?$O(B @samp{$}) $B$r$D$1$?$b$N$G$"$k(B.
  $B$3$NC1J85Z$SN`;w$N(B @code{return} $BJ8(B, @code{break} $BJ8$J$I$,J8$N:G>.9=@.(B
- $BC10L$H$J$k(B. @code{if} $BJ8$d(B @code{for} $BJ8$NDj5A(B (@xref{$BJ8K!$N>\:Y(B}) $B$r8+$l(B
+ $BC10L$H$J$k(B. @code{if} $BJ8$d(B @code{for} $BJ8$NDj5A(B (@ref{$BJ8K!$N>\:Y(B}) $B$r8+$l(B
  $B$P$o$+$kDL$j(B, $B$=$l$i$NK\BN$O(B, $BC1$J$k0l$D$NJ8$H$7$FDj5A$5$l$F$$$k(B. $BDL>o(B
  $B$O(B, $BK\BN$K$OJ#?t$NJ8$,=q$1$k$3$H$,I,MW$H$J$k(B.  $B$3$N$h$&$J>l9g(B,
  @samp{@{} $B$H(B @samp{@}} $B$GJ8$NJB$S$r3g$C$F(B, $B0l$D$NJ8$H$7$F07$&$3$H$,$G(B
  $B$-$k(B. $B$3$l$rJ#J8$H8F$V(B.
+ \E
+ \BEG
+ A `@code{return} statement' and `@code{break} statement' are also
+ primitives to construct `statements.'
+ As you can see the syntactic definition of `@code{if} statement' and
+ `@code{for} statement', each of their bodies consists of a single
+ `statement.'  Usually, you need several statements in such a body.
+ To solve this contradictory requirement, you may use the `compound
+ statement.'  A `compound statement' is a sequence of `statement's
+ enclosed by a left brace @samp{@{} and a right brace @samp{@}}.
+ Thus, you can use multiple statement as if it were a single statement.
+ \E
  @example
  if ( I == 0 ) @{
-Line 339  if ( I == 0 ) @{
+Line 703  if ( I == 0 ) @{
 Line 339  if ( I == 0 ) @{
 Line 703  if ( I == 0 ) @{
  @end example
  @noindent
+ \BJP
  @samp{@}} $B$N8e$m$K$O=*C<5-9f$OI,MW$J$$(B. $B$J$<$J$i(B, @samp{@{} $BJ8JB$S(B
  @samp{@}}$B$,4{$KJ8$H$J$C$F$$$F(B, @code{if} $BJ8$NMW@A$rK~$?$7$F$$$k$+$i$G(B
  $B$"$k(B.
+ \E
+ \BEG
+ No terminator symbol is necessary after @samp{@}},
+ because @samp{@{} statement sequence @samp{@}} already forms a statement,
+ and it satisfies the syntactical requirement of the
+ `@code{if} statement.'
+ \E
+ \BJP
  @node return $BJ8(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection @code{return} $BJ8(B
+ \E
+ \BEG
+ @node return statement,,, Writing user defined functions
+ @subsection @code{return} statement
+ \E
  @noindent
- @code{return} $BJ8$O(B,
+ \JP @code{return} $BJ8$O(B,
+ \EG There are two forms of @code{return} statement.
  @example
- return $B<0(B;
+ \JP return $B<0(B;
+ \EG return expression;
  return;
  @end example
  @noindent
+ \BJP
  $B$N(B 2 $B$D$N7A<0$,$"$k(B. $B$$$:$l$bH!?t$+$iH4$1$k$?$a$NJ8$G$"$k(B. $BA0<T$O(B
  $BH!?t$NCM$H$7$F(B $B<0(B $B$rJV$9(B. $B8e<T$G$O(B, $BH!?t$NCM$H$7$F2?$,JV$5$l$k$+(B
  $B$O$o$+$i$J$$(B.
+ \E
+ \BEG
+ Both forms are used for exiting from a function.
+ The former returns the value of the expression as a function value.
+ The function value of the latter is not defined.
+ \E
+ \BJP
  @node if $BJ8(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection @code{if} $BJ8(B
+ \E
+ \BEG
+ @node if statement,,, Writing user defined functions
+ @subsection @code{if} statement
+ \E
  @noindent
- @code{if} $BJ8$K$O(B
+ \JP @code{if} $BJ8$K$O(B
+ \EG There are two forms of @code{if} statement.
  @example
+ \BJP
  if ( $B<0(B )             if ( $B<0(B )
      $BJ8(B       $B5Z$S(B         $BJ8(B
  else
      $BJ8(B
+ \E
+ \BEG
+ if ( expression )             if ( expression )
+      statement       and           statement
+ else
+      statement
+ \E
  @end example
  @noindent
+ \BJP
  $B$N(B 2 $B<oN`$,$"$k(B. $B$3$l$i$NF0:n$OL@$i$+$G$"$k$,(B, $BJ8$N0LCV$K(B @code{if} $BJ8(B
  $B$,Mh$?>l9g$KCm0U$rMW$9$k(B. $B<!$NNc$r9M$($F$_$h$&(B.
+ \E
+ \BEG
+ The interpretation of these forms are obvious.  However, be careful
+ when another @code{if} statement comes at the place for `statement'.
+ Let us examine the following example.
+ \E
  @example
+ \BJP
  if ( $B<0(B )
      if ( $B<0(B ) $BJ8(B
  else
      $BJ8(B
+ \E
+ \BEG
+ if ( expression1 )
+     if ( expression2 ) statement1
+ else
+     statement2
+ \E
  @end example
  @noindent
+ \BJP
  $B$3$N>l9g(B, $B;z2<$2$+$i$O(B, @code{else} $B0J2<$O(B, $B:G=i$N(B @code{if} $B$KBP1~$9$k(B
  $B$h$&$K8+$($k$,(B, $B%Q!<%6$O(B, $B<+F0E*$K(B 2 $BHVL\$N(B @code{if} $B$KBP1~$9$k$HH=CG$9$k(B.
  $B$9$J$o$A(B, 2 $B<oN`$N(B @code{if} $BJ8$r5v$7$?$?$a$K(B, $BJ8K!$K[#Kf@-$,8=$l(B, $B$=$l$r(B
  $B2r>C$9$k$?$a$K(B, @code{else} $B0J2<$O(B, $B:G$b6a$$(B @code{if} $B$KBP1~$9$k$H(B
  $B$$$&5,B'$,E,MQ$5$l$k$N$G$"$k(B. $B=>$C$F(B, $B$3$NNc$O(B,
+ \E
+ \BEG
+ One might guess @code{statement2} after @code{else} corresponds with the
+ first @code{if ( expression1 )} by its appearance of indentation.
+ But, as a matter of fact, the @code{Asir} parser decides that it
+ correspond with the second @code{if ( expression2 )}.
+ Ambiguity due to such two kinds of forms of @code{if} statement is
+ thus solved by introducing a rule that a statement preceded by an
+ @code{else} matches to the nearest preceding @code{if}.
+ Therefore, rearrangement of the above example for improving readability
+ according to the actual interpretation gives the following.
+ \E
  @example
+ \BJP
  if ( $B<0(B ) @{
      if ( $B<0(B ) $BJ8(B else $BJ8(B
  @}
+ \E
+ \BEG
+ if ( expression1 ) @{
+     if ( expression2 ) statement1 else statement2
+ @}
+ \E
  @end example
  @noindent
- $B$H$$$&0UL#$H$J$k(B. $B;z2<$2$KBP1~$5$;$k$?$a$K$O(B,
+ \JP $B$H$$$&0UL#$H$J$k(B. $B;z2<$2$KBP1~$5$;$k$?$a$K$O(B,
+ \BEG
+ On the other hand, in order to reflect the indentation, it must be
+ written as the following.
+ \E
  @example
+ \BJP
  if ( $B<0(B ) @{
      if ( $B<0(B ) $BJ8(B
  @} else
      $BJ8(B
+ \E
+ \BEG
+ if ( expression1 ) @{
+     if ( expression2 ) statement1
+ @} else
+     statement2
+ \E
  @end example
  @noindent
- $B$H$7$J$1$l$P$J$i$J$$(B.
+ \JP $B$H$7$J$1$l$P$J$i$J$$(B.
+ \BJP
  @node $B%k!<%W(B break return continue,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $B%k!<%W(B, @code{break}, @code{return}, @code{continue}
+ \E
+ \BEG
+ @node loop break return continue,,, Writing user defined functions
+ @subsection @code{loop}, @code{break}, @code{return}, @code{continue}
+ \E
  @noindent
+ \BJP
  $B%k!<%W$r9=@.$9$kJ8$O(B, @code{while} $BJ8(B, @code{for} $BJ8(B, @code{do} $BJ8(B
  $B$N(B 3 $B<oN`$,$"$k(B.
+ \E
+ \BEG
+ There are three kinds of statements for loops (repetitions):
+ the @code{while} statement, the @code{for} statement, and  the
+ @code{do} statement.
+ \E
  @itemize @bullet
  @item
- @code{while} $BJ8(B
+ \JP @code{while} $BJ8(B
+ \EG @code{while} statement
+ @*
+ \JP $B7A<0$O(B,
+ \EG It has the following form.
- $B7A<0$O(B,
  @example
- while ( $B<0(B ) $BJ8(B
+ \JP while ( $B<0(B ) $BJ8(B
+ \EG while ( expression ) statement
  @end example
  @noindent
+ \BJP
  $B$G(B, $B$3$l$O(B, $B<0(B $B$rI>2A$7$F(B, $B$=$NCM$,(B 0 $B$G$J$$8B$j(B $BJ8(B $B$r<B9T$9$k$H$$$&(B
  $B0UL#$H$J$k(B. $B$?$H$($P(B $B<0(B $B$,(B 1 $B$J$i$P(B, $BC1=c$JL58B%k!<%W$H$J$k(B.
+ \E
+ \BEG
+ This statement specifies that @code{statement} is repeatedly evaluated
+ as far as the @code{expression} evaluates to a non-zero value.
+ If the expression 1 is given to the @code{expression}, it forms an
+ infinite loop.
+ \E
  @item
- @code{for} $BJ8(B
+ \JP @code{for} $BJ8(B
+ \EG @code{for} statement
+ @*
+ \JP $B7A<0$O(B,
+ \EG It has the following form.
- $B7A<0$O(B,
  @example
- for ( $B<0JB$S(B-1; $B<0(B; $B<0JB$S(B-2 ) $BJ8(B
+ \JP for ( $B<0JB$S(B-1; $B<0(B; $B<0JB$S(B-2 ) $BJ8(B
+ \EG for ( expr list-1; expr; expr list-2 ) statement
  @end example
- $B$G(B, $B$3$l$O(B
+ \JP $B$G(B, $B$3$l$O(B
+ \EG This is equivalent to the program
  @example
+ \BJP
  $B<0JB$S(B-1 ($B$rC1J8JB$S$K$7$?$b$N(B)
  while ( $B<0(B ) @{
      $BJ8(B
      $B<0JB$S(B-2 ($B$rC1J8JB$S$K$7$?$b$N(B)
  @}
+ \E
+ \BEG
+ expr list-1 (transformed into a sequence of simple statement)
+ while ( expr ) @{
+     statement
+     expr list-2 (transformed into a sequence of simple statement)
+ @}
+ \E
  @end example
- $B$HEy2A$G$"$k(B.
+ \JP $B$HEy2A$G$"$k(B.
  @item
- @code{do} $BJ8(B
+ \JP @code{do} $BJ8(B
+ \EG @code{do} statement
+ @*
  @example
+ \BJP
  do @{
      $BJ8(B
  @} while ( $B<0(B )
+ \E
+ \BEG
+ do @{
+     statement
+ @} while ( expression )
+ \E
  @end example
+ \BJP
  $B$O(B, $B@h$K(B $BJ8$r<B9T$7$F$+$i>r7o<0$K$h$kH=Dj$r9T$&=j$,(B @code{while} $BJ8(B
  $B$H0[$J$C$F$$$k(B.
+ \E
+ \BEG
+ This statement differs from @code{while} statement by the location of
+ the termination condition: This statement first execute the
+ @code{statement} and then check the condition, whereas @code{while}
+ statement does it in the reverse order.
+ \E
  @end itemize
  @noindent
+ \BJP
  $B%k!<%W$rH4$1=P$9<jCJ$H$7$F(B,
  @code{break} $BJ85Z$S(B @code{return} $BJ8$,$"$k(B. $B$^$?(B, $B%k!<%W$N@)8f$r(B
  $B$"$k0LCV$K0\$9<jCJ$H$7$F(B @code{continue} $BJ8$,$"$k(B.
+ \E
+ \BEG
+ As means for exiting from loops, there are @code{break} statement and
+ @code{return} statement.  The @code{continue} statement allows to move
+ the control to a certain point of the loop.
+ \E
  @itemize @bullet
  @item
  @code{break}
+ @*
- @code{break} $BJ8$O(B, $B$=$l$r0O$`%k!<%W$r0l$D$@$1H4$1$k(B.
+ \JP @code{break} $BJ8$O(B, $B$=$l$r0O$`%k!<%W$r0l$D$@$1H4$1$k(B.
+ \EG The @code{break} statement is used to exit the inner most loop.
  @item
  @code{return}
+ @*
+ \BJP
  @code{return} $BJ8$O(B, $B0lHL$KH!?t$+$iH4$1$k$?$a$NJ8$G$"$j(B,
  $B%k!<%W$NCf$+$i$G$bM-8z$G$"$k(B.
+ \E
+ \BEG
+ The @code{return} statement is usually used to exit from a function call
+ and it is also effective in a loop.
+ \E
  @item
  @code{continue}
+ @*
+ \BJP
  @code{continue} $BJ8$O(B, $B%k!<%W$NK\BN$NJ8$NKvC<$K@)8f$r0\$9(B.
  $BNc$($P(B @code{for} $BJ8$G$O(B, $B:G8e$N<0JB$S$N<B9T$r9T$$(B, @code{while}
  $BJ8$G$O>r7o<0$NH=Dj$K0\$k(B.
+ \E
+ \BEG
+ The @code{continue} statement is used to move the control to the end
+ point of the loop body.
+ For example, the last expression list will be evaluated in a @code{for}
+ statement, and the termination condition will be evaluated in a
+ @code{while} statement.
+ \E
  @end itemize
+ \BJP
+ @node $B9=B$BNDj5A(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
+ @subsection $B9=B$BNDj5A(B
+ \E
+ \BEG
+ @node structure definition,,, Writing user defined functions
+ @subsection structure definition
+ \E
+ \BJP
+ $B9=B$BN$H$O(B, $B3F@.J,$NMWAG$,L>A0$G%"%/%;%9$G$-$k8GDjD9G[Ns$H;W$C$F$h$$(B.
+ $B3F9=B$BN$OL>A0$G6hJL$5$l$k(B. $B9=B$BN$O(B, @code{struct} $BJ8$K$h$j@k8@$5$l$k(B.
+ $B$"$k7?$N9=B$BN$O(B, $BAH$_9~$_4X?t(B @code{newstruct} $B$K$h$j@8@.$5$l$k(B.
+ $B9=B$BN$N3F%a%s%P$O(B, $B1i;;;R(B @code{->} $B$K$h$j%"%/%;%9$9$k(B.
+ $B%a%s%P$,9=B$BN$N>l9g(B, @code{->} $B$K$h$k;XDj$OF~$l;R$K$G$-$k(B.
+ \E
+ \BEG
+ A structure data type is a fixed length array and each component of the array
+ is accessed by its name. Each type of structure is distinguished by its name.
+ A structure data type is declared by @code{struct} statement.
+ A structure object is generated by a builtin function @code{newstruct}.
+ Each member of a structure is accessed by an operatator @code{->}.
+ If a member of a structure is again a structure, then the specification
+ by @code{->} can be nested.
+ \E
+ @example
+ [1] struct rat @{num,denom@};
+ [2] A = newstruct(rat);
+ @{0,0@}
+ [3] A->num = 1;
+ [4] A->den = 2;
+ [5] A;
+ @{1,2@}
+ @end example
+ \BJP
  @node $B$5$^$6$^$J<0(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $B$5$^$6$^$J<0(B
+ \E
+ \BEG
+ @node various expressions,,, Writing user defined functions
+ @subsection various expressions
+ \E
  @noindent
- $B<g$J<0$N9=@.MWAG$H$7$F$O(B, $B<!$N$h$&$J$b$N$,$"$k(B.
+ \JP $B<g$J<0$N9=@.MWAG$H$7$F$O(B, $B<!$N$h$&$J$b$N$,$"$k(B.
+ \EG Major elements to construct expressions are the following:
  @itemize @bullet
  @item
- $B2C8:>h=|(B, $BQQ(B
+ \JP $B2C8:>h=|(B, $BQQ(B
+ \EG addition, subtraction, multiplication, division, exponentiation
+ @*
+ \BJP
  $BQQ$O(B, @samp{^} $B$K$h$jI=$9(B. $B=|;;(B @samp{/} $B$O(B, $BBN$H$7$F$N1i;;$KMQ$$$k(B.
  $BNc$($P(B, @code{2/3} $B$OM-M}?t$N(B @code{2/3} $B$rI=$9(B.
  $B@0?t=|;;(B, $BB?9`<0=|;;(B ($B>jM>$r4^$`1i;;(B) $B$K$OJLESAH$_9~$_H!?t$,MQ0U$5$l$F$$$k(B.
+ \E
+ \BEG
+ The exponentiation is denoted by @samp{^}. (This differs from C language.)
+ Division denoted by @samp{/} is used to operate in a field, for example,
+ @code{2/3} results in a rational number @code{2/3}.
+ For integer division and polynomial division, both including remainder
+ operation, built-in functions are provided.
+ \E
  @example
  x+1  A^2*B*afo X/3
  @end example
  @item
- $B%$%s%G%C%/%9$D$-$NJQ?t(B
+ \JP $B%$%s%G%C%/%9$D$-$NJQ?t(B
+ \EG programming variables with indices
+ @*
+ \BJP
  $B%Y%/%H%k(B, $B9TNs(B, $B%j%9%H$NMWAG$O%$%s%G%C%/%9$rMQ$$$k$3$H$K$h$j<h$j=P$;$k(B.
  $B%$%s%G%C%/%9$O(B 0 $B$+$i;O$^$k$3$H$KCm0U$9$k(B. $B<h$j=P$7$?MWAG$,%Y%/%H%k(B,
  $B9TNs(B, $B%j%9%H$J$i(B, $B$5$i$K%$%s%G%C%/%9$r$D$1$k$3$H$bM-8z$G$"$k(B.
+ \E
+ \BEG
+ An element of a vector, a matrix or a list can be referred to by
+ indexing.
+ Note that the indices begin with number 0.  When the referred element
+ is again a vector, a matrix or a list, repeated indexing is also
+ effective.
+ \E
  @example
  V[0] M[1][2]
  @end example
  @item
- $BHf3S1i;;(B
+ \JP $BHf3S1i;;(B
+ \EG comparison operation
+ @*
+ \BJP
  $BEy$7$$(B (@samp{==}), $BEy$7$/$J$$(B (@samp{!=}), $BBg>.(B (@samp{>}, @samp{<},
  @samp{>=}, @samp{<=}) $B$N(B 2 $B9`1i;;$,$"$k(B. $B??$J$i$PM-M}?t$N(B 1, $B56$J$i$P(B
 $B$rCM$K;}$D(B.
+ \E
+ \BEG
+ There are comparison operations
+ @samp{==} for equivalence, @samp{!=} for non-equivalence,
+ @samp{>}, @samp{<},@samp{>=}, and @samp{<=} for larger or smaller.
+ The results of these operations are either value 1 for the truth,
+ or 0 for the false.
+ \E
  @item
- $BO@M}<0(B
+ \JP $BO@M}<0(B
+ \EG logical expression
+ @*
+ \BJP
  $BO@M}@Q(B (@samp{&&}), $BO@M}OB(B (@samp{||}) $B$N(B 2 $B9`1i;;$H(B, $BH]Dj(B (@samp{!})
  $B$,MQ0U$5$l$F$$$k(B. $BCM$O$d$O$j(B 1, 0 $B$G$"$k(B.
+ \E
+ \BEG
+ There are two binary logical operations
+ @samp{&&} for logical @samp{conjunction}(and),
+ @samp{||} for logical @samp{disjunction}(or),
+ and one unary logical operation @samp{!} for logical @samp{negation}(not).
+ The results of these operations are either value 1 for the truth,
+ and 0 for the false.
+ \E
  @item
- $BBeF~(B
+ \JP $BBeF~(B
+ \EG assignment
+ @*
+ \BJP
  $BDL>o$NBeF~$O(B @samp{=} $B$G9T$&(B. $B$3$N$[$+(B, $B;;=Q1i;;;R$HAH$_9g$o$;$F(B
  $BFC<l$JBeF~$r9T$&$3$H$b$G$-$k(B.
+ \E
+ \BEG
+ Value assignment of a program variable is usually done by @samp{=}.
+ There are special assignments combined with arithmetic operations.
+ \E
  (@samp{+=}, @samp{-=}, @samp{*=}, @samp{/=}, @samp{^=})
  @example
- A = 2  A *= 3 ($B$3$l$O(B A = A*3 $B$HF1$8(B; $B$=$NB>$N1i;;;R$bF1MM(B)
+ \JP A = 2  A *= 3 ($B$3$l$O(B A = A*3 $B$HF1$8(B; $B$=$NB>$N1i;;;R$bF1MM(B)
+ \EG A = 2  A *= 3 (the same as A = A*3; The others are alike.)
  @end example
  @item
- $BH!?t8F$S=P$7(B
+ \JP $BH!?t8F$S=P$7(B
+ \EG function call
- $BH!?t8F$S=P$7$b<0$N0l<o$G$"$k(B.
+ @*
+ \JP $BH!?t8F$S=P$7$b<0$N0l<o$G$"$k(B.
+ \EG A function call is also an expression.
  @item
  @samp{++}, @samp{--}
+ @*
- $B$3$l$i$O(B, $BJQ?t$NA08e$K$D$$$F(B, $B$=$l$>$l<!$N$h$&$JA`:n(B, $BCM$rI=$9(B.
+ \JP $B$3$l$i$O(B, $BJQ?t$NA08e$K$D$$$F(B, $B$=$l$>$l<!$N$h$&$JA`:n(B, $BCM$rI=$9(B.
+ \BEG
+ These operators are attached to or before a program variable,
+ and denote special operations and values.
+ \E
  @example
+ \BJP
  A++  $BCM$O85$N(B A $B$NCM(B, A = A+1
  A--  $BCM$O85$N(B A $B$NCM(B, A = A-1
  ++A  A = A+1, $BCM$OJQ2=8e$NCM(B
  --A  A = A-1, $BCM$OJQ2=8e$NCM(B
+ \E
+ \BEG
+ A++  the expression value is the previous value of A, and A = A+1
+ A--  the expression value is the previous value of A, and A = A-1
+ ++A  A = A+1, and the value is the one after increment of A
+ --A  A = A-1, and the value is the one after decrement of A
+ \E
  @end example
  @end itemize
+ \BJP
  @node $B%W%j%W%m%;%C%5(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
  @subsection $B%W%j%W%m%;%C%5(B
+ \E
+ \BEG
+ @node preprocessor,,, Writing user defined functions
+ @subsection preprocessor
+ \E
  @noindent
+ \BJP
  @b{Asir} $B$N%f!<%68@8l$O(B C $B8@8l$rLO$7$?$b$N$G$"$k(B. C $B$NFCD'$H$7$F(B,
  $B%W%j%W%m%;%C%5(B @code{cpp} $B$K$h$k%^%/%mE83+(B, $B%U%!%$%k$N%$%s%/%k!<%I(B
  $B$,$"$k$,(B, @b{Asir} $B$K$*$$$F$b%f!<%68@8l%U%!%$%k$NFI$_9~$_$N:](B
  @code{cpp} $B$rDL$7$F$+$iFI$_9~$`$3$H$H$7$?(B. $B$3$l$K$h$j%f!<%68@8l(B
  $B%U%!%$%kCf$G(B @code{#include}, @code{#define}, @code{#if} $B$J$I$,;H$($k(B.
+ \E
+ \BEG
+ he @b{Asir} user language imitates C language.  A typical features of
+ C language include macro expansion and file inclusion by the
+ preprocessor @code{cpp}.  Also, @b{Asir} read in user program files
+ through @code{cpp}.  This enables @b{Asir} user to use
+ @code{#include}, @code{#define}, @code{#if} etc. in his programs.
+ \E
  @itemize @bullet
  @item
  @code{#include}
+ @*
+ \BJP
  @code{cpp} $B$KFC$K0z?t$rEO$5$J$$$?$a(B, $B%$%s%/%k!<%I%U%!%$%k$O(B,
  @code{#include} $B$,=q$+$l$F$$$k%U%!%$%k$HF1$8%G%#%l%/%H%j$G%5!<%A$5$l$k(B.
+ \E
+ \BEG
+ Include files are searched within the same directory as the file
+ containing @code{#include} so that no arguments are passed to @code{cpp}.
+ \E
  @item
  @code{#define}
+ @*
+ \JP $B$3$l$O(B, C $B$K$*$1$k$N$HA4$/F1MM$KMQ$$$k$3$H$,$G$-$k(B.
+ \EG This can be used just as in C language.
- $B$3$l$O(B, C $B$K$*$1$k$N$HA4$/F1MM$KMQ$$$k$3$H$,$G$-$k(B.
  @item
  @code{#if}
+ @*
+ \BJP
  @code{/*}, @code{*/} $B$K$h$k%3%a%s%H$OF~$l;R$K$G$-$J$$$N$G(B, $B%W%m%0%i%`(B
  $B$NBg$-$JItJ,$r%3%a%s%H%"%&%H$9$k:]$K(B, @code{#if 0}, @code{#endif}
  $B$r;H$&$HJXMx$G$"$k(B.
+ \E
+ \BEG
+ This is conveniently used to comment out a large part of a user program
+ that may contain comments by @code{/*} and @code{*/},
+ because such comments cannot be nested.
+ \E
  @end itemize
  @noindent
- $B<!$NNc$O(B, @samp{defs.h} $B$K$"$k%^%/%mDj5A$G$"$k(B.
+ \JP $B<!$NNc$O(B, @samp{defs.h} $B$K$"$k%^%/%mDj5A$G$"$k(B.
+ \EG the following are the macro definitions in @samp{defs.h}.
  @example
  #define ZERO 0
-Line 632  A--  $BCM$O85$N(B A $B$NCM(B, A = A-1
+Line 1294  A--  $BCM$O85$N(B A $B$NCM(B, A = A-1
 Line 632  A--  $BCM$O85$N(B A $B$NCM(B, A = A-1
 Line 1294  A--  $BCM$O85$N(B A $B$NCM(B, A = A-1
  #define MAX(a,b) ((a)>(b)?(a):(b))
  @end example
+ \BJP
+ @node $B%*%W%7%g%s;XDj(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
+ @subsection $B%*%W%7%g%s;XDj(B
+ \E
+ \BEG
+ @node option,,, Writing user defined functions
+ @subsection option
+ \E
+ \BJP
+ $B%f!<%6Dj5A4X?t$,(B @var{N} $BJQ?t$G@k8@$5$l$?>l9g(B, $B$=$N4X?t$O(B, @var{N}
+ $BJQ?t$G$N8F$S=P$7$N$_$,5v$5$l$k(B.
+ \E
+ \BEG
+ If a user defined function is declared with @var{N} arguments,
+ then the function is callable with @var{N} arguments only.
+ \E
+ @example
+ [0] def factor(A) @{ return fctr(A); @}
+ [1] factor(x^5-1,3);
+ evalf : argument mismatch in factor()
+ return to toplevel
+ @end example
+ \BJP
+ $BITDj8D0z?t$N4X?t$r%f!<%68@8l$G5-=R$7$?$$>l9g(B, $B%j%9%H(B, $BG[Ns$rMQ$$$k$3$H$G(B
+ $B2DG=$H$J$k$,(B, $B<!$N$h$&$J$h$jJ,$+$j$d$9$$J}K!$b2DG=$G$"$k(B.
+ \E
+ \BEG
+ A function with indefinite number of arguments can be realized by
+ using a list or an array as its argument. Another method is available
+ as follows:
+ \E
+ @example
+ % cat factor
+ def factor(F)
+ @{
+     Mod = getopt(mod);
+     ModType = type(Mod);
+     if ( ModType == 1 ) /* 'mod' is not specified. */
+         return fctr(F);
+     else if ( ModType == 0 ) /* 'mod' is a number */
+         return modfctr(F,Mod);
+ @}
+ @end example
+ @example
+ [0] load("factor")$
+ [1] factor(x^5-1);
+ [[1,1],[x-1,1],[x^4+x^3+x^2+x+1,1]]
+ [2] factor(x^5-1|mod=11);
+ [[1,1],[x+6,1],[x+2,1],[x+10,1],[x+7,1],[x+8,1]]
+ @end example
+ \BJP
+$BHVL\$N(B @code{factor()} $B$N8F$S=P$7$K$*$$$F(B, $B4X?tDj5A$N:]$K@k8@$5$l$?0z(B
+ $B?t(B @code{x^5-1}$B$N8e$m$K(B @code{|mod=11} $B$,CV$+$l$F$$$k(B. $B$3$l$O(B, $B4X?t<B9T;~(B
+ $B$K(B, @code{mod} $B$H$$$&(B keyword $B$KBP$7$F(B @code{11} $B$H$$$&CM$r3d$jEv$F$k$3$H(B
+ $B$r;XDj$7$F$$$k(B. $B$3$l$r%*%W%7%g%s;XDj$H8F$V$3$H$K$9$k(B. $B$3$NCM$O(B
+ @code{getopt(mod)} $B$G<h$j=P$9$3$H$,$G$-$k(B. 1 $BHVL\$N8F$S=P$7$N$h$&$K(B
+ @code{mod} $B$KBP$9$k%*%W%7%g%s;XDj$,$J$$>l9g$K$O(B, @code{getopt(mod)} $B$O7?(B
+ $B<1JL;R(B -1 $B$N%*%V%8%'%/%H$rJV$9(B. $B$3$l$K$h$j(B, $B;XDj$,$J$$>l9g$NF0:n$r(B if $BJ8(B
+ $B$K$h$j5-=R$G$-$k(B. @samp{|} $B$N8e$m$K$O(B, $BG$0U8D$N%*%W%7%g%s$r(B, @samp{,}
+ $B$G6h@Z$C$F;XDj$9$k$3$H$,$G$-$k(B.
+ \E
+ \BEG
+ In the second call of @code{factor()}, @code{|mod=11} is placed
+ after the argument @code{x^5-1}, which appears in the declaration of
+ @code{factor()}. This means that the value @code{11} is assigned to
+ the keyword @code{mod} when the function is executed. The value
+ can be retrieved by @code{getopt(mod)}. We call such machinery
+ @var{option}. If the option for @var{mod} is not specified,
+ @code{getopt(mod)} returns an object whose type is -1. By this
+ feature, one can describe the behaviour of the function when
+ the option is not specified by @var{if} statements.
+ After @samp{|} one can append any number of options seperated by @samp{,}.
+ \E
+ @example
+ [100] xxx(1,2,x^2-1,[1,2,3]|proc=1,index=5);
+ @end example
+ \BJP
+ @node $B%b%8%e!<%k(B,,, $B%f!<%6Dj5AH!?t$N=q$-J}(B
+ @subsection $B%b%8%e!<%k(B
+ \E
+ \BEG
+ @node module,,, Writing user defined functions
+ @subsection module
+ \E
+ \BJP
+ $B%i%$%V%i%j$GDj5A$5$l$F$$$k4X?t(B, $BJQ?t$r%+%W%;%k2=$9$k;EAH$_$,(B
+ $B%b%8%e!<%k(B (module) $B$G$"$k(B.
+ $B$O$8$a$K%b%8%e!<%k$rMQ$$$?%W%m%0%i%`$NNc$r$"$2$h$&(B.
+ \E
+ \BEG
+ Function names and variables in a library may be
+ encapsulated by module.
+ Let us see an example of using module
+ \E
+ @example
+ module stack;
+ static Sp $
+ Sp = 0$
+ static Ssize$
+ Ssize = 100$
+ static Stack $
+ Stack = newvect(Ssize)$
+ localf push $
+ localf pop $
+ def push(A) @{
+   if (Sp >= Ssize) @{print("Warning: Stack overflow\nDiscard the top"); pop();@}
+   Stack[Sp] = A;
+   Sp++;
+ @}
+ def pop() @{
+   local A;
+   if (Sp <= 0) @{print("Stack underflow"); return 0;@}
+   Sp--;
+   A = Stack[Sp];
+   return A;
+ @}
+ endmodule;
+ def demo() @{
+   stack.push(1);
+   stack.push(2);
+   print(stack.pop());
+   print(stack.pop());
+ @}
+ @end example
+ \BJP
+ $B%b%8%e!<%k$O(B @code{module} $B%b%8%e!<%kL>(B  $B!A(B @code{endmodule}$B$G0O$`(B.
+ $B%b%8%e!<%k$NCf$@$1$G;H$&Bg0hJQ?t$O(B @code{static} $B$G@k8@$9$k(B.
+ $B$3$NJQ?t$O%b%8%e!<%k$N30$+$i$O;2>H$b$G$-$J$$$7JQ99$b$G$-$J$$(B.
+ $B%b%8%e!<%k$N30$NBg0hJQ?t$O(B @code{extern} $B$G@k8@$9$k(B.
+ \E
+ \BEG
+ Module is encapsulated by the sentences
+ @code{module} module name
+ and
+ @code{endmodule}.
+ A variable of a module is declared with the key word @code{static}.
+ The static variables cannot be refered nor changed out of the module,
+ but it can be refered and changed in any functions in the module.
+ A global variable which can be refered and changed at any place
+ is declared with the key word @code{extern}.
+ \E
+ \BJP
+ $B%b%8%e!<%kFbIt$GDj5A$9$k4X?t$O(B @code{localf} $B$rMQ$$$F@k8@$7$J$$$H$$$1$J$$(B.
+ $B>e$NNc$G$O(B @code{push} $B$H(B @code{pop} $B$r@k8@$7$F$$$k(B.
+ $B$3$N@k8@$OI,?\$G$"$k(B.
+ \E
+ \BEG
+ Any function defined in a module must be declared forward
+ with the keyword @code{localf}.
+ In the example above, @code{push} and @code{pop} are declared.
+ This declaration is necessary.
+ \E
+ \BJP
+ $B%b%8%e!<%k(B @code{moduleName} $B$GDj5A$5$l$?4X?t(B @code{functionName} $B$r(B
+ $B%b%8%e!<%k$N30$+$i8F$V$K$O(B
+      @code{moduleName.functionName($B0z?t(B1, $B0z?t(B2, ... )}
+ $B$J$k7A<0$G$h$V(B.
+ $B%b%8%e!<%k$NCf$+$i$O(B, $B4X?tL>$N$_$G$h$$(B.
+ $B<!$NNc$G$O(B, $B%b%8%e!<%k$N30$+$i%b%8%e!<%k(B @code{stack} $B$GDj5A$5$l$?4X?t(B @code{push},
+ @code{pop} $B$r8F$s$G$$$k(B.
+ \E
+ \BEG
+ A function @code{functionName} defined in a module @code{moduleName}
+ can be called by the expression
+ @code{moduleName.functioName(arg1, arg2, ...)}
+ out of the module.
+ Inside the module, @code{moduleName.} is not necessary.
+ In the example below, the functions @code{push} and @code{pop} defined
+ in the module @code{stack} are called out of the module.
+ \E
+ @example
+  stack.push(2);
+  print( stack.pop() );
+ @end example
+ \BJP
+ $B%b%8%e!<%k$GMQ$$$k4X?tL>$O6I=jE*$G$"$k(B.
+ $B$D$^$j%b%8%e!<%k$N30$dJL$N%b%8%e!<%k$GDj5A$5$l$F$$$k4X?tL>$HF1$8L>A0$,(B
+ $BMxMQ$G$-$k(B.
+ \E
+ \BEG
+ Any function name defined in a module is local.
+ In other words, the same function name may be used out of the module
+ to define a different function.
+ \E
+ \BJP
+ $B%b%8%e!<%k5!G=$OBg5,LO%i%$%V%i%j$N3+H/$rA[Dj$7$F$$$k(B.
+ $B%i%$%V%i%j$rI,MW$K1~$8$FJ,3d%m!<%I$9$k$K$O(B, $B4X?t(B @code{module_definedp} $B$rMQ$$$k$N$,(B
+ $BJXMx$G$"$k(B.
+ $B%G%^%s%I%m!<%I$O$?$H$($P<!$N$h$&$K9T$J$($PNI$$(B.
+ \E
+ \BEG
+ The module structure of asir is introduced to develop large libraries.
+ In order to load libraries on demand, the command @code{module_definedp}
+ will be useful.
+ The below is an example of demand loading.
+ \E
+ @example
+ if (!module_definep("stack")) load("stack.rr") $
+ @end example
+ \BJP
+ asir $B$G$O6I=jJQ?t$N@k8@$OITMW$G$"$C$?(B.
+ $B$7$+$7%b%8%e!<%k(B stack $B$NNc$r8+$l$PJ,$+$k$h$&$K(B, @code{local A;} $B$J$k7A<0$G(B
+ $B6I=jJQ?t$r@k8@$G$-$k(B.
+ $B%-!<%o!<%I(B @code{local} $B$rMQ$$$k$H(B, $B@k8@5!G=$,M-8z$H$J$k(B.
+ $B@k8@5!G=$rM-8z$K$9$k$H(B, $B@k8@$5$l$F$J$$JQ?t$O%m!<%I$NCJ3,$G(B
+ $B%(%i!<$r5/$3$9(B.
+ $BJQ?tL>$N%?%$%W%_%9$K$h$kM=4|$7$J$$%H%i%V%k$rKI$0$K$O(B,
+ $B@k8@5!G=$rM-8z$K$7$F%W%m%0%i%`$9$k$N$,$h$$(B.
+ \E
+ \BEG
+ It is not necessary to declare local variables in asir.
+ As you see in the example of the stack module,
+ we may declare local variables by the key word @code{local}.
+ Once this key word is used, asir requires to declare all the
+ variables.
+ In order to avoid some troubles to develop a large libraries,
+ it is recommended to use @code{local} declarations.
+ \E
+ \BJP
+ $B%b%8%e!<%kFb$N4X?t$r$=$N%b%8%e!<%k$,Dj5A$5$l$kA0$K(B
+ $B8F$S=P$9$h$&$J4X?t$r=q$/$H$-$K$O(B, $B$=$N4X?t$NA0$G%b%8%e!<%k$r<!$N$h$&$K(B
+ $B%W%m%H%?%$%W@k8@$7$F$*$/I,MW$,$"$k(B.
+ \E
+ \BEG
+ When we need to call a function in a module before the module is defined,
+ we must make a prototype declaration as the example below.
+ \E
+ @example
+ /* Prototype declaration of the module stack */
+ module stack;
+ localf push $
+ localf pop $
+ endmodule;
+ def demo() @{
+   print("----------------");
+   stack.push(1);
+   print(stack.pop());
+   print("---------------");
+ @}
+ module stack;
+   /* The body of the module stack */
+ endmodule;
+ @end example

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