| version 1.2, 1999/12/21 02:47:33 |
version 1.11, 2009/03/24 08:00:50 |
|
|
| @comment $OpenXM$ |
@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/array.texi,v 1.10 2005/02/10 04:59:21 noro Exp $ |
| \BJP |
\BJP |
| @node $BG[Ns(B,,, $BAH$_9~$_H!?t(B |
@node $BG[Ns(B,,, $BAH$_9~$_H!?t(B |
| @section $BG[Ns(B |
@section $BG[Ns(B |
|
|
| \E |
\E |
| |
|
| @menu |
@menu |
| * newvect:: |
* newvect vector vect:: |
| |
* ltov:: |
| * vtol:: |
* vtol:: |
| * newmat:: |
* newbytearray:: |
| |
* newmat matrix:: |
| * size:: |
* size:: |
| * det:: |
* det nd_det invmat:: |
| |
|
| * qsort:: |
* qsort:: |
| @end menu |
@end menu |
| |
|
| \JP @node newvect,,, $BG[Ns(B |
\JP @node newvect vector vect,,, $BG[Ns(B |
| \EG @node newvect,,, Arrays |
\EG @node newvect vector vect,,, Arrays |
| @subsection @code{newvect} |
@subsection @code{newvect}, @code{vector}, @code{vect} |
| @findex newvect |
@findex newvect |
| |
@findex vector |
| |
@findex vect |
| |
|
| @table @t |
@table @t |
| @item newvect(@var{len}[,@var{list}]) |
@item newvect(@var{len}[,@var{list}]) |
| |
@item vector(@var{len}[,@var{list}]) |
| \JP :: $BD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B. |
\JP :: $BD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B. |
| \EG :: Creates a new vector object with its length @var{len}. |
\EG :: Creates a new vector object with its length @var{len}. |
| |
@item vect([@var{elements}]) |
| |
\JP :: @var{elements} $B$rMWAG$H$9$k%Y%/%H%k$r@8@.$9$k(B. |
| |
\EG :: Creates a new vector object by @var{elements}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
|
|
| @item list |
@item list |
| \JP $B%j%9%H(B |
\JP $B%j%9%H(B |
| \EG list |
\EG list |
| |
@item elements |
| |
\JP $BMWAG$NJB$S(B |
| |
\EG elements of the vector |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| \BJP |
\BJP |
| @item |
@item |
| $BD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B. $BBh(B 2 $B0z?t$,$J$$>l9g(B, |
@code{vect} $B$OMWAG$NJB$S$+$i%Y%/%H%k$r@8@.$9$k(B. |
| |
@item |
| |
@code{vector} $B$O(B @code{newvect} $B$NJLL>$G$"$k(B. |
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@item |
| |
@code{newvect} $B$OD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B. $BBh(B 2 $B0z?t$,$J$$>l9g(B, |
| $B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 2 $B0z?t$,$"$k>l9g(B, |
$B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 2 $B0z?t$,$"$k>l9g(B, |
| $B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B%j%9%H$N(B |
$B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B%j%9%H$N(B |
| $B3FMWAG$K$h$j=i4|2=$5$l$k(B. $B3FMWAG$O(B, $B@hF,$+$i=g$K(B |
$B3FMWAG$K$h$j=i4|2=$5$l$k(B. $B3FMWAG$O(B, $B@hF,$+$i=g$K(B |
|
|
| $B$r=q$-49$($k$3$H$,$G$-$k(B. |
$B$r=q$-49$($k$3$H$,$G$-$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| |
@item |
| |
@code{vect} creates a new vector object by its elements. |
| |
@item |
| |
@code{vector} is an alias of @code{newvect}. |
| @item |
@item |
| Creates a new vector object with its length @var{len} and its elements |
@code{newvect} creates a new vector object with its length @var{len} and its elements |
| all cleared to value 0. |
all cleared to value 0. |
| If the second argument, a list, is given, the vector is initialized by |
If the second argument, a list, is given, the vector is initialized by |
| the list elements. |
the list elements. |
| Line 134 separated simply by a `blank space', while those of a |
|
| Line 154 separated simply by a `blank space', while those of a |
|
| [5,6] |
[5,6] |
| [4] size(A); |
[4] size(A); |
| [5] |
[5] |
| [5] def afo(V) @{ V[0] = x; @} |
[5] length(A); |
| [6] afo(A)$ |
5 |
| [7] A; |
[6] vect(1,2,3,4,[5,6]); |
| |
[ 1 2 3 4 [5,6] ] |
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[7] def afo(V) @{ V[0] = x; @} |
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[8] afo(A)$ |
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[9] A; |
| [ x 2 3 4 [5,6] ] |
[ x 2 3 4 [5,6] ] |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{newmat}, @fref{size}. |
@fref{newmat}, @fref{size}, @fref{ltov}, @fref{vtol}. |
| @end table |
@end table |
| |
|
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\JP @node ltov,,, $BG[Ns(B |
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\EG @node ltov,,, Arrays |
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@subsection @code{ltov} |
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@findex ltov |
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|
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@table @t |
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@item ltov(@var{list}) |
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\JP :: $B%j%9%H$r%Y%/%H%k$KJQ49$9$k(B. |
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\EG :: Converts a list into a vector. |
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@end table |
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|
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@table @var |
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@item return |
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\JP $B%Y%/%H%k(B |
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\EG vector |
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@item list |
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\JP $B%j%9%H(B |
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\EG list |
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@end table |
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|
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@itemize @bullet |
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\BJP |
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@item |
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$B%j%9%H(B @var{list} $B$rF1$8D9$5$N%Y%/%H%k$KJQ49$9$k(B. |
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@item |
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$B$3$N4X?t$O(B @code{newvect(length(@var{list}), @var{list})} $B$KEy$7$$(B. |
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\E |
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\BEG |
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@item |
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Converts a list @var{list} into a vector of same length. |
| |
See also @code{newvect()}. |
| |
\E |
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@end itemize |
| |
|
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@example |
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[3] A=[1,2,3]; |
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[4] ltov(A); |
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[ 1 2 3 ] |
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@end example |
| |
|
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@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{newvect}, @fref{vtol}. |
| |
@end table |
| |
|
| \JP @node vtol,,, $BG[Ns(B |
\JP @node vtol,,, $BG[Ns(B |
| \EG @node vtol,,, Arrays |
\EG @node vtol,,, Arrays |
| @subsection @code{vtol} |
@subsection @code{vtol} |
| Line 193 A conversion from a list to a vector is done by @code{ |
|
| Line 263 A conversion from a list to a vector is done by @code{ |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| |
@fref{newvect}, @fref{ltov}. |
| |
@end table |
| |
|
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\JP @node newbytearray,,, $BG[Ns(B |
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\EG @node newbytearray,,, Arrays |
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@subsection @code{newbytearray} |
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@findex newbytearray |
| |
|
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@table @t |
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@item newbytearray(@var{len},[@var{listorstring}]) |
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\JP :: $BD9$5(B @var{len} $B$N(B byte array $B$r@8@.$9$k(B. |
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\EG :: Creates a new byte array. |
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@end table |
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|
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@table @var |
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@item return |
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byte array |
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@item len |
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\JP $B<+A3?t(B |
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\EG non-negative integer |
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@item listorstring |
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\JP $B%j%9%H$^$?$OJ8;zNs(B |
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\EG list or string |
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@end table |
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|
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@itemize @bullet |
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@item |
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\JP @code{newvect} $B$HF1MM$K$7$F(B byte array $B$r@8@.$9$k(B. |
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\EG This function generates a byte array. The specification is |
| |
similar to that of @code{newvect}. |
| |
@item |
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\JP $BJ8;zNs$G=i4|CM$r;XDj$9$k$3$H$b2DG=$G$"$k(B. |
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\EG The initial value can be specified by a character string. |
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@item |
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\JP byte array $B$NMWAG$N%"%/%;%9$OG[Ns$HF1MM$G$"$k(B. |
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\EG One can access elements of a byte array just as an array. |
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@end itemize |
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|
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@example |
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[182] A=newbytearray(3); |
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|00 00 00| |
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[183] A=newbytearray(3,[1,2,3]); |
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|01 02 03| |
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[184] A=newbytearray(3,"abc"); |
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|61 62 63| |
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[185] A[0]; |
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97 |
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[186] A[1]=123; |
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123 |
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[187] A; |
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|61 7b 63| |
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@end example |
| |
|
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@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{newvect}. |
@fref{newvect}. |
| @end table |
@end table |
| |
|
| \JP @node newmat,,, $BG[Ns(B |
\JP @node newmat matrix,,, $BG[Ns(B |
| \EG @node newmat,,, Arrays |
\EG @node newmat matrix,,, Arrays |
| @subsection @code{newmat} |
@subsection @code{newmat}, @code{matrix} |
| @findex newmat |
@findex newmat |
| |
@findex matrix |
| |
|
| @table @t |
@table @t |
| @item newmat(@var{row},@var{col} [,@var{[[a,b,}...@var{],[c,d,}...@var{],}...@var{]}]) |
@item newmat(@var{row},@var{col} [,[[@var{a},@var{b},...],[@var{c},@var{d},...],...]]) |
| |
@item matrix(@var{row},@var{col} [,[[@var{a},@var{b},...],[@var{c},@var{d},...],...]]) |
| \JP :: @var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B. |
\JP :: @var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B. |
| \EG :: Creates a new matrix with @var{row} rows and @var{col} columns. |
\EG :: Creates a new matrix with @var{row} rows and @var{col} columns. |
| @end table |
@end table |
| Line 211 A conversion from a list to a vector is done by @code{ |
|
| Line 339 A conversion from a list to a vector is done by @code{ |
|
| @item return |
@item return |
| \JP $B9TNs(B |
\JP $B9TNs(B |
| \EG matrix |
\EG matrix |
| @item row,col |
@item row col |
| \JP $B<+A3?t(B |
\JP $B<+A3?t(B |
| \EG non-negative integer |
\EG non-negative integer |
| @item a,b,c,d |
@item a b c d |
| \JP $BG$0U(B |
\JP $BG$0U(B |
| \EG arbitrary |
\EG arbitrary |
| @end table |
@end table |
| Line 222 A conversion from a list to a vector is done by @code{ |
|
| Line 350 A conversion from a list to a vector is done by @code{ |
|
| @itemize @bullet |
@itemize @bullet |
| \BJP |
\BJP |
| @item |
@item |
| |
@code{matrix} $B$O(B @code{newmat} $B$NJLL>$G$"$k(B. |
| |
@item |
| @var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B. $BBh(B 3 $B0z?t$,$J$$>l9g(B, |
@var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B. $BBh(B 3 $B0z?t$,$J$$>l9g(B, |
| $B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 3 $B0z?t$,$"$k>l9g(B, |
$B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 3 $B0z?t$,$"$k>l9g(B, |
| $B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B3F9T$,(B, $B%j%9%H$N(B |
$B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B3F9T$,(B, $B%j%9%H$N(B |
| Line 238 A conversion from a list to a vector is done by @code{ |
|
| Line 368 A conversion from a list to a vector is done by @code{ |
|
| $B$r=q$-49$($k$3$H$,$G$-$k(B. |
$B$r=q$-49$($k$3$H$,$G$-$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| |
@item |
| |
@code{matrix} is an alias of @code{newmat}. |
| @item |
@item |
| If the third argument, a list, is given, the newly created matrix |
If the third argument, a list, is given, the newly created matrix |
| is initialized so that each element of the list (again a list) |
is initialized so that each element of the list (again a list) |
| Line 280 return to toplevel |
|
| Line 412 return to toplevel |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{newvect}, @fref{size}, @fref{det}. |
@fref{newvect}, @fref{size}, @fref{det nd_det invmat}. |
| @end table |
@end table |
| |
|
| \JP @node size,,, $BG[Ns(B |
\JP @node size,,, $BG[Ns(B |
| Line 314 or a list containing row size and column size of the g |
|
| Line 446 or a list containing row size and column size of the g |
|
| @itemize @bullet |
@itemize @bullet |
| \BJP |
\BJP |
| @item |
@item |
| @var{vect} $BKt$O(B, @var{mat} $B$N%5%$%:$r%j%9%H$G=PNO$9$k(B. |
@var{vect} $B$ND9$5(B, $B$^$?$O(B @var{mat} $B$NBg$-$5$r%j%9%H$G=PNO$9$k(B. |
| @item |
@item |
| @var{list} $B$N%5%$%:$O(B @code{length()}$B$r(B, $BM-M}<0$K8=$l$kC19`<0$N?t$O(B @code{nmono()} $B$rMQ$$$k(B. |
@var{vect} $B$ND9$5$O(B @code{length()} $B$G5a$a$k$3$H$b$G$-$k(B. |
| |
@item |
| |
@var{list} $B$ND9$5$O(B @code{length()}$B$r(B, $BM-M}<0$K8=$l$kC19`<0$N?t$O(B @code{nmono()} $B$rMQ$$$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Line 335 in a rational expression. |
|
| Line 469 in a rational expression. |
|
| [ 0 0 0 0 ] |
[ 0 0 0 0 ] |
| [1] size(A); |
[1] size(A); |
| [4] |
[4] |
| [2] B = newmat(2,3,[[1,2,3],[4,5,6]]); |
[2] length(A); |
| |
4 |
| |
[3] B = newmat(2,3,[[1,2,3],[4,5,6]]); |
| [ 1 2 3 ] |
[ 1 2 3 ] |
| [ 4 5 6 ] |
[ 4 5 6 ] |
| [3] size(B); |
[4] size(B); |
| [2,3] |
[2,3] |
| @end example |
@end example |
| |
|
| Line 348 in a rational expression. |
|
| Line 484 in a rational expression. |
|
| @fref{car cdr cons append reverse length}, @fref{nmono}. |
@fref{car cdr cons append reverse length}, @fref{nmono}. |
| @end table |
@end table |
| |
|
| \JP @node det,,, $BG[Ns(B |
\JP @node det nd_det invmat,,, $BG[Ns(B |
| \EG @node det,,, Arrays |
\EG @node det nd_det invmat,,, Arrays |
| @subsection @code{det} |
@subsection @code{det}, @code{nd_det}, @code{invmat} |
| @findex det |
@findex det |
| |
@findex nd_det |
| |
@findex invmat |
| |
|
| @table @t |
@table @t |
| @item det(@var{mat}[,@var{mod}]) |
@item det(@var{mat}[,@var{mod}]) |
| |
@itemx nd_det(@var{mat}[,@var{mod}]) |
| \JP :: @var{mat} $B$N9TNs<0$r5a$a$k(B. |
\JP :: @var{mat} $B$N9TNs<0$r5a$a$k(B. |
| \EG :: Determinant of @var{mat}. |
\EG :: Determinant of @var{mat}. |
| |
@item invmat(@var{mat}) |
| |
\JP :: @var{mat} $B$N5U9TNs$r5a$a$k(B. |
| |
\EG :: Inverse matrix of @var{mat}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| \JP $B<0(B |
\JP @code{det}: $B<0(B, @code{invmat}: $B%j%9%H(B |
| \EG expression |
\EG @code{det}: expression, @code{invmat}: list |
| @item mat |
@item mat |
| \JP $B9TNs(B |
\JP $B9TNs(B |
| \EG matrix |
\EG matrix |
| Line 374 in a rational expression. |
|
| Line 516 in a rational expression. |
|
| @itemize @bullet |
@itemize @bullet |
| \BJP |
\BJP |
| @item |
@item |
| $B9TNs(B @var{mat} $B$N9TNs<0$r5a$a$k(B. |
@code{det} $B$*$h$S(B @code{nd_det} $B$O9TNs(B @var{mat} $B$N9TNs<0$r5a$a$k(B. |
| |
@code{invmat} $B$O9TNs(B @var{mat} $B$N5U9TNs$r5a$a$k(B. $B5U9TNs$O(B @code{[$BJ,Jl(B, $BJ,;R(B]} |
| |
$B$N7A$GJV$5$l(B, @code{$BJ,Jl(B}$B$,9TNs(B, @code{$BJ,Jl(B/$BJ,;R(B} $B$,5U9TNs$H$J$k(B. |
| @item |
@item |
| $B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N9TNs<0$r5a$a$k(B. |
$B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N9TNs<0$r5a$a$k(B. |
| @item |
@item |
| $BJ,?t$J$7$N%,%&%9>C5nK!$K$h$C$F$$$k$?$a(B, $BB?JQ?tB?9`<0$r@.J,$H$9$k(B |
$BJ,?t$J$7$N%,%&%9>C5nK!$K$h$C$F$$$k$?$a(B, $BB?JQ?tB?9`<0$r@.J,$H$9$k(B |
| $B9TNs$KBP$7$F$O>.9TNs<0E83+$K$h$kJ}K!$N$[$&$,8zN($,$h$$>l9g$b$"$k(B. |
$B9TNs$KBP$7$F$O>.9TNs<0E83+$K$h$kJ}K!$N$[$&$,8zN($,$h$$>l9g$b$"$k(B. |
| |
@item |
| |
@code{nd_det} $B$OM-M}?t$^$?$OM-8BBN>e$NB?9`<09TNs$N9TNs<0(B |
| |
$B7W;;@lMQ$G$"$k(B. $B%"%k%4%j%:%`$O$d$O$jJ,?t$J$7$N%,%&%9>C5nK!$@$,(B, |
| |
$B%G!<%?9=B$$*$h$S>h=|;;$N9)IW$K$h$j(B, $B0lHL$K(B @code{det} $B$h$j9bB.$K(B |
| |
$B7W;;$G$-$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Determinant of matrix @var{mat}. |
@code{det} and @code{nd_det} compute the determinant of matrix @var{mat}. |
| |
@code{invmat} computes the inverse matrix of matrix @var{mat}. |
| |
@code{invmat} returns a list @code{[num,den]}, where @code{num} |
| |
is a matrix and @code{num/den} represents the inverse matrix. |
| @item |
@item |
| The computation is done over GF(@var{mod}) if @var{mod} is specitied. |
The computation is done over GF(@var{mod}) if @var{mod} is specitied. |
| @item |
@item |
| The fraction free Gaussian algorithm is employed. For matrices with |
The fraction free Gaussian algorithm is employed. For matrices with |
| multi-variate polynomial entries, minor expansion algorithm sometimes |
multi-variate polynomial entries, minor expansion algorithm sometimes |
| is more efficient than the fraction free Gaussian algorithm. |
is more efficient than the fraction free Gaussian algorithm. |
| |
@item |
| |
@code{nd_det} can be used for computing the determinant of a matrix with |
| |
polynomial entries over the rationals or finite fields. The algorithm |
| |
is an improved vesion of the fraction free Gaussian algorithm |
| |
and it computes the determinant faster than @code{det}. |
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 405 is more efficient than the fraction free Gaussian algo |
|
| Line 562 is more efficient than the fraction free Gaussian algo |
|
| [ 1 u u^2 u^3 u^4 ] |
[ 1 u u^2 u^3 u^4 ] |
| [ 1 v v^2 v^3 v^4 ] |
[ 1 v v^2 v^3 v^4 ] |
| [95] fctr(det(A)); |
[95] fctr(det(A)); |
| [[1,1],[u-v,1],[-z+v,1],[-z+u,1],[-y+u,1],[y-v,1],[-y+z,1],[-x+u,1],[-x+z,1], |
[[1,1],[u-v,1],[-z+v,1],[-z+u,1],[-y+u,1],[y-v,1],[-y+z,1],[-x+u,1], |
| [-x+v,1],[-x+y,1]] |
[-x+z,1],[-x+v,1],[-x+y,1]] |
| |
[96] A = newmat(3,3)$ |
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[97] for(I=0;I<3;I++)for(J=0,B=A[I],W=V[I];J<3;J++)B[J]=W^J; |
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[98] A; |
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[ 1 x x^2 ] |
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[ 1 y y^2 ] |
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[ 1 z z^2 ] |
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[99] invmat(A); |
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[[ -z*y^2+z^2*y z*x^2-z^2*x -y*x^2+y^2*x ] |
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[ y^2-z^2 -x^2+z^2 x^2-y^2 ] |
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[ -y+z x-z -x+y ],(-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y] |
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[100] A*B[0]; |
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[ (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 0 ] |
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[ 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 ] |
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[ 0 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y ] |
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[101] map(red,A*B[0]/B[1]); |
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[ 1 0 0 ] |
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[ 0 1 0 ] |
| |
[ 0 0 1 ] |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
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