| version 1.1.1.1, 1999/12/08 05:47:44 |
version 1.3, 2002/09/03 01:50:59 |
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@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/poly.texi,v 1.2 1999/12/21 02:47:34 noro Exp $ |
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\BJP |
| @node $BB?9`<0$*$h$SM-M}<0$N1i;;(B,,, $BAH$_9~$_H!?t(B |
@node $BB?9`<0$*$h$SM-M}<0$N1i;;(B,,, $BAH$_9~$_H!?t(B |
| @section $BB?9`<0(B, $BM-M}<0$N1i;;(B |
@section $BB?9`<0(B, $BM-M}<0$N1i;;(B |
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\E |
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\BEG |
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@node Polynomials and rational expressions,,, Built-in Function |
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@section operations with polynomials and rational expressions |
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\E |
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|
| @menu |
@menu |
| * var:: |
* var:: |
|
|
| * red:: |
* red:: |
| @end menu |
@end menu |
| |
|
| @node var,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node var,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node var,,, Polynomials and rational expressions |
| @subsection @code{var} |
@subsection @code{var} |
| @findex var |
@findex var |
| |
|
| @table @t |
@table @t |
| @item var(@var{rat}) |
@item var(@var{rat}) |
| :: @var{rat} $B$N<gJQ?t(B. |
\JP :: @var{rat} $B$N<gJQ?t(B. |
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\EG :: Main variable (indeterminate) of @var{rat}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BITDj85(B |
\JP $BITDj85(B |
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\EG indeterminate |
| @item rat |
@item rat |
| $BM-M}<0(B |
\JP $BM-M}<0(B |
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\EG rational expression |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B<gJQ?t$K4X$7$F$O(B, @xref{Asir $B$G;HMQ2DG=$J7?(B}. |
$B<gJQ?t$K4X$7$F$O(B, @xref{Asir $B$G;HMQ2DG=$J7?(B}. |
| @item |
@item |
|
|
| |
|
| @code{x}, @code{y}, @code{z}, @code{u}, @code{v}, @code{w}, @code{p}, @code{q}, @code{r}, @code{s}, @code{t}, @code{a}, @code{b}, @code{c}, @code{d}, @code{e}, |
@code{x}, @code{y}, @code{z}, @code{u}, @code{v}, @code{w}, @code{p}, @code{q}, @code{r}, @code{s}, @code{t}, @code{a}, @code{b}, @code{c}, @code{d}, @code{e}, |
| @code{f}, @code{g}, @code{h}, @code{i}, @code{j}, @code{k}, @code{l}, @code{m}, @code{n}, @code{o},$B0J8e$OJQ?t$N8=$l$?=g(B. |
@code{f}, @code{g}, @code{h}, @code{i}, @code{j}, @code{k}, @code{l}, @code{m}, @code{n}, @code{o},$B0J8e$OJQ?t$N8=$l$?=g(B. |
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\E |
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\BEG |
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@item |
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See @ref{Types in Asir} for main variable. |
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@item |
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Indeterminates (variables) are ordered by default as follows. |
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|
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@code{x}, @code{y}, @code{z}, @code{u}, @code{v}, @code{w}, @code{p}, @code{q}, |
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@code{r}, @code{s}, @code{t}, @code{a}, @code{b}, @code{c}, @code{d}, @code{e}, |
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@code{f}, @code{g}, @code{h}, @code{i}, @code{j}, @code{k}, @code{l}, @code{m}, |
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@code{n}, @code{o}. The other variables will be ordered after the above noted variables |
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so that the first comer will be ordered prior to the followers. |
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\E |
| @end itemize |
@end itemize |
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|
| @example |
@example |
|
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| @end example |
@end example |
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|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
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\EG @item References |
| @fref{ord}, @fref{vars}. |
@fref{ord}, @fref{vars}. |
| @end table |
@end table |
| |
|
| @node vars,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node vars,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
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\EG @node vars,,, Polynomials and rational expressions |
| @subsection @code{vars} |
@subsection @code{vars} |
| @findex vars |
@findex vars |
| |
|
| @table @t |
@table @t |
| @item vars(@var{obj}) |
@item vars(@var{obj}) |
| :: @var{obj} $B$K4^$^$l$kJQ?t$N%j%9%H(B. |
\JP :: @var{obj} $B$K4^$^$l$kJQ?t$N%j%9%H(B. |
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\EG :: A list of variables (indeterminates) in an expression @var{obj}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B%j%9%H(B |
\JP $B%j%9%H(B |
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\EG list |
| @item obj |
@item obj |
| $BG$0U(B |
\JP $BG$0U(B |
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\EG arbitrary |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
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\BJP |
| @item |
@item |
| $BM?$($i$l$?<0$K4^$^$l$kJQ?t$N%j%9%H$rJV$9(B. |
$BM?$($i$l$?<0$K4^$^$l$kJQ?t$N%j%9%H$rJV$9(B. |
| @item |
@item |
| $BJQ?t=g=x$N9b$$$b$N$+$i=g$KJB$Y$k(B. |
$BJQ?t=g=x$N9b$$$b$N$+$i=g$KJB$Y$k(B. |
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\E |
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\BEG |
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@item |
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Returns a list of variables (indeterminates) contained in a given expression. |
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@item |
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Lists variables according to the variable ordering. |
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\E |
| @end itemize |
@end itemize |
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|
| @example |
@example |
|
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| @end example |
@end example |
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|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
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\EG @item References |
| @fref{var}, @fref{uc}, @fref{ord}. |
@fref{var}, @fref{uc}, @fref{ord}. |
| @end table |
@end table |
| |
|
| @node uc,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node uc,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
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\EG @node uc,,, Polynomials and rational expressions |
| @subsection @code{uc} |
@subsection @code{uc} |
| @findex uc |
@findex uc |
| |
|
| @table @t |
@table @t |
| @item uc() |
@item uc() |
| :: $B?7$?$JITDj85$r@8@.$9$k(B. |
\JP :: $BL$Dj78?tK!$N$?$a$NITDj85$r@8@.$9$k(B. |
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\EG :: Create a new indeterminate for an undermined coeficient. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| @code{vtype} $B$,(B 1 $B$NITDj85(B |
\JP @code{vtype} $B$,(B 1 $B$NITDj85(B |
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\EG indeterminate with its @code{vtype} 1. |
| @end table |
@end table |
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|
| @itemize @bullet |
@itemize @bullet |
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\BJP |
| @item |
@item |
| @code{uc()} $B$r<B9T$9$k$?$S$K(B, @code{_0}, @code{_1}, @code{_2},... $B$H$$$&(B |
@code{uc()} $B$r<B9T$9$k$?$S$K(B, @code{_0}, @code{_1}, @code{_2},... $B$H$$$&(B |
| $BITDj85$r@8@.$9$k(B. |
$BITDj85$r@8@.$9$k(B. |
|
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| @code{strtov()} $B$rMQ$$$k(B. |
@code{strtov()} $B$rMQ$$$k(B. |
| @item |
@item |
| @code{uc()} $B$G@8@.$5$l$?ITDj85$NITDj85$H$7$F$N7?(B (@code{vtype}) $B$O(B 1 $B$G$"$k(B. |
@code{uc()} $B$G@8@.$5$l$?ITDj85$NITDj85$H$7$F$N7?(B (@code{vtype}) $B$O(B 1 $B$G$"$k(B. |
| (@xref{$BITDj85$N7?(B}) |
(@xref{$BITDj85$N7?(B}.) |
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\E |
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\BEG |
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@item |
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At every evaluation of command @code{uc()}, a new indeterminate in |
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the sequence of indeterminates @code{_0}, @code{_1}, @code{_2}, @dots{} |
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is created successively. |
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@item |
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Indeterminates created by @code{uc()} cannot be input on the keyboard. |
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By this property, you are free, no matter how many indeterminates you |
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will create dynamically by a program, from collision of created names |
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with indeterminates input from the keyboard or from program files. |
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@item |
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Functions, @code{rtostr()} and @code{strtov()}, are used to create |
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ordinary indeterminates (indeterminates having 0 for their @code{vtype}). |
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@item |
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Kernel sub-type of indeterminates created by @code{uc()} is 1. |
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(@code{vtype(uc())}=1) |
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\E |
| @end itemize |
@end itemize |
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|
| @example |
@example |
| Line 143 _0^2+2*_1*_0+_1^2 |
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| Line 204 _0^2+2*_1*_0+_1^2 |
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| @end example |
@end example |
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|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
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\EG @item References |
| @fref{vtype}, @fref{rtostr}, @fref{strtov}. |
@fref{vtype}, @fref{rtostr}, @fref{strtov}. |
| @end table |
@end table |
| |
|
| @node coef,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node coef,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
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\EG @node coef,,, Polynomials and rational expressions |
| @subsection @code{coef} |
@subsection @code{coef} |
| @findex coef |
@findex coef |
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|
| @table @t |
@table @t |
| @item coef(@var{poly},@var{deg}[,@var{var}]) |
@item coef(@var{poly},@var{deg}[,@var{var}]) |
| :: @var{poly} $B$N(B @var{var} ($B>JN,;~$O<gJQ?t(B) $B$K4X$9$k(B @var{deg} $B<!$N78?t(B. |
\JP :: @var{poly} $B$N(B @var{var} ($B>JN,;~$O<gJQ?t(B) $B$K4X$9$k(B @var{deg} $B<!$N78?t(B. |
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\BEG |
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:: The coefficient of a polynomial @var{poly} at degree @var{deg} |
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with respect to the variable @var{var} (main variable if unspecified). |
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\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
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\EG polynomial |
| @item poly |
@item poly |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
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\EG polynomial |
| @item var |
@item var |
| $BITDj85(B |
\JP $BITDj85(B |
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\EG indeterminate |
| @item deg |
@item deg |
| $B<+A3?t(B |
\JP $B<+A3?t(B |
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\EG non-negative integer |
| @end table |
@end table |
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|
| @itemize @bullet |
@itemize @bullet |
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\BJP |
| @item |
@item |
| @var{poly} $B$N(B @var{var} $B$K4X$9$k(B @var{deg} $B<!$N78?t$r=PNO$9$k(B. |
@var{poly} $B$N(B @var{var} $B$K4X$9$k(B @var{deg} $B<!$N78?t$r=PNO$9$k(B. |
| @item |
@item |
| Line 175 _0^2+2*_1*_0+_1^2 |
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| Line 247 _0^2+2*_1*_0+_1^2 |
|
| @item |
@item |
| @var{var} $B$,<gJQ?t$G$J$$;~(B, @var{var} $B$,<gJQ?t$N>l9g$KHf3S$7$F(B |
@var{var} $B$,<gJQ?t$G$J$$;~(B, @var{var} $B$,<gJQ?t$N>l9g$KHf3S$7$F(B |
| $B8zN($,Mn$A$k(B. |
$B8zN($,Mn$A$k(B. |
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\E |
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\BEG |
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@item |
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The coefficient of a polynomial @var{poly} at degree @var{deg} |
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with respect to the variable @var{var}. |
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@item |
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The default value for @var{var} is the main variable, i.e., |
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@t{var(@var{poly})}. |
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@item |
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For multi-variate polynomials, access to coefficients depends on |
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the specified indeterminates. For example, taking coef for the main |
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variable is much faster than for other variables. |
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\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 187 y^3+3*z*y^2+3*z^2*y+z^3 |
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| Line 272 y^3+3*z*y^2+3*z^2*y+z^3 |
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| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
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\EG @item References |
| @fref{var}, @fref{deg mindeg}. |
@fref{var}, @fref{deg mindeg}. |
| @end table |
@end table |
| |
|
| @node deg mindeg,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node deg mindeg,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
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\EG @node deg mindeg,,, Polynomials and rational expressions |
| @subsection @code{deg}, @code{mindeg} |
@subsection @code{deg}, @code{mindeg} |
| @findex deg |
@findex deg |
| @findex mindeg |
@findex mindeg |
| |
|
| @table @t |
@table @t |
| @item deg(@var{poly},@var{var}) |
@item deg(@var{poly},@var{var}) |
| :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B. |
\JP :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B. |
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\EG :: The degree of a polynomial @var{poly} with respect to variable. |
| @item mindeg(@var{poly},@var{var}) |
@item mindeg(@var{poly},@var{var}) |
| :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:GDc<!?t(B. |
\JP :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:GDc<!?t(B. |
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\BEG |
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:: The least exponent of the terms with non-zero coefficients in |
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a polynomial @var{poly} with respect to the variable @var{var}. |
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In this manual, this quantity is sometimes referred to the minimum |
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degree of a polynomial for short. |
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\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B<+A3?t(B |
\JP $B<+A3?t(B |
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\EG non-negative integer |
| @item poly |
@item poly |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
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\EG polynomial |
| @item var |
@item var |
| $BITDj85(B |
\JP $BITDj85(B |
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\EG indeterminate |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BM?$($i$l$?B?9`<0$NJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B, $B:GDc<!?t$r=PNO$9$k(B. |
$BM?$($i$l$?B?9`<0$NJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B, $B:GDc<!?t$r=PNO$9$k(B. |
| @item |
@item |
| $BJQ?t(B @var{var} $B$r>JN,$9$k$3$H$O=PMh$J$$(B. |
$BJQ?t(B @var{var} $B$r>JN,$9$k$3$H$O=PMh$J$$(B. |
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\E |
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\BEG |
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@item |
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The least exponent of the terms with non-zero coefficients in |
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a polynomial @var{poly} with respect to the variable @var{var}. |
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In this manual, this quantity is sometimes referred to the minimum |
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degree of a polynomial for short. |
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@item |
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Variable @var{var} must be specified. |
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\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 228 y^3+3*z*y^2+3*z^2*y+z^3 |
|
| Line 336 y^3+3*z*y^2+3*z^2*y+z^3 |
|
| 1 |
1 |
| @end example |
@end example |
| |
|
| @node nmono,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node nmono,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node nmono,,,Polynomials and rational expressions |
| @subsection @code{nmono} |
@subsection @code{nmono} |
| @findex nmono |
@findex nmono |
| |
|
| @table @t |
@table @t |
| @item nmono(@var{rat}) |
@item nmono(@var{rat}) |
| :: @var{rat} $B$NC19`<0$N9`?t(B. |
\JP :: @var{rat} $B$NC19`<0$N9`?t(B. |
| |
\EG :: Number of monomials in rational expression @var{rat}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B<+A3?t(B |
\JP $B<+A3?t(B |
| |
\EG non-negative integer |
| @item rat |
@item rat |
| $BM-M}<0(B |
\JP $BM-M}<0(B |
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\EG rational expression |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BB?9`<0$rE83+$7$?>uBV$G$N(B 0 $B$G$J$$78?t$r;}$DC19`<0$N9`?t$r5a$a$k(B. |
$BB?9`<0$rE83+$7$?>uBV$G$N(B 0 $B$G$J$$78?t$r;}$DC19`<0$N9`?t$r5a$a$k(B. |
| @item |
@item |
| $BM-M}<0$N>l9g$O(B, $BJ,;R$HJ,Jl$N9`?t$NOB$,JV$5$l$k(B. |
$BM-M}<0$N>l9g$O(B, $BJ,;R$HJ,Jl$N9`?t$NOB$,JV$5$l$k(B. |
| @item |
@item |
| $BH!?t7A<0(B (@xref{$BITDj85$N7?(B}) $B$O(B, $B0z?t$,2?$G$"$C$F$bC19`$H$_$J$5$l$k(B. (1 $B8D$NITDj85$HF1$8(B. ) |
$BH!?t7A<0(B (@ref{$BITDj85$N7?(B}) $B$O(B, $B0z?t$,2?$G$"$C$F$bC19`$H$_$J$5$l$k(B. (1 $B8D$NITDj85$HF1$8(B. ) |
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\E |
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\BEG |
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@item |
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Number of monomials with non-zero number coefficients in the full |
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expanded form of the given polynomial. |
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@item |
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For a rational expression, the sum of the numbers of monomials |
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of the numerator and denominator. |
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@item |
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A function form is regarded as a single indeterminate no matter how |
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complex arguments it has. |
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\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 263 y^3+3*z*y^2+3*z^2*y+z^3 |
|
| Line 388 y^3+3*z*y^2+3*z^2*y+z^3 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{vtype}. |
@fref{vtype}. |
| @end table |
@end table |
| |
|
| @node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node ord,,, Polynomials and rational expressions |
| @subsection @code{ord} |
@subsection @code{ord} |
| @findex ord |
@findex ord |
| |
|
| @table @t |
@table @t |
| @item ord([@var{varlist}]) |
@item ord([@var{varlist}]) |
| :: $BJQ?t=g=x$N@_Dj(B |
\JP :: $BJQ?t=g=x$N@_Dj(B |
| |
\EG :: It sets the ordering of indeterminates (variables). |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BJQ?t$N%j%9%H(B |
\JP $BJQ?t$N%j%9%H(B |
| |
\EG list of indeterminates |
| @item varlist |
@item varlist |
| $BJQ?t$N%j%9%H(B |
\JP $BJQ?t$N%j%9%H(B |
| |
\EG list of indeterminates |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B0z?t$,$"$k$H$-(B, $B0z?t$NJQ?t%j%9%H$r@hF,$K=P$7(B, $B;D$j$NJQ?t$,$=$N8e$K(B |
$B0z?t$,$"$k$H$-(B, $B0z?t$NJQ?t%j%9%H$r@hF,$K=P$7(B, $B;D$j$NJQ?t$,$=$N8e$K(B |
| $BB3$/$h$&$KJQ?t=g=x$r@_Dj$9$k(B. $B0z?t$N$"$k$J$7$K4X$o$i$:(B, @code{ord()} |
$BB3$/$h$&$KJQ?t=g=x$r@_Dj$9$k(B. $B0z?t$N$"$k$J$7$K4X$o$i$:(B, @code{ord()} |
| Line 296 y^3+3*z*y^2+3*z^2*y+z^3 |
|
| Line 427 y^3+3*z*y^2+3*z^2*y+z^3 |
|
| $B$"$k$$$O(B, $B?7$?$JJQ?t$,8=$l$?;~E@$K9T$o$l$k(B |
$B$"$k$$$O(B, $B?7$?$JJQ?t$,8=$l$?;~E@$K9T$o$l$k(B |
| $B$Y$-$G$"$k(B. $B0[$J$kJQ?t=g=x$N$b$H$G@8@.$5$l$?<0$I$&$7$N1i;;(B |
$B$Y$-$G$"$k(B. $B0[$J$kJQ?t=g=x$N$b$H$G@8@.$5$l$?<0$I$&$7$N1i;;(B |
| $B$,9T$o$l$?>l9g(B, $BM=4|$;$L7k2L$,@8$:$k$3$H$b$"$jF@$k(B. |
$B$,9T$o$l$?>l9g(B, $BM=4|$;$L7k2L$,@8$:$k$3$H$b$"$jF@$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
When an argument is given, |
| |
this function rearranges the ordering of variables (indeterminates) |
| |
so that the indeterminates in the argument @var{varlist} precede |
| |
and the other indeterminates follow in the system's variable ordering. |
| |
Regardless of the existence of an argument, it always returns the |
| |
final variable ordering. |
| |
|
| |
@item |
| |
Note that no change will be made to the variable ordering of internal |
| |
forms of objects which already exists in the system, no matter what |
| |
reordering you specify. Therefore, the reordering should be limited to |
| |
the time just after starting @b{Asir}, or to the time when one has |
| |
decided himself to start a totally new computation which has no relation |
| |
with the previous results. |
| |
Note that unexpected results may be obtained from operations between |
| |
objects which are created under different variable ordering. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 311 _w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l, |
|
| Line 462 _w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l, |
|
| cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)] |
cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)] |
| @end example |
@end example |
| |
|
| @node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node sdiv sdivm srem sremm sqr sqrm,,, Polynomials and rational expressions |
| @subsection @code{sdiv}, @code{sdivm}, @code{srem}, @code{sremm}, @code{sqr}, @code{sqrm} |
@subsection @code{sdiv}, @code{sdivm}, @code{srem}, @code{sremm}, @code{sqr}, @code{sqrm} |
| @findex sdiv |
@findex sdiv |
| @findex sdivm |
@findex sdivm |
| Line 323 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(- |
|
| Line 475 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(- |
|
| @table @t |
@table @t |
| @item sdiv(@var{poly1},@var{poly2}[,@var{v}]) |
@item sdiv(@var{poly1},@var{poly2}[,@var{v}]) |
| @itemx sdivm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) |
@itemx sdivm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) |
| :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&$r5a$a$k(B. |
\JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&$r5a$a$k(B. |
| |
\BEG |
| |
:: Quotient of @var{poly1} divided by @var{poly2} provided that the |
| |
division can be performed within polynomial arithmetic over the |
| |
rationals. |
| |
\E |
| @item srem(@var{poly1},@var{poly2}[,@var{v}]) |
@item srem(@var{poly1},@var{poly2}[,@var{v}]) |
| @item sremm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) |
@item sremm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) |
| :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>jM>$r5a$a$k(B. |
\JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>jM>$r5a$a$k(B. |
| |
\BEG |
| |
:: Remainder of @var{poly1} divided by @var{poly2} provided that the |
| |
division can be performed within polynomial arithmetic over the |
| |
rationals. |
| |
\E |
| @item sqr(@var{poly1},@var{poly2}[,@var{v}]) |
@item sqr(@var{poly1},@var{poly2}[,@var{v}]) |
| @item sqrm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) |
@item sqrm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) |
| |
\BJP |
| :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&(B, $B>jM>$r(B |
:: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&(B, $B>jM>$r(B |
| $B5a$a$k(B. |
$B5a$a$k(B. |
| |
\E |
| |
\BEG |
| |
:: Quotient and remainder of @var{poly1} divided by @var{poly2} provided |
| |
that the division can be performed within polynomial arithmetic over |
| |
the rationals. |
| |
\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : $BB?9`<0(B, @code{sqr()}, @code{sqrm()} : @code{[$B>&(B,$B>jM>(B]} $B$J$k%j%9%H(B |
\JP @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : $BB?9`<0(B, @code{sqr()}, @code{sqrm()} : @code{[$B>&(B,$B>jM>(B]} $B$J$k%j%9%H(B |
| |
\EG @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : polynomial @code{sqr()}, @code{sqrm()} : a list @code{[quotient,remainder]} |
| @item poly1 poly2 |
@item poly1 poly2 |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @item v |
@item v |
| $BITDj85(B |
\JP $BITDj85(B |
| |
\EG indeterminate |
| @item mod |
@item mod |
| $BAG?t(B |
\JP $BAG?t(B |
| |
\EG prime |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @var{poly1} $B$r(B @var{poly2} $B$N<gJQ?t(B @t{var}(@var{poly2}) |
@var{poly1} $B$r(B @var{poly2} $B$N<gJQ?t(B @t{var}(@var{poly2}) |
| ( $B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) $B$K4X$9$kB?9`<0$H8+$F(B, |
( $B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) $B$K4X$9$kB?9`<0$H8+$F(B, |
| Line 368 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(- |
|
| Line 542 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(- |
|
| $B@0?t=|;;$N>&(B, $B>jM>$O(B @code{idiv}, @code{irem} $B$rMQ$$$k(B. |
$B@0?t=|;;$N>&(B, $B>jM>$O(B @code{idiv}, @code{irem} $B$rMQ$$$k(B. |
| @item |
@item |
| $B78?t$KBP$9$k>jM>1i;;$O(B @code{%} $B$rMQ$$$k(B. |
$B78?t$KBP$9$k>jM>1i;;$O(B @code{%} $B$rMQ$$$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Regarding @var{poly1} as an uni-variate polynomial in the main variable |
| |
of @var{poly2}, |
| |
i.e. @t{var(@var{poly2})} (@var{v} if specified), @code{sdiv()} and |
| |
@code{srem()} compute |
| |
the polynomial quotient and remainder of @var{poly1} divided by @var{poly2}. |
| |
@item @code{sdivm()}, @code{sremm()}, @code{sqrm()} execute the same |
| |
operation over GF(@var{mod}). |
| |
@item |
| |
Division operation of polynomials is performed by the following steps: |
| |
(1) obtain the quotient of leading coefficients; let it be Q; |
| |
(2) remove the leading term of @var{poly1} by subtracting, from |
| |
@var{poly1}, the product of Q with some powers of main variable |
| |
and @var{poly2}; obtain a new @var{poly1}; |
| |
(3) repeat the above step until the degree of @var{poly1} become smaller |
| |
than that of @var{poly2}. |
| |
For fulfillment, by operating in polynomials, of this procedure, the |
| |
divisions at step (1) in every repetition must be an exact division of |
| |
polynomials. This is the true meaning of what we say |
| |
``division can be performed within polynomial arithmetic |
| |
over the rationals.'' |
| |
@item |
| |
There are typical cases where the division is possible: |
| |
leading coefficient of @var{poly2} is a rational number; |
| |
@var{poly2} is a factor of @var{poly1}. |
| |
@item |
| |
Use @code{sqr()} to get both the quotient and remainder at once. |
| |
@item |
| |
Use @code{idiv()}, @code{irem()} for integer quotient. |
| |
@item |
| |
For remainder operation on all integer coefficients, use @code{%}. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 391 return to toplevel |
|
| Line 599 return to toplevel |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{idiv irem}, @fref{%}. |
@fref{idiv irem}, @fref{%}. |
| @end table |
@end table |
| |
|
| @node tdiv,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node tdiv,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node tdiv,,, Polynomials and rational expressions |
| @subsection @code{tdiv} |
@subsection @code{tdiv} |
| @findex tdiv |
@findex tdiv |
| |
|
| @table @t |
@table @t |
| @item tdiv(@var{poly1},@var{poly2}) |
@item tdiv(@var{poly1},@var{poly2}) |
| :: @var{poly1} $B$,(B @var{poly2} $B$G3d$j@Z$l$k$+$I$&$+D4$Y$k(B. |
\JP :: @var{poly1} $B$,(B @var{poly2} $B$G3d$j@Z$l$k$+$I$&$+D4$Y$k(B. |
| |
\EG :: Tests whether @var{poly2} divides @var{poly1}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B3d$j@Z$l$k$J$i$P>&(B, $B3d$j@Z$l$J$1$l$P(B 0 |
\JP $B3d$j@Z$l$k$J$i$P>&(B, $B3d$j@Z$l$J$1$l$P(B 0 |
| |
\EG Quotient if @var{poly2} divides @var{poly1}, 0 otherwise. |
| @item poly1 poly2 |
@item poly1 poly2 |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @var{poly2} $B$,(B @var{poly1} $B$rB?9`<0$H$7$F3d$j@Z$k$+$I$&$+D4$Y$k(B. |
@var{poly2} $B$,(B @var{poly1} $B$rB?9`<0$H$7$F3d$j@Z$k$+$I$&$+D4$Y$k(B. |
| @item |
@item |
| $B$"$kB?9`<0$,4{Ls0x;R$G$"$k$3$H$O$o$+$C$F$$$k$,(B, $B$=$N=EJ#EY$,$o$+$i$J$$(B |
$B$"$kB?9`<0$,4{Ls0x;R$G$"$k$3$H$O$o$+$C$F$$$k$,(B, $B$=$N=EJ#EY$,$o$+$i$J$$(B |
| $B>l9g$K(B, @code{tdiv()} $B$r7+$jJV$78F$V$3$H$K$h$j=EJ#EY$,$o$+$k(B. |
$B>l9g$K(B, @code{tdiv()} $B$r7+$jJV$78F$V$3$H$K$h$j=EJ#EY$,$o$+$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Tests whether @var{poly2} divides @var{poly1} in polynomial ring. |
| |
@item |
| |
One application of this function: Consider the case where a polynomial |
| |
is certainly an irreducible factor of the other polynomial, but |
| |
the multiplicity of the factor is unknown. Application of @code{tdiv()} |
| |
to the polynomials repeatedly yields the multiplicity. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 432 x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18* |
|
| Line 656 x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18* |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{sdiv sdivm srem sremm sqr sqrm}. |
@fref{sdiv sdivm srem sremm sqr sqrm}. |
| @end table |
@end table |
| |
|
| @node %,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node %,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node %,,, Polynomials and rational expressions |
| @subsection @code{%} |
@subsection @code{%} |
| @findex % |
@findex % |
| |
|
| @table @t |
@table @t |
| @item @var{poly} % @var{m} |
@item @var{poly} % @var{m} |
| :: $B@0?t$K$h$k>jM>(B |
\JP :: $B@0?t$K$h$k>jM>(B |
| |
\EG :: integer remainder to all integer coefficients of the polynomial. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B@0?t$^$?$OB?9`<0(B |
\JP $B@0?t$^$?$OB?9`<0(B |
| |
\EG integer or polynomial |
| @item poly |
@item poly |
| $B@0?t$^$?$O@0?t78?tB?9`<0(B |
\JP $B@0?t$^$?$O@0?t78?tB?9`<0(B |
| |
\EG integer or polynomial with integer coefficients |
| @item m |
@item m |
| $B@0?t(B |
\JP $B@0?t(B |
| |
\EG intger |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @var{poly} $B$N3F78?t$r(B @var{m} $B$G3d$C$?>jM>$GCV$-49$($?B?9`<0$rJV$9(B. |
@var{poly} $B$N3F78?t$r(B @var{m} $B$G3d$C$?>jM>$GCV$-49$($?B?9`<0$rJV$9(B. |
| @item |
@item |
| Line 464 x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18* |
|
| Line 695 x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18* |
|
| @code{irem()} $B$HF1MM$KMQ$$$k$3$H$,$G$-$k(B. |
@code{irem()} $B$HF1MM$KMQ$$$k$3$H$,$G$-$k(B. |
| @item |
@item |
| @var{poly} $B$N78?t(B, @var{m} $B$H$b@0?t$G$"$kI,MW$,$"$k$,(B, $B%A%'%C%/$O9T$J$o$l$J$$(B. |
@var{poly} $B$N78?t(B, @var{m} $B$H$b@0?t$G$"$kI,MW$,$"$k$,(B, $B%A%'%C%/$O9T$J$o$l$J$$(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Returns a polynomial whose coefficients are remainders of the |
| |
coefficients of the input polynomial divided by @var{m}. |
| |
@item |
| |
The resulting coefficients are all normalized to non-negative integers. |
| |
@item |
| |
An integer is allowed for @var{poly}. This can be used for an |
| |
alternative for @code{irem()} except that the result is normalized to |
| |
a non-negative integer. |
| |
@item |
| |
Coefficients of @var{poly} and @var{m} must all be integers, though the |
| |
type checking is not done. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 478 x^5+2*x^4+x^3+x^2+2*x+1 |
|
| Line 724 x^5+2*x^4+x^3+x^2+2*x+1 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{idiv irem}. |
@fref{idiv irem}. |
| @end table |
@end table |
| |
|
| @node subst psubst,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node subst psubst,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node subst psubst,,, Polynomials and rational expressions |
| @subsection @code{subst}, @code{psubst} |
@subsection @code{subst}, @code{psubst} |
| @findex subst |
@findex subst |
| @findex psubst |
@findex psubst |
| Line 490 x^5+2*x^4+x^3+x^2+2*x+1 |
|
| Line 738 x^5+2*x^4+x^3+x^2+2*x+1 |
|
| @table @t |
@table @t |
| @item subst(@var{rat}[,@var{varn},@var{ratn}]*) |
@item subst(@var{rat}[,@var{varn},@var{ratn}]*) |
| @item psubst(@var{rat}[,@var{var},@var{rat}]*) |
@item psubst(@var{rat}[,@var{var},@var{rat}]*) |
| |
\BJP |
| :: @var{rat} $B$N(B @var{varn} $B$K(B @var{ratn} $B$rBeF~(B |
:: @var{rat} $B$N(B @var{varn} $B$K(B @var{ratn} $B$rBeF~(B |
| (@var{n=1,2},... $B$G:8$+$i1&$K=g<!BeF~$9$k(B). |
(@var{n=1,2},... $B$G:8$+$i1&$K=g<!BeF~$9$k(B). |
| |
\E |
| |
\BEG |
| |
:: Substitute @var{ratn} for @var{varn} in expression @var{rat}. |
| |
(@var{n=1,2},@dots{}. |
| |
Substitution will be done successively from left to right |
| |
if arguments are repeated.) |
| |
\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BM-M}<0(B |
\JP $BM-M}<0(B |
| |
\EG rational expression |
| @item rat,ratn |
@item rat,ratn |
| $BM-M}<0(B |
\JP $BM-M}<0(B |
| |
\EG rational expression |
| @item varn |
@item varn |
| $BITDj85(B |
\JP $BITDj85(B |
| |
\EG indeterminate |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BM-M}<0$NFCDj$NITDj85$K(B, $BDj?t$"$k$$$OB?9`<0(B, $BM-M}<0$J$I$rBeF~$9$k$N$KMQ$$$k(B. |
$BM-M}<0$NFCDj$NITDj85$K(B, $BDj?t$"$k$$$OB?9`<0(B, $BM-M}<0$J$I$rBeF~$9$k$N$KMQ$$$k(B. |
| @item |
@item |
| Line 523 x^5+2*x^4+x^3+x^2+2*x+1 |
|
| Line 783 x^5+2*x^4+x^3+x^2+2*x+1 |
|
| $B$J$k$Y$/J,Jl(B, $BJ,;R$,Bg$-$/$J$i$J$$$h$&$KG[N8$9$k$3$H$b$7$P$7$PI,MW$H$J$k(B. |
$B$J$k$Y$/J,Jl(B, $BJ,;R$,Bg$-$/$J$i$J$$$h$&$KG[N8$9$k$3$H$b$7$P$7$PI,MW$H$J$k(B. |
| @item |
@item |
| $BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B. |
$BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Substitutes rational expressions for specified kernels in a rational |
| |
expression. |
| |
@item |
| |
@t{subst}(@var{rat},@var{var1},@var{rat1},@var{var2},@var{rat2},@dots{}) |
| |
has the same effect as |
| |
@t{subst}(@t{subst}(@var{rat},@var{var1},@var{rat1}),@var{var2},@var{rat2},@dots{}). |
| |
@item |
| |
Note that repeated substitution is done from left to right successively. |
| |
You may get different result by changing the specification order. |
| |
@item |
| |
Ordinary @code{subst()} performs |
| |
substitution at all levels of a scalar algebraic expression creeping |
| |
into arguments of function forms recursively. |
| |
Function @code{psubst()} regards such a function form as an independent |
| |
indeterminate, and does not attempt to apply substitution to its |
| |
arguments. (The name comes after Partial SUBSTitution.) |
| |
@item |
| |
Since @b{Asir} does not reduce common divisors of a rational expression |
| |
automatically, substitution of a rational expression to an expression |
| |
may cause unexpected increase of computation time. |
| |
Thus, it is often necessary to write a special function to meet the |
| |
individual problem so that the denominator and the numerator do not |
| |
become too large. |
| |
@item |
| |
The same applies to substitution by rational numbers. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| sin(x)*t |
sin(x)*t |
| @end example |
@end example |
| |
|
| @node diff,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node diff,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node diff,,, Polynomials and rational expressions |
| @subsection @code{diff} |
@subsection @code{diff} |
| @findex diff |
@findex diff |
| |
|
| @table @t |
@table @t |
| @item diff(@var{rat}[,@var{varn}]*) |
@item diff(@var{rat}[,@var{varn}]*) |
| @item diff(@var{rat},@var{varlist}) |
@item diff(@var{rat},@var{varlist}) |
| :: @var{rat} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=g<!HyJ,$9$k(B. |
\JP :: @var{rat} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=g<!HyJ,$9$k(B. |
| |
\BEG |
| |
:: Differentiate @var{rat} successively by @var{var}'s for the first |
| |
form, or by variables in @var{varlist} for the second form. |
| |
\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B<0(B |
\JP $B<0(B |
| |
\EG expression |
| @item rat |
@item rat |
| $BM-M}<0(B ($B=iEyH!?t$r4^$s$G$b$h$$(B) |
\JP $BM-M}<0(B ($B=iEyH!?t$r4^$s$G$b$h$$(B) |
| |
\EG rational expression which contains elementary functions. |
| @item varn |
@item varn |
| $BITDj85(B |
\JP $BITDj85(B |
| |
\EG indeterminate |
| @item varlist |
@item varlist |
| $BITDj85$N%j%9%H(B |
\JP $BITDj85$N%j%9%H(B |
| |
\EG list of indeterminates |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BM?$($i$l$?=iEyH!?t$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G(B |
$BM?$($i$l$?=iEyH!?t$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G(B |
| $B=g<!HyJ,$9$k(B. |
$B=g<!HyJ,$9$k(B. |
| @item |
@item |
| $B:8B&$NITDj85$h$j(B, $B=g$KHyJ,$7$F$$$/(B. $B$D$^$j(B, @t{diff}(@var{rat},@t{x,y}) $B$O(B, |
$B:8B&$NITDj85$h$j(B, $B=g$KHyJ,$7$F$$$/(B. $B$D$^$j(B, @t{diff}(@var{rat},@t{x,y}) $B$O(B, |
| @t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}) $B$HF1$8$G$"$k(B. |
@t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}) $B$HF1$8$G$"$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Differentiate @var{rat} successively by @var{var}'s for the first |
| |
form, or by variables in @var{varlist} for the second form. |
| |
@item |
| |
differentiation is performed by the specified indeterminates (variables) |
| |
from left to right. |
| |
@t{diff}(@var{rat},@t{x,y}) is the same as |
| |
@t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}). |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| sin(x) |
sin(x) |
| @end example |
@end example |
| |
|
| @node res,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node res,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node res,,, Polynomials and rational expressions |
| @subsection @code{res} |
@subsection @code{res} |
| @findex res |
@findex res |
| |
|
| @table @t |
@table @t |
| @item res(@var{var},@var{poly1},@var{poly2}[,@var{mod}]) |
@item res(@var{var},@var{poly1},@var{poly2}[,@var{mod}]) |
| :: @var{var} $B$K4X$9$k(B @var{poly1} $B$H(B @var{poly2} $B$N=*7k<0(B. |
\JP :: @var{var} $B$K4X$9$k(B @var{poly1} $B$H(B @var{poly2} $B$N=*7k<0(B. |
| |
\EG :: Resultant of @var{poly1} and @var{poly2} with respect to @var{var}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @item var |
@item var |
| $BITDj85(B |
\JP $BITDj85(B |
| |
\EG indeterminate |
| @item poly1,poly2 |
@item poly1,poly2 |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @item mod |
@item mod |
| $BAG?t(B |
\JP $BAG?t(B |
| |
\EG prime |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BFs$D$NB?9`<0(B @var{poly1} $B$H(B @var{poly2} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k(B |
$BFs$D$NB?9`<0(B @var{poly1} $B$H(B @var{poly2} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k(B |
| $B=*7k<0$r5a$a$k(B. |
$B=*7k<0$r5a$a$k(B. |
|
|
| $BItJ,=*7k<0%"%k%4%j%:%`$K$h$k(B. |
$BItJ,=*7k<0%"%k%4%j%:%`$K$h$k(B. |
| @item |
@item |
| $B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N7W;;$r9T$&(B. |
$B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N7W;;$r9T$&(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Resultant of two polynomials @var{poly1} and @var{poly2} |
| |
with respect to @var{var}. |
| |
@item |
| |
Sub-resultant algorithm is used to compute the resultant. |
| |
@item |
| |
The computation is done over GF(@var{mod}) if @var{mod} is specified. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| -x^3-x^2-y^3 |
-x^3-x^2-y^3 |
| @end example |
@end example |
| |
|
| @node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node fctr sqfr,,, Polynomials and rational expressions |
| @subsection @code{fctr}, @code{sqfr} |
@subsection @code{fctr}, @code{sqfr} |
| @findex fctr |
@findex fctr |
| @findex sqfr |
@findex sqfr |
| |
|
| @table @t |
@table @t |
| @item fctr(@var{poly}) |
@item fctr(@var{poly}) |
| :: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. |
\JP :: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. |
| |
\EG :: Factorize polynomial @var{poly} over the rationals. |
| @item sqfr(@var{poly}) |
@item sqfr(@var{poly}) |
| :: @var{poly} $B$rL5J?J}J,2r$9$k(B. |
\JP :: @var{poly} $B$rL5J?J}J,2r$9$k(B. |
| |
\EG :: Gets a square-free factorization of polynomial @var{poly}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B%j%9%H(B |
\JP $B%j%9%H(B |
| |
\EG list |
| @item poly |
@item poly |
| $BM-M}?t78?t$NB?9`<0(B |
\JP $BM-M}?t78?t$NB?9`<0(B |
| |
\EG polynomial with rational coefficients |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BM-M}?t78?t$NB?9`<0(B @var{poly} $B$r0x?tJ,2r$9$k(B. @code{fctr()} $B$O4{Ls0x;RJ,2r(B, |
$BM-M}?t78?t$NB?9`<0(B @var{poly} $B$r0x?tJ,2r$9$k(B. @code{fctr()} $B$O4{Ls0x;RJ,2r(B, |
| @code{sqfr()} $B$OL5J?J}0x;RJ,2r(B. |
@code{sqfr()} $B$OL5J?J}0x;RJ,2r(B. |
|
|
| @item |
@item |
| @b{$B?t78?t(B} $B$O(B, (@var{poly}/@b{$B?t78?t(B}) $B$,(B, $B@0?t78?t$G(B, $B78?t$N(B GCD $B$,(B 1 $B$H$J$k(B |
@b{$B?t78?t(B} $B$O(B, (@var{poly}/@b{$B?t78?t(B}) $B$,(B, $B@0?t78?t$G(B, $B78?t$N(B GCD $B$,(B 1 $B$H$J$k(B |
| $B$h$&$JB?9`<0$K$J$k$h$&$KA*$P$l$F$$$k(B. (@code{ptozp()} $B;2>H(B) |
$B$h$&$JB?9`<0$K$J$k$h$&$KA*$P$l$F$$$k(B. (@code{ptozp()} $B;2>H(B) |
| |
\E |
| |
\BEG |
| |
@item |
| |
Factorizes polynomial @var{poly} over the rationals. |
| |
@code{fctr()} for irreducible factorization; |
| |
@code{sqfr()} for square-free factorization. |
| |
@item |
| |
The result is represented by a list, whose elements are a pair |
| |
represented as |
| |
|
| |
[[@b{num},1],[@b{factor},@b{multiplicity}],...]. |
| |
@item |
| |
Products of all @b{factor}^@b{multiplicity} and @b{num} is equal to |
| |
@var{poly}. |
| |
@item |
| |
The number @b{num} is determined so that (@var{poly}/@b{num}) is an |
| |
integral polynomial and its content (GCD of all coefficients) is 1. |
| |
(@xref{ptozp}.) |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 670 x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4 |
|
| Line 1022 x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{ufctrhint}. |
@fref{ufctrhint}. |
| @end table |
@end table |
| |
|
| @node ufctrhint,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node ufctrhint,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node ufctrhint,,, Polynomials and rational expressions |
| @subsection @code{ufctrhint} |
@subsection @code{ufctrhint} |
| @findex ufctrhint |
@findex ufctrhint |
| |
|
| @table @t |
@table @t |
| @item ufctrhint(@var{poly},@var{hint}) |
@item ufctrhint(@var{poly},@var{hint}) |
| :: $B<!?t>pJs$rMQ$$$?(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B |
\JP :: $B<!?t>pJs$rMQ$$$?(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B |
| |
\BEG |
| |
:: Factorizes uni-variate polynomial @var{poly} over the rational number |
| |
field when the degrees of its factors are known to be some integer |
| |
multiples of @var{hint}. |
| |
\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B%j%9%H(B |
\JP $B%j%9%H(B |
| |
\EG list |
| @item poly |
@item poly |
| $BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B |
\JP $BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B |
| |
\EG uni-variate polynomial with rational coefficients |
| @item hint |
@item hint |
| $B<+A3?t(B |
\JP $B<+A3?t(B |
| |
\EG non-negative integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B3F4{Ls0x;R$N<!?t$,(B @var{hint} $B$NG\?t$G$"$k$3$H$,$o$+$C$F$$$k>l9g$K(B |
$B3F4{Ls0x;R$N<!?t$,(B @var{hint} $B$NG\?t$G$"$k$3$H$,$o$+$C$F$$$k>l9g$K(B |
| @var{poly} $B$N4{Ls0x;RJ,2r$r(B @code{fctr()} $B$h$j8zN(NI$/9T$&(B. |
@var{poly} $B$N4{Ls0x;RJ,2r$r(B @code{fctr()} $B$h$j8zN(NI$/9T$&(B. |
| @var{poly} $B$,(B, @var{d} $B<!$N3HBgBN>e$K$*$1$k(B |
@var{poly} $B$,(B, @var{d} $B<!$N3HBgBN>e$K$*$1$k(B |
| $B$"$kB?9`<0$N%N%k%`(B (@xref{$BBe?tE*?t$K4X$9$k1i;;(B}) $B$GL5J?J}$G$"$k>l9g(B, |
$B$"$kB?9`<0$N%N%k%`(B (@ref{$BBe?tE*?t$K4X$9$k1i;;(B}) $B$GL5J?J}$G$"$k>l9g(B, |
| $B3F4{Ls0x;R$N<!?t$O(B @var{d} $B$NG\?t$H$J$k(B. $B$3$N$h$&$J>l9g$K(B |
$B3F4{Ls0x;R$N<!?t$O(B @var{d} $B$NG\?t$H$J$k(B. $B$3$N$h$&$J>l9g$K(B |
| $BMQ$$$i$l$k(B. |
$BMQ$$$i$l$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
By any reason, if the degree of all the irreducible factors of @var{poly} |
| |
is known to be some multiples of @var{hint}, factors can be computed |
| |
more efficiently by the knowledge than @code{fctr()}. |
| |
@item |
| |
When @var{hint} is 1, @code{ufctrhint()} is the same as @code{fctr()} for |
| |
uni-variate polynomials. |
| |
An typical application where @code{ufctrhint()} is effective: |
| |
Consider the case where @var{poly} is a norm (@ref{Algebraic numbers}) |
| |
of a certain polynomial over an extension field with its extension |
| |
degree @var{d}, and it is square free; Then, every irreducible factor |
| |
has a degree that is a multiple of @var{d}. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 740 t^9-15*t^6-87*t^3-125 |
|
| Line 1118 t^9-15*t^6-87*t^3-125 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{fctr sqfr}. |
@fref{fctr sqfr}. |
| @end table |
@end table |
| |
|
| @node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node modfctr,,, Polynomials and rational expressions |
| @subsection @code{modfctr} |
@subsection @code{modfctr} |
| @findex modfctr |
@findex modfctr |
| |
|
| @table @t |
@table @t |
| @item modfctr(@var{poly},@var{mod}) |
@item modfctr(@var{poly},@var{mod}) |
| :: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B |
\JP :: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B |
| |
\EG :: Univariate factorizer over small finite fields |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B%j%9%H(B |
\JP $B%j%9%H(B |
| |
\EG list |
| @item poly |
@item poly |
| $B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B |
\JP $B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B |
| |
\EG univariate polynomial with integer coefficients |
| @item mod |
@item mod |
| $B<+A3?t(B |
\JP $B<+A3?t(B |
| |
\EG non-negative integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| 2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B |
2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B |
| @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. |
@var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. |
| Line 770 t^9-15*t^6-87*t^3-125 |
|
| Line 1155 t^9-15*t^6-87*t^3-125 |
|
| $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B. |
$B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B. |
| @item |
@item |
| @b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B. |
@b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B. |
| |
@item |
| |
$BBg$-$J0L?t$r;}$DM-8BBN>e$N0x?tJ,2r$K$O(B @code{fctr_ff} $B$rMQ$$$k(B. |
| |
(@ref{$BM-8BBN$K4X$9$k1i;;(B},@pxref{fctr_ff}). |
| |
\E |
| |
\BEG |
| |
@item |
| |
This function factorizes a univarate polynomial @var{poly} over |
| |
the finite prime field of characteristic @var{mod}, where |
| |
@var{mod} must be smaller than 2^31. |
| |
@item |
| |
The result is represented by a list, whose elements are a pair |
| |
represented as |
| |
|
| |
[[@b{num},1],[@b{factor},@b{multiplicity}],...]. |
| |
@item |
| |
Products of all @b{factor}^@b{multiplicity} and @b{num} is equal to |
| |
@var{poly}. |
| |
@item |
| |
To factorize polynomials over large finite fields, use |
| |
@code{fctr_ff} (@pxref{Finite fields},@ref{fctr_ff}). |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 780 t^9-15*t^6-87*t^3-125 |
|
| Line 1186 t^9-15*t^6-87*t^3-125 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{fctr sqfr}. |
@fref{fctr sqfr}. |
| @end table |
@end table |
| |
|
| @node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node ptozp,,, Polynomials and rational expressions |
| @subsection @code{ptozp} |
@subsection @code{ptozp} |
| @findex ptozp |
@findex ptozp |
| |
|
| @table @t |
@table @t |
| @item ptozp(@var{poly}) |
@item ptozp(@var{poly}) |
| :: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B. |
\JP :: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B. |
| |
\BEG |
| |
:: Converts a polynomial @var{poly} with rational coefficients into |
| |
an integral polynomial such that GCD of all its coefficients is 1. |
| |
\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @item poly |
@item poly |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BM?$($i$l$?B?9`<0(B @var{poly} $B$KE,Ev$JM-M}?t$r3]$1$F(B, $B@0?t78?t$+$D(B |
$BM?$($i$l$?B?9`<0(B @var{poly} $B$KE,Ev$JM-M}?t$r3]$1$F(B, $B@0?t78?t$+$D(B |
| $B78?t$N(B GCD $B$,(B 1 $B$K$J$k$h$&$K$9$k(B. |
$B78?t$N(B GCD $B$,(B 1 $B$K$J$k$h$&$K$9$k(B. |
| Line 812 t^9-15*t^6-87*t^3-125 |
|
| Line 1227 t^9-15*t^6-87*t^3-125 |
|
| $BJ,;RB?9`<0$N78?t$OM-M}?t$N$^$^$G$"$j(B, $BM-M}<0$NJ,;R$r5a$a$k(B |
$BJ,;RB?9`<0$N78?t$OM-M}?t$N$^$^$G$"$j(B, $BM-M}<0$NJ,;R$r5a$a$k(B |
| @code{nm()} $B$G$O(B, $BJ,?t78?tB?9`<0$O(B, $BJ,?t78?t$N$^$^$N7A$G=PNO$5$l$k$?$a(B, |
@code{nm()} $B$G$O(B, $BJ,?t78?tB?9`<0$O(B, $BJ,?t78?t$N$^$^$N7A$G=PNO$5$l$k$?$a(B, |
| $BD>$A$K@0?t78?tB?9`<0$rF@$k;v$O=PMh$J$$(B. |
$BD>$A$K@0?t78?tB?9`<0$rF@$k;v$O=PMh$J$$(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Converts the given polynomial by multiplying some rational number |
| |
into an integral polynomial such that GCD of all its coefficients is 1. |
| |
@item |
| |
In general, operations on polynomials can be |
| |
performed faster for integer coefficients than for rational number |
| |
coefficients. Therefore, this function is conveniently used to improve |
| |
efficiency. |
| |
@item |
| |
Function @code{red} does not convert rational coefficients of the |
| |
numerator. |
| |
You cannot obtain an integral polynomial by direct use of the function |
| |
@code{nm()}. The function @code{nm()} returns the numerator of its |
| |
argument, and a polynomial with rational coefficients is |
| |
the numerator of itself and will be returned as it is. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 822 t^9-15*t^6-87*t^3-125 |
|
| Line 1255 t^9-15*t^6-87*t^3-125 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{nm dn}. |
@fref{nm dn}. |
| @end table |
@end table |
| |
|
| @node prim cont,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node prim cont,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node prim cont,,, Polynomials and rational expressions |
| @subsection @code{prim}, @code{cont} |
@subsection @code{prim}, @code{cont} |
| @findex prim |
@findex prim |
| |
|
| @table @t |
@table @t |
| @item prim(@var{poly}[,@var{v}]) |
@item prim(@var{poly}[,@var{v}]) |
| :: @var{poly} $B$N86;OE*ItJ,(B (primitive part). |
\JP :: @var{poly} $B$N86;OE*ItJ,(B (primitive part). |
| |
\EG :: Primitive part of @var{poly}. |
| @item cont(@var{poly}[,@var{v}]) |
@item cont(@var{poly}[,@var{v}]) |
| :: @var{poly} $B$NMFNL(B (content). |
\JP :: @var{poly} $B$NMFNL(B (content). |
| |
\EG :: Content of @var{poly}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return poly |
@item return poly |
| $BM-M}?t78?tB?9`<0(B |
\JP $BM-M}?t78?tB?9`<0(B |
| |
\EG polynomial over the rationals |
| @item v |
@item v |
| $BITDj85(B |
\JP $BITDj85(B |
| |
\EG indeterminate |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @var{poly} $B$N<gJQ?t(B ($B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) |
@var{poly} $B$N<gJQ?t(B ($B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) |
| $B$K4X$9$k86;OE*ItJ,(B, $BMFNL$r5a$a$k(B. |
$B$K4X$9$k86;OE*ItJ,(B, $BMFNL$r5a$a$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
The primitive part and the content of a polynomial @var{poly} |
| |
with respect to its main variable (@var{v} if specified). |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{var}, @fref{ord}. |
@fref{var}, @fref{ord}. |
| @end table |
@end table |
| |
|
| @node gcd gcdz,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node gcd gcdz,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node gcd gcdz,,, Polynomials and rational expressions |
| @subsection @code{gcd}, @code{gcdz} |
@subsection @code{gcd}, @code{gcdz} |
| @findex gcd |
@findex gcd |
| |
|
| @table @t |
@table @t |
| @item gcd(@var{poly1},@var{poly2}[,@var{mod}]) |
@item gcd(@var{poly1},@var{poly2}[,@var{mod}]) |
| @item gcdz(@var{poly1},@var{poly2}) |
@item gcdz(@var{poly1},@var{poly2}) |
| :: @var{poly1} $B$H(B @var{poly2} $B$N(B gcd. |
\JP :: @var{poly1} $B$H(B @var{poly2} $B$N(B gcd. |
| |
\EG :: The polynomial greatest common divisor of @var{poly1} and @var{poly2}. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @item poly1,poly2 |
@item poly1,poly2 |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @item mod |
@item mod |
| $BAG?t(B |
\JP $BAG?t(B |
| |
\EG prime |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BFs$D$NB?9`<0$N:GBg8xLs<0(B (GCD) $B$r5a$a$k(B. |
$BFs$D$NB?9`<0$N:GBg8xLs<0(B (GCD) $B$r5a$a$k(B. |
| @item |
@item |
|
|
| @code{gcd()}, @code{gcdz()} Extended Zassenhaus $B%"%k%4%j%:%`$K$h$k(B. |
@code{gcd()}, @code{gcdz()} Extended Zassenhaus $B%"%k%4%j%:%`$K$h$k(B. |
| $BM-8BBN>e$N(B GCD $B$O(B PRS $B%"%k%4%j%:%`$K$h$C$F$$$k$?$a(B, $BBg$-$JLdBj(B, |
$BM-8BBN>e$N(B GCD $B$O(B PRS $B%"%k%4%j%:%`$K$h$C$F$$$k$?$a(B, $BBg$-$JLdBj(B, |
| GCD $B$,(B 1 $B$N>l9g$J$I$K$*$$$F8zN($,0-$$(B. |
GCD $B$,(B 1 $B$N>l9g$J$I$K$*$$$F8zN($,0-$$(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Functions @code{gcd()} and @code{gcdz()} return the greatest common divisor |
| |
(GCD) of the given two polynomials. |
| |
@item |
| |
Function @code{gcd()} returns an integral polynomial GCD over the |
| |
rational number field. The coefficients are normalized such that |
| |
their GCD is 1. It returns 1 in case that the given polynomials are |
| |
mutually prime. |
| |
@item |
| |
Function @code{gcdz()} works for arguments of integral polynomials, |
| |
and returns a polynomial GCD over the integer ring, that is, |
| |
it returns @code{gcd()} multiplied by the contents of all coefficients |
| |
of the two input polynomials. |
| |
@item |
| |
@code{gcd()} computes the GCD over GF(@var{mod}) if @var{mod} is specified. |
| |
@item |
| |
Polynomial GCD is computed by an improved algorithm based |
| |
on Extended Zassenhaus algorithm. |
| |
@item |
| |
GCD over a finite field is computed by PRS algorithm and it may not be |
| |
efficient for large inputs and co-prime inputs. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 916 x^3+y*x^2+y^2*x+y^3 |
|
| Line 1393 x^3+y*x^2+y^2*x+y^3 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{igcd igcdcntl}. |
@fref{igcd igcdcntl}. |
| @end table |
@end table |
| |
|
| @node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
\JP @node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B |
| |
\EG @node red,,, Polynomials and rational expressions |
| @subsection @code{red} |
@subsection @code{red} |
| @findex red |
@findex red |
| |
|
| @table @t |
@table @t |
| @item red(@var{rat}) |
@item red(@var{rat}) |
| :: @var{rat} $B$rLsJ,$7$?$b$N(B. |
\JP :: @var{rat} $B$rLsJ,$7$?$b$N(B. |
| |
\EG :: Reduced form of @var{rat} by canceling common divisors. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BM-M}<0(B |
\JP $BM-M}<0(B |
| |
\EG rational expression |
| @item rat |
@item rat |
| $BM-M}<0(B |
\JP $BM-M}<0(B |
| |
\EG rational expression |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @b{Asir} $B$OM-M}?t$NLsJ,$r>o$K<+F0E*$K9T$&(B. |
@b{Asir} $B$OM-M}?t$NLsJ,$r>o$K<+F0E*$K9T$&(B. |
| $B$7$+$7(B, $BM-M}<0$K$D$$$F$ODLJ,$O9T$&$,(B, |
$B$7$+$7(B, $BM-M}<0$K$D$$$F$ODLJ,$O9T$&$,(B, |
| Line 953 GCD $B$OBgJQ=E$$1i;;$J$N$G(B, $BB>$NJ}K!$G=|$1$k6&D |
|
| Line 1436 GCD $B$OBgJQ=E$$1i;;$J$N$G(B, $BB>$NJ}K!$G=|$1$k6&D |
|
| $BK>$^$7$$(B. $B$^$?(B, $BJ,Jl(B, $BJ,;R$,Bg$-$/$J$C$F$+$i$N$3$NH!?t$N8F$S=P$7$O(B, |
$BK>$^$7$$(B. $B$^$?(B, $BJ,Jl(B, $BJ,;R$,Bg$-$/$J$C$F$+$i$N$3$NH!?t$N8F$S=P$7$O(B, |
| $BHs>o$K;~4V$,3]$+$k>l9g$,B?$$(B. $BM-M}<01i;;$r9T$&>l9g$O(B, $B$"$kDxEY(B |
$BHs>o$K;~4V$,3]$+$k>l9g$,B?$$(B. $BM-M}<01i;;$r9T$&>l9g$O(B, $B$"$kDxEY(B |
| $BIQHK$K(B, $BLsJ,$r9T$&I,MW$,$"$k(B. |
$BIQHK$K(B, $BLsJ,$r9T$&I,MW$,$"$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
@b{Asir} automatically performs cancellation of common divisors of rational numb |
| |
ers. |
| |
But, without an explicit command, it does not cancel common polynomial divisors |
| |
of rational expressions. |
| |
(Reduction of rational expressions to a common denominator will be always done.) |
| |
Use command @t{red()} to perform this cancellation. |
| |
@item |
| |
Cancel the common divisors of the numerator and the denominator of |
| |
a rational expression @var{rat} by computing their GCD. |
| |
@item |
| |
The denominator polynomial of the result is an integral polynomial |
| |
which has no common divisors in its coefficients, |
| |
while the numerator may have rational coefficients. |
| |
@item |
| |
Since GCD computation is a very hard operation, it is desirable to |
| |
detect and remove by any means common divisors as far as possible. |
| |
Furthermore, a call to this function after swelling of the denominator |
| |
and the numerator shall usually take a very long time. Therefore, |
| |
often, to some extent, reduction of common divisors is inevitable for |
| |
operations of rational expressions. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 969 x^2+(-y-z)*x+y^2-z*y+z^2 |
|
| Line 1476 x^2+(-y-z)*x+y^2-z*y+z^2 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{nm dn}, @fref{gcd gcdz}, @fref{ptozp}. |
@fref{nm dn}, @fref{gcd gcdz}, @fref{ptozp}. |
| @end table |
@end table |
| |
|
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