=================================================================== RCS file: /home/cvs/OpenXM/src/asir-doc/parts/builtin/poly.texi,v retrieving revision 1.1.1.1 retrieving revision 1.2 diff -u -p -r1.1.1.1 -r1.2 --- OpenXM/src/asir-doc/parts/builtin/poly.texi 1999/12/08 05:47:44 1.1.1.1 +++ OpenXM/src/asir-doc/parts/builtin/poly.texi 1999/12/21 02:47:34 1.2 @@ -1,5 +1,12 @@ +@comment $OpenXM$ +\BJP @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 +\E +\BEG +@node Polynomials and rational expressions,,, Built-in Function +@section operations with polynomials and rational expressions +\E @menu * var:: @@ -24,23 +31,28 @@ * red:: @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} @findex var @table @t @item var(@var{rat}) -:: @var{rat} $B$NH(B +\JP @item $B;2>H(B +\EG @item References @fref{ord}, @fref{vars}. @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 +\EG @node vars,,, Polynomials and rational expressions @subsection @code{vars} @findex vars @table @t @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. +\EG :: A list of variables (indeterminates) in an expression @var{obj}. @end table @table @var @item return -$B%j%9%H(B +\JP $B%j%9%H(B +\EG list @item obj -$BG$0U(B +\JP $BG$0U(B +\EG arbitrary @end table @itemize @bullet +\BJP @item $BM?$($i$l$?<0$K4^$^$l$kJQ?t$N%j%9%H$rJV$9(B. @item $BJQ?t=g=x$N9b$$$b$N$+$i=g$KJB$Y$k(B. +\E +\BEG +@item +Returns a list of variables (indeterminates) contained in a given expression. +@item +Lists variables according to the variable ordering. +\E @end itemize @example @@ -97,25 +135,30 @@ abc @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{var}, @fref{uc}, @fref{ord}. @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 +\EG @node uc,,, Polynomials and rational expressions @subsection @code{uc} @findex uc @table @t @item uc() -:: $B?7$?$JITDj85$r@8@.$9$k(B. +\JP :: $BL$Dj78?tK!$N$?$a$NITDj85$r@8@.$9$k(B. +\EG :: Create a new indeterminate for an undermined coeficient. @end table @table @var @item return -@code{vtype} $B$,(B 1 $B$NITDj85(B +\JP @code{vtype} $B$,(B 1 $B$NITDj85(B +\EG indeterminate with its @code{vtype} 1. @end table @itemize @bullet +\BJP @item @code{uc()} $B$rH(B +\JP @item $B;2>H(B +\EG @item References @fref{vtype}, @fref{rtostr}, @fref{strtov}. @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 +\EG @node coef,,, Polynomials and rational expressions @subsection @code{coef} @findex coef @table @t @item coef(@var{poly},@var{deg}[,@var{var}]) -:: @var{poly} $B$N(B @var{var} ($B>JN,;~$OJN,;~$Ol9g$KHf3S$7$F(B $B8zN($,Mn$A$k(B. +\E +\BEG +@item +The coefficient of a polynomial @var{poly} at degree @var{deg} +with respect to the variable @var{var}. +@item +The default value for @var{var} is the main variable, i.e., +@t{var(@var{poly})}. +@item +For multi-variate polynomials, access to coefficients depends on +the specified indeterminates. For example, taking coef for the main +variable is much faster than for other variables. +\E @end itemize @example @@ -187,36 +272,59 @@ y^3+3*z*y^2+3*z^2*y+z^3 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{var}, @fref{deg mindeg}. @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 +\EG @node deg mindeg,,, Polynomials and rational expressions @subsection @code{deg}, @code{mindeg} @findex deg @findex mindeg @table @t @item deg(@var{poly},@var{var}) -:: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:G9bJN,$9$k$3$H$O=PMh$J$$(B. +\E +\BEG +@item +The least exponent of the terms with non-zero coefficients in +a polynomial @var{poly} with respect to the variable @var{var}. +In this manual, this quantity is sometimes referred to the minimum +degree of a polynomial for short. +@item +Variable @var{var} must be specified. +\E @end itemize @example @@ -228,29 +336,46 @@ y^3+3*z*y^2+3*z^2*y+z^3 1 @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} @findex nmono @table @t @item nmono(@var{rat}) -:: @var{rat} $B$NC19`<0$N9`?t(B. +\JP :: @var{rat} $B$NC19`<0$N9`?t(B. +\EG :: Number of monomials in rational expression @var{rat}. @end table @table @var @item return -$B<+A3?t(B +\JP $B<+A3?t(B +\EG non-negative integer @item rat -$BM-M}<0(B +\JP $BM-M}<0(B +\EG rational expression @end table @itemize @bullet +\BJP @item $BB?9`<0$rE83+$7$?>uBV$G$N(B 0 $B$G$J$$78?t$r;}$DC19`<0$N9`?t$r5a$a$k(B. @item $BM-M}<0$N>l9g$O(B, $BJ,;R$HJ,Jl$N9`?t$NOB$,JV$5$l$k(B. @item $BH!?t7A<0(B (@xref{$BITDj85$N7?(B}) $B$O(B, $B0z?t$,2?$G$"$C$F$bC19`$H$_$J$5$l$k(B. (1 $B8D$NITDj85$HF1$8(B. ) +\E +\BEG +@item +Number of monomials with non-zero number coefficients in the full +expanded form of the given polynomial. +@item +For a rational expression, the sum of the numbers of monomials +of the numerator and denominator. +@item +A function form is regarded as a single indeterminate no matter how +complex arguments it has. +\E @end itemize @example @@ -263,27 +388,33 @@ y^3+3*z*y^2+3*z^2*y+z^3 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{vtype}. @end table -@node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\JP @node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\EG @node ord,,, Polynomials and rational expressions @subsection @code{ord} @findex ord @table @t @item ord([@var{varlist}]) -:: $BJQ?t=g=x$N@_Dj(B +\JP :: $BJQ?t=g=x$N@_Dj(B +\EG :: It sets the ordering of indeterminates (variables). @end table @table @var @item return -$BJQ?t$N%j%9%H(B +\JP $BJQ?t$N%j%9%H(B +\EG list of indeterminates @item varlist -$BJQ?t$N%j%9%H(B +\JP $BJQ?t$N%j%9%H(B +\EG list of indeterminates @end table @itemize @bullet +\BJP @item $B0z?t$,$"$k$H$-(B, $B0z?t$NJQ?t%j%9%H$r@hF,$K=P$7(B, $B;D$j$NJQ?t$,$=$N8e$K(B $BB3$/$h$&$KJQ?t=g=x$r@_Dj$9$k(B. $B0z?t$N$"$k$J$7$K4X$o$i$:(B, @code{ord()} @@ -296,6 +427,26 @@ y^3+3*z*y^2+3*z^2*y+z^3 $B$"$k$$$O(B, $B?7$?$JJQ?t$,8=$l$?;~E@$K9T$o$l$k(B $B$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. +\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 @example @@ -311,7 +462,8 @@ _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)] @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} @findex sdiv @findex sdivm @@ -323,28 +475,50 @@ cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(- @table @t @item sdiv(@var{poly1},@var{poly2}[,@var{v}]) @itemx sdivm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) -:: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$Gl9g$K>&$r5a$a$k(B. +\JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$Gl9g$K>&$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 sremm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) -:: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$Gl9g$K>jM>$r5a$a$k(B. +\JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$Gl9g$K>jM>$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 sqrm(@var{poly1},@var{poly2},@var{mod}[,@var{v}]) +\BJP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$Gl9g$K>&(B, $B>jM>$r(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 @table @var @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$k%j%9%H(B +\EG @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : polynomial @code{sqr()}, @code{sqrm()} : a list @code{[quotient,remainder]} @item poly1 poly2 -$BB?9`<0(B +\JP $BB?9`<0(B +\EG polynomial @item v -$BITDj85(B +\JP $BITDj85(B +\EG indeterminate @item mod -$BAG?t(B +\JP $BAG?t(B +\EG prime @end table @itemize @bullet +\BJP @item @var{poly1} $B$r(B @var{poly2} $B$Nl9g$K$O(B @var{v}) $B$K4X$9$kB?9`<0$H8+$F(B, @@ -368,6 +542,40 @@ 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$$$k(B. @item $B78?t$KBP$9$k>jM>1i;;$O(B @code{%} $B$rMQ$$$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 @example @@ -391,32 +599,48 @@ return to toplevel @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{idiv irem}, @fref{%}. @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} @findex tdiv @table @t @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$k(B. +\EG :: Tests whether @var{poly2} divides @var{poly1}. @end table @table @var @item return -$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$P(B 0 +\EG Quotient if @var{poly2} divides @var{poly1}, 0 otherwise. @item poly1 poly2 -$BB?9`<0(B +\JP $BB?9`<0(B +\EG polynomial @end table @itemize @bullet +\BJP @item @var{poly2} $B$,(B @var{poly1} $B$rB?9`<0$H$7$F3d$j@Z$k$+$I$&$+D4$Y$k(B. @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 $B>l9g$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 @example @@ -432,29 +656,36 @@ 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 @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{sdiv sdivm srem sremm sqr sqrm}. @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{%} @findex % @table @t @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 @table @var @item return -$B@0?t$^$?$OB?9`<0(B +\JP $B@0?t$^$?$OB?9`<0(B +\EG integer or polynomial @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 -$B@0?t(B +\JP $B@0?t(B +\EG intger @end table @itemize @bullet +\BJP @item @var{poly} $B$N3F78?t$r(B @var{m} $B$G3d$C$?>jM>$GCV$-49$($?B?9`<0$rJV$9(B. @item @@ -464,6 +695,21 @@ 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$$$k$3$H$,$G$-$k(B. @item @var{poly} $B$N78?t(B, @var{m} $B$H$b@0?t$G$"$kI,MW$,$"$k$,(B, $B%A%'%C%/$O9T$J$o$l$J$$(B. +\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 @example @@ -478,11 +724,13 @@ x^5+2*x^4+x^3+x^2+2*x+1 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{idiv irem}. @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} @findex subst @findex psubst @@ -490,20 +738,32 @@ x^5+2*x^4+x^3+x^2+2*x+1 @table @t @item subst(@var{rat}[,@var{varn},@var{ratn}]*) @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{n=1,2},... $B$G:8$+$i1&$K=gl9g$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 @example @@ -544,34 +833,55 @@ sint(t)*t sin(x)*t @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} @findex diff @table @t @item diff(@var{rat}[,@var{varn}]*) @item diff(@var{rat},@var{varlist}) -:: @var{rat} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=ge$G$N7W;;$r9T$&(B. +\E +\BEG +@item +Resultant of two polynomials @var{poly1} and @var{poly2} +with respect to @var{var}. +@item +Sub-resultant algorithm is used to compute the resultant. +@item +The computation is done over GF(@var{mod}) if @var{mod} is specified. +\E @end itemize @example @@ -620,26 +947,32 @@ sin(x) -x^3-x^2-y^3 @end example -@node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\JP @node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\EG @node fctr sqfr,,, Polynomials and rational expressions @subsection @code{fctr}, @code{sqfr} @findex fctr @findex sqfr @table @t @item fctr(@var{poly}) -:: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. +\JP :: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. +\EG :: Factorize polynomial @var{poly} over the rationals. @item sqfr(@var{poly}) -:: @var{poly} $B$rL5J?J}J,2r$9$k(B. +\JP :: @var{poly} $B$rL5J?J}J,2r$9$k(B. +\EG :: Gets a square-free factorization of polynomial @var{poly}. @end table @table @var @item return -$B%j%9%H(B +\JP $B%j%9%H(B +\EG list @item poly -$BM-M}?t78?t$NB?9`<0(B +\JP $BM-M}?t78?t$NB?9`<0(B +\EG polynomial with rational coefficients @end table @itemize @bullet +\BJP @item $BM-M}?t78?t$NB?9`<0(B @var{poly} $B$r0x?tJ,2r$9$k(B. @code{fctr()} $B$O4{Ls0x;RJ,2r(B, @code{sqfr()} $B$OL5J?J}0x;RJ,2r(B. @@ -650,6 +983,25 @@ sin(x) @item @b{$B?t78?t(B} $B$O(B, (@var{poly}/@b{$B?t78?t(B}) $B$,(B, $B@0?t78?t$G(B, $B78?t$N(B GCD $B$,(B 1 $B$H$J$k(B $B$h$&$JB?9`<0$K$J$k$h$&$KA*$P$l$F$$$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 @example @@ -670,29 +1022,40 @@ x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{ufctrhint}. @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} @findex ufctrhint @table @t @item ufctrhint(@var{poly},@var{hint}) -:: $BpJs$rMQ$$$?(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B +\JP :: $BpJs$rMQ$$$?(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 @table @var @item return -$B%j%9%H(B +\JP $B%j%9%H(B +\EG list @item poly -$BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B +\JP $BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B +\EG uni-variate polynomial with rational coefficients @item hint -$B<+A3?t(B +\JP $B<+A3?t(B +\EG non-negative integer @end table @itemize @bullet +\BJP @item $B3F4{Ls0x;R$Nl9g$K(B @var{poly} $B$N4{Ls0x;RJ,2r$r(B @code{fctr()} $B$h$j8zN(NI$/9T$&(B. @@ -700,6 +1063,21 @@ x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4 $B$"$kB?9`<0$N%N%k%`(B (@xref{$BBe?tE*?t$K4X$9$k1i;;(B}) $B$GL5J?J}$G$"$k>l9g(B, $B3F4{Ls0x;R$Nl9g$K(B $BMQ$$$i$l$k(B. +\E +\BEG +@item +By any reason, if the degree of all the irreducible factors of @var{poly} +is known to be some multiples of @var{hint}, factors can be computed +more efficiently by the knowledge than @code{fctr()}. +@item +When @var{hint} is 1, @code{ufctrhint()} is the same as @code{fctr()} for +uni-variate polynomials. +An typical application where @code{ufctrhint()} is effective: +Consider the case where @var{poly} is a norm (@xref{Algebraic numbers}) +of a certain polynomial over an extension field with its extension +degree @var{d}, and it is square free; Then, every irreducible factor +has a degree that is a multiple of @var{d}. +\E @end itemize @example @@ -740,29 +1118,36 @@ t^9-15*t^6-87*t^3-125 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{fctr sqfr}. @end table -@node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\JP @node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\EG @node modfctr,,, Polynomials and rational expressions @subsection @code{modfctr} @findex modfctr @table @t @item modfctr(@var{poly},@var{mod}) -:: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B +\JP :: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B +\EG :: Univariate factorizer over small finite fields @end table @table @var @item return -$B%j%9%H(B +\JP $B%j%9%H(B +\EG list @item poly -$B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B +\JP $B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B +\EG univariate polynomial with integer coefficients @item mod -$B<+A3?t(B +\JP $B<+A3?t(B +\EG non-negative integer @end table @itemize @bullet +\BJP @item 2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. @@ -770,6 +1155,27 @@ t^9-15*t^6-87*t^3-125 $B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B. @item @b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B. +@item +$BBg$-$J0L?t$r;}$DM-8BBN>e$N0x?tJ,2r$K$O(B @code{fctr_ff} $B$rMQ$$$k(B. +(@ref{$BM-8BBN$K4X$9$k1i;;(B},@pxref{fctr_ff}). +\E +\BEG +@item +This function factorizes a univarate polynomial @var{poly} over +the finite prime field of characteristic @var{mod}, where +@var{mod} must be smaller than 2^31. +@item +The result is represented by a list, whose elements are a pair +represented as + +[[@b{num},1],[@b{factor},@b{multiplicity}],...]. +@item +Products of all @b{factor}^@b{multiplicity} and @b{num} is equal to +@var{poly}. +@item +To factorize polynomials over large finite fields, use +@code{fctr_ff} (@pxref{Finite fields},@ref{fctr_ff}). +\E @end itemize @example @@ -780,27 +1186,36 @@ t^9-15*t^6-87*t^3-125 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{fctr sqfr}. @end table -@node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\JP @node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\EG @node ptozp,,, Polynomials and rational expressions @subsection @code{ptozp} @findex ptozp @table @t @item ptozp(@var{poly}) -:: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B. +\JP :: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B. +\BEG +:: Converts a polynomial @var{poly} with rational coefficients into +an integral polynomial such that GCD of all its coefficients is 1. +\E @end table @table @var @item return -$BB?9`<0(B +\JP $BB?9`<0(B +\EG polynomial @item poly -$BB?9`<0(B +\JP $BB?9`<0(B +\EG polynomial @end table @itemize @bullet +\BJP @item $BM?$($i$l$?B?9`<0(B @var{poly} $B$KE,Ev$JM-M}?t$r3]$1$F(B, $B@0?t78?t$+$D(B $B78?t$N(B GCD $B$,(B 1 $B$K$J$k$h$&$K$9$k(B. @@ -812,6 +1227,24 @@ t^9-15*t^6-87*t^3-125 $BJ,;RB?9`<0$N78?t$OM-M}?t$N$^$^$G$"$j(B, $BM-M}<0$NJ,;R$r5a$a$k(B @code{nm()} $B$G$O(B, $BJ,?t78?tB?9`<0$O(B, $BJ,?t78?t$N$^$^$N7A$G=PNO$5$l$k$?$a(B, $BD>$A$K@0?t78?tB?9`<0$rF@$k;v$O=PMh$J$$(B. +\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 @example @@ -822,32 +1255,45 @@ t^9-15*t^6-87*t^3-125 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{nm dn}. @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} @findex prim @table @t @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}]) -:: @var{poly} $B$NMFNL(B (content). +\JP :: @var{poly} $B$NMFNL(B (content). +\EG :: Content of @var{poly}. @end table @table @var @item return poly -$BM-M}?t78?tB?9`<0(B +\JP $BM-M}?t78?tB?9`<0(B +\EG polynomial over the rationals @item v -$BITDj85(B +\JP $BITDj85(B +\EG indeterminate @end table @itemize @bullet +\BJP @item @var{poly} $B$Nl9g$K$O(B @var{v}) $B$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 @example @@ -862,30 +1308,37 @@ y-z @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{var}, @fref{ord}. @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} @findex gcd @table @t @item gcd(@var{poly1},@var{poly2}[,@var{mod}]) @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 @table @var @item return -$BB?9`<0(B +\JP $BB?9`<0(B +\EG polynomial @item poly1,poly2 -$BB?9`<0(B +\JP $BB?9`<0(B +\EG polynomial @item mod -$BAG?t(B +\JP $BAG?t(B +\EG prime @end table @itemize @bullet +\BJP @item $BFs$D$NB?9`<0$N:GBg8xLs<0(B (GCD) $B$r5a$a$k(B. @item @@ -902,6 +1355,30 @@ y-z @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$$$k$?$a(B, $BBg$-$JLdBj(B, GCD $B$,(B 1 $B$N>l9g$J$I$K$*$$$F8zN($,0-$$(B. +\E +\BEG +@item +Functions @code{gcd()} and @code{gcdz()} return the greatest common divisor +(GCD) of the given two polynomials. +@item +Function @code{gcd()} returns an integral polynomial GCD over the +rational number field. The coefficients are normalized such that +their GCD is 1. It returns 1 in case that the given polynomials are +mutually prime. +@item +Function @code{gcdz()} works for arguments of integral polynomials, +and returns a polynomial GCD over the integer ring, that is, +it returns @code{gcd()} multiplied by the contents of all coefficients +of the two input polynomials. +@item +@code{gcd()} computes the GCD over GF(@var{mod}) if @var{mod} is specified. +@item +Polynomial GCD is computed by an improved algorithm based +on Extended Zassenhaus algorithm. +@item +GCD over a finite field is computed by PRS algorithm and it may not be +efficient for large inputs and co-prime inputs. +\E @end itemize @example @@ -916,27 +1393,33 @@ x^3+y*x^2+y^2*x+y^3 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{igcd igcdcntl}. @end table -@node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\JP @node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B +\EG @node red,,, Polynomials and rational expressions @subsection @code{red} @findex red @table @t @item red(@var{rat}) -:: @var{rat} $B$rLsJ,$7$?$b$N(B. +\JP :: @var{rat} $B$rLsJ,$7$?$b$N(B. +\EG :: Reduced form of @var{rat} by canceling common divisors. @end table @table @var @item return -$BM-M}<0(B +\JP $BM-M}<0(B +\EG rational expression @item rat -$BM-M}<0(B +\JP $BM-M}<0(B +\EG rational expression @end table @itemize @bullet +\BJP @item @b{Asir} $B$OM-M}?t$NLsJ,$r>o$K<+F0E*$K9T$&(B. $B$7$+$7(B, $BM-M}<0$K$D$$$F$ODLJ,$O9T$&$,(B, @@ -953,6 +1436,30 @@ 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, $BHs>o$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. +\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 @example @@ -969,7 +1476,8 @@ x^2+(-y-z)*x+y^2-z*y+z^2 @end example @table @t -@item $B;2>H(B +\JP @item $B;2>H(B +\EG @item References @fref{nm dn}, @fref{gcd gcdz}, @fref{ptozp}. @end table