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Diff for /OpenXM/src/asir-doc/parts/builtin/poly.texi between version 1.2 and 1.6

version 1.2, 1999/12/21 02:47:34

version 1.6, 2003/11/27 15:56:08

Line 1

@comment $OpenXM$

@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/poly.texi,v 1.5 2003/04/20 08:01:29 noro Exp $

\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

Line 63

\BEG

@item

@xref{Types in Asir} for main variable.

See @ref{Types in Asir} for main variable.

@item

Indeterminates (variables) are ordered by default as follows.

Line 171 Lists variables according to the variable ordering.

@code{strtov()} $B$rMQ$$$k(B.

@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.

(@xref{$BITDj85$N7?(B})

(@xref{$BITDj85$N7?(B}.)

\BEG

@item

Line 363 Variable @var{var} must be specified.

@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. )

$BH!?t7A<0(B (@ref{$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. )

\BEG

@item

Line 451 objects which are created under different variable ord

@example

[0] ord();

[x,y,z,u,v,w,p,q,r,s,t,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,_w,_p,

[x,y,z,u,v,w,p,q,r,s,t,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,

_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),(_x)^(_y),

_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,

log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),

exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),

tanh(_x),(_x^2+1)^(-1/2),(_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)]

[1] ord([dx,dy,dz,a,b,c]);

[dx,dy,dz,a,b,c,x,y,z,u,v,w,p,q,r,s,t,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,

[dx,dy,dz,a,b,c,x,y,z,u,v,w,p,q,r,s,t,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,

_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),

_z,_u,_v,_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,

(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),

_o,exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),

cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_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

\JP @node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

Line 645 to the polynomials repeatedly yields the multiplicity.

Line 647 to the polynomials repeatedly yields the multiplicity.

@example

[11] Y=(x+y+z)^5*(x-y-z)^3;

x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^5

x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6

+(6*y^5+30*z*y^4+60*z^2*y^3+60*z^3*y^2+30*z^4*y+6*z^5)*x^3+(2*y^6+12*z*y^5

+(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^5

+30*z^2*y^4+40*z^3*y^3+30*z^4*y^2+12*z^5*y+2*z^6)*x^2+(-2*y^7-14*z*y^6

+(6*y^5+30*z*y^4+60*z^2*y^3+60*z^3*y^2+30*z^4*y+6*z^5)*x^3

-42*z^2*y^5-70*z^3*y^4-70*z^4*y^3-42*z^5*y^2-14*z^6*y-2*z^7)*x-y^8-8*z*y^7

+(2*y^6+12*z*y^5+30*z^2*y^4+40*z^3*y^3+30*z^4*y^2+12*z^5*y+2*z^6)*x^2

-28*z^2*y^6-56*z^3*y^5-70*z^4*y^4-56*z^5*y^3-28*z^6*y^2-8*z^7*y-z^8

+(-2*y^7-14*z*y^6-42*z^2*y^5-70*z^3*y^4-70*z^4*y^3-42*z^5*y^2

-14*z^6*y-2*z^7)*x-y^8-8*z*y^7-28*z^2*y^6-56*z^3*y^5-70*z^4*y^4

-56*z^5*y^3-28*z^6*y^2-8*z^7*y-z^8

[12] for(I=0,F=x+y+z,T=Y; T=tdiv(T,F); I++);

[13] I;

Line 740 x^5+2*x^4+x^3+x^2+2*x+1

Line 744 x^5+2*x^4+x^3+x^2+2*x+1

@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=g<!BeF~$9$k(B).

(@var{n}=1,2,... $B$G:8$+$i1&$K=g<!BeF~$9$k(B).

\BEG

:: Substitute @var{ratn} for @var{varn} in expression @var{rat}.

(@var{n=1,2},@dots{}.

(@var{n}=1,2,@dots{}.

Substitution will be done successively from left to right

if arguments are repeated.)

Line 754 if arguments are repeated.)

Line 758 if arguments are repeated.)

@item return

\JP $BM-M}<0(B

\EG rational expression

@item rat,ratn

@item rat ratn

\JP $BM-M}<0(B

\EG rational expression

@item varn

Line 783 if arguments are repeated.)

Line 787 if arguments are repeated.)

$B$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

$BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B.

@item

@code{subst}$B$N0z?t(B@var{rat}$B$,%j%9%H(B,$BG[Ns(B,$B9TNs(B,$B$"$k$$$OJ,;6I=8=B?9`<0$G(B

$B$"$C$?>l9g$K$O(B, $B$=$l$>$l$NMWAG$^$?$O78?t$KBP$7$F:F5"E*$K(B@code{subst}$B$r(B

$B9T$&(B.

\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{})

@t{subst}(@var{r},@var{v1},@var{r1},@var{v2},@var{r2},@dots{})

has the same effect as

@t{subst}(@t{subst}(@var{rat},@var{var1},@var{rat1}),@var{var2},@var{rat2},@dots{}).

@t{subst}(@t{subst}(@var{r},@var{v1},@var{r1}),@var{v2},@var{r2},@dots{}).

@item

Note that repeated substitution is done from left to right successively.

You may get different result by changing the specification order.

Line 913 sin(x)

Line 921 sin(x)

@item var

\JP $BITDj85(B

\EG indeterminate

@item poly1,poly2

@item poly1 poly2

\JP $BB?9`<0(B

\EG polynomial

@item mod

Line 1060 multiples of @var{hint}.

Line 1068 multiples of @var{hint}.

$B3F4{Ls0x;R$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} $B$,(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

$BMQ$$$i$l$k(B.

Line 1073 more efficiently by the knowledge than @code{fctr()}.

Line 1081 more efficiently by the knowledge than @code{fctr()}.

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})

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}.

Line 1085 has a degree that is a multiple of @var{d}.

Line 1093 has a degree that is a multiple of @var{d}.

t^9-15*t^6-87*t^3-125

0msec

[11] N=res(t,subst(A,t,x-2*t),A);

-x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69+1720634125410*x^66

-x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69

-88249977024390*x^63-4856095669551930*x^60+1999385245240571421*x^57

+1720634125410*x^66-88249977024390*x^63-4856095669551930*x^60

-15579689952590251515*x^54+15956967531741971462865*x^51

+1999385245240571421*x^57-15579689952590251515*x^54

+15956967531741971462865*x^51

...

+140395588720353973535526123612661444550659875*x^6

+10122324287343155430042768923500799484375*x^3

Line 1105 t^9-15*t^6-87*t^3-125

Line 1114 t^9-15*t^6-87*t^3-125

[[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],

[x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3

+296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],

[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],

[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3

+31524548679,1],

[x^18+10773*x^12+2784051*x^6+307546875,1]]

167.050sec + gc : 1.890sec

[14] ufctrhint(N,9);

[[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],

[x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3

+296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],

[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],

[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3

+31524548679,1],

[x^18+10773*x^12+2784051*x^6+307546875,1]]

119.340sec + gc : 1.300sec

@end example

Line 1130 t^9-15*t^6-87*t^3-125

Line 1141 t^9-15*t^6-87*t^3-125

@table @t

@item modfctr(@var{poly},@var{mod})

\JP :: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B

\JP :: $BM-8BBN>e$G$NB?9`<0$N0x?tJ,2r(B

\EG :: Univariate factorizer over small finite fields

\EG :: Factorizer over small finite fields

@end table

@table @var

Line 1139 t^9-15*t^6-87*t^3-125

Line 1150 t^9-15*t^6-87*t^3-125

\JP $B%j%9%H(B

\EG list

@item poly

\JP $B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B

\JP $B@0?t78?t$NB?9`<0(B

\EG univariate polynomial with integer coefficients

\EG Polynomial with integer coefficients

@item mod

\JP $B<+A3?t(B

\EG non-negative integer

Line 1149 t^9-15*t^6-87*t^3-125

Line 1160 t^9-15*t^6-87*t^3-125

@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

2^29 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$GB?9`<0(B

@var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.

@item

$B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B.

Line 1161 t^9-15*t^6-87*t^3-125

Line 1172 t^9-15*t^6-87*t^3-125

\BEG

@item

This function factorizes a univarate polynomial @var{poly} over

This function factorizes a polynomial @var{poly} over

the finite prime field of characteristic @var{mod}, where

@var{mod} must be smaller than 2^31.

@var{mod} must be smaller than 2^29.

@item

The result is represented by a list, whose elements are a pair

represented as

Line 1183 To factorize polynomials over large finite fields, use

Line 1194 To factorize polynomials over large finite fields, use

[[1,1],[x+1513477736,1],[x+2055628767,1],[x+91854880,1],

[x+634005911,1],[x+1513477735,1],[x+634005912,1],

[x^4+1759639395*x^2+2045307031,1]]

[1] modfctr(2*x^6+(y^2+z*y)*x^4+2*z*y^3*x^2+(2*z^2*y^2+z^3*y)*x+z^4,3);

[[2,1],[2*x^3+z*y*x+z^2,1],[2*x^3+y^2*x+2*z^2,1]]

@end example

@table @t

Line 1329 y-z

Line 1342 y-z

@item return

\JP $BB?9`<0(B

\EG polynomial

@item poly1,poly2

@item poly1 poly2

\JP $BB?9`<0(B

\EG polynomial

@item mod

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