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Diff for /OpenXM/src/asir-doc/parts/builtin/poly.texi between version 1.2 and 1.7

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

version 1.7, 2003/12/23 10:41:10

Line 1

@comment $OpenXM$

@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/poly.texi,v 1.6 2003/11/27 15:56:08 ohara 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 21

* %::

* subst psubst::

* diff::

* ediff::

* res::

* fctr sqfr::

* modfctr::

Line 63

Line 64

\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.

Line 172 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.

Line 364 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

Line 452 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 648 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 745 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 759 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 788 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 895 from left to right.

Line 904 from left to right.

sin(x)

@end example

\JP @node ediff,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node ediff,,, Polynomials and rational expressions

@subsection @code{ediff}

@findex ediff

@table @t

@item ediff(@var{poly}[,@var{varn}]*)

@item ediff(@var{poly},@var{varlist})

\JP :: @var{poly} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=g<!%*%$%i!<HyJ,$9$k(B.

\BEG

:: Differentiate @var{poly} successively by Euler operators of @var{var}'s for the first

form, or by Euler operators of variables in @var{varlist} for the second form.

@end table

@table @var

@item return

\JP $BB?9`<0(B

\EG polynomial

@item poly

\JP $BB?9`<0(B

\EG polynomial

@item varn

\JP $BITDj85(B

\EG indeterminate

@item varlist

\JP $BITDj85$N%j%9%H(B

\EG list of indeterminates

@end table

@itemize @bullet

\BJP

@item

$B:8B&$NITDj85$h$j(B, $B=g$K%*%$%i!<HyJ,$7$F$$$/(B. $B$D$^$j(B, @t{ediff}(@var{poly},@t{x,y}) $B$O(B,

@t{ediff}(@t{ediff}(@var{poly},@t{x}),@t{y}) $B$HF1$8$G$"$k(B.

\BEG

@item

differentiation is performed by the specified indeterminates (variables)

from left to right.

@t{ediff}(@var{poly},@t{x,y}) is the same as

@t{ediff}(@t{ediff}(@var{poly},@t{x}),@t{y}).

@end itemize

@example

[0] ediff((x+2*y)^2,x);

2*x^2+4*y*x

[1] ediff((x+2*y)^2,x,y);

4*y*x

@end example

\JP @node res,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node res,,, Polynomials and rational expressions

@subsection @code{res}

Line 913 sin(x)

Line 974 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 1121 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 1134 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 1146 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 1167 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 1194 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 1203 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 1213 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 1225 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 1247 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 1395 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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