Return to gtt_ekn-ja.texi CVS log

Up to [local] / OpenXM / src / asir-contrib / packages / doc / gtt_ekn

Annotation of OpenXM/src/asir-contrib/packages/doc/gtt_ekn/gtt_ekn-ja.texi, Revision 1.2

1.2     ! takayama    1: %% $OpenXM: OpenXM/src/asir-contrib/packages/doc/gtt_ekn/gtt_ekn-ja.texi,v 1.1 2016/03/21 00:16:10 takayama Exp $
1.1       takayama    2: %% ptex gtt_ekn.texi   (.texi $B$^$G$D$1$k(B. platex $B$G$J$/(B ptex)
                      3: %% $B0J2<%3%a%s%H$O(B @comment $B$G;O$a$k(B.  \input texinfo $B0J9_$OIaDL$N(B tex $BL?Na$O;H$($J$$(B.
                      4: \input texinfo
                      5: @iftex
                      6: @catcode`@#=6
                      7: @def@fref#1{@xrefX[#1,,@code{#1},,,]}
                      8: @def@b#1{{@bf@gt #1}}
                      9: @catcode`@#=@other
                     10: @end iftex
                     11: @overfullrule=0pt
                     12: @c -*-texinfo-*-
                     13: @comment %**start of header
                     14: @comment --- $B$*$^$8$J$$=*$j(B ---
                     15:
                     16: @comment --- GNU info $B%U%!%$%k$NL>A0(B ---
                     17: @setfilename xyzman
                     18:
                     19: @comment --- $B%?%$%H%k(B ---
                     20: @settitle 2$B85J,3dI=(BHGM
                     21:
                     22: @comment %**end of header
                     23: @comment %@setchapternewpage odd
                     24:
                     25: @comment --- $B$*$^$8$J$$(B ---
                     26: @ifinfo
                     27: @macro fref{name}
                     28: @ref{\name\,,@code{\name\}}
                     29: @end macro
                     30: @end ifinfo
                     31:
                     32: @iftex
                     33: @comment @finalout
                     34: @end iftex
                     35:
                     36: @titlepage
                     37: @comment --- $B$*$^$8$J$$=*$j(B ---
                     38:
                     39: @comment --- $B%?%$%H%k(B, $B%P!<%8%g%s(B, $BCx<TL>(B, $BCx:n8"I=<((B ---
                     40: @title 2$B85J,3dI=(BHGM$B4X?t(B
                     41: @subtitle Risa/Asir 2$B85J,3dI=(BHGM$B4X?t@bL@=q(B
                     42: @subtitle 1.0 $BHG(B
1.2     ! takayama   43: @subtitle 2016 $BG/(B 3 $B7n(B 22 $BF|(B
1.1       takayama   44:
                     45: @author  by Y.Goto, Y.Tachibana, N.Takayama
                     46: @page
                     47: @vskip 0pt plus 1filll
                     48: Copyright @copyright{} Risa/Asir committers
                     49: 2004--2010. All rights reserved.
                     50: @end titlepage
                     51:
                     52: @comment --- $B$*$^$8$J$$(B ---
                     53: @synindex vr fn
                     54: @comment --- $B$*$^$8$J$$=*$j(B ---
                     55:
                     56: @comment --- @node $B$O(B GNU info, HTML $BMQ(B ---
                     57: @comment --- @node  $B$N0z?t$O(B node-name,  next,  previous,  up ---
                     58: @node Top,, (dir), (dir)
                     59:
                     60: @comment --- @menu $B$O(B GNU info, HTML $BMQ(B ---
                     61: @comment --- chapter $BL>$r@53N$KJB$Y$k(B ---
                     62: @comment --- $B$3$NJ8=q$G$O(B chapter XYZ, Chapter Index $B$,$"$k(B.
                     63: @comment ---  Chapter XYZ $B$K$O(B section XYZ$B$K$D$$$F(B, section XYZ$B$K4X$9$k4X?t$,$"$k(B.
                     64: @menu
                     65: * 2$B85J,3dI=(BHGM$B$N4X?t@bL@=q$K$D$$$F(B::
                     66: * 2$B85J,3dI=(BHGM$B$N4X?t(B::
                     67: * Index::
                     68: @end menu
                     69:
                     70: @comment --- chapter $B$N3+;O(B ---
                     71: @comment --- $B?F(B chapter $BL>$r@53N$K(B. $B?F$,$J$$>l9g$O(B Top ---
                     72: @node 2$B85J,3dI=(BHGM$B$N4X?t@bL@=q$K$D$$$F(B,,, Top
                     73: @chapter 2$B85J,3dI=(BHGM$B$N4X?t@bL@=q$K$D$$$F(B
                     74:
                     75: $B$3$N@bL@=q$G$O(B
                     76: HGM(holonomic gradient method) $B$rMQ$$$?(B2$B85J,3dI=$N4X?t$K$D$$$F@bL@$9$k(B.
                     77: ChangeLog $B$N9`L\$O(B www.openxm.org $B$N(B cvsweb $B$G(B
                     78: $B%=!<%9%3!<%I$rFI$`;~$N=u$1$K$J$k>pJs$,=q$+$l$F$$$k(B.
                     79:
                     80: $BK\J8Cf$G0zMQ$7$F$$$kJ88%$rNs5s$9$k(B.
                     81: @itemize @bullet
                     82: @item [GM2016]
                     83: Y.Goto, K.Matsumoto, Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1,k+n+2) and their applications, arxiv:1602.01637 (version 1)
                     84: @item [T2016]
                     85: Y.Tachibana, $B:9J,%[%m%N%_%C%/8{G[K!$N%b%8%e%i!<%a%=%C%I$K$h$k7W;;$N9bB.2=(B,
                     86: 2016, $B?@8MBg3X=$;NO@J8(B.
                     87: @item [GTT2016]
                     88: Y.Goto, Y.Tachibana, N.Takayama, 2$B85J,3dI=$KBP$9$k:9J,%[%m%N%_%C%/8{G[K!$N<BAu(B,
                     89: $B?tM}8&9V5fO?(B($B7G:\M=Dj(B).
                     90: @item [TKT2015]
                     91: N.Takayama, S.Kuriki, A.Takemura,
                     92:          $A$-hypergeometric distributions and Newton polytopes.
                     93:          arxiv:1510.02269
                     94: @end itemize
                     95:
                     96: $B$3$N%^%K%e%"%k$G@bL@$9$k4X?t$rMQ$$$?%W%m%0%i%`Nc$O(B
                     97: gtt_ekn/test-t1.rr
                     98: $B$J$I(B.
                     99:
                    100: @node 2$B85J,3dI=(BHGM$B$N4X?t(B,,, Top
                    101: @chapter 2$B85J,3dI=(BHGM$B$N4X?t(B
                    102:
                    103: @comment --- section ``$B<B83E*4X?t(B'' $B$N(B subsection xyz_abc
                    104: @comment --- subsection xyz_pqr xyz_stu $B$,$"$k(B.
                    105: @menu
                    106: * gtt_ekn.gmvector::
                    107: * gtt_ekn.nc::
                    108: * gtt_ekn.lognc::
                    109: * gtt_ekn.expectation::
                    110: * gtt_ekn.setup::
                    111: * gtt_ekn.upAlpha::
                    112: @end menu
                    113:
                    114: @node $BD64v2?4X?t(BE(k,n),,, 2$B85J,3dI=(BHGM$B$N4X?t(B
                    115: @section $BD64v2?4X?t(BE(k,n)
                    116:
                    117: @comment **********************************************************
                    118: @comment --- $B"~"~"~"~(B  $B$N@bL@(B
                    119: @comment --- $B8D!9$N4X?t$N@bL@$N3+;O(B ---
                    120: @comment --- section $BL>$r@53N$K(B ---
                    121: @node gtt_ekn.gmvector,,, $BD64v2?4X?t(BE(k,n)
                    122: @subsection @code{gtt_ekn.gmvector}
                    123: @comment --- $B:w0zMQ%-!<%o!<%I(B
                    124: @findex gtt_ekn.gmvector
                    125:
                    126: @table @t
                    127: @item gtt_ekn.gmvector(@var{beta},@var{p})
                    128: :: $B<~JUOB(B @var{beta}, $B%;%k$N3NN((B @var{p} $B$NFs85J,3dI=$KIU?o$9$kD64v2?4X?t(B
                    129: E(k,n) $B$NCM$*$h$S$=$NHyJ,$NCM$rLa$9(B.
                    130: @item gtt_ekn.ekn_cBasis_2(@var{beta},@var{p})
                    131: $B$NJLL>$G$"$k(B.
                    132: @end table
                    133:
                    134: @comment --- $B0z?t$N4JC1$J@bL@(B ---  $B0J2<$^$@=q$$$F$J$$(B.
                    135: @table @var
                    136: @item return
                    137: $B%Y%/%H%k(B, $BD64v2?4X?t$NCM$H$=$NHyJ,(B. $B>\$7$/$O2<5-(B.
                    138: @item beta
                    139: $B9TOB(B, $BNsOB$N%j%9%H(B. $B@.J,$O$9$Y$F@5$G$"$k$3$H(B.
                    140: @item p
                    141: $BFs85J,3dI=$N%;%k$N3NN($N%j%9%H(B
                    142: @end table
                    143:
                    144: @comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
                    145: @comment --- @itemize$B!A(B@end itemize $B$O2U>r=q$-(B ---
                    146: @comment --- @bullet $B$O9uE@IU$-(B ---
                    147: @itemize @bullet
                    148: @item
                    149: gmvector $B$O(B Gauss-Manin vector $B$NN,$G$"$k(B [GM2016].
                    150: @item
1.2     ! takayama  151: gmvector $B$NLa$jCM$O(B
        !           152: [GM2016] $B$N(B 6$B>O(B p.23 $B$N%Y%/%H%k(B S$B$G$"$k(B.
        !           153: $B$3$l$O(B
        !           154: [GM2016] $B$N#4>O$GDj5A$5$l$F$$$k%Y%/%H%k(B F $B$NDj?tG\$G$"$j(B,
        !           155: $B$=$NDj?t$O(B
        !           156: $BBh0l@.J,$,(B [GM2016] $B$N#6>O$GDj5A$5$l$F$$$k5i?t(B S $B$NCM$HEy$7$/(B
        !           157: $B$J$k$h$&$K7h$a$i$l$F$$$k(B.
1.1       takayama  158: @item
                    159:  r1 x r2 $BJ,3dI=$r9M$($k(B.
                    160:  m+1=r1, n+1=r2 $B$H$*$/(B.
                    161:  $B@55,2=Dj?t(B Z $B$OJ,3dI=(B u $B$r(B (m+1) $B!_(B (n+1) $B9TNs$H$9$k$H$-(B p^u/u! $B$NOB$G$"$k(B.
                    162:  $B$3$3$GOB$O9TOBNsOB$,(B @var{beta} $B$G$"$k$h$&$J(B u $BA4BN$G$H$k(B
                    163:  [TKT2015], [GM2016].
1.2     ! takayama  164:  S $B$O$3$NB?9`<0(B Z $B$N(B p $B$r(B
1.1       takayama  165: @verbatim
                    166:   [[1,y11,...,y1n],
                    167:    [1,y21,...,y2n],...,
                    168:    [1,ym1, ...,ymn],
                    169:    [1,1, ..., 1]]
                    170: @end verbatim
                    171: $B!!(B(1 $B$,(B L $B;z7?$KJB$V(B),
                    172: $B$H@55,2=$7$?5i?t$G$"$k(B.
                    173: @item
                    174: 2x(n+1)$BJ,3dI=$G(B, gmvector $B$NLa$jCM$r(B Lauricella  F_D $B$G=q$/$3$H$,(B
                    175: $B0J2<$N$h$&$K$G$-$k(B
                    176: (b[2][1]-b[1][1] >= 0 $B$N>l9g(B).
                    177: $B$3$3$G(B b[1][1], b[1][2] $B$O(B, $B$=$l$>$l(B 1 $B9TL\$N9TOB(B, 2 $B9TL\$N9TOB(B,
                    178: b[2][i] $B$O(B i $BNsL\$NNsOB$G$"$k(B.
                    179: @comment ekn/Talks/2015-12-3-goto.tex
                    180: @verbatim
                    181: S=F_D(-b[1,1], [-b[2,2],...,-b[2,n+1]], b[2,1]-b[1,1]+1 ; y)/C,
                    182: @end verbatim
                    183: C=b[1,1]! b[2,2]! ... b[2][n+1]! (b[2,1]-b[1,1])!
                    184: $B$H$*$/(B.
                    185: 1/C $B$O(B L $B;z7?$NJ,3dI=(B
                    186: @verbatim
                    187: [[b[1,1],       0,      ..., 0       ],
                    188:  [b[2,1]-b[1,1],b[2,2], ..., b[2,n+1]]]
                    189: @end verbatim
                    190: $B$KBP1~(B.
                    191: gmvector $B$O(B
                    192: @verbatim
                    193: [S,(y11/a2) d_11 S,(y12/a3) d_12 S, ..., (y1n/a_(n+1)) d_1n S]
                    194: @end verbatim
                    195: $B$G$"$k(B.
                    196: $B$3$3$G(B d_ij $B$O(B yij $B$K$D$$$F$NHyJ,(B,
                    197: @verbatim
                    198:   [a0,     a1, ...                      ,a_(n+2)]
                    199: = [-b[1,2],-b[1,1],b[2,2], ..., b[2,n+1],b[2,1]]
                    200: @end verbatim
                    201: $B$G$"$k(B.
                    202: @item
                    203: $B<~JUOB(B @var{beta}$B$N;~$N@55,2=Dj?t$N%;%k3NN((B @var{p} $B$KBP$9$kCM$O(B $BB?9`<0$KB`2=$7$?(B E(k,n) $B$NCM$GI=8=$G$-$k(B. $BJ88%(B [TKT2015], [GM2016] $B;2>H(B.
                    204: @item
                    205: option crt=1 (crt = Chinese remainder theorem) $B$rM?$($k$H(B, $BJ,;67W;;$r$*$3$J$&(B
                    206: [T2016].
                    207: $BJ,;67W;;MQ$N3F<o%Q%i%a!<%?$N@_Dj$O(B
                    208: gtt_ekn.setup $B$G9T$J$&(B.
                    209: @end itemize
                    210:
                    211: @comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
                    212: $BNc(B: $B<!$O(B2 x 2 $BJ,3dI=$G9TOB$,(B [5,1],  $BNsOB$,(B [3,3], $B3F%;%k$N3NN($,(B
                    213: [[1/2,1/3],[1/7,1/5]] $B$N>l9g$N(B gmvector $B$NCM$G$"$k(B.
                    214: @example
                    215: [3000] load("gtt_ekn.rr");
                    216: [3001] ekn_gtt.gmvector([[5,1],[3,3]],[[1/2,1/3],[1/7,1/5]])
                    217: [775/27783]
                    218: [200/9261]
                    219: @end example
                    220:
                    221: $B;29M(B: 2 x m $BJ,3dI=(B(Lauricella FD)$B$K$D$$$F$O%Q%C%1!<%8(B tk_fd $B$G$b2<5-$N$h$&$KF1Ey$J(B
                    222: $B7W;;$,$G$-$k(B.
                    223: $B<iHwHO0O$N0[$J$k%W%m%0%i%`F1;N$NHf3S(B, debug $BMQ;29M(B.
                    224: @example
                    225: [3080] import("tk_fd.rr");
                    226: [3081] A=tk_fd.marginal2abc([4,5],[2,4,3]);
                    227: [-4,[-4,-3],-1]  // 2$BJQ?t(B FD $B$N%Q%i%a!<%?(B. a,[b1,b2],c
                    228: [3082] tk_fd.fd_hessian2(A[0],A[1],A[2],[1/2,1/3]);
                    229: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    230: [4483/124416,[ 1961/15552 185/1728 ],
                    231:  [ 79/288 259/864 ]
                    232:  [ 259/864 47/288 ]]
                    233: // $BLaCM$O(B [F=F_D, gradient(F), Hessian(F)]
                    234:
                    235: // ekn_gt $B$G$NNc$HF1$8%Q%i%a!<%?(B.
                    236: [3543] A=tk_fd.marginal2abc([5,1],[3,3]);
                    237: [-5,[-3],-1]
                    238: [3544] tk_fd.fd_hessian2(A[0],A[1],A[2],[(1/3)*(1/7)/((1/2)*(1/5))]);
                    239: Computing Dmat(ca) for parameters B=[-3],X=[ 10/21 ]
                    240: [775/27783,[ 20/147 ],[ 17/42 ]]
                    241: @end example
                    242:
                    243: $B;29M(B: $B0lHL$N(B A $BJ,I[$N@55,2=Dj?t$K$D$$$F$N(B Hessian $B$N7W;;$O<B83E*(B package ot_hessian_ahg.rr
                    244: $B$G<BAu$N%F%9%H$,$5$l$F$$$k(B. ($B$3$l$O$^$@L$40@.$N%F%9%HHG$J$N$G=PNO7A<0Ey$b>-MhE*$K$OJQ99$5$l$k(B.)
                    245: @example
                    246: import("ot_hgm_ahg.rr");
                    247: import("ot_hessian_ahg.rr");
                    248: def  htest4() @{
                    249:   extern C11_A;
                    250:   extern C11_Beta;
                    251:   Hess=newmat(7,7);
                    252:   A =C11_A;
                    253:   Beta0= [b0,b1,b2,b3];
                    254:   BaseIdx=[4,5,6];
                    255:   X=[x0,x1,x2,x3,x4,x5,x6];
                    256:   for (I=0; I<7; I++) for (J=0; J<7; J++) @{
                    257:     Idx = [I,J];
                    258:     H=hessian_simplify(A,Beta0,X,BaseIdx,Idx);
                    259:     Hess[I][J]=H;
                    260:     printf("[I,J]=%a, Hessian_ij=%a\n",Idx,H);
                    261:   @}
                    262:   return(Hess);
                    263: @}
                    264: [2917] C11_A;
                    265: [[0,0,0,1,1,1,1],[1,0,0,1,0,1,0],[0,1,1,0,1,0,1],[1,1,0,1,1,0,0]]
                    266: [2918] C11_Beta;
                    267: [166,36,290,214]
                    268: [2919] Ans=htest4$
                    269: [2920] Ans[0][0];
                    270: [[((b1-b0-1)*x4)/(x0^2),[4]],[((b1-b0-1)*x6)/(x0^2),[6]],
                    271:  [(b1^2+(-2*b0-1)*b1+b0^2+b0)/(x0^2),[]],[(x6)/(x0),[6,0]],[(x4)/(x0),[4,0]]]
                    272: @end example
                    273:
                    274: @comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
                    275: @table @t
                    276: @item $B;2>H(B
                    277: @ref{gtt_ekn.setup}
                    278: @ref{gtt_ekn.pfaffian_basis}
                    279: @end table
                    280:
                    281: @comment --- ChangeLog $B$r=q$/(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B
                    282: @noindent
                    283: ChangeLog
                    284: @itemize @bullet
                    285: @item
                    286:  $B$3$N4X?t$O(B
                    287: [GM2016] $B$N%"%k%4%j%:%`$*$h$S(B
                    288: [T2016] $B$K$h$k(B modular method $B$rMQ$$$?9bB.2=$r<BAu$7$?$b$N$G$"$k(B.
                    289: @item
                    290:  $BJQ99$r<u$1$?%U%!%$%k$O(B
                    291:  OpenXM/src/asir-contrib/packages/src/gtt_ekn.rr 1.1, gtt_ekn/ekn_pfaffian_8.rr
                    292: @end itemize
                    293:
                    294:
                    295: @comment **********************************************************
                    296: @node gtt_ekn.nc,,, $BD64v2?4X?t(BE(k,n)
                    297: @subsection @code{gtt_ekn.nc}
                    298: @comment --- $B:w0zMQ%-!<%o!<%I(B
                    299: @findex gtt_ekn.nc
                    300:
                    301: @table @t
                    302: @item gtt_ekn.nc(@var{beta},@var{p})
                    303: :: $B<~JUOB(B @var{beta}, $B%;%k$N3NN((B @var{p} $B$NFs85J,3dI=$N>r7oIU$-3NN($N@55,2=Dj?t(B Z
                    304: $B$*$h$S$=$NHyJ,$NCM$rLa$9(B.
                    305: @end table
                    306:
                    307: @comment --- $B0z?t$N4JC1$J@bL@(B ---  $B0J2<$^$@=q$$$F$J$$(B.
                    308: @table @var
                    309: @item return
                    310: $B%Y%/%H%k(B [Z,[[d_11 Z, d_12 Z, ...], ..., [d_m1 Z, d_m2 Z, ...., d_mn Z]]]
                    311: @item beta
                    312: $B9TOB(B, $BNsOB$N%j%9%H(B. $B@.J,$O$9$Y$F@5$G$"$k$3$H(B.
                    313: @item p
                    314: $BFs85J,3dI=$N%;%k$N3NN($N%j%9%H(B
                    315: @end table
                    316:
                    317: @comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
                    318: @comment --- @itemize$B!A(B@end itemize $B$O2U>r=q$-(B ---
                    319: @comment --- @bullet $B$O9uE@IU$-(B ---
                    320: @itemize @bullet
                    321: @item
                    322:  r1 x r2 $BJ,3dI=$r9M$($k(B.
                    323:  m=r1, n=r2 $B$H$*$/(B.
                    324:  $B@55,2=Dj?t(B Z $B$OJ,3dI=(B u $B$r(B m $B!_(B n $B9TNs$H$9$k$H$-(B p^u/u! $B$NOB$G$"$k(B.
                    325:  $B$3$3$GOB$O9TOBNsOB$,(B @var{beta} $B$G$"$k$h$&$J(B u $BA4BN$G$H$k(B
                    326:  [TKT2015], [GM2016].
                    327:  p^u $B$O(B p_ij^u_ij $B$N@Q(B, u! $B$O(B u_ij! $B$N@Q$G$"$k(B.
                    328:  d_ij Z $B$G(B Z $B$NJQ?t(B p_ij $B$K$D$$$F$NJPHyJ,$rI=$9(B.
                    329: @item
                    330: nc $B$O(B gmvector $B$NCM$r85$K(B, [GM2016] $B$N(B Prop
                    331:  7.1 $B$K4p$E$$$F(B Z $B$NCM$r7W;;$9$k(B.
                    332: @item
                    333: option crt=1 (crt = Chinese remainder theorem) $B$rM?$($k$H(B, $BJ,;67W;;$r$*$3$J$&(B.
                    334: $BJ,;67W;;MQ$N3F<o%Q%i%a!<%?$N@_Dj$O(B
                    335: gtt_ekn.setup $B$G9T$J$&(B.
                    336: @end itemize
                    337:
                    338: @comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
                    339: $BNc(B: 2x3 $BJ,3dI=$G$N(B Z $B$H$=$NHyJ,$N7W;;(B.
                    340: @example
                    341: [2237] gtt_ekn.nc([[4,5],[2,4,3]],[[1,1/2,1/3],[1,1,1]]);
                    342: [4483/124416,[ 353/7776 1961/15552 185/1728 ]
                    343: [ 553/20736 1261/15552 1001/13824 ]]
                    344: @end example
                    345:
                    346: $B;29M(B: 2 x m $BJ,3dI=(B(Lauricella FD)$B$K$D$$$F$O%Q%C%1!<%8(B tk_fd $B$G$b2<5-$N$h$&$KF1Ey$J(B
                    347: $B7W;;$,$G$-$k(B.
                    348: @example
                    349: [3076] import("tk_fd.rr");
                    350: [3077] A=tk_fd.marginal2abc([4,5],[2,4,3]);
                    351: [-4,[-4,-3],-1]
                    352: [3078] tk_fd.ahmat_abc(A[0],A[1],A[2],[[1,1/2,1/3],[1,1,1]]);
                    353: RS=[ 4 5 ], CSnew=[ 2 4 3 ], Ynew=[ 1 1/2 1/3 ]
                    354: [ 1 1 1 ]
                    355: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    356: [4483/124416,[[353/7776,1961/15552,185/1728],
                    357:               [553/20736,1261/15552,1001/13824]]]
                    358: // $BLaCM$O(B [Z, [[d_11 Z, d_12 Z, d_13 Z],
                    359: //             [d_21 Z, d_22 Z, d_23 Z]]] $B$NCM(B.
                    360: //           $B$3$3$G(B d_ij $B$O(B i,j $B@.J,$K$D$$$F$NHyJ,$rI=$9(B.
                    361: @end example
                    362:
                    363: @comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
                    364: @table @t
                    365: @item $B;2>H(B
                    366: @ref{gtt_ekn.setup}
                    367: @ref{gtt_ekn.lognc}
                    368: @end table
                    369:
                    370: @comment --- ChangeLog $B$r=q$/(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B
                    371: @noindent
                    372: ChangeLog
                    373: @itemize @bullet
                    374: @item
                    375:  $BJQ99$r<u$1$?%U%!%$%k$O(B
                    376:  OpenXM/src/asir-contrib/packages/src/gtt_ekn.rr 1.1, gtt_ekn/ekn_eval.rr
                    377: @end itemize
                    378:
                    379:
                    380: @comment **********************************************************
                    381: @node gtt_ekn.lognc,,, $BD64v2?4X?t(BE(k,n)
                    382: @subsection @code{gtt_ekn.lognc}
                    383: @comment --- $B:w0zMQ%-!<%o!<%I(B
                    384: @findex gtt_ekn.lognc
                    385:
                    386: @table @t
                    387: @item gtt_ekn.lognc(@var{beta},@var{p})
                    388: :: $B<~JUOB(B @var{beta}, $B%;%k$N3NN((B @var{p} $B$NFs85J,3dI=$N>r7oIU$-3NN($N@55,2=Dj?t(B Z
                    389: $B$N(B log $B$N6a;wCM$*$h$S$=$NHyJ,$N6a;wCM$rLa$9(B.
                    390: @end table
                    391:
                    392: @comment --- $B0z?t$N4JC1$J@bL@(B ---  $B0J2<$^$@=q$$$F$J$$(B.
                    393: @table @var
                    394: @item return
                    395: $B%Y%/%H%k(B [log(Z), [[d_11 log(Z), d_12 log(Z), ...], [d_21 log(Z),...], ... ]
                    396: @item beta
                    397: $B9TOB(B, $BNsOB$N%j%9%H(B. $B@.J,$O$9$Y$F@5$G$"$k$3$H(B.
                    398: @item p
                    399: $BFs85J,3dI=$N%;%k$N3NN($N%j%9%H(B
                    400: @end table
                    401:
                    402: @comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
                    403: @comment --- @itemize$B!A(B@end itemize $B$O2U>r=q$-(B ---
                    404: @comment --- @bullet $B$O9uE@IU$-(B ---
                    405: @itemize @bullet
                    406: @item
                    407: $B>r7oIU$-:GL`?dDj$KMxMQ$9$k(B [TKT2015].
                    408: @item option crt=1 (crt = Chinese remainder theorem) $B$rM?$($k$H(B, $BJ,;67W;;$r$*$3$J$&(B.
                    409: $BJ,;67W;;MQ$N3F<o%Q%i%a!<%?$N@_Dj$O(B
                    410: gtt_ekn.setup $B$G9T$J$&(B.
                    411: @end itemize
                    412:
                    413: @comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
                    414: $BNc(B: 2 $B!_(B 3 $BJ,3dI=$G$NNc(B. $BBh0l@.J,$N$_6a;wCM(B.
                    415: @example
                    416: [2238] gtt_ekn.lognc([[4,5],[2,4,3]],[[1,1/2,1/3],[1,1,1]]);
                    417: [-3.32333832422461674630,[ 5648/4483 15688/4483 13320/4483 ]
                    418: [ 3318/4483 10088/4483 9009/4483 ]]
                    419: @end example
                    420:
                    421: $B;29M(B: 2 x m $BJ,3dI=(B(Lauricella FD)$B$K$D$$$F$O%Q%C%1!<%8(B tk_fd $B$G$b2<5-$N$h$&$KF1Ey$J(B
                    422: $B7W;;$,$G$-$k(B.
                    423: @example
                    424: [3076] import("tk_fd.rr");
                    425: [3077] A=tk_fd.marginal2abc([4,5],[2,4,3]);
                    426: [-4,[-4,-3],-1]
                    427: [3078] tk_fd.log_ahmat_abc(A[0],A[1],A[2],[[1,1/2,1/3],[1,1,1]]);
                    428: RS=[ 4 5 ], CSnew=[ 2 4 3 ], Ynew=[ 1 1/2 1/3 ]
                    429: [ 1 1 1 ]
                    430: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    431: [-3.32333832422461674639485797719209322217260539267246045320,
                    432:  [[1.2598706, 3.499442, 2.971224],
                    433:   [0.7401293, 2.250278, 2.009591]]]
                    434: // $BLaCM$O(B [log(Z),
                    435: //          [[d_11 log(Z), d_12 log(Z), d_13 log(Z)],
                    436: //           [d_21 log(Z), d_22 log(Z), d_23 log(Z)]]]
                    437: // $B$N6a;wCM(B.
                    438: @end example
                    439:
                    440: @comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
                    441: @table @t
                    442: @item $B;2>H(B
                    443: @ref{gtt_ekn.setup}
                    444: @ref{gtt_ekn.nc}
                    445: @end table
                    446:
                    447: @comment --- ChangeLog $B$r=q$/(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B
                    448: @noindent
                    449: ChangeLog
                    450: @itemize @bullet
                    451: @item
                    452:  $BJQ99$r<u$1$?%U%!%$%k$O(B
                    453:  OpenXM/src/asir-contrib/packages/src/gtt_ekn.rr 1.1.
                    454: @end itemize
                    455:
                    456: @comment **********************************************************
                    457: @node gtt_ekn.expectation,,, $BD64v2?4X?t(BE(k,n)
                    458: @subsection @code{gtt_ekn.expectation}
                    459: @comment --- $B:w0zMQ%-!<%o!<%I(B
                    460: @findex gtt_ekn.expectation
                    461:
                    462: @table @t
                    463: @item gtt_ekn.expectation(@var{beta},@var{p})
                    464: :: $B<~JUOB(B @var{beta}, $B%;%k$N3NN((B @var{p} $B$NFs85J,3dI=$N4|BTCM$r7W;;$9$k(B.
                    465: @end table
                    466:
                    467: @comment --- $B0z?t$N4JC1$J@bL@(B ---  $B0J2<$^$@=q$$$F$J$$(B.
                    468: @table @var
                    469: @item return
                    470: $BFs85J,3dI=$N3F%;%k$N4|BTCM$N%j%9%H(B.
                    471: @item beta
                    472: $B9TOB(B, $BNsOB$N%j%9%H(B. $B@.J,$O$9$Y$F@5$G$"$k$3$H(B.
                    473: @item p
                    474: $BFs85J,3dI=$N%;%k$N3NN($N%j%9%H(B
                    475: @end table
                    476:
                    477: @comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
                    478: @comment --- @itemize$B!A(B@end itemize $B$O2U>r=q$-(B ---
                    479: @comment --- @bullet $B$O9uE@IU$-(B ---
                    480: @itemize @bullet
                    481: @item
                    482: [GM2016] $B$N(B Algorithm 7.8 $B$N<BAu(B.
                    483: @item option crt=1 (crt = Chinese remainder theorem) $B$rM?$($k$H(B, $BJ,;67W;;$r$*$3$J$&(B.
                    484: $BJ,;67W;;MQ$N3F<o%Q%i%a!<%?$N@_Dj$O(B
                    485: gtt_ekn.setup $B$G9T$J$&(B.
                    486: @item option index $B$rM?$($k$H(B, $B;XDj$5$l$?@.J,$N4|BTCM$N$_7W;;$9$k(B.
                    487: $B$?$H$($P(B 2 x 2 $BJ,3dI=$G(B index=[[0,0],[1,1]] $B$H;XDj$9$k$H(B, 1 $B$N$"$k@.J,$N4|BTCM$N$_7W;;$9$k(B.
                    488: @end itemize
                    489:
                    490: @comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
                    491:
                    492: 2$B!_(B2, 3$B!_(B3 $B$NJ,3dI=$N4|BTCM7W;;Nc(B.
                    493: @example
                    494: [2235] gtt_ekn.expectation([[1,4],[2,3]],[[1,1/3],[1,1]]);
                    495: [ 2/3 1/3 ]
                    496: [ 4/3 8/3 ]
                    497: [2236] gtt_ekn.expectation([[4,5],[2,4,3]],[[1,1/2,1/3],[1,1,1]]);
                    498: [ 5648/4483 7844/4483 4440/4483 ]
                    499: [ 3318/4483 10088/4483 9009/4483 ]
                    500:
                    501: [2442] gtt_ekn.expectation([[4,14,9],[11,6,10]],[[1,1/2,1/3],[1,1/5,1/7],[1,1,1]]);
                    502: [ 207017568232262040/147000422096729819 163140751505489940/147000422096729819
                    503:                                         217843368649167296/147000422096729819 ]
                    504: [ 1185482401011137878/147000422096729819 358095302885438604/147000422096729819
                    505:                                          514428205457640984/147000422096729819 ]
                    506: [ 224504673820628091/147000422096729819 360766478189450370/147000422096729819
                    507:                                         737732646860489910/147000422096729819 ]
                    508: @end example
                    509:
                    510: $B;29M(B: 2 x m $BJ,3dI=(B(Lauricella FD)$B$K$D$$$F$O%Q%C%1!<%8(B tk_fd $B$G$b2<5-$N$h$&$KF1Ey$J(B
                    511: $B7W;;$,$G$-$k(B.
                    512: @example
                    513: [3076] import("tk_fd.rr");
                    514: [3077] A=tk_fd.marginal2abc([4,5],[2,4,3]);
                    515: [-4,[-4,-3],-1]
                    516: [3078] tk_fd.expectation_abc(A[0],A[1],A[2],[[1,1/2,1/3],[1,1,1]]);
                    517: RS=[ 4 5 ], CSnew=[ 2 4 3 ], Ynew=[ 1 1/2 1/3 ]
                    518: [ 1 1 1 ]
                    519: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    520: [[5648/4483,7844/4483,4440/4483],
                    521:  [3318/4483,10088/4483,9009/4483]]
                    522: // $B3F%;%k$N4|BTCM(B.
                    523: @end example
                    524:
                    525: $B;29M(B: $B0lHL$N(B A $BJ,I[$N7W;;$O(B ot_hgm_ahg.rr. $B$^$@<B83E*$J$?$a(B, module $B2=$5$l$F$$$J$$(B.
                    526: ot_hgm_ahg.rr $B$K$D$$$F$N;29MJ88%(B:
                    527: K.Ohara, N.Takayama, Pfaffian Systems of A-Hypergeometric Systems II --- Holonomic Gradient Method, arxiv:1505.02947
                    528: @example
                    529: [3237] import("ot_hgm_ahg.rr");
                    530: // 2 x 2 $BJ,3dI=(B.
                    531: [3238] hgm_ahg_expected_values_contiguity([[0,0,1,1],[1,0,1,0],[0,1,0,1]],
                    532:         [9,6,8],[1/2,1/3,1/5,1/7],[x1,x2,x3,x4]|geometric=1);
                    533: oohg_native=0, oohg_curl=1
                    534: [1376777025/625400597,1750225960/625400597,
                    535:  2375626557/625400597,3252978816/625400597]
                    536: // 2 x 2 $BJ,3dI=$N4|BTCM(B.
                    537:
                    538: // 2 x 3 $BJ,3dI=(B.
                    539: [3238] hgm_ahg_expected_values_contiguity(
                    540:  [[0,0,0,1,1,1],[1,0,0,1,0,0],[0,1,0,0,1,0],[0,0,1,0,0,1]],
                    541:  [5,2,4,3],[1,1/2,1/3,1,1,1],[x1,x2,x3,x4,x5,x6]|geometric=1);
                    542: [5648/4483,7844/4483,4440/4483,3318/4483,10088/4483,9009/4483]
                    543: // 2 x 3 $BJ,3dI=$N4|BTCM(B. $B>e$HF1$8LdBj(B.
                    544: @end example
                    545:
                    546: 3 x 3 $BJ,3dI=(B. $B9=B$E*(B0$B$,0l$D(B.
                    547: @example
                    548: /*
                    549:   dojo, p.221 $B$N%G!<%?(B.  $B@.@S(B3$B0J2<$N@8EL$O=8$a$F$R$H$D$K(B.
                    550:   2 1 1
                    551:   8 3 3
                    552:   0 2 6
                    553:
                    554:   row sum: 4,14,8
                    555:   column sum: 10,6,10
                    556:   0 $B$r0l$D4^$`$N$G(B, (3,6) $B7?$N(B A $B$+$i(B 7 $BNsL\$rH4$/(B.
                    557: */
                    558:
                    559: A=[[0,0,0,1,1,1, 0,0],
                    560:    [0,0,0,0,0,0, 1,1],
                    561:    [1,0,0,1,0,0, 0,0],
                    562:    [0,1,0,0,1,0, 1,0],
                    563:    [0,0,1,0,0,1, 0,1]];
                    564: B=[14,8,10,6,10];
                    565: hgm_ahg_expected_values_contiguity(A,B,[1,1/2,1/3,1,1/5,1/7,1,1],
                    566: $B!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(B[x1,x2,x3,x4,x5,x6,x7,x8]|geometric=1);
                    567:
                    568: // $BEz(B.
                    569: [14449864949304/9556267369631,
                    570:  10262588586540/9556267369631, 13512615942680/9556267369631,
                    571:  81112808747006/9556267369631,
                    572:  21816297744346/9556267369631, 30858636683482/9556267369631,
                    573:
                    574:  25258717886900/9556267369631,51191421070148/9556267369631]
                    575: @end example
                    576:
                    577: 3 x 3 $BJ,3dI=(B.
                    578: @example
                    579: /*
                    580:  $B>e$N%G!<%?$G(B 0 $B$r(B 1 $B$KJQ99(B.
                    581:   2 1 1
                    582:   8 3 3
                    583:   1 2 6
                    584:
                    585:   row sum: 4,14,9
                    586:   column sum: 11,6,10
                    587: */
                    588: A=[[0,0,0,1,1,1,0,0,0],
                    589:    [0,0,0,0,0,0,1,1,1],
                    590:    [1,0,0,1,0,0,1,0,0],
                    591:    [0,1,0,0,1,0,0,1,0],
                    592:    [0,0,1,0,0,1,0,0,1]];
                    593: B=[14,9,11,6,10];
                    594: hgm_ahg_expected_values_contiguity(A,B,[1,1/2,1/3,1,1/5,1/7,1,1,1],
                    595:                               [x1,x2,x3,x4,x5,x6,x7,x8]|geometric=1);
                    596:
                    597: // $B4|BTCM(B, $BEz(B.   x9 $B$r;XDj$7$F$$$J$$$N$G(B, 9$BHVL\$N4|BTCM$O=PNO$7$F$J$$(B.
                    598: [207017568232262040/147000422096729819,
                    599:  163140751505489940/147000422096729819,217843368649167296/147000422096729819,
                    600:  1185482401011137878/147000422096729819,
                    601:  358095302885438604/147000422096729819,514428205457640984/147000422096729819,
                    602:  224504673820628091/147000422096729819,360766478189450370/147000422096729819]
                    603:
                    604: // Z $B$d$=$NHyJ,$N7W;;$O(B hgm_ahg_contiguity $B4X?t$,$*$3$J$&$,(B, $B$3$l$N4J0W%$%s%?!<%U%'!<%9$O(B
                    605: // $B$^$@=q$$$F$J$$(B.
                    606: @end example
                    607:
                    608:
                    609:
                    610: @comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
                    611: @table @t
                    612: @item $B;2>H(B
                    613: @ref{gtt_ekn.setup}
                    614: @ref{gtt_ekn.nc}
                    615: @end table
                    616:
                    617: @comment --- ChangeLog $B$r=q$/(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B
                    618: @noindent
                    619: ChangeLog
                    620: @itemize @bullet
                    621: @item
                    622:  $BJQ99$r<u$1$?%U%!%$%k$O(B
                    623:  OpenXM/src/asir-contrib/packages/src/gtt_ekn.rr 1.1.
                    624: @end itemize
                    625:
                    626:
                    627: @comment **********************************************************
                    628: @comment --- $B"~"~"~"~(B  $B$N@bL@(B
                    629: @comment --- $B8D!9$N4X?t$N@bL@$N3+;O(B ---
                    630: @comment --- section $BL>$r@53N$K(B ---
                    631: @node gtt_ekn.setup,,, $BD64v2?4X?t(BE(k,n)
                    632: @subsection @code{gtt_ekn.setup}
                    633: @comment --- $B:w0zMQ%-!<%o!<%I(B
                    634: @findex gtt_ekn.setup
                    635:
                    636: @table @t
                    637: @item gtt_ekn.setup()
                    638: :: $BJ,;67W;;MQ$N4D6-@_Dj$r$*$3$J$&(B. $B8=:_$N4D6-$rJs9p$9$k(B.
                    639: @end table
                    640:
                    641: @comment --- $B0z?t$N4JC1$J@bL@(B ---  $B0J2<$^$@=q$$$F$J$$(B.
                    642: @table @var
                    643: @item return
                    644:
                    645: @end table
                    646:
                    647: @comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
                    648: @comment --- @itemize$B!A(B@end itemize $B$O2U>r=q$-(B ---
                    649: @comment --- @bullet $B$O9uE@IU$-(B ---
                    650: @itemize @bullet
                    651: @item $B;HMQ$9$k%W%m%;%9$HAG?t$N8D?t(B, $B:G>.$NAG?t$rI=<($9$k(B. $B=`Hw$5$l$F$$$J$$>l9g$O$=$N;]$rI=<((B.
1.2     ! takayama  652: @item $B$3$N%Q%C%1!<%8$G$NJ,;67W;;$OJ#?t$N(Bcpu$B$rEk:\$7$?7W;;5!$G<B9T$5$l$k$3$H$rA[Dj$7$F$$$k(B.
        !           653: @item option nps ($B$^$?$O(B number_of_processes)$B$rM?$($k$H;XDj$7$??t$@$1%W%m%;%9$rMQ0U$9$k(B.
        !           654: @item option nprm ($B$^$?$O(B number_of_primes)$B$rM?$($k$H(Bnprm$B$,J8;zNs$N>l9g;XDj$5$l$?AG?t%j%9%H$N%U%!%$%k$rFI$_9~$`(B. nprm$B$,<+A3?t$N>l9g$5$i$K(Boption minp (minp =MINimum Prime)$B$rM?$($k$H(Bminp$B$h$jBg$-$JAG?t$r(Bnprm$B8D@8@.$9$k(B. $B$=$N:](Boption fgp ($B$^$?$O(B file_of_generated_primes)$B$rM?$($k$H@8@.$7$?AG?t%j%9%H$r%U%!%$%kL>$r(Bfgp$B$H$7$FJ]B8$9$k(B.
        !           655: @item $B>e5-$N(Boption $B$r;XDj$7$J$+$C$?>l9g<!$N%G%U%)%k%HCM$,MQ$$$i$l$k(B. nps=1. nprm=10. fgp=0.
        !           656: @item option report=1$B$rM?$($k$H8=:_$N4D6-$NJs9p$N$_$r9T$&(B. setup(|report=1)$B$NJLL>$H$7$F(Breport$B4X?t$r;HMQ$9$k$3$H$b$G$-$k(B.
1.1       takayama  657: @end itemize
                    658:
                    659: @comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
                    660: $BNc(B: $BAG?t$N%j%9%H$r@8@.$7$F%U%!%$%k(B p.txt $B$X=q$-=P$9(B.
                    661: @example
                    662: gtt_ekn.setup(|nps=2,nprm=20,minp=10^10,fgp="p.txt")$
                    663: @end example
                    664:
                    665:
                    666: @comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
                    667: @table @t
                    668: @item $B;2>H(B
                    669: @ref{gtt_ekn.nc}
                    670: @ref{gtt_ekn.gmvector}
                    671: @end table
                    672:
                    673: @comment --- ChangeLog $B$r=q$/(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B
                    674: @noindent
                    675: ChangeLog
                    676: @itemize @bullet
                    677: @item
                    678:  $BJQ99$r<u$1$?%U%!%$%k$O(B
                    679:  OpenXM/src/asir-contrib/packages/src/gtt_ekn.rr 1.1,
                    680:  gtt_ekn/g_mat_fac.rr
                    681:
                    682: @end itemize
                    683:
                    684: @comment **********************************************************
                    685: @comment --- $B"~"~"~"~(B  $B$N@bL@(B
                    686: @comment --- $B8D!9$N4X?t$N@bL@$N3+;O(B ---
                    687: @comment --- section $BL>$r@53N$K(B ---
                    688: @node gtt_ekn.upAlpha,,, $BD64v2?4X?t(BE(k,n)
                    689: @subsection @code{gtt_ekn.upAlpha}
                    690: @comment --- $B:w0zMQ%-!<%o!<%I(B
                    691: @findex gtt_ekn.upAlpha
                    692:
                    693: @table @t
                    694: @item gtt_ekn.upAlpha(@var{i},@var{k},@var{n})
                    695: ::
                    696: @end table
                    697:
                    698: @comment --- $B0z?t$N4JC1$J@bL@(B ---  $B0J2<$^$@=q$$$F$J$$(B.
                    699: @table @var
                    700: @item i  a_i $B$r(B a_i+1 $B$HJQ2=$5$;$k(B contiguity relation.
                    701: @item k  E(k+1,n+k+2)$B7?$ND64v2?4X?t$N(B k. $BJ,3dI=$G$O(B (k+1)$B!_(B(n+1).
                    702: @item n  E(k+1,n+k+2)$B7?$ND64v2?4X?t$N(B n. $BJ,3dI=$G$O(B (k+1)$B!_(B(n+1).
                    703: @item return  contiguity relation $B$N(B pfaffian_basis $B$K$D$$$F$N9TNsI=8=$rLa$9(B. [GM2016] $B$N(B Cor 6.3.
                    704: @end table
                    705:
                    706: @comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
                    707: @comment --- @itemize$B!A(B@end itemize $B$O2U>r=q$-(B ---
                    708: @comment --- @bullet $B$O9uE@IU$-(B ---
                    709: @itemize @bullet
                    710: @item
                    711:  upAlpha $B$O!!(B[GM2016] $B$N(B Cor 6.3 $B$N9TNs(B U_i $B$rLa$9(B.
                    712: @item $B4XO"$9$k3F4X?t$N4J7i$J@bL@$HNc$b2C$($k(B.
                    713: @item a_i $B$r(B a_i-1 $B$HJQ2=$5$;$?$$>l9g$O4X?t(B downAlpha $B$rMQ$$$k(B.
                    714: @item a_i $B$HJ,3dI=$N<~JUOB$r8+$k$K$O(B, $B4X?t(B marginaltoAlpha([$B9TOB(B,$BNsOB(B]) $B$rMQ$$$k(B.
                    715: @item
                    716:    pfaffian_basis $B$O(B [GM2016] $B$N#4>O$N%Y%/%H%k(B F $B$KBP1~$9$kJPHyJ,$rLa$9(B.
                    717: @end itemize
                    718:
                    719: @comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
                    720: $BNc(B: $B0J2<$NNc$O(B 2$B!_(B2$BJ,3dI=(B(E(2,4)), 2$B!_(B3$BJ,3dI=(B(E(2,5))$B$N>l9g$G$"$k(B.
                    721: [2225] $B$^$G$O=PNO$rN,$7$F$$$k(B.
                    722: @example
                    723: [2221] gtt_ekn.marginaltoAlpha([[1,4],[2,3]]);
                    724: [[a_0,-4],[a_1,-1],[a_2,3],[a_3,2]]
                    725: [2222] gtt_ekn.upAlpha(1,1,1);  // E(2,4) $B$N(B a_1 $BJ}8~$N(B
                    726:                                 //     contiguity $B$rI=8=$9$k9TNs(B
                    727: [2223] gtt_ekn.upAlpha(2,1,1);  // E(2,4) $B$N(B a_2 $BJ}8~(B
                    728: [2224] gtt_ekn.upAlpha(3,1,1);  // E(2,4) $B$N(B a_3 $BJ}8~(B
                    729: [2225] function f(x_1_1);
                    730: [2232] gtt_ekn.pfaffian_basis(f(x_1_1),1,1);
                    731: [ f(x_1_1) ]
                    732: [ (f{1}(x_1_1)*x_1_1)/(a_2) ]
                    733: [2233] function f(x_1_1,x_1_2);
                    734: f() redefined.
                    735: [2234] gtt_ekn.pfaffian_basis(f(x_1_1,x_1_2),1,2); // E(2,5), 2*3 $BJ,3dI=(B
                    736: [ f(x_1_1,x_1_2) ]
                    737: [ (f{1,0}(x_1_1,x_1_2)*x_1_1)/(a_2) ]
                    738: [ (f{0,1}(x_1_1,x_1_2)*x_1_2)/(a_3) ]
                    739: @end example
                    740:
                    741:
                    742: @comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
                    743: @table @t
                    744: @item $B;2>H(B
                    745: @ref{gtt_ekn.nc}
                    746: @ref{gtt_ekn.gmvector}
                    747: @end table
                    748:
                    749: @comment --- ChangeLog $B$r=q$/(B. $B%=!<%9%3!<%I$N0LCV(B. $BJQ99F|;~(B $B$J$I(B CVS$B%5!<%P$r8+$k$?$a(B
                    750: @noindent
                    751: ChangeLog
                    752: @itemize @bullet
                    753: @item
                    754:  $B$3$N4X?t$O(B [GM2016]
                    755: $B$GM?$($i$l$?%"%k%4%j%:%`$K=>$$(B contiguity relation $B$rF3=P$9$k(B.
                    756: @item
                    757:  $BJQ99$r<u$1$?%U%!%$%k$O(B
                    758:  OpenXM/src/asir-contrib/packages/src/gtt_ekn/ekn_pfaffian_8.rr 1.1.
                    759: @end itemize
                    760:
                    761:
                    762:
                    763: @comment --- $B$*$^$8$J$$(B ---
                    764: @node Index,,, Top
                    765: @unnumbered Index
                    766: @printindex fn
                    767: @printindex cp
                    768: @iftex
                    769: @vfill @eject
                    770: @end iftex
                    771: @summarycontents
                    772: @contents
                    773: @bye
                    774: @comment --- $B$*$^$8$J$$=*$j(B ---
                    775:
                    776:
                    777: // 2 x m $BJ,3dI=$K$*$$$F;w$?5!G=$rM-$9$k4X?t$NMxMQNc$r;29M$^$G$K5-:\$9$k(B;
                    778: // $B@55,2=Dj?t$H$=$NHyJ,4XO"(B.
                    779: // $B$=$N(B1.
                    780: [3077] A=tk_fd.marginal2abc([4,5],[2,4,3]);
                    781: [-4,[-4,-3],-1]
                    782: [3078] tk_fd.ahmat_abc(A[0],A[1],A[2],[[1,1/2,1/3],[1,1,1]]);
                    783: RS=[ 4 5 ], CSnew=[ 2 4 3 ], Ynew=[ 1 1/2 1/3 ]
                    784: [ 1 1 1 ]
                    785: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    786: [4483/124416,[[353/7776,1961/15552,185/1728],[553/20736,1261/15552,1001/13824]]]
                    787: // $BLaCM$O(B [Z, [[d_11 Z, d_12 Z, d_13 Z],[d_21 Z, d_22 Z, d_23 Z]]] $B$NCM(B.
                    788:
                    789: // $B$=$N(B2.
                    790: [3079] tk_fd.log_ahmat_abc(A[0],A[1],A[2],[[1,1/2,1/3],[1,1,1]]);
                    791: RS=[ 4 5 ], CSnew=[ 2 4 3 ], Ynew=[ 1 1/2 1/3 ]
                    792: [ 1 1 1 ]
                    793: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    794: [-3.32333832422461674639485797719209322217260539267246045320,
                    795:  [[1.25987062235110417131385233102832924380994869507026544724,3.49944233772027660049074280615659156814633058219942003122,2.97122462636627258532232879768012491635065804149007361142],
                    796:   [0.740129377648895828686147668971670756190051304929734552754,2.25027883113986169975462859692170421592683470890028998438,2.00959179121124247155922373410662502788311398616997546285]]]
                    797: // $BLaCM$O(B [log(Z),
                    798: //          [[d_11 log(Z), d_12 log(Z), d_13 log(Z)],
                    799: //           [d_21 log(Z), d_22 log(Z), d_23 log(Z)]]]
                    800: // $B$N6a;wCM(B.
                    801:
                    802: // $B$=$N(B3.
                    803: [3082] fd_hessian2(A[0],A[1],A[2],[1/2,1/3]);
                    804: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    805: [4483/124416,[ 1961/15552 185/1728 ],
                    806:  [ 79/288 259/864 ]
                    807:  [ 259/864 47/288 ]]
                    808: // $BLaCM$O(B [F=F_D, gradient(F), Hessian(F)]
                    809:
                    810: // $B;29M(B.
                    811: // ygahvec $B$G6R4X?tJ,$ND4@0(B. $BFHN)$7$?4X?t$O$J$$$h$&$@(B.
                    812:
                    813: //-----------------------------------------------------------------------
                    814: // 2 x m $BJ,3dI=$K$*$$$F;w$?5!G=$rM-$9$k4X?t$NMxMQNc$r;29M$^$G$K5-:\$9$k(B;
                    815: // $B4|BTCM4XO"(B.
                    816: [3079] A=tk_fd.marginal2abc([4,5],[2,4,3]);
                    817: [-4,[-4,-3],-1]
                    818: [3080] tk_fd.expectation_abc(A[0],A[1],A[2],[[1,1/2,1/3],[1,1,1]]);
                    819: RS=[ 4 5 ], CSnew=[ 2 4 3 ], Ynew=[ 1 1/2 1/3 ]
                    820: [ 1 1 1 ]
                    821: Computing Dmat(ca) for parameters B=[-4,-3],X=[ 1/2 1/3 ]
                    822: [[5648/4483,7844/4483,4440/4483],
                    823:  [3318/4483,10088/4483,9009/4483]]
                    824: // $B3F%;%k$N4|BTCM(B.
                    825:
                    826: //-----------------------------------------------------------------------
                    827: // ot_hgm_ahg.rr $B$NNc(B.  $B<B83E*$J$?$a(B module $B2=$5$l$F$$$J$$(B.
                    828: [3237] import("ot_hgm_ahg.rr");
                    829: // 2 x 2 $BJ,3dI=(B.
                    830: [3238] hgm_ahg_expected_values_contiguity([[0,0,1,1],[1,0,1,0],[0,1,0,1]],
                    831:         [9,6,8],[1/2,1/3,1/5,1/7],[x1,x2,x3,x4]|geometric=1);
                    832: oohg_native=0, oohg_curl=1
                    833: [1376777025/625400597,1750225960/625400597,2375626557/625400597,3252978816/625400597]
                    834: // 2 x 2 $BJ,3dI=$N4|BTCM(B.
                    835:
                    836: // 2 x 3 $BJ,3dI=(B.
                    837: [3238] hgm_ahg_expected_values_contiguity(
                    838:  [[0,0,0,1,1,1],[1,0,0,1,0,0],[0,1,0,0,1,0],[0,0,1,0,0,1]],
                    839:  [5,2,4,3],[1,1/2,1/3,1,1,1],[x1,x2,x3,x4,x5,x6]|geometric=1);
                    840: [5648/4483,7844/4483,4440/4483,3318/4483,10088/4483,9009/4483]
                    841: // 2 x 3 $BJ,3dI=$N4|BTCM(B. $B>e$HF1$8LdBj(B.
                    842:
                    843: /*
                    844:   dojo, p.221.  $B@.@S(B3$B0J2<$N@8EL$O=8$a$F$R$H$D$K(B.
                    845:   2 1 1
                    846:   8 3 3
                    847:   0 2 6
                    848:
                    849:   row sum: 4,14,8
                    850:   column sum: 10,6,10
                    851:   0 $B$r0l$D4^$`$N$G(B, (3,6) $B7?$N(B A $B$+$i(B 7 $BNsL\$rH4$/(B.
                    852: */
                    853: // 3 x 3 $BJ,3dI=(B. $B9=B$E*(B0$B$,0l$D(B.
                    854:
                    855: A=[[0,0,0,1,1,1, 0,0],
                    856:    [0,0,0,0,0,0, 1,1],
                    857:    [1,0,0,1,0,0, 0,0],
                    858:    [0,1,0,0,1,0, 1,0],
                    859:    [0,0,1,0,0,1, 0,1]];
                    860: B=[14,8,10,6,10];
                    861: hgm_ahg_expected_values_contiguity(A,B,[1,1/2,1/3,1,1/5,1/7,1,1],[x1,x2,x3,x4,x5,x6,x7,x8]|geometric=1);
                    862:
                    863: // $BEz(B.
                    864: [14449864949304/9556267369631,10262588586540/9556267369631,13512615942680/9556267369631,
                    865:  81112808747006/9556267369631,21816297744346/9556267369631,30858636683482/9556267369631,
                    866:                               25258717886900/9556267369631,51191421070148/9556267369631]
                    867:
                    868:
                    869: /*
                    870:  $B>e$N%G!<%?$G(B 0 $B$r(B 1 $B$KJQ99(B.
                    871:   2 1 1
                    872:   8 3 3
                    873:   1 2 6
                    874:
                    875:   row sum: 4,14,9
                    876:   column sum: 11,6,10
                    877: */
                    878: // 3 x 3 $BJ,3dI=(B.
                    879: A=[[0,0,0,1,1,1,0,0,0],
                    880:    [0,0,0,0,0,0,1,1,1],
                    881:    [1,0,0,1,0,0,1,0,0],
                    882:    [0,1,0,0,1,0,0,1,0],
                    883:    [0,0,1,0,0,1,0,0,1]];
                    884: B=[14,9,11,6,10];
                    885: hgm_ahg_expected_values_contiguity(A,B,[1,1/2,1/3,1,1/5,1/7,1,1,1],[x1,x2,x3,x4,x5,x6,x7,x8]|geometric=1);
                    886:
                    887: // $B4|BTCM(B, $BEz(B.
                    888: [207017568232262040/147000422096729819,163140751505489940/147000422096729819,217843368649167296/147000422096729819,
                    889:  1185482401011137878/147000422096729819,358095302885438604/147000422096729819,514428205457640984/147000422096729819,
                    890:  224504673820628091/147000422096729819,360766478189450370/147000422096729819]
                    891:
                    892: // Z $B$d$=$NHyJ,$N7W;;$O(B hgm_ahg_contiguity $B4X?t$,$*$3$J$&$,(B, $B$3$l$N4J0W%$%s%?!<%U%'!<%9$O(B
                    893: // $B$^$@=q$$$F$J$$(B.
                    894:
                    895:
                    896: 4. x_ij $B$O(B [GM2016] $B$N#1>O$G(B,
                    897:  $B$?$H$($P(B 3x3 $B$N;~(B [[1,1,1],[x_11,x_12,1],[x_21,x_22,1]]
                    898: $B$H$J$C$F$$$k$,(B, [GM2016] $B$N(B Prop 7.1 $B$NBP1~$G$O(B,
                    899:    p = [[1,x_11,x_12],[1,x_21,x_22],[1,1,1]] $B$H$J$C$F$$$k$N$GCm0U(B.