HGM functions for two way contingency tables.

HGM functions for two way contingency tables on Risa/Asir

Version 3.0

June 12, 2019

by Y.Goto, Y.Tachibana, N.Takayama

Copyright © Risa/Asir committers 2004–2019. All rights reserved.


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1 About this document

This document explains Risa/Asir functions for two way contingency tables by HGM(holonomic gradient method). Loading the package:

import("gtt_ekn3.rr");

The package gtt_ekn3.rr is a major version up of gtt_ekn.rr. In order to download the latest asir-contrib package, please use the asir_contrib_update() as follows.

import("names.rr");
asir_contrib_update(|update=1);

References cited in this document.

The changelogs are described only in the Japanese version of this document.


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2 Functions of HGM for two way contingency tables


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2.1 Hypergeometric function E(k,n)


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2.1.1 gtt_ekn3.gmvector

gtt_ekn3.gmvector(beta,p)

:: It returns the value of the hypergeometric function E(k,n) and its derivatives associated to the two way contingency table with the marginal sum beta, parameter p (cell probability).

It is an alias of gtt_ekn3.ekn_cBasis_2(beta,p)

return

vector, see below.

beta

a list of the row sum and the colum sum.

p

the parameter.

Example: A 2 x 2 contingency table. The row sum is [5,1] and column sum is [3,3]. The parameter (cell probability) is [[1/2,1/3],[1/7,1/5]].

[3000] import("gtt_ekn3.rr");
[3001] gtt_ekn3.gmvector([[5,1],[3,3]],[[1/2,1/3],[1/7,1/5]])
[775/27783]
[200/9261]

Example: Interval option.

[4009] P=gtt_ekn3.prob1(5,5,100);
[[[100,200,300,400,500],[100,200,300,400,500]],
 [[1,1/2,1/3,1/5,1/7],[1,1/11,1/13,1/17,1/19],
  [1,1/23,1/29,1/31,1/37],[1,1/41,1/43,1/47,1/53],[1,1,1,1,1]]]

[4010] util_timing(quote(gtt_ekn3.gmvector(P[0],P[1])[1];
[cpu,72.852,gc,0,memory,4462742364,real,72.856]

[4011] util_timing(quote(gtt_ekn3.gmvector(P[0],P[1]|interval=100)))[1];
[cpu,67.484,gc,0,memory,3535280544,real,67.4844]
Refer to

gtt_ekn3.setup @ref{gtt_ekn3.pfaffian_basis}


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2.1.2 gtt_ekn3.nc

gtt_ekn3.nc(beta,p)

:: It returns the normalizing constant Z and its derivatives for the two way contingency tables with the marginal sum beta and the parameter (cell probability) p. See, e.g., [TKT2015], [TGKT] as to the definition of $Z$.

return

A list [Z,[[d_11 Z, d_12 Z, ...], ..., [d_m1 Z, d_m2 Z, ...., d_mn Z]]] where d_ij Z denotes the partial derivative of Z with respect to the parameter p_ij.

beta

The row sum and the column sum.

p

The parameter (cell probability).

Example: A 2x3 contingency table.

[2237] gtt_ekn3.nc([[4,5],[2,4,3]],[[1,1/2,1/3],[1,1,1]]);
[4483/124416,[ 353/7776 1961/15552 185/1728 ]
[ 553/20736 1261/15552 1001/13824 ]]
Refer to

gtt_ekn3.setup gtt_ekn3.lognc


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2.1.3 gtt_ekn3.lognc

gtt_ekn3.lognc(beta,p)

:: It returns the logarithm of Z.

return

A list [log(Z), [[d_11 log(Z), d_12 log(Z), ...], [d_21 log(Z),...], ... ]

Example: A 2x3 contingency table. The first element is an approximate value of log(Z). The rests are exact values when the arguments of lognc are rational numbers.

[2238] gtt_ekn3.lognc([[4,5],[2,4,3]],[[1,1/2,1/3],[1,1,1]]);
[-3.32333832422461674630,[ 5648/4483 15688/4483 13320/4483 ]
[ 3318/4483 10088/4483 9009/4483 ]]
Refer to

gtt_ekn3.setup gtt_ekn3.nc


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2.1.4 gtt_ekn3.expectation

gtt_ekn3.expectation(beta,p)

:: It returns the expectation of the hypergeometric distribution with the mariginal sum beta and the parameter p.

return

The expectation of each cell.

Examples of the evaluation of expectations for 2 x 2 and 3 x 3 contingency tables.

[2235] gtt_ekn3.expectation([[1,4],[2,3]],[[1,1/3],[1,1]]);
[ 2/3 1/3 ]
[ 4/3 8/3 ]
[2236] gtt_ekn3.expectation([[4,5],[2,4,3]],[[1,1/2,1/3],[1,1,1]]);
[ 5648/4483 7844/4483 4440/4483 ]
[ 3318/4483 10088/4483 9009/4483 ]

[2442] gtt_ekn3.expectation([[4,14,9],[11,6,10]],[[1,1/2,1/3],[1,1/5,1/7],[1,1,1]]);
[ 207017568232262040/147000422096729819 163140751505489940/147000422096729819 
                                        217843368649167296/147000422096729819 ]
[ 1185482401011137878/147000422096729819 358095302885438604/147000422096729819 
                                         514428205457640984/147000422096729819 ]
[ 224504673820628091/147000422096729819 360766478189450370/147000422096729819 
                                        737732646860489910/147000422096729819 ]
Refer to

gtt_ekn3.setup gtt_ekn3.nc


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2.1.5 gtt_ekn3.setup

gtt_ekn3.setup()

:: It sets parameters for a distributed computation or report the current values of the parameters.

return

0

Example: Generating a list of primes and outputing them to the file p.txt.

gtt_ekn3.setup(|nps=2,nprm=20,minp=10^10,fgp="p.txt")$

Example: Evaluating the gmvector by the Chinese remainder theorem (crt).

[2867] gtt_ekn3.setup(|nprm=20,minp=10^20);
[2868] N=2; T2=gtt_ekn3.gmvector([[36*N,13*N-1],[38*N-1,11*N]],
                                [[1,(1-1/N)/56],[1,1]] | crt=1)$
Refer to

gtt_ekn3.nc gtt_ekn3.gmvector


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2.1.6 gtt_ekn3.upAlpha, gtt_ekn3.downAlpha

gtt_ekn3.upAlpha(i,k,n)
gtt_ekn3.downAlpha(i,k,n)

::

i

It indicates the direction of the contiguity relation to get. In other words, the contiguity relation from a_i to a_i+1 (from a_i to a_i-1, the downAlpha case) is obtained.

k, n

The contiguity relation for the hypergeometric function E(k+1,n+k+2) standing for the (k+1)×(n+1) contingency table is obtained.

return

The matrix representation of the contiguity relation with respect to the pfaffian_basis (see gtt_ekn3.pfaffian_basis). See also Cor 6.3 of [GM2016].

Example: 2x2 contingency table (E(2,4)), 2x3 contingency table (E(2,5)). Outputs of [2221] — [2225] are left out.

[2221] gtt_ekn3.marginaltoAlpha([[1,4],[2,3]]);
[[a_0,-4],[a_1,-1],[a_2,3],[a_3,2]]
[2222] gtt_ekn3.upAlpha(1,1,1);  // contiguity relation of E(2,4) 
                                //    for the a_1 direction
[2223] gtt_ekn3.upAlpha(2,1,1);  // E(2,4),  a_2 direction
[2224] gtt_ekn3.upAlpha(3,1,1);  // E(2,4),  a_3 direction
[2225] function f(x_1_1);
[2232] gtt_ekn3.pfaffian_basis(f(x_1_1),1,1);
[ f(x_1_1) ]
[ (f{1(x_1_1)*x_1_1)/(a_2) ]
[2233] function f(x_1_1,x_1_2);
f() redefined.
[2234] gtt_ekn3.pfaffian_basis(f(x_1_1,x_1_2),1,2); // E(2,5), 2*3 contingency table
[ f(x_1_1,x_1_2) ]
[ (f{1,0(x_1_1,x_1_2)*x_1_1)/(a_2) ]
[ (f{0,1(x_1_1,x_1_2)*x_1_2)/(a_3) ]

[2235]   RuleA=[[a_2,1/3],[a_3,1/2]]$ RuleX=[[x_1_1,1/5]]$
  base_replace(gtt_ekn3.upAlpha(1,1,1),append(RuleA,RuleX))
 -gtt_ekn3.upAlpha(1,1,1 | arule=RuleA, xrule=RuleX);

[ 0 0 ]
[ 0 0 ]

Refer to

gtt_ekn3.nc gtt_ekn3.gmvector


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2.1.7 gtt_ekn3.cmle

gtt_ekn3.cmle(u)

:: It finds a parameter p (cell probability) which maximizes P(U=u | row sum, column sum = these of U) for given observed data u. The value of p is an approximate value.

u

The observed data.

return

An estimated parameter p

Example: 2x4 contingency table.

U=[[1,1,2,3],[1,3,1,1]];
gtt_ekn3.cmle(U);
 [[ 1 1 2 3 ]
  [ 1 3 1 1 ],[[7,6],[2,4,3,4]],   // Data, row sum, column sum
 [ 1 67147/183792 120403/64148 48801/17869 ]  // p obtained.
 [ 1 1 1 1 ]]
Refer to

gtt_ekn3.expectation


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2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1

gtt_ekn3.set_debug_level(m)

:: It sets the level of debug messages.

gtt_ekn3.contiguity_mat_list_2

:: It returns a list of contiguity directions to be used.

gtt_ekn3.show_path()

:: It returns the path to apply contiguity relations. See [TGKT].

gtt_ekn3.get_svalue()

:: It returns the values of the static variables.

gtt_ekn3.assert1(N)

:: It checks the system by 2x2 contingency tables. N is proportional to the marginal sum.

gtt_ekn3.assert2(N)

:: It checks the system by 3x3 contingency tables.

gtt_ekn3.assert3(R1, R2, Size)

:: It checks the distributed computation system by R1 x R2 contingency tables.

gtt_ekn3.prob1(R1,R2,Size)

:: It returns a test data for R1 x R2 contingency tables in the format [marginal sum, parameter p]. The marginal sum is proportional to Size. See benchmark tests in [TGKT].

Example:

[2846] gtt_ekn3.set_debug_level(0x4);
[2847] N=2; T2=gtt_ekn3.gmvector([[36*N,13*N-1],[38*N-1,11*N]],
                                [[1,(1-1/N)/56],[1,1]])$
[2848] 
level&0x4: g_mat_fac_test([ 113/112 ]
[ 1/112 ],[ (t+225/112)/(t^2+4*t+4) (111/112*t+111/112)/(t^2+4*t+4) ]
[ (1/112)/(t^2+4*t+4) (111/112*t+111/112)/(t^2+4*t+4) ],0,20,1,t)
Note: we do not use g_mat_fac_itor. Call gtt_ekn3.setup(); to use the crt option.
level&0x4: g_mat_fac_test([ 67/62944040755546030080000 ]
[ 1/125888081511092060160000 ],[ (t+24)/(t^2+25*t+46) (2442)/(t^2+25*t+46) ]
[ (1)/(t^2+25*t+46) (-111*t-111)/(t^2+25*t+46) ],0,73,1,t)
level&0x4: g_mat_fac_test ------  snip

Example:

[2659] gtt_ekn3.nc([[4,5,6],[2,4,9]],[[1,1/2,1/3],[1,1/5,1/7],[1,1,1]])$
[2660] L=matrix_transpose(gtt_ekn3.show_path())$
[2661] L[2];
[2 1]

This means that the contiguity relations for the directions [2 1] (a_2, a_1) are used to evaluate the normalizing constant Z. L[0] is the contiguity matrix, L[1] is a list of the steps to apply for corresponding relations.

Example: Finding a path without evaluations of gmvectors.

A=gtt_ekn3.marginaltoAlpha_list([[400,410,1011],[910,411,500]])$
[2666] gtt_ekn3.contiguity_mat_list_2(A,2,2)$
[2667] L=matrix_transpose(gtt_ekn3.show_path())$
[2668] L[2];
[ 2 1 5 4 3 ]
[2669] gtt_ekn3.contiguity_mat_list_3(A,2,2)$ // new alg in [TGKT]
[2670] L=matrix_transpose(gtt_ekn3.show_path())$
[2671] L[2];
[2 1]  // shorter

Example: When assert2() returns 0 matrices, then the results of g_mat_fac_plain and g_mat_fac_int agree. In other words, the system is OK.

[8859] gtt_ekn3.assert2(1);
Marginal=[[130,170,353],[90,119,444]]
P=[[17/100,1,10],[7/50,1,33/10],[1,1,1]]
Try g_mat_fac_test_int: Note: we do not use g_mat_fac_itor. Call gtt_ekn3.setup(); to use the crt option.
Timing (int) =0.413916 (CPU) + 0.590723 (GC) = 1.00464 (total), real time=0.990672

Try g_mat_fac_test_plain: Note: we do not use g_mat_fac_itor. Call gtt_ekn3.setup(); to use the crt option.
Timing (rational) =4.51349 (CPU) + 6.32174 (GC) = 10.8352 (total)
diff of both method = 
[ 0 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]
[8860] 

[8863] gtt_ekn3.setup(|nprm=100,minp=10^50);
Number of processes = 1.
Number of primes = 100.
Min of plist = 100000000000000000000000000000000000000000000000151.
0
[8864] gtt_ekn3.assert2(1 | crt=1);
Marginal=[[130,170,353],[90,119,444]]
P=[[17/100,1,10],[7/50,1,33/10],[1,1,1]]
Try [[crt,1]]
----  snip

Example: 3x5 contingency table. The parameter p (cell probability) is a list of 1/(prime number) ’s.

[9054] L=gtt_ekn3.prob1(3,5,10 | factor=1, factor_row=3); 
[[[10,20,420],[30,60,90,120,150]],[[1,1/2,1/3,1/5,1/7],[1,1/11,1/13,1/17,1/19],[1,1,1,1,1]]]
[9055] number_eval(gtt_ekn3.expectation(L[0],L[1]));
[ 1.65224223218613 ... snip ]

Example:

[5779] import("gtt_ekn3.rr"); load("gtt_ekn3/ekn_eval-timing.rr");
[5780] gtt_ekn3.assert3(5,5,100 | nps=32, interval=100);
 -- snip
Parallel method: Number of process=32, File name tmp-gtt_ekn3/p300.txt is written.
Number of processes = 32.
  -- snip
initialPoly of path=3: [ 2.184 0 124341044 2.1831 ] [CPU(s),0,*,real(s)]
contiguity_mat_list_3 of path=3: [ 0.04 0 630644 9.6774 ] [CPU(s),0,*,real(s)]
Note: interval option will lead faster evaluation. We do not use g_mat_fac_itor (crt). Call gtt_ekn3.setup(); to use the crt option.
g_mat_fac of path=3: [ 21.644 0 1863290168 21.6457 ] [CPU(s),0,*,real(s)]
Done. Saved in 2.ab
Diff (should be 0)=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,..., 0,0,0]
Refer to

gtt_ekn3.nc


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3 Modular method


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3.1 Chinese remainder theorem and itor


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3.1.1 gtt_ekn3.chinese_itor

gtt_ekn3.chinese_itor(data,idlist)

:: It performs a rational reconstruction by the Chinese remainder theorem (itor = integer to rational).

return

[val, n], the vector val is the value by the rational reconstruction. n = n1*n2*...

data

[[val1,n1],[val2,n2], ...], val1, val2 are values evaluated in mod n1, mod n2, ... respectively. The relations val mod n1 = val1, val mod n2 = val2,.. are satisfied.

idlist

The list of server id’s for itor.

Example: [3!, 5^3*3!]=[6,750] is the return value. The relations 6 mod 109 =6, 750 mod 109=96 stand for [[6,96],109], ...

gtt_ekn3.setup(|nps=2,nprm=3,minp=101,fgp="p_small.txt");
SS=gtt_ekn3.get_svalue();
SS[0];
  [103,107,109]   // list of primes
SS[1];
  [0,2]           // list of server ID's
gtt_ekn3.chinese_itor([[[ 6,96 ],109],[[ 6,29 ],103],[[ 6,1 ],107]],SS[1]);
  [[ 6 750 ],1201289]

// The argument may be a scalar.
gtt_ekn3.chinese_itor([[96,109],[29,103]],SS[1]);
  [[ 750 ],11227]
Refer to

gtt_ekn3.setup


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4 Binary splitting


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4.1 Matrix factorial


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4.1.1 gtt_ekn3.init_bsplit, gtt_ekn3.init_dm_bsplit, gtt_ekn3.setup_dm_bsplit

gtt_ekn3.init_bsplit(|minsize=16,levelmax=1);

:: It sets parameters for the binary splitting to evaluate the matrix factorial M(1) M(2) ... M(n) where M(k) is a matrix with a parameter k.

gtt_ekn3.init_dm_bsplit(|bsplit_x=0, bsplit_reduce=0)

:: It sets parameters for the binary splitting by a distributed computation.

gtt_ekn3.setup_dm_bsplit(C)

:: It starts C processes for the binary splitting.

Option minsize.

When the size of the matrix factorial is less than the minsize, the binary splitting is not used and sequential multiplication is used instead.

Option levelmax.

The maximum of recursions of the recursive binary splitting in the distributed computation. See gtt_ekn3/dm_bsplit.rr C should be set to levelmax-1. When levalmax=1, the distributed computation is not performed.

Option bsplit_x.

When bsplit_x=1, a window attached to every process is opened.

Example: A comparison of bs=1 and no bs.

[4618] cputime(1)$
[4619] gtt_ekn3.expectation(Marginal=[[1950,2550,5295],[1350,1785,6660]],
                          P=[[17/100,1,10],[7/50,1,33/10],[1,1,1]]|bs=1)$
4.912sec(4.914sec)
[4621] V2=gtt_ekn3.expectation(Marginal,P)$
6.752sec(6.756sec)

Example: Note that distributed computations are often slower than computations on a single process in our implementation of the binary splitting. The option bsplit_x=1 opens a debug windows, it makes things slower. The function gtt_ekn3.test_bs_dist() is a test function of the binary splitting by a distributed computation.

[3669] C=4$ gtt_ekn3.init_bsplit(|minsize=16,levelmax=C+1)$ 
gtt_ekn3.init_dm_bsplit(|bsplit_x=1)$
[3670] [3671] [3672] gtt_ekn3.setup_dm_bsplit(C);
[0,0]
[3673] gtt_ekn3.assert2(10|bs=1)$
Refer to

gtt_ekn3.gmvector gtt_ekn3.expectation gtt_ekn3.assert1 gtt_ekn3.assert2


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Index

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Index Entry  Section

G
gtt_ekn3.assert1 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.assert2 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.assert3 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.chinese_itor Chinese remainder theorem and itor 3.1.1 gtt_ekn3.chinese_itor
gtt_ekn3.cmle 2.1.7 gtt_ekn3.cmle
gtt_ekn3.contiguity_mat_list_2 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.downAlpha 2.1.6 gtt_ekn3.upAlpha, gtt_ekn3.downAlpha
gtt_ekn3.expectation 2.1.4 gtt_ekn3.expectation
gtt_ekn3.get_svalue 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.gmvector 2.1.1 gtt_ekn3.gmvector
gtt_ekn3.init_bsplit matrix factorial 4.1.1 gtt_ekn3.init_bsplit, gtt_ekn3.init_dm_bsplit, gtt_ekn3.setup_dm_bsplit
gtt_ekn3.init_dm_bsplit matrix factorial 4.1.1 gtt_ekn3.init_bsplit, gtt_ekn3.init_dm_bsplit, gtt_ekn3.setup_dm_bsplit
gtt_ekn3.lognc 2.1.3 gtt_ekn3.lognc
gtt_ekn3.nc 2.1.2 gtt_ekn3.nc
gtt_ekn3.prob1 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.setup 2.1.5 gtt_ekn3.setup
gtt_ekn3.setup_dm_bsplit matrix factorial 4.1.1 gtt_ekn3.init_bsplit, gtt_ekn3.init_dm_bsplit, gtt_ekn3.setup_dm_bsplit
gtt_ekn3.set_debug_level 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.show_path 2.1.8 gtt_ekn3.set_debug_level, gtt_ekn3.show_path, gtt_ekn3.get_svalue, gtt_ekn3.assert1, gtt_ekn3.assert2, gtt_ekn3.assert3, gtt_ekn3.prob1
gtt_ekn3.upAlpha 2.1.6 gtt_ekn3.upAlpha, gtt_ekn3.downAlpha

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Table of Contents


[Top] [Contents] [Index] [ ? ]

Short Table of Contents


[Top] [Contents] [Index] [ ? ]

About This Document

This document was generated on September 18, 2019 using texi2html 5.0.

The buttons in the navigation panels have the following meaning:

Button Name Go to From 1.2.3 go to
[ << ] FastBack Beginning of this chapter or previous chapter 1
[ < ] Back Previous section in reading order 1.2.2
[ Up ] Up Up section 1.2
[ > ] Forward Next section in reading order 1.2.4
[ >> ] FastForward Next chapter 2
[Top] Top Cover (top) of document  
[Contents] Contents Table of contents  
[Index] Index Index  
[ ? ] About About (help)  

where the Example assumes that the current position is at Subsubsection One-Two-Three of a document of the following structure:


This document was generated on September 18, 2019 using texi2html 5.0.