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Table of Contents
*****************

Numdiff User Manual
1 Copying
2 Acknowledgments
3 Overview
  3.1 Output format
  3.2 Overview mode
  3.3 Output of the filter
4 Installing
5 Invoking numdiff
6 Selecting lines and fields for the comparison
7 Invoking ndselect
8 Using the filter of numdiff
9 Warnings
Appendix A GNU Free Documentation License
Index


Numdiff User Manual
*******************

_"...und der eignen Kraft vertrauend steigt ein frei Geschlecht empor!"_


This manual describes how to install and use Numdiff, a program which
compares putatively similar files line by line and field by field,
ignoring small numeric differences or/and different numeric formats.

   Copyright (C) 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013
Ivano Primi  <ivprimi(at)libero(dot)it>

     Permission is granted to copy, distribute and/or modify this
     document under the terms of the GNU Free Documentation License,
     Version 1.3 or any later version published by the Free Software
     Foundation; with no Invariant Sections, no Front-Cover Texts, and
     no Back-Cover Texts.  A copy of the license is included in *note
     GNU Free Documentation License::.

1 Copying
*********

Numdiff (also written numdiff) is free software: you can redistribute
it and/or modify it under the terms of the GNU General Public License
as published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.

   Numdiff is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
General Public License for more details.

   You should have received a copy of the GNU General Public License
along with this program.  If not, see `http://www.gnu.org/licenses/'.

2 Acknowledgments
*****************

I want to thank Mr. Norman Clerman <norm(dot)opcon(at)fuse(dot)net> for
several suggestions he gave me to improve the readability and the
effectiveness of the output produced by Numdiff.  He also pointed out
the need to implement a filter to resynchronize the lines between two
files in case of addition or deletion of one or more lines.  I have to
give him credit for the urge to prepare the versions 4.x and 5.x of
Numdiff.

   Moreover, I want to thank my friends Mariapia Palombaro, since she
removed some errors while reviewing the first version of this document,
and Paolo Caramanica, who suggested me to add more information to the
output of the option `-S'.

3 Overview
**********

Computer users often find occasion to ask how two files differ.
Perhaps one file is a newer version of the other file.  Or maybe the
two files started out as identical copies but were changed by different
people.

   There are several ways to think about the differences between two
files.  One way to think of the differences is as a series of lines
that were deleted from, inserted in, or changed in one file to produce
the other file.  The well-known `diff' program compares two files line
by line, finds groups of lines that differ, and reports each group of
differing lines.  Without particular options, the `diff' program
considers any change in the amount or in the type of the characters as a
relevant difference. However, trough some command line options it also
provides ways to suppress certain kinds of differences that are not
important to the user.  For instance, `diff' provides ways to ignore
differences in the amount of white space between words or lines, or
differences in alphabetic case.

   Another way to think of the differences is as a series of words that
were deleted from, inserted in, or changed in one file to produce the
other file.  Here "word" refers to a sequence of non white-space
characters delimited by a couple of white-spaces, one before and the
other one after the word.

   The less known `wdiff' program by Franc,ois Pinard
<pinard(at)iro(dot)umontreal(dot)ca> compares words in two files and
reports the differences.

   At last, one can think of the differences between two files as a
sequence of pairs of bytes that can be either identical or different.
The `cmp' program reports the differences between two files byte by
byte, instead of line by line or word by word.  As a result, it is
often more useful than `diff' or `wdiff' for comparing binary files.

   However, none of these approaches turns out to be good when you want
to compare a couple of text files composed partially or entirely by
numerical fields.  Indeed, when you compare a couple of such files,
what you want to obtain usually is a list of the numerical fields in
the second file which *numerically* differ from the corresponding
fields in the first file.  But, as you probably knows, a number can be
written using different notations and programs like `diff' or `wdiff'
can not recognize whether a difference between two numeric fields is
only due to the notation or is actually a difference of numerical
values.

   For instance, 11.23   and   11.2300000 are the same number but
represented in different ways.  While, if you are interested in the
numerical values, it is obvious that the difference in the
representation is not meaningful and then it should be ignored, however
`diff' and `wdiff' consider the previous one as a relevant difference
and there is no way for you to tell these programs to ignore it!

   Another example of this type is given by 98765.4321  and
9.87654321E04 where the difference is only due to the use of the
scientific notation in place of the ordinary decimal notation.

   Moreover, depending on your country you could stick to different
conventions in writing numbers.  For instance, the amount "three
hundred millions and fifty-two thousands of dollars and forty-six cents"
is usually written by an Italian accountant as 300.052.000,46$ while an
American accountant would write 300,052,000.46$.  Of course,
300.052.000,46$  and  300,052,000.46$ represent the same amount of
money but `diff' and `wdiff' would report a difference, which probably
is not what you want in a similar case.

   At last, sometimes you could want to ignore even differences in
numerical values as long as they do not overcome a certain threshold.
In other words, you could desire to suppress all "small" numerical
differences too.

   For instance, it could happen that you want to ignore all numerical
differences whose absolute value is not greater than 0.0001. If this is
the case, then the numerical fields 33  and  33.00009 must be
considered equal, while 33  and  33.00011 must be reported as different.

   However, `diff' and `wdiff' can not be used to ignore "small"
numerical differences, since they do not even know what a numerical
difference is.

   What I have been saying till now explains why I decided to implement
a new program with the capability to "appropriately" compare files
containing numerical fields.  In writing this program I was inspired by
`ndiff', a GPL'ed software by Nelson H. Baabe of the Salt Lake City
University. The author of `ndiff' had the same good reasons as me to
write `ndiff'. `ndiff' is actually a good tool and I used it for a
while. But I did not completely like the way it works and so `numdiff'
was born.  Although `ndiff' inspired `numdiff', they are completely
different from the viewpoint of the source code: `numdiff' has been
entirely written from scratch with the addition of code coming from GNU
bc, GNU diff and GNUlib.  In addition, the last versions of Numdiff
offer much more features than `ndiff' does.

   `numdiff' can be used to compare putatively similar files line by
line and field by field, ignoring small numeric differences or/and
different numeric formats.  `numdiff' takes two mandatory arguments,
the paths of the two files to compare, and, after splitting them into
lines and the lines into fields according to a given list of field
delimiters, it compares every field of every line of the first file with
the corresponding field of the second file.  What _corresponding_ here
exactly means depends on the options passed to the program on the
command line. With no options, corresponding means the field of the
second file at the same position, where position refers both to the
line number and to the location within the line.  If the compared
fields are both legal numerical values, then `numdiff' performs a
numerical comparison between them, else it performs a literal
comparison, i.e. the usual byte-by-byte comparison.  In case of literal
comparison, two fields are regarded as equal if they are formed by the
same sequence of characters.  In case of numerical comparison and
without specific command line options, two fields are regarded as equal
if their numerical difference is zero.  Mind that, if you do not
explicitly specify a list of field delimiters by means of the option
`-s' or `-D', `numdiff' takes as field delimiters the characters
newline (`\n', ASCII code 0x0A), horizontal tabulation (`\t', ASCII
code 0x09), and blank (` ', ASCII code 0x20).

   For example, if the file `list1' contains the data
accident        123     23Joshua          34.55           +3+4i	        water
dog       -3455.321     cat             2.345678e-9     .0005-6.23e2i

and file `list2' contains the data

Accident        123     23456          34.5500         +3.0001+4i
dog       -3455.320098  Cat     +2.345678e-9            -6.23e2i    $$$
A new line

then the output of the command `numdiff list1 list2' will be:

     ----------------
     ##1       #:1   <== accident
     ##1       #:1   ==> Accident
     @                                                     @@
     ##1       #:3   <== 23Joshua
     ##1       #:3   ==> 23456
     @                                                     @@
     ##1       #:5   <== +3+4i
     ##1       #:5   ==> +3.0001+4i
     @ Absolute error = 1.0000000000e-4, Relative error = 2.0000000000e-5
     ##1       #>6   <== water
     ##1             ==>
     @ Line 1 in file "list2" is shorter than expected!
     ----------------
     ##2       #:2   <== -3455.321
     ##2       #:2   ==> -3455.320098
     @ Absolute error = 9.0200000000e-4, Relative error = 2.6104672633e-7
     ##2       #:3   <== cat
     ##2       #:3   ==> Cat
     @                                                     @@
     ##2       #:5   <== .0005-6.23e2i
     ##2       #:5   ==> -6.23e2i
     @ Absolute error = 5.0000000000e-4, Relative error = 8.0256821830e-7
     ##2             <==
     ##2       #>6   ==> $$$
     @ Line 2 in file "list1" is shorter than expected!
     ----------------
     ##3             <==
     ##3       #>1   ==> A new line
     @ Line 3 in file "list1" is shorter than expected!
     ----------------
               <==
     ##4       ==>


     +++  File "list1" differs from file "list2"

   At the same time `numdiff' will print the following error message on
stderr:
     ***  End of file "list1" reached while trying to read line 4.
          File "list2" has more lines than file "list1",
          line 4 is the last one read from file "list2"

   It is worth remarking that `numdiff' can recognize complex numbers,
provided that they are written in the form a+bi or a-bi with no extra
characters between the values a, b and the sign + or - (the symbol i,
used to represent the imaginary unit, can be changed by a suitable
command line option, *note Invoking numdiff::).  If you do not know
what complex numbers are, do not worry!  In this case probably you will
never manage files containing complex numbers and so you can happily
continue to ignore them. :)

   We consider now an example which shows how Numdiff can resynchronize
the lines between two files in case of addition or deletion of one or
more lines.  The versions of Numdiff prior to 5 did not work well if
one of the two files to compare contains in the middle some lines more
or less than the other one.  For instance, if you have one file that is
1000 lines long that you are comparing to a second file 1001 lines
long, and except for that one extra line, located, let us say, at line
500, the files are identical, then `numdiff' version 4.x does *not*
show only the one line difference: once the files are out of
synchronization `numdiff' 4.x reports every line as different.  Since
version 5 it is possible in such cases to activate a filter which
handles additions and deletions of lines.  There are several options
ruling how the filter actually works and I will give later a detailed
explanation on how to use them to obtain each time the wished result.
The simplest way to activate the filter consists in using the option
`-z @'.  If `bill1' and `bill2' are given by
 Month          Expenses
-------------------------
 Jan09          $  233.56
 Feb09          $  850.77
 Mar09          $   12.55
 Apr09          $  524.00
 May09          $   78.25
 Jun09          $  230.00
 Jul09          $  443.10
 Aug09          $   67.65
 Sep09          $   10.00
 Oct09          $  201.45
 Nov09          $  110.00
 Dec09          $  200.27
-------------------------
Total           $ 2961.60

and

 Month          Expenses
 Jan09          $  234.00
 Mar09          $   13.00
 May09          $   78.25
 Jul09          $  443.10
 Sep09          $   10.00
 Nov09          $  110.00
 Jan10          $  200.00
-------------------------

Total           $ 1088.35

respectively, then the differences between the two files are:
   * the insertion of the separator ------------------------- in
     `bill1' before the list of the months,

   * the deletion in `bill2' of the lines related to the expenses for
     the months February, April, June, August, October, December,

   * small changes in `bill2' to the expenses of the months January
     2009 and March,

   * the presence in `bill2' of an entry for January 2010 just before
     the separator -------------------------,

   * the addition of an empty line to `bill2' after the separator
     -------------------------,

   * and the different values for the total sum of the expenses.

   The output of the command `numdiff -z @ -V bill1 bill2' (I have
added here the option `-V' to let Numdiff show  which couples of lines
it is comparing each time) is exactly then what you expect:

     ----------------
     ##2       <== -------------------------
               ==>

     ----------------
     ##3       <==  Jan09          $  233.56
     ##2       ==>  Jan09          $  234.00

     ##3       #:3   <== 233.56
     ##2       #:3   ==> 234.00
     @ Absolute error = 4.4000000000e-1, Relative error = 1.8838842268e-3
     ----------------
     ##4       <==  Feb09          $  850.77
               ==>

     ----------------
     ##5       <==  Mar09          $   12.55
     ##3       ==>  Mar09          $   13.00

     ##5       #:3   <== 12.55
     ##3       #:3   ==> 13.00
     @ Absolute error = 4.5000000000e-1, Relative error = 3.5856573705e-2
     ----------------
     ##6       <==  Apr09          $  524.00
               ==>

     ----------------
     ##8       <==  Jun09          $  230.00
               ==>

     ----------------
     ##10      <==  Aug09          $   67.65
               ==>

     ----------------
     ##12      <==  Oct09          $  201.45
               ==>

     ----------------
     ##14      <==  Dec09          $  200.27
     ##8       ==>  Jan10          $  200.00

     ##14      #:1   <== Dec09
     ##8       #:1   ==> Jan10
     @                                                     @@
     ##14      #:3   <== 200.27
     ##8       #:3   ==> 200.00
     @ Absolute error = 2.7000000000e-1, Relative error = 1.3500000000e-3
     ----------------
               <==
     ##10      ==>

     ----------------
     ##16      <== Total           $ 2961.60
     ##11      ==> Total           $ 1088.35

     ##16      #:3   <== 2961.60
     ##11      #:3   ==> 1088.35
     @ Absolute error = 1.8732500000e+3, Relative error = 1.7211834428e+0

     +++  File "bill1" differs from file "bill2"
   Numdiff has reported correctly the following differences:
   * the second line of file `bill1', i.e.  the one containing the
     separator, has no correspondance, or, if you prefer, has been
     deleted from file `bill2'.

   * The lines related to the months January and March 2009 have been
     slightly modified in `bill2', namely the values of the expenses
     are slightly different. Notice that the line with the expenses for
     January 2009 is the third one in file `bill1' and the second one
     in file `bill2'. This information is printed by Numdiff in the form
          ##3       <==  Jan09          $  233.56
          ##2       ==>  Jan09          $  234.00
     Analogously
          ##5       <==  Mar09          $   12.55
          ##3       ==>  Mar09          $   13.00
     says that the line for March is the fifth one in `bill1' and the
     third one in `bill2'.

   * The line related to the total amount of the expenses appears also
     differently in the two files, since the amount of the expenses is
     different.  Notice that this line is the 16th one in file `bill1'
     and the 11th one in file `bill2'.

   * The lines related to the months February, April, June, August and
     October, i.e. the lines no. 4, 6, 8, 10 and 12 of `bill1', are not
     present in `bill2'.

   * The line of `bill1' with the expenses for December 2009 is
     replaced in `bill2' by the line containing the value of the
     expenses for January 2010.

   * The tenth line of `bill2', i.e. the empty line after the
     separator, is not present in `bill1'.  With respect to `bill1'
     this line represents then an addition.

   If you compare `bill1' and `bill2' without using the option `-z @',
the result is completely misleading. This is the output of `numdiff -V
bill1 bill2':

     ----------------
     ##2       <== -------------------------
     ##2       ==>  Jan09          $  234.00

     ##2       #:1   <== -------------------------
     ##2       #:1   ==> Jan09
     @                                                     @@
     ##2             <==
     ##2       #>2   ==> $  234.00
     @ Line 2 in file "bill1" is shorter than expected!
     ----------------
     ##3       <==  Jan09          $  233.56
     ##3       ==>  Mar09          $   13.00

     ##3       #:1   <== Jan09
     ##3       #:1   ==> Mar09
     @                                                     @@
     ##3       #:3   <== 233.56
     ##3       #:3   ==> 13.00
     @ Absolute error = 2.2056000000e+2, Relative error = 1.6966153846e+1
     ----------------
     ##4       <==  Feb09          $  850.77
     ##4       ==>  May09          $   78.25

     ##4       #:1   <== Feb09
     ##4       #:1   ==> May09
     @                                                     @@
     ##4       #:3   <== 850.77
     ##4       #:3   ==> 78.25
     @ Absolute error = 7.7252000000e+2, Relative error = 9.8724600639e+0
     ----------------
     ##5       <==  Mar09          $   12.55
     ##5       ==>  Jul09          $  443.10

     ##5       #:1   <== Mar09
     ##5       #:1   ==> Jul09
     @                                                     @@
     ##5       #:3   <== 12.55
     ##5       #:3   ==> 443.10
     @ Absolute error = 4.3055000000e+2, Relative error = 3.4306772908e+1
     ----------------
     ##6       <==  Apr09          $  524.00
     ##6       ==>  Sep09          $   10.00

     ##6       #:1   <== Apr09
     ##6       #:1   ==> Sep09
     @                                                     @@
     ##6       #:3   <== 524.00
     ##6       #:3   ==> 10.00
     @ Absolute error = 5.1400000000e+2, Relative error = 5.1400000000e+1
     ----------------
     ##7       <==  May09          $   78.25
     ##7       ==>  Nov09          $  110.00

     ##7       #:1   <== May09
     ##7       #:1   ==> Nov09
     @                                                     @@
     ##7       #:3   <== 78.25
     ##7       #:3   ==> 110.00
     @ Absolute error = 3.1750000000e+1, Relative error = 4.0575079872e-1
     ----------------
     ##8       <==  Jun09          $  230.00
     ##8       ==>  Jan10          $  200.00

     ##8       #:1   <== Jun09
     ##8       #:1   ==> Jan10
     @                                                     @@
     ##8       #:3   <== 230.00
     ##8       #:3   ==> 200.00
     @ Absolute error = 3.0000000000e+1, Relative error = 1.5000000000e-1
     ----------------
     ##9       <==  Jul09          $  443.10
     ##9       ==> -------------------------

     ##9       #:1   <== Jul09
     ##9       #:1   ==> -------------------------
     @                                                     @@
     ##9       #>2   <== $  443.10
     ##9             ==>
     @ Line 9 in file "bill2" is shorter than expected!
     ----------------
     ##10      <==  Aug09          $   67.65
     ##10      ==>

     ##10      #>1   <== Aug09          $   67.65
     ##10            ==>
     @ Line 10 in file "bill2" is shorter than expected!
     ----------------
     ##11      <==  Sep09          $   10.00
     ##11      ==> Total           $ 1088.35

     ##11      #:1   <== Sep09
     ##11      #:1   ==> Total
     @                                                     @@
     ##11      #:3   <== 10.00
     ##11      #:3   ==> 1088.35
     @ Absolute error = 1.0783500000e+3, Relative error = 1.0783500000e+2
     ----------------
     ##12      <==  Oct09          $  201.45
               ==>


     ***  End of file "bill2" reached while trying to read line 12.
          File "bill1" has more lines than file "bill2",
          line 12 is the last one read from file "bill1"


     +++  File "bill1" differs from file "bill2"
   Numdiff compares now the first, second, third line of `bill1' with
the first, second, third line of `bill2' and so on.  But probably this
is not what you want in such a case: what is reasonable here is to
compare entries related to the same month and not lines having the same
location, i.e. the same line number.

   Numdiff offers also an option to run just the filter and see how it
resynchronizes the two given files without doing any comparison between
corresponding lines.  The output of `numdiff -z @ -f bill1 bill2' is
      Month          Expenses          Month          Expenses
     -------------------------     <
      Jan09          $  233.56         Jan09          $  234.00
      Feb09          $  850.77     <
      Mar09          $   12.55         Mar09          $   13.00
      Apr09          $  524.00     <
      May09          $   78.25         May09          $   78.25
      Jun09          $  230.00     <
      Jul09          $  443.10         Jul09          $  443.10
      Aug09          $   67.65     <
      Sep09          $   10.00         Sep09          $   10.00
      Oct09          $  201.45     <
      Nov09          $  110.00         Nov09          $  110.00
      Dec09          $  200.27     |   Jan10          $  200.00
     -------------------------        -------------------------
                                   >
     Total           $ 2961.60        Total           $ 1088.35


     +++  File "bill1" differs from file "bill2"

and shows that the filter is doing its job in the right way,
associating the lines according to the month and not to the line number.
Running just the filter is extremely useful in all situations when you
are not sure if the filter is working as you wish.  You have indeed to
instruct the filter in the right way to let it work correctly, and this
requires the use of different options depending on the structure of the
files to compare.  Since to guess the right options can be sometime
tricky, running just the filter and see the result is the best way to
be certain that you are setting up everything correctly.  Later, *note
Filtering::, I will explain in detail
   * what the filter does behind the scenes to understand if and how it
     has to resynchronize the files to compare,

   * and how the related options affect the action of the filter.

   By the way, it is even possible to use `-f' without any other
additional option for the filter, like in `numdiff -f bill1 bill2', but
the result is more or less the same you would obtain by performing a
byte-by-byte comparison with removal of the field delimiters.

   The option `-f' can be followed by an argument in the form of an
integer number whose meaning will be explained later, *note Use of the
option -f::.

   Even if the output of `numdiff' is self-explanatory, in the next
section I will explain in details all you have to know about it.

3.1 Output format
=================

Let us go back to our first example.  If the files `list1' and `list2'
contain the data
accident        123     23Joshua          34.55           +3+4i	        water
dog       -3455.321     cat             2.345678e-9     .0005-6.23e2i

and

Accident        123     23456          34.5500         +3.0001+4i
dog       -3455.320098  Cat     +2.345678e-9            -6.23e2i    $$$
A new line

respectively, then the output of the command `numdiff list1 list2' will
be:

     ----------------
     ##1       #:1   <== accident
     ##1       #:1   ==> Accident
     @                                                     @@
     ##1       #:3   <== 23Joshua
     ##1       #:3   ==> 23456
     @                                                     @@
     ##1       #:5   <== +3+4i
     ##1       #:5   ==> +3.0001+4i
     @ Absolute error = 1.0000000000e-4, Relative error = 2.0000000000e-5
     ##1       #>6   <== water
     ##1             ==>
     @ Line 1 in file "list2" is shorter than expected!
     ----------------
     ##2       #:2   <== -3455.321
     ##2       #:2   ==> -3455.320098
     @ Absolute error = 9.0200000000e-4, Relative error = 2.6104672633e-7
     ##2       #:3   <== cat
     ##2       #:3   ==> Cat
     @                                                     @@
     ##2       #:5   <== .0005-6.23e2i
     ##2       #:5   ==> -6.23e2i
     @ Absolute error = 5.0000000000e-4, Relative error = 8.0256821830e-7
     ##2             <==
     ##2       #>6   ==> $$$
     @ Line 2 in file "list1" is shorter than expected!
     ----------------
     ##3             <==
     ##3       #>1   ==> A new line
     @ Line 3 in file "list1" is shorter than expected!
     ----------------
               <==
     ##4       ==>


     +++  File "list1" differs from file "list2"

   `numdiff' prints a report on the standard output for every field of
the first file which differs from the corresponding field of the second
file.

   First this report indicates the locations of the fields, namely the
numbers of the lines where the fields appear and their positions within
the line. The position in the line is "1" for the first field of a
line, "2" for the second field, "3" for the third one and so on: fields
are numerated starting from the left hand of the line and proceeding
towards the right hand.  For each report the line number is introduced
by the symbol "##", while the field number by "#:".  Then `numdiff'
shows in what the difference consists.  For instance,
     ##1       #:1   <== accident
     ##1       #:1   ==> Accident
     @                                                     @@

means that the first field of the first line is "accident" in the first
file, while in the second file it appears as "Accident".  This
difference could be then canceled by removing "accident" from the first
file and inserting "Accident" in place of it.  The arrows "<==" and
"==>" try to visualize this idea.  Analogously,
     ##2       #:2   <== -3455.321
     ##2       #:2   ==> -3455.320098
     @ Absolute error = 9.0200000000e-4, Relative error = 2.6104672633e-7

means that the second field of the second line is "-3455.321" in the
first file and "-3455.320098" in the second one.  Since the contents of
the field are numerical in both files, `numdiff' also prints the
absolute and relative errors.

   The absolute error (or absolute difference) is given by the absolute
value of the difference between the values appearing in the two files.

   The relative error (or relative difference) is actually defined in a
more complicated way.  If _n1_ is the value appearing in the first file
and _n2_ is the value in the second file, then the absolute error is
given by the formula _A=|n1-n2|_, while the relative error _R_ is given
by:

   * _R = 0_ if _n1_ and _n2_ are equal,

   * _Inf_ (infinity) if _n2_ differs from _n1_ and at least one of
     them is zero,

   * _R = A/ min(|n1|, |n2|)_ if _n1_ and _n2_ are both non zero and
     _n2_ differs from _n1_. _min(|n1|, |n2|)_ denotes the minimum
     between the absolute value of _n1_ and the absolute value of _n2_.

With these definitions of absolute and relative error it turns out that
_A(n2, n1) = A(n1, n2)_ and _R(n2, n1) = R(n1, n2)_. In other words,
the absolute/relative error does not change if you only change the
order of the compared values.  Since version 5 it is actually possible
to let Numdiff compute the relative error always with respect to the
value from the first file or always with respect to the value from the
second file, instead of using the preceding formula.  This can be done
through the option `-F', *note Alternative formulas for the computation
of the relative difference::.

   If at least one of the compared fields is not numerical, then the
output line reporting absolute and relative errors is replaced by the
separator:
@                                                     @@

   It can happen that a line in one of the two files to compare contains
more fields than the corresponding line of the other file.  When this
is the case, `numdiff' reports this difference by telling that a
certain line (identified by its line number) appears to be shorter than
expected, just as in
     ##1       #>6   <== water
     ##1             ==>
     @ Line 1 in file "list2" is shorter than expected!

   or in
     ##3             <==
     ##3       #>1   ==> A new line
     @ Line 3 in file "list1" is shorter than expected!

   In addition, `numdiff' shows the tail of the longer line, using the
notation "#>n" to indicate the number n of the first field of the
longer line for which there is no corresponding field in the shorter
line.  For example,
     ##1       #>6   <== water
     ##1             ==>
     @ Line 1 in file "list2" is shorter than expected!

means that none of the fields of the first line starting from the sixth
one has a corresponding field in the second file (`list2').  In this
context, the symbol <<*>> (when it appears) is used to denote the
End-Of-File, i.e. a line or the tail of a line which is located at the
end of the corresponding file and does not have a terminating _newline_
character.

   It can also happen that one of the two files to compare has less
lines than the other one.  In this case, if no special option is passed
to the program, `numdiff' prints the number of the first line which
appears in only one of the two files and a message on the standard
error telling in which of the two files the end has been prematurely
reached:
     ***  End of file "list1" reached while trying to read line 4.
          File "list2" has more lines than file "list1",
          line 4 is the last one read from file "list2"

   Unless the option `-q' is used (*note Invoking numdiff::), `numdiff'
prints on standard output a message reporting the final status of the
comparison.  This message says either the two files are equal or they
are different, just as in the example we are considering:
     +++  File "list1" differs from file "list2"

3.2 Overview mode
=================

Since version 5.6 an alternative way to display the differences between
two files is available, which can be activated through the option `-O'.
If this option is present on the command line, `numdiff' prints a
side-by-side report instead of the usual one.

   For example, if `sheet1' contains the text
A    1    1
B    2    4
C    3    9
D    4   16
E    5   25
F    6   36
G    7   49
H    8   64
I    9   81
J   10  100

and `sheet2' the following lines

A    1      1
B    2      4
C    3.3    9.03
D    4     16
E    5.5   25.05
F    6.6   36
G    7.7   49.49
H    8     64
I    9.9   81.09

then `numdiff -O sheet1 sheet2' prints this report
     A    1    1                            A    1      1
     B    2    4                            B    2      4
     C    3    9                         :!:C    3.3    9.03
     D    4   16                            D    4     16
     E    5   25                         :!:E    5.5   25.05
     F    6   36                         :!:F    6.6   36
     G    7   49                         :!:G    7.7   49.49
     H    8   64                            H    8     64
     I    9   81                         :!:I    9.9   81.09
     J   10  100                         :<:

     ***  End of file "sheet2" reached while trying to read line 10.
          File "sheet1" has more lines than file "sheet2",
          line 10 is the last one read from file "sheet1"


     +++  File "sheet1" differs from file "sheet2"

   On the left side you can see the lines coming from the file
specified as first on the command line, i.e. `sheet1', on the right
side the lines from the second file of the command line, in this case
`sheet2'. In the middle there is a gutter which contains one of these
markers:
`white spaces'
     The corresponding lines are in common.  That is, either the lines
     are identical, or the difference is ignored because of one of the
     options `-s', `-D', `-I', `-X', `-a', `-r', `-P' and `-N'.

`:!:'
     The corresponding lines have at least one field which differs.

`:<:'
     The files differ and only the first file contains the line.

`:>:'
     The files differ and only the second file contains the line.

   In the case of `sheet1' and `sheet2' a message is printed after the
report saying that the end of the second file has been prematurely
reached. The two files do not have indeed the same number of lines and
the filter has not been activated.

   The option `-O' can take an optional argument, which allows to set
the width of the output and eventually to suppress common lines, *note
Invoking numdiff::.  The default value for the width of the
side-by-side report is 130. No wonder then that the command `numdiff
-O40 sheet1 sheet2' displays a report with shorter lines:
     A    1    1           A    1      1
     B    2    4           B    2      4
     C    3    9       :!: C    3.3    9.03
     D    4   16           D    4     16
     E    5   25       :!: E    5.5   25.05
     F    6   36       :!: F    6.6   36
     G    7   49       :!: G    7.7   49.49
     H    8   64           H    8     64
     I    9   81       :!: I    9.9   81.09
     J   10  100       :<:

     ***  End of file "sheet2" reached while trying to read line 10.
          File "sheet1" has more lines than file "sheet2",
          line 10 is the last one read from file "sheet1"


     +++  File "sheet1" differs from file "sheet2"

   A negative argument makes that only the differences are listed in
the side-by-side report, as shown by the output of the command `numdiff
-O40 sheet1 sheet2':
     C    3    9       :!: C    3.3    9.03
     E    5   25       :!: E    5.5   25.05
     F    6   36       :!: F    6.6   36
     G    7   49       :!: G    7.7   49.49
     I    9   81       :!: I    9.9   81.09
     J   10  100       :<:

     ***  End of file "sheet2" reached while trying to read line 10.
          File "sheet1" has more lines than file "sheet2",
          line 10 is the last one read from file "sheet1"


     +++  File "sheet1" differs from file "sheet2"

   If you set the width of the report to a too small value, it can
happen that some or even all lines from the compared files appear
truncated as in the output of `numdiff -O24 sheet1 sheet2':
     A    1         A    1
     B    2         B    2
     C    3    :!:  C    3.3
     D    4         D    4
     E    5    :!:  E    5.5
     F    6    :!:  F    6.6
     G    7    :!:  G    7.7
     H    8         H    8
     I    9    :!:  I    9.9
     J   10    :<:

     ***  End of file "sheet2" reached while trying to read line 10.
          File "sheet1" has more lines than file "sheet2",
          line 10 is the last one read from file "sheet1"


     +++  File "sheet1" differs from file "sheet2"

   If you set the width of the report to a very small value, Numdiff
ignores it and uses the default value, i.e. 130.

   Notice that the numeric argument must immediately follow the option
`-O', intermediate spaces are not allowed. This is also the case for
the optional argument of `-f', while the options of Numdiff which
require a mandatory argument permit the presence of intermediate spaces
between them and the argument.

   The option `-O' can be used together with any other option of
Numdiff except for `-f', `-q', `-U', `-E', `-V' and `-b'. When `-O' is
in use, `-U', `-E', `-V' and `-b' are ignored. If `-q' is present on
the command line together with `-O', then `-O' is ignored. Finally, if
both `-f' and `-O' are present, then the behavior depends on the order:
the option which appears first on the command line is the one which
matters.

   Therefore, the command `numdiff -O40 -f sheet1 sheet2' displays the
same report as `numdiff -O40 sheet1 sheet2', while the output of
`numdiff -f -O40 sheet1 sheet2' is given by
     A    1    1                            A    1      1
     B    2    4                            B    2      4
     C    3    9                          | C    3.3    9.03
     D    4   16                            D    4     16
     E    5   25                          | E    5.5   25.05
     F    6   36                          | F    6.6   36
     G    7   49                          | G    7.7   49.49
     H    8   64                            H    8     64
     I    9   81                          | I    9.9   81.09
     J   10  100                          <


     +++  File "sheet1" differs from file "sheet2"

and coincides then with the one of `numdiff -f sheet1 sheet2'.

   The option `-O' can be used together with the filter to cope with
the addition/deletion of lines.  If the file `sheet3' contains the text
A    1      1
C    3.3    9.03
E    5.5   25.05
G    7.7   49.49
I    9.9   81.09
J    10   100.00
K    0      0.02

then `numdiff -O40 sheet1 sheet3' prints a wrong report, as in the
example with files `bill1' and `bill2':
     A    1    1           A    1      1
     B    2    4     :!:   C    3.3    9.03
     C    3    9     :!:   E    5.5   25.05
     D    4   16     :!:   G    7.7   49.49
     E    5   25     :!:   I    9.9   81.09
     F    6   36     :!:   J    10   100.00
     G    7   49     :!:   K    0      0.02
     H    8   64     :<:

     ***  End of file "sheet3" reached while trying to read line 8.
          File "sheet1" has more lines than file "sheet3",
          line 8 is the last one read from file "sheet1"


     +++  File "sheet1" differs from file "sheet3"

   On the other hand, the presence of `-z @' makes Numdiff always
compare fields of corresponding lines, as shown by the output of the
command `numdiff -O40 -z @ sheet1 sheet3':
     A    1    1           A    1      1
     B    2    4     :<:
     C    3    9     :!:   C    3.3    9.03
     D    4   16     :<:
     E    5   25     :!:   E    5.5   25.05
     F    6   36     :<:
     G    7   49     :!:   G    7.7   49.49
     H    8   64     :<:
     I    9   81     :!:   I    9.9   81.09
     J   10  100           J    10   100.00
                     :>:   K    0      0.02

     +++  File "sheet1" differs from file "sheet3"

   Side-by-side format is easy to read, but it has limitations.  It
generates much wider output than usual, and truncates lines that are
too long to fit.  Also, it relies on lining up output quite heavily, so
its output looks particularly bad if you use varying width fonts,
nonstandard tab stops, or nonprinting characters.

3.3 Output of the filter
========================

The output produced just by running the filter (option `-f') is a
side-by-side difference listing of the compared files like the one
displayed by GNU sdiff.  The files are listed in two columns with a
gutter between them.  The gutter contains one of the following markers:
`white space'
     The corresponding lines are in common.  That is, either the lines
     are identical, or the difference is ignored because of one of the
     options `-s', `-D', `-I', `-X', `-z' and `-Z'.

`|'
     The corresponding lines differ, and they are either both complete
     or both incomplete.

`<'
`('
     The files differ and only the first file contains the line.

`>'
`)'
     The files differ and only the second file contains the line.

`\'
     The corresponding lines differ, and only the first line is
     incomplete.

`/'
     The corresponding lines differ, and only the second line is
     incomplete.

   An input line is incomplete if its last character is not a newline.
This can happen only if the line is the last one of its file.  When an
output line of the side by side difference listing represents two
differing lines, one might be incomplete while the other is not.  In
this case the gutter is marked `\' if the line from the first file is
incomplete, `/' if the line from the second file is.

   Like `-O', the option `-f' can take an optional argument which
allows to set the width of the output and eventually to suppress common
lines, *note Invoking numdiff:: and *note Use of the option -f::.

   More generally, the user can always make `numdiff' avoid to print,
partially or totally, the messages that it would otherwise send to
standard output.  This can be achieved by some suitable command line
options, *note Invoking numdiff::.

4 Installing
************

To successfully compile, build and install Numdiff some tools are
required. The first one is an ANSI C compiler.  This compiler should at
least accept the option `-o' to write its output to a specified file,
the option `-D' for macros predefinition, the option `-l' to search for
a specified library, and the options `-I' and `-L' to add a given
directory to the search path for include and library files respectively.

   In addition, you need a POSIX implementation of the `make' utility
(I used both GNU make and smake by Joerg Schilling to compile Numdiff)
and a POSIX implementation of the commands `rm' and `find'.  At last,
you need a proper installation of GNU Texinfo (in order to install the
info documentation) and a shell sh-compatible.

   Numdiff has been successfully compiled and tested on:

   * Slackware(R) GNU/Linux 10.2 with the version 3.3.6 of the GNU C
     Compiler (GCC),

   * Slackware GNU/Linux 11 with GCC 3.4.6,

   * Slackware GNU/Linux 12.2 with GCC 4.2.4,

   * Slackware GNU/Linux 13 with GCC 4.3.3,

   * Debian(R) GNU/Linux 4.0 with GCC 4.1.2 20061115 (prerelease)
     (Debian 4.1.1-21),

   * Debian GNU/Linux 6.0.3 with GCC 4.4.5 (Debian 4.4.5-8),

   * Debian GNU/Linux 7.1 with GCC 4.7.2 (Debian 4.7.2-5),

   * SunOS(R) 5.8 with GCC 2.95.3, and

   * SunOS 5.10 (i386) with the version 5.9 of the Sun C compiler.

   Configuration, building and installation of Numdiff can be performed
through the standard three steps:
             ./configure
             make
             make install

If you leave enabled the Natural Language Support and you also want to
install the localization files (at the moment only the Italian
localization is supplied), then, after `make', you will have to type
and run
             make install-nls

   By default, `make install' will install all the files in
`/usr/local/bin', `/usr/local/info' etc.  You can specify an
installation prefix other than `/usr/local' using the option `--prefix'
in the `configure' step, for instance `--prefix=$HOME':
             ./configure --prefix=$HOME

For better control, you can use the options `--bindir', `--infodir',
and so on.  Type `./configure --help' to obtain the complete list of
all the available options.

   Anyway, the documentation files, including a full User Manual
available in several formats (HTML, PDF and plain ASCII text), will
always be put in `DOCDIR/numdiff', where DOCDIR is the path specified
by the option `--docdir' or, if this option has not been given to
`configure', `PREFIX/local/doc'.  Here PREFIX is the installation
prefix specified by the option `--prefix' or the default `/usr/local'.

   Once Numdiff has been installed you can remove all the files
previously installed by a simple `make uninstall'.  If you have also
installed the localization files trough `make install-nls', then, in
order to remove also these ones, use `make uninstall-nls' in place of
`make uninstall'.

   Between the options accepted by `configure' there are
`--enable-debug', `--enable-optimization', `--enable-nls' and
`--enable-gmp'.

   The option `--enable-debug' turns on debugging when compiling the
source code. This is obtained by passing to the compiler the `-g'
option, but you can change this default debugging flag (which could not
even be recognized by your compiler) by setting the environment
variable `DBGFLAGS' before calling `configure'.

   The option `--enable-optimization' turns on basic optimization when
compiling the source code. This is obtained by passing to the compiler
the `-O' option, but you can change this default flag (which could not
even be recognized by your compiler) by setting the environment
variable `OPTFLAGS' before calling `configure'.

   The option `--enable-nls' turns on Natural Language Support.  But
you do not need to use it explicitly, since Natural Language Support is
enabled by default.  However, you can disable it by using
`--disable-nls'.  Disabling Natural Language Support is suggested
whenever you want to install Numdiff on a system where the GNU gettext
library is not present. In this case the installation of Numdiff can be
accomplished, for example, through
             ./configure --disable-nls
             make
             make install

Since version 5.2.0 Numdiff uses to perform all computations the GNU
Multiple Precision Arithmetic Library (also called GNU MP or GMP), if
this library is available at build-time.  The old internal support for
multiple precision arithmetic is a fall-back in case GNU MP is absent.
However, it is possible to use the internal support for multiple
precision arithmetic even when GNU MP is available: it is sufficient to
pass the option `--enable-gmp=no' or `--disable-gmp' to the  configure
script before building the program, like in
             ./configure --disable-gmp
             make
             make install

Enabling the old internal support for multiple precision arithmetic is
deprecated, *note with GNU MP is better::.  The latest version of GNU
MP is available at `ftp://ftp.gnu.org/gnu/gmp/'. See the GNU MP web page
at `http://gmplib.org/' for up-to-date information on GNU MP.

5 Invoking numdiff
******************

*SYNOPSIS*

        numdiff -h|--help|-v|--version
   or
        numdiff [-s IFS][-D DELIMS][-a THRVAL[:RANGE|:RANGE1:RANGE2]]
        [-r THRVAL[:RANGE|:RANGE1:RANGE2]][-2][-F NUM][-# NUM][-P][-N][-I]
        [-c CURRNAME][-d C1C2][-t C1C2][-g N1N2][-p C1C2][-n C1C2][-e C1C2]
        [-i C1C2][-X 1:RANGE][-X 2:RANGE][-E][-U][-b][-V][-O[NUM]][-q][-S]
        [-z 1:RANGE][-z 2:RANGE][-Z 1:RANGE][-Z 2:RANGE][-m][-H][-f[NUM]]
        [-T][-B][-l PATH][-o PATH] FILE1 FILE2

where FILE1 and FILE2 are the names of the two files to compare and
RANGE, RANGE1 and RANGE2 stay for a positive integer value or for a
range of integer values, like `1-', `3-5' or `-7'.

   In the first case `numdiff' prints a short help (not so short
actually :)) or/and version number, Copyright, License notice,
NO-Warranty disclaimer and some information about the way it was built.
In the second case `numdiff' compares the files specified by the two
mandatory arguments which follow the list of the options.  The complete
paths of the files should be given, a directory name is not accepted.
In addition, the two arguments cannot refer to the same file but one of
them can be -, which refers to stdin.

*OPTIONS*

`-s, --separators=IFS'
     Specify the set of characters to use as delimiters while splitting
     the input lines into fields (The default set of characters is
     space, tab and newline).  If IFS is prefixed with `1:' or `2:'
     then use the given delimiter set only for the lines from the first
     or the second file respectively

`-D, --delimiters=DELIMS'
     Specify the set of strings to use as delimiters while splitting
     the input lines into fields (The default set of characters is
     space, tab and newline).  If DELIMS is prefixed with `1:' or `2:'
     then use the given delimiter set only for the lines from the first
     or the second file respectively

`-a, --absolute-tolerance=THRVAL[:RANGE|:RANGE1:RANGE2]'
     Set to THRVAL the maximum absolute difference permitted before
     that two numeric fields are regarded as different (The default
     value is zero).  If a RANGE is given, use the specified threshold
     only when comparing fields whose indexes lie in RANGE.  If both
     RANGE1 and RANGE2 are given and have the same length, then use the
     specified threshold when comparing a field of FILE1 lying in
     RANGE1 with the corresponding field of FILE2 in RANGE2

`-r, --relative-tolerance=THRVAL[:RANGE|:RANGE1:RANGE2]'
     Set to THRVAL the maximum relative difference permitted before
     that two numeric fields are regarded as different (The default
     value is zero).  If a RANGE is given, use the specified threshold
     only when comparing fields whose indexes lie in RANGE.  If both
     RANGE1 and RANGE2 are given and have the same length, then use the
     specified threshold when comparing a field of FILE1 lying in
     RANGE1 with the corresponding field of FILE2 in RANGE2

`-2, --strict'
     Consider two numerical values as equal only if both absolute and
     relative difference do not exceed the corresponding tolerance
     threshold

`-F, --formula=NUM'
     Use the formula indicated by NUM to compute the relative errors.
     If NUM is 0 use the classic formula.  If NUM is 1 compute the
     relative errors by considering the values in FILE1 as sample
     values.  If NUM is 2 compute the relative errors by considering
     the values in FILE2 as sample values.

`-#, --digits=NUM'
     Set to NUM the number of digits in the significands used in
     multiple precision arithmetic

`-P, --positive-differences'
     Ignore all differences due to numeric fields of the second file
     that are less than the corresponding numeric fields in the first
     file

`-N, --negative-differences'
     Ignore all differences due to numeric fields of the second file
     that are greater than the corresponding numeric fields in the
     first file

`-I, --ignore-case'
     Ignore changes in case while doing literal comparisons

`-c, --currency=CURRNAME'
     Set to CURRNAME the currency name for the two files to compare.
     CURRNAME must be prefixed with `1:' or `2:' to specify the
     currency name only for the first or the second file

`-d, --decimal-point=C1C2'
     Specify the characters representing the decimal point in the two
     files to compare

`-t, --thousands-separator=C1C2'
     Specify the characters representing the thousands separator in the
     two files to compare

`-g, --group-length=N1N2'
     Specify the number of digits forming each group of thousands in
     the two files to compare

`-p, --plus-prefix=C1C2'
     Specify the (optional) prefixes for positive values used in the
     two files to compare

`-n, --minus-prefix=C1C2'
     Specify the prefixes for negative values used in the two files to
     compare

`-e, --exponent-letter=C1C2'
     Specify the exponent letters used in the two files to compare

`-i, --imaginary-unit=C1C2'
     Specify the characters representing the imaginary unit in the two
     files to compare

`-X, --exclude=1:RANGE'
     Select the fields of the first file that have to be ignored

`-X, --exclude=2:RANGE'
     Select the fields of the second file that have to be ignored

`-E, --essential'
     While printing the differences between the two compared files show
     only the numerical ones

`-U, --dummy'
     While printing the differences between the two compared files
     neglect all the numerical ones (dummy mode)

`-b, --brief'
     Suppress all messages concerning the differences discovered in the
     structures of the two files

`-V, --verbose'
     For every couple of lines which differ in at least one field print
     an header to show how these lines appear in the two compared files

`-O, --overview[=NUM]'
     Display a side by side difference listing of the two files showing
     which lines are present only in one file, which lines are present
     in both files but with one or more differing fields, and which
     lines are identical.  If NUM is zero or is not specified, output
     at most 130 columns per line.  If NUM is a positive number, output
     at most NUM columns per line.  If NUM is a negative number, do not
     output common lines and display at most -NUM columns per line.

`-q, --quiet, --silent'
     Suppress all the standard output

`-S, --statistics'
     Add some statistics to the standard output

`-z, --blur-if-numerical=1:RANGE'
     Select the fields of the first file that have to be blurred during
     the synchronization procedure only if they turn out to be numeric

`-z, --blur-if-numerical=2:RANGE'
     Select the fields of the second file that have to be blurred
     during the synchronization procedure only if they turn out to be
     numeric

`-Z, --blur-unconditionally=1:RANGE'
     Select the fields of the first file that have to be
     unconditionally blurred during the synchronization procedure

`-Z, --blur-unconditionally=2:RANGE'
     Select the fields of the second file that have to be
     unconditionally blurred during the synchronization procedure

`-m, --minimal'
     During synchronization try hard to find a smaller set of changes

`-H, --speed-large-files'
     During synchronization assume large files and many scattered small
     changes

`-f, --test-filter[=NUM]'
     Run only the filter and then show the results of its attempt to
     synchronize the two files.  If NUM is zero or is not specified,
     output at most 130 columns per line.  If NUM is a positive number,
     output at most NUM columns per line.  If NUM is a negative number,
     do not output common lines and display at most -NUM columns per
     line.

`-T, --expand-tabs'
     Expand tabs to spaces in output while displaying the results of the
     synchronization procedure (meaningful only together with option
     `-O' or `-f')

`-B, --binary'
     Treat both files as binary files (only meaningful under Doz/Windoz)

`-l, --warnings-to=PATH'
     Redirect warning and error messages from stderr to the indicated
     file

`-o, --output=PATH'
     Redirect output from stdout to the indicated file

`-h, --help'
     Show help message and predefined settings

`-v, --version'
     Show version number, Copyright, Distribution Terms and NO-Warranty

*DIAGNOSTICS*

   The exit status is 1 if the two given files differ, 0 if they are
equal, -1 (255) in case of error.

*DEFAULT NUMERIC FORMAT (for both files to compare):* Currency name = ""

   Decimal point = `.'

   Thousands separator = `,'

   Number of digits in each thousands group = 3

   Leading positive sign = `+'

   Leading negative sign = `-'

   Prefix for decimal exponent = `e'

   Symbol used to denote the imaginary unit = `i'

*SOME EXPLANATIONS*

   The options `-U', `-E', `-b' and `-q' are used to hide part of the
standard output of the program according to some rule.

   The option `-U' triggers the dummy mode. In this mode `numdiff' does
not print the numerical differences. A numerical difference occurs
whenever the compared fields turn out to be both of numerical type, but
the field from the second file has a value which differs from the one
of the field from the first file.  The dummy mode is so called since in
this mode `numdiff' does not perform the job for which I created it.

   The option `-E' triggers the essential mode. In this mode `numdiff'
only prints the numerical differences between the files and, if there
are some, the differences in the structure. The latter ones occur
either when one of the files contains a line for which there is no
corresponding line in the other file, or when, comparing two lines, it
turns out that one of them contains a field for which there exists no
corresponding field in the other line.  If you are not running any
filter or cutting out any fields through the option `-X', then the
differences in the structure simply consist either in a different
number of lines in the two files, or in a different number of fields on
a line.

   The option `-b' triggers the brief mode. In this mode `numdiff' does
not print the differences in the structure of the two files (see above
for an explanation about what they are).

   The option `-q' triggers the quiet mode.  In this mode `numdiff'
does not print anything on the standard output. The quiet mode is
useful if you only want to know whether a couple of files are equal or
not.  This information can be obtained by looking at the exit status of
the program.

   The option `-O' activates the overview mode, which makes `numdiff'
print a side-by-side report in the form described in section *note
Overview mode::.  The optional numeric argument after `-O' must
immediately follow, intermediate spaces are not allowed.  The option
`-O' can be used together with any other option of Numdiff except for
`-f', `-q', `-U', `-E', `-V' and `-b'. When `-O' is in use, `-U', `-E',
`-V' and `-b' are ignored.  If `-q' is present on the command line
together with `-O', then `-O' is ignored. Finally, if both `-f' and
`-O' are present, then the behavior depends on the order: the option
which appears first on the command line is the one which matters.

   The option `-V' triggers the verbose mode. In this mode `numdiff'
produces a richer report by printing an header whenever the compared
lines differ. The header shows how and where these lines appear in the
compared files.  For instance, if the files `data1' and `data2' contain
the data
12      33
22      44.5
0.008   1.002
221.12  -34.56  water
2101.21 boats

and

12      33
22.3      44.5
0.008   1.202
221.12  -34.56
2101.21 boats   dogs

respectively, then the command `numdiff -V data1 data2' will print the
following output:
     ----------------
     ##2       <== 22      44.5
     ##2       ==> 22.3      44.5

     ##2       #:1   <== 22
     ##2       #:1   ==> 22.3
     @ Absolute error = 3.0000000000e-1, Relative error = 1.3636363636e-2
     ----------------
     ##3       <== 0.008   1.002
     ##3       ==> 0.008   1.202

     ##3       #:2   <== 1.002
     ##3       #:2   ==> 1.202
     @ Absolute error = 2.0000000000e-1, Relative error = 1.9960079840e-1
     ----------------
     ##4       <== 221.12  -34.56  water
     ##4       ==> 221.12  -34.56

     ##4       #>3   <== water
     ##4             ==>
     @ Line 4 in file "data2" is shorter than expected!
     ----------------
     ##5       <== 2101.21 boats
     ##5       ==> 2101.21 boats   dogs

     ##5             <==
     ##5       #>3   ==> dogs
     @ Line 5 in file "data1" is shorter than expected!

     +++  File "data1" differs from file "data2"

   You must care that the options `-b' and `-V' will be overridden if
`-q' is also set.

   The amount of additional information printed by `-V' is trivially
influenced by the options that alter the way `numdiff' performs the
comparisons between fields (for instance `-a', `-r', `-2', `-N', `-P',
`-U', `-E', `-I', `-X').

   In the headers printed by `numdiff' when in "verbose mode" can also
appear the symbol <<*>>. This symbol, if present, is always located at
the end of a line to mean that the line is at the end of the
corresponding file and does not have a terminating *newline* character.

   The option `-S' adds to the standard output of `numdiff' a
statistical report with the following information:
   * the number of numeric comparisons the program has done (this
     quantity, like the successive ones, is influenced by the options
     `-P' and `-N') and the number of those comparisons whose outcome
     is a relevant numerical difference.

   * the largest absolute error in the set of relevant numerical
     differences and the corresponding relative error,

   * the largest relative error in the set of relevant numerical
     differences together with the corresponding absolute error,

   * the sum and the arithmetic mean of all absolute errors,

   * the sum and the arithmetic mean of the relevant absolute errors,

   * the square root of the sum of the squares of all absolute errors,

   * the square root of the sum of the squares of the relevant absolute
     errors,

   * the quadratic mean of all absolute errors, and

   * the quadratic mean of the relevant absolute errors.
   By relevant numerical differences and relevant absolute errors I
mean those ones appearing in the output of `numdiff' when the options
`-U', `-f', `-O' and `-q' are not used.  The information printed by
`-S' is not removed when this option is used together with `-q'.

   The options `-a', `-r', `-2', `-P' and `-N' affect the way `numdiff'
performs the comparisons between numerical values.  Without any of
these options, `numdiff' considers two numerical fields as equal when
their difference is zero.

   The option `-a' can be used to order that two numerical fields must
be considered equal as long as their absolute difference does not
exceed a certain threshold, which is specified by the argument that
follows the `-a' option.  This argument can take several forms. The
basic form consists of a single numerical value, the extended form adds
the specification of one or two ranges of integer values.

   Independently of the form of the argument, if the absolute
difference between two fields does not exceed the given threshold value,
the fields are considered equal; otherwise, `numdiff' prints the
difference in its report, unless some other option, for example `-P' or
`-N', makes the difference unimportant.  If nothing else follows the
threshold value, what has been just explained applies to all comparisons
between numerical fields.  To see this in practice, if the file
`many_columns1' contains the text
A	1	 1.2	  1	 0.1	  11.0         1.0e-1
B	2        2.4      4      0.4      24.0         1.0e-2
C	3        3.6      9      0.9      39.0         1.0e-3
D	4	 4.8     16      1.6     416.0         1.0e-4

and the contents of the file `many_columns2' are given by

A	1.1	1.08	 1.01	 0.1	 11.011        -1.0e-1
B	2.2     2.16     4.04    0.4     24.024        -1.0e-2
C	3.3     3.24     9.09    0.9     39.039        -1.0e-3
D	4.4	4.32    16.16    1.6    416.039        -1.0e-4

then the output of the command `numdiff -a 0.5 many_columns1
many_columns2' is

     +++  Files "many_columns1" and "many_columns2" are equal

The highest absolute difference between a field from `many_columns1'
and the corresponding field from `many_columns2' is given indeed by
|4.32 - 4.8| = |-0.48| = 0.48, and then all numeric differences between
the two files remain below the threshold value 0.5.

   On the other hand, the command `numdiff -a 0.35 many_columns1
many_columns2' prints the report
     ----------------
     ##3       #:3   <== 3.6
     ##3       #:3   ==> 3.24
      Absolute error = 3.6000000000e-1, Relative error = 1.1111111111e-1
     ----------------
     ##4       #:2   <== 4
     ##4       #:2   ==> 4.4
      Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1
     ##4       #:3   <== 4.8
     ##4       #:3   ==> 4.32
      Absolute error = 4.8000000000e-1, Relative error = 1.1111111111e-1

     +++  File "many_columns1" differs from file "many_columns2"

since the absolute differences |3.24-3.6| = |-0.36| = 0.36, |4.4-4| =
|0.4| = 0.4 and |4.32 - 4.8| = 0.48 exceed the value 0.35, and the
other differences are below this threshold.

   If you want that the specified threshold value applies only when
comparing some particular fields, you have to use the extended form for
the argument of `-a'.  This means that after the threshold value one or
two ranges of integer numbers must follow, each preceded by a colon
(`:').  If you specify only one range of numbers after the threshold
value, `numdiff' uses the given threshold only when comparing fields
whose positions lie in the specified range.  Remember that the
positions of the fields on a line are numbered starting from the left
hand of the line and proceeding towards the right hand.  For example,
`-a 0.01:2-5' sets the threshold value to 0.01 only for the comparisons
between numerical fields which occupy on their lines a position between
the second and the fifth one inclusive.  For the other comparisons the
threshold value is left unchanged and is in particular equal to zero if
it has not been explicitly set.  If the files `many_columns1' and
`many_columns2' are the same as before, then the command `numdiff -a
0.5:3-6 many_columns1 many_columns2' displays the following report
     ----------------
     ##1       #:2   <== 1
     ##1       #:2   ==> 1.1
      Absolute error = 1.0000000000e-1, Relative error = 1.0000000000e-1
     ##1       #:7   <== 1.0e-1
     ##1       #:7   ==> -1.0e-1
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e+0
     ----------------
     ##2       #:2   <== 2
     ##2       #:2   ==> 2.2
      Absolute error = 2.0000000000e-1, Relative error = 1.0000000000e-1
     ##2       #:7   <== 1.0e-2
     ##2       #:7   ==> -1.0e-2
      Absolute error = 2.0000000000e-2, Relative error = 2.0000000000e+0
     ----------------
     ##3       #:2   <== 3
     ##3       #:2   ==> 3.3
      Absolute error = 3.0000000000e-1, Relative error = 1.0000000000e-1
     ##3       #:7   <== 1.0e-3
     ##3       #:7   ==> -1.0e-3
      Absolute error = 2.0000000000e-3, Relative error = 2.0000000000e+0
     ----------------
     ##4       #:2   <== 4
     ##4       #:2   ==> 4.4
      Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1
     ##4       #:7   <== 1.0e-4
     ##4       #:7   ==> -1.0e-4
      Absolute error = 2.0000000000e-4, Relative error = 2.0000000000e+0

     +++  File "many_columns1" differs from file "many_columns2"

since the threshold value 0.5 applies now only when comparing fields in
third, fourth, fifth and sixth position, while for the other
comparisons the threshold value is the default one, i.e. zero.  If you
want to specify a non null threshold also for the fields in second and
seventh position, you can do it by using the option `-a' more times.
The command `numdiff -a 0.5:3-6 -a 0.25:2 -a 4e-3:7 many_columns1
many_columns2' sets the threshold value to 0.25 for the comparisons
between the fields in second position, and to 4e-3 for the comparisons
of the fields in seventh position. No wonder then, that the command
prints exactly this report:
     ----------------
     ##1       #:7   <== 1.0e-1
     ##1       #:7   ==> -1.0e-1
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e+0
     ----------------
     ##2       #:7   <== 1.0e-2
     ##2       #:7   ==> -1.0e-2
      Absolute error = 2.0000000000e-2, Relative error = 2.0000000000e+0
     ----------------
     ##3       #:2   <== 3
     ##3       #:2   ==> 3.3
      Absolute error = 3.0000000000e-1, Relative error = 1.0000000000e-1
     ----------------
     ##4       #:2   <== 4
     ##4       #:2   ==> 4.4
      Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1

     +++  File "many_columns1" differs from file "many_columns2"

Observe that :2 and :7 are abbreviations of :2-2 and :7-7, respectively.
It is even possible to use range expressions like `M-' or `-N'. The
first expression corresponds to all fields starting from the Mth one
(inclusive) till to the end of line, the second selects all fields from
the first one till to the Nth one, both inclusive.

   If you specify two ranges of numbers after the threshold value and
they have the same length (the length of a range is the difference
between its maximum and its minimum), `numdiff' uses the given
threshold when comparing a field of the first file lying in the first
range with the corresponding field of the second file from the second
range.  For example, `-a 1e-4:3-5:4-6' sets the threshold value to
0.0001 only for the numerical comparisons of the third, fourth, and
fifth field of each line from the first file with the fourth, fifth and
sixth field respectively of the corresponding line from the second file.
For the other comparisons the threshold value is left unchanged and is
in particular equal to zero if it has not been explicitly set. This way
to restrict the application of a threshold value is useful in
conjunction with the option `-X', which makes `numdiff' ignore one or
more fields from one of the compared files.

   The file `many_columns3':
A    I	1.1	1.08	 1.01	 0.1	 11.011        -1.0e-1
B   II	2.2     2.16     4.04    0.4     24.024        -1.0e-2
C  III	3.3     3.24     9.09    0.9     39.039        -1.0e-3
D   IV	4.4	4.32    16.16    1.6    416.039        -1.0e-4

has one column more than the file `many_columns1', namely the second
one. When comparing `many_columns1' with `many_columns3' it is natural
then to ignore the second column of the second file. This can be
achieved by passing the argument 2:2 to the option `-X' (for a full
description of the use of this option, *note Restriction of the
comparison to particular fields::).  Ignoring the second field of each
line of `many_columns3' implies that the fields in the third column of
this file are compared with the corresponding fields of the second
column of `many_columns1', the fields in the fourth column of
`many_columns3' are compared with the ones in the third column of
`many_columns1', and so on.  Therefore, if you want to set a threshold
value only for the comparisons between some particular fields you have
to consider that `-X 2:2' makes `numdiff' compare the first, second,
third, fourth, fifth, sixth, and seventh field of each line of
`many_columns1' with the first, third, fourth, fifth, sixth, seventh,
and eighth field respectively of the corresponding line of
`many_columns3'.  Therefore, the command `numdiff -X 2:2 -a 0.5:3-6
many_columns1 many_columns3' will use 0.5 as threshold value only when
comparing the third, fourth, and fifth field of a line from
`many_columns1' with the fourth, fifth, and sixth field respectively of
the corresponding line of `many_columns3'.  This explains why the
report of `numdiff -X 2:2 -a 0.5:3-6 many_columns1 many_columns3'
     ----------------
     ##1       #:2   <== 1
     ##1       #:3   ==> 1.1
      Absolute error = 1.0000000000e-1, Relative error = 1.0000000000e-1
     ##1       #:6   <== 11.0
     ##1       #:7   ==> 11.011
      Absolute error = 1.1000000000e-2, Relative error = 1.0000000000e-3
     ##1       #:7   <== 1.0e-1
     ##1       #:8   ==> -1.0e-1
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e+0
     ----------------
     ##2       #:2   <== 2
     ##2       #:3   ==> 2.2
      Absolute error = 2.0000000000e-1, Relative error = 1.0000000000e-1
     ##2       #:6   <== 24.0
     ##2       #:7   ==> 24.024
      Absolute error = 2.4000000000e-2, Relative error = 1.0000000000e-3
     ##2       #:7   <== 1.0e-2
     ##2       #:8   ==> -1.0e-2
      Absolute error = 2.0000000000e-2, Relative error = 2.0000000000e+0
     ----------------
     ##3       #:2   <== 3
     ##3       #:3   ==> 3.3
      Absolute error = 3.0000000000e-1, Relative error = 1.0000000000e-1
     ##3       #:6   <== 39.0
     ##3       #:7   ==> 39.039
      Absolute error = 3.9000000000e-2, Relative error = 1.0000000000e-3
     ##3       #:7   <== 1.0e-3
     ##3       #:8   ==> -1.0e-3
      Absolute error = 2.0000000000e-3, Relative error = 2.0000000000e+0
     ----------------
     ##4       #:2   <== 4
     ##4       #:3   ==> 4.4
      Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1
     ##4       #:6   <== 416.0
     ##4       #:7   ==> 416.039
      Absolute error = 3.9000000000e-2, Relative error = 9.3750000000e-5
     ##4       #:7   <== 1.0e-4
     ##4       #:8   ==> -1.0e-4
      Absolute error = 2.0000000000e-4, Relative error = 2.0000000000e+0

     +++  File "many_columns1" differs from file "many_columns3"

does not show the same difference listing of the command `numdiff -a
0.5:3-6 many_columns1 many_columns2'.

   If what you want is to obtain the same difference listing of
`numdiff -a 0.5:3-6 many_columns1 many_columns2', then the right
command is `numdiff -X 2:2 -a 0.5:3-6:4-7 many_columns1 many_columns3'.
The report printed by this last command is indeed
     ----------------
     ##1       #:2   <== 1
     ##1       #:3   ==> 1.1
      Absolute error = 1.0000000000e-1, Relative error = 1.0000000000e-1
     ##1       #:7   <== 1.0e-1
     ##1       #:8   ==> -1.0e-1
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e+0
     ----------------
     ##2       #:2   <== 2
     ##2       #:3   ==> 2.2
      Absolute error = 2.0000000000e-1, Relative error = 1.0000000000e-1
     ##2       #:7   <== 1.0e-2
     ##2       #:8   ==> -1.0e-2
      Absolute error = 2.0000000000e-2, Relative error = 2.0000000000e+0
     ----------------
     ##3       #:2   <== 3
     ##3       #:3   ==> 3.3
      Absolute error = 3.0000000000e-1, Relative error = 1.0000000000e-1
     ##3       #:7   <== 1.0e-3
     ##3       #:8   ==> -1.0e-3
      Absolute error = 2.0000000000e-3, Relative error = 2.0000000000e+0
     ----------------
     ##4       #:2   <== 4
     ##4       #:3   ==> 4.4
      Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1
     ##4       #:7   <== 1.0e-4
     ##4       #:8   ==> -1.0e-4
      Absolute error = 2.0000000000e-4, Relative error = 2.0000000000e+0

     +++  File "many_columns1" differs from file "many_columns3"

and up to the positions of the fields from `many_columns3' coincides
with the one of `numdiff -a 0.5:3-6 many_columns1 many_columns2'.

   The option `-a' can appear more times on the command line.  In case
of conflicts, the last setting is the one which matters.  If you look
at the report of the command `numdiff -a 0.5:3-6 -a 0.08:4
many_columns1 many_columns2'
     ----------------
     ##1       #:2   <== 1
     ##1       #:2   ==> 1.1
      Absolute error = 1.0000000000e-1, Relative error = 1.0000000000e-1
     ##1       #:7   <== 1.0e-1
     ##1       #:7   ==> -1.0e-1
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e+0
     ----------------
     ##2       #:2   <== 2
     ##2       #:2   ==> 2.2
      Absolute error = 2.0000000000e-1, Relative error = 1.0000000000e-1
     ##2       #:7   <== 1.0e-2
     ##2       #:7   ==> -1.0e-2
      Absolute error = 2.0000000000e-2, Relative error = 2.0000000000e+0
     ----------------
     ##3       #:2   <== 3
     ##3       #:2   ==> 3.3
      Absolute error = 3.0000000000e-1, Relative error = 1.0000000000e-1
     ##3       #:4   <== 9
     ##3       #:4   ==> 9.09
      Absolute error = 9.0000000000e-2, Relative error = 1.0000000000e-2
     ##3       #:7   <== 1.0e-3
     ##3       #:7   ==> -1.0e-3
      Absolute error = 2.0000000000e-3, Relative error = 2.0000000000e+0
     ----------------
     ##4       #:2   <== 4
     ##4       #:2   ==> 4.4
      Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1
     ##4       #:4   <== 16
     ##4       #:4   ==> 16.16
      Absolute error = 1.6000000000e-1, Relative error = 1.0000000000e-2
     ##4       #:7   <== 1.0e-4
     ##4       #:7   ==> -1.0e-4
      Absolute error = 2.0000000000e-4, Relative error = 2.0000000000e+0

     +++  File "many_columns1" differs from file "many_columns2"

you see that 0.08 and not 0.5 is taken as threshold value for the
comparison of the fields in fourth position.

   Finally, if `-a' is not present on the command line, then the
default threshold value of zero applies to all comparisons of numerical
fields and any non null absolute difference is considered as
significant, unless some other option, for example `-P' or `-N', makes
`numdiff' ignore it.

   The option `-r' can be used to order that two numerical fields must
be considered equal as long as their relative difference does not
exceed a certain threshold, which is specified by the argument that
follows the `-r' option.  As for the option `-a', the argument of `-r'
can have several forms. These forms are the same accepted by `-a' and
have the same meanings, but the threshold value applies to the relative
difference, not to the absolute one.

   The relative difference is normally defined in this way.  If _n1_ is
a value from the file specified as first on the command line and _n2_
is the corresponding value from the second file, then the absolute
difference is given by the formula _A=|n1-n2|_.  The relative
difference _R_ is given by:

   * _R = 0_ if _n1_ and _n2_ are equal,

   * _Inf_ (infinity) if _n2_ differs from _n1_ and at least one of
     them is zero,

   * _R = A/ min(|n1|, |n2|)_ if _n1_ and _n2_ are both non zero and
     _n2_ differs from _n1_. _min(|n1|, |n2|)_ denotes the minimum
     between the absolute value of _n1_ and the absolute value of _n2_.

With this definition of relative difference it turns out that _R(n2,
n1) = R(n1, n2)_: the relative difference does not change if you only
change the ordering of the compared files on the command line.

   However there are cases when this default definition of relative
error makes no sense. This can happen for instance when one of the
files is a sample file containing a list of expected data, which could
have been computed theoretically or come from experiments in a
laboratory.  In this case it is more natural to define the relative
difference as the ratio between the absolute difference and the absolute
value of the number coming from the sample file.  If you use the option
`-F' together with the argument 1 (or 2), then Numdiff always compute
the relative difference as the ratio between the absolute difference
and the absolute value of the number from the first file (the second
file, respectively).  More precisely, with `-F 1' the relative
difference _R_ is computed according to these rules:
   * _R = 0_ if _n1_ and _n2_ are equal,

   * _Inf_ (infinity) if _n2_ differs from _n1_ and _n1_ is zero,

   * _R = |n1-n2|/ |n1|_ if _n1_ is not zero and _n2_ differs from _n1_.

With `-F 2' the rules become:
   * _R = 0_ if _n1_ and _n2_ are equal,

   * _Inf_ (infinity) if _n2_ differs from _n1_ and _n2_ is zero,

   * _R = |n1-n2|/ |n2|_ if _n2_ is not zero and _n2_ differs from _n1_.

With the last two sets of rules it is not anymore true that _R(n2, n1)
= R(n1, n2)_: the relative difference changes, in the general case,
together with the ordering of the files on the command line.  As a
simple example, suppose that FILE1 and FILE2 contain
1    9.9  0.5  440

and
1.2  8    0.51 400

respectively. Then `numdiff FILE1 FILE2' displays
     ----------------
     ##1       #:1   <== 1
     ##1       #:1   ==> 1.2
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e-1
     ##1       #:2   <== 9.9
     ##1       #:2   ==> 8
      Absolute error = 1.9000000000e+0, Relative error = 2.3750000000e-1
     ##1       #:3   <== 0.5
     ##1       #:3   ==> 0.51
      Absolute error = 1.0000000000e-2, Relative error = 2.0000000000e-2
     ##1       #:4   <== 440
     ##1       #:4   ==> 400
      Absolute error = 4.0000000000e+1, Relative error = 1.0000000000e-1

     +++  File "file1" differs from file "file2"

`numdiff -F 1 FILE1 FILE2' prints
     ----------------
     ##1       #:1   <== 1
     ##1       #:1   ==> 1.2
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e-1
     ##1       #:2   <== 9.9
     ##1       #:2   ==> 8
      Absolute error = 1.9000000000e+0, Relative error = 1.9191919192e-1
     ##1       #:3   <== 0.5
     ##1       #:3   ==> 0.51
      Absolute error = 1.0000000000e-2, Relative error = 2.0000000000e-2
     ##1       #:4   <== 440
     ##1       #:4   ==> 400
      Absolute error = 4.0000000000e+1, Relative error = 9.0909090909e-2

     +++  File "file1" differs from file "file2"

the output of `numdiff -F 2 FILE1 FILE2' is
     ----------------
     ##1       #:1   <== 1
     ##1       #:1   ==> 1.2
      Absolute error = 2.0000000000e-1, Relative error = 1.6666666667e-1
     ##1       #:2   <== 9.9
     ##1       #:2   ==> 8
      Absolute error = 1.9000000000e+0, Relative error = 2.3750000000e-1
     ##1       #:3   <== 0.5
     ##1       #:3   ==> 0.51
      Absolute error = 1.0000000000e-2, Relative error = 1.9607843137e-2
     ##1       #:4   <== 440
     ##1       #:4   ==> 400
      Absolute error = 4.0000000000e+1, Relative error = 1.0000000000e-1

     +++  File "file1" differs from file "file2"

`numdiff -F 1 -r 0.195 FILE1 FILE2' displays
     ----------------
     ##1       #:1   <== 1
     ##1       #:1   ==> 1.2
      Absolute error = 2.0000000000e-1, Relative error = 2.0000000000e-1

     +++  File "file1" differs from file "file2"

and, finally, `numdiff -F 2 -r 0.195 FILE1 FILE2' displays
     ----------------
     ##1       #:2   <== 9.9
     ##1       #:2   ==> 8
      Absolute error = 1.9000000000e+0, Relative error = 2.3750000000e-1

     +++  File "file1" differs from file "file2"

The option `-2' is only meaningful when the user specifies a non-zero
tolerance threshold for both absolute and relative difference. Without
this option `numdiff' considers two numerical fields equal as long as
at least one between absolute and relative difference does not exceed
the corresponding threshold.  With the option `-2' `numdiff' regards
two numerical fields as equal only if both absolute and relative
difference do not exceed the thresholds of tolerance specified for
those fields.  For example, if FILE1 contains the unique line
100

and FILE2 the line
100.00012

then the output of the command `numdiff FILE1 FILE2' will be

     ----------------
     ##1       #:1   <== 100
                     ==> 100.00012
     @ Absolute error = 1.2000000000e-4, Relative error = 1.2000000000e-6

     +++  File "FILE1" differs from file "FILE2"

The output of the commands `numdiff -a 1.0e-4 FILE1 FILE2' and `numdiff
-r 1.0e-6 FILE1 FILE2' will be the same as above, but `numdiff -a
1.0e-4 -r 1.3e-6 FILE1 FILE2' and `numdiff -a 1.3e-4 -r 1.0e-6 FILE1
FILE2' will print the message

     +++  Files "FILE1" and "FILE2" are equal

since the relative difference is 1.2e-6 < 1.3e-6, the absolute
difference is 1.2e-4 < 1.3e-4, and it is sufficient that one of them
does not exceed its tolerance threshold.

   On the other hand, the commands `numdiff -a 1.0e-4 -r 1.3e-6 -2
FILE1 FILE2' and `numdiff -a 1.3e-4 -r 1.0e-6 -2 FILE1 FILE2' will
print the message

     ----------------
     ##1       #:1   <== 100
                     ==> 100.00012
     @ Absolute error = 1.2000000000e-4, Relative error = 1.2000000000e-6

     +++  File "FILE1" differs from file "FILE2"

since the option `-2' makes `numdiff' regard two values as equal only
if both absolute and relative difference do not exceed the
corresponding threshold of tolerance.

   The option `-P' makes `numdiff' consider two values equal whenever
the second one, i.e. the value coming from the file specified as last
on the command line, is less or equal than the first one, which is the
value coming from the file specified as first on the command line.  If
the values to compare are complex numbers, saying that the second one
is less or equal than the first one means that both real and imaginary
part of the second value are not greater than the real part and,
respectively, the imaginary part of the first value.

   Finally, the option `-N' makes `numdiff' consider two values equal
whenever the second one, i.e. the value coming from the file specified
as last on the command line, is greater or equal than the first one,
which is the value coming from the file specified as first on the
command line.  If the values to compare are complex numbers, saying that
the second one is greater or equal than the first one means that both
real and imaginary part of the second value are not less than the real
part and, respectively, the imaginary part of the first value.

   The options `-B', `-I', `-l', `-o', `-h' and `-v' do not require
further explanations.  The options `-l' and `-o' are only supplied for
the users of some poorly designed operating systems (like MSDog or
MSWindoze), whose default shell does not allow the redirection of
standard error and standard output.  The option `-I' has no effect on
the outcome of numerical comparisons but affects the action of the
filter, *note Filtering::.

   The option `-s' requires as argument a set of characters, which will
be taken as field delimiters.  It is better to quote the set of the
delimiters, just as in the next examples:

     numdiff -s ' \t\n,;:.' FILE1 FILE2
     numdiff -s ' \t\n\r\f\v"\:;' FILE1 FILE2
     numdiff -s `` \t\n''' FILE1 FILE2

If you want to include in the set of delimiters also some special
characters, e.g the *blank*, then you must quote it.  I recommend you
to always use the single quote character (') to enclose the list of the
delimiters, since in this way you will prevent any substitution or
handling of characters by the shell.

`numdiff' recognizes and interprets the following sequences of
characters within the argument passed to the option `-s':

   * `\a'     alert (bell),

   * `\b'     backspace,

   * `\f'     form feed,

   * `\n'     newline,

   * `\r'     carriage return,

   * `\s'     blank,

   * `\t'     horizontal tab,

   * `\v'     vertical tab,

   * `\\'     backslash,

   * `\NNN'   the eight-bit character whose value is the octal value
     NNN (one to three digits),

   * `\xHH'   the eight-bit character whose value is the hexadecimal
     value HH (one or two digits).

By passing the string ` \t\n,;:.' as argument for the option `-s', one
tells `numdiff' to use as field delimiters the characters *blank*,
*horizontal tab*, *newline*, *comma*, *semicolon*, *colon* and *dot*.
Passing ` \t\n' as argument to the option `-s' is the same as not using
at all the option `-s', since *blank*, *horizontal tab* and *newline*
are the default field delimiters.

   In the list of field delimiters the character *backslash* (`\') is
always treated in a special way.  If it forms, combined with the
subsequent character(s), one of the backslash escape sequences listed
above, then it is considered to be an escape character and the whole
escape sequence is decoded as shown above. Otherwise, the *backslash*
is just ignored.

   Therefore, the delimiters specified by the command line

     numdiff -s' \t\n\\\"' FILE1 FILE2

are *blank*, *horizontal tab*, *newline*, *backslash* and *double
quote*, since `\\' and `\"' are interpreted by `numdiff' as `\' and `"'.

   Even if I have recommended to enclose the set of delimiters in
single quotes, there are cases in which you will be constrained to use
the double quote character (`"') to enclose the set of field
delimiters, e.g. if the single quote character is used as field
delimiter, like in one of the precedent examples.  However you must
take into account that in this case the shell could make some
substitutions on the command line before executing `numdiff'.  For
instance, if your shell is GNU bash, then (citing the man page of GNU
bash)

     Enclosing characters in double quotes preserves the literal value
     of all characters within the quotes, with the exception of `$',
     ``', and `\'.  The characters `$' and ``' retain their special
     meaning within double quotes.  The backslash retains its special
     meaning only when followed by one of the following characters:
     `$', ``', `"', `\', or *<newline>*.  A double quote may be quoted
     within double quotes by preceding it with a backslash ...  The
     special parameters * and @ have special meaning when in double
     quotes ...

Therefore, if the set of delimiters is formed by ` ', `\t', `\n', `\'
and `"', and you decide to enclose them in double quotes, the `numdiff'
command line should be

     numdiff -s'' \t\n\\\\\"'' FILE1 FILE2

and not

     numdiff -s'' \t\n\\\"'' FILE1 FILE2

In this last case the shell would indeed replace the string

   ` \t\n\\\"'

by

   ` \t\n\"'

and then `numdiff' would take ` ', `\t', `\n' and `"' as field
delimiters.

   `numdiff' requires the presence of the *newline* in the set of
characters passed to `-s'. The absence of the *newline* in the set of
delimiters causes the issue of a suitable warning message and the
termination of the program.

   If you run Numdiff with the option `-B' (`--binary') on files
created under MSDog/MSWindoze, then you should put the *carriage
return* in the set of field delimiters.  Otherwise, this character
would be included in all the fields which stay at the end of a line and
this would cause some undesirable effects. For instance, a number put
at the end of a line would not be regarded as a numerical field by
`numdiff', since `numdiff' would consider the final *carriage return*
as part of the field and this one would be then qualified as
non-numerical.

   You can specify different delimiters for the two files to compare by
putting the prefix `1:' or `2:' in front of the set of characters
passed to `-s'.  If the argument of `-s' begins with `1:', the
characters after this prefix are used as field delimiters only for the
file passed as first on the command line.  Analogously, if the prefix
is `2:', then the characters after it are used as field delimiters only
for the file specified as second on the command line.  You can also
provide an explicit set of delimiters for just one of the files to
compare, in which case `numdiff' uses the default field delimiters
*blank*, *tab* and *newline* for the other file.  Therefore, with
`numdiff -s '1:: \n' FILE1 FILE2' the program will take *colon*, *blank*
and *newline* as delimiters for FILE1, and *blank*, *tab* and *newline*
as delimiters for FILE2.  The recommendations about quoting the set of
delimiters are valid also in presence of a prefix.

   Starting from version 5.8 `numdiff' allows to specify whole strings
as field delimiters instead of single characters.  To this purpose the
option `-D' is provided.  Assume that file `register1' and file
`register2' contain
--A:    +1.0---
--B:    -2.0---
--C:    +3.0---
--D:    -4.0---
--E:    +5.0---
--F:    -6.0---

and
--a:    +1.1---
--b:    -2.2---
--c:    +3.3---
--d:    -4.4---
--e:    +5.5---
--f:    -6.6---

respectively. Assume in addition, you would like that the dashes at the
begin and at the end of every line are treated as delimiters and then
neglected during the line by line comparison. To obtain this you cannot
just specify the character - (minus) as delimiter via the option `-s':
if you do it, then the negative numbers appearing in the two files will
be treated as positive, since the minus sign will be regarded as a
delimiter. By means of the option `-D' you can tell `numdiff' to
consider the strings -- and --- as field delimiters but not the single
character -.  To see this in practice, look at the output of the command
`numdiff -D ': -- --- \s \n' register1 register2':
     ----------------
     ##1       #:1   <== A
     ##1       #:1   ==> a
                                                          @
     ##1       #:2   <== +1.0
     ##1       #:2   ==> +1.1
      Absolute error = 1.0000000000e-1, Relative error = 1.0000000000e-1
     ----------------
     ##2       #:1   <== B
     ##2       #:1   ==> b
                                                          @
     ##2       #:2   <== -2.0
     ##2       #:2   ==> -2.2
      Absolute error = 2.0000000000e-1, Relative error = 1.0000000000e-1
     ----------------
     ##3       #:1   <== C
     ##3       #:1   ==> c
                                                          @
     ##3       #:2   <== +3.0
     ##3       #:2   ==> +3.3
      Absolute error = 3.0000000000e-1, Relative error = 1.0000000000e-1
     ----------------
     ##4       #:1   <== D
     ##4       #:1   ==> d
                                                          @
     ##4       #:2   <== -4.0
     ##4       #:2   ==> -4.4
      Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1
     ----------------
     ##5       #:1   <== E
     ##5       #:1   ==> e
                                                          @
     ##5       #:2   <== +5.0
     ##5       #:2   ==> +5.5
      Absolute error = 5.0000000000e-1, Relative error = 1.0000000000e-1
     ----------------
     ##6       #:1   <== F
     ##6       #:1   ==> f
                                                          @
     ##6       #:2   <== -6.0
     ##6       #:2   ==> -6.6
      Absolute error = 6.0000000000e-1, Relative error = 1.0000000000e-1

     +++  File "register1" differs from file "register2"

The argument `-D ': -- --- \s \n'' instructs `numdiff' to regard every
occurrence of a colon (:), of a blank (\s), of a newline (\n), as well
as every occurrence of the strings -- and --- as field delimiters. The
minus sign in front of the negative numbers is then handled as it
should be.

   In general the argument to the option `-D' is a blank separated
sequence of one or more strings each of which contains no blank.  Thus,
the general form of the argument to the option `-D' is
     STRING1 STRING2 ... STRINGN

where STRING1, STRING2, and so on are sequences of one or more
characters (strings) containing no blank.

   Mind that at least one of these strings must be `\n'.  In addition,
if a string contains the newline character, this must be the only one:
strings like `#\n', `%%\n', or `\s\n' are not allowed (entering such a
string makes the program terminate after issuing a suitable warning
message).

   Since the blank character has a special meaning for the shell, if
the argument of `-D' is formed by two or more strings it should be
quoted either with a single (`'') or with a double quote (`"').
Quoting is also adviced if one of the strings passed to `-D' contains a
character (or a sequence of characters) having a special meaning for
the shell.

   For the usage of single and double quoting to delimit the argument
of `-D' the same warnings and recommendations apply as for the argument
of `-s'.

   If you want to set as delimiter a string which contains one or more
blanks, then you have to make use of the escape sequence `\s', like in
the example above: within the argument of `-D' the blank character is
always interpreted as a separator of adjacent delimiters.

   More generally, when writing the argument of `-D' the same escape
sequences are allowed as for the argument of `-s'.  This turns out to
be particularly useful whenever a multibyte character is used as
delimiter in (one of) the files to compare.  As example consider the
comparison between `ledger1':
                  In                            Out

     Jan      1200.00Euro                 1000.00Euro
     Feb       800.40Euro                  650.00Euro
     Mar      1620.50Euro                 1500.00Euro
     Apr       760.00Euro                  900.00Euro

     Total    4380.90Euro                 4050.00Euro

     Difference: +330.90Euro

and `ledger2':
                  In                            Out

     Jan      1100.00Euro                 1000.00Euro
     Feb       800.40Euro                  750.00Euro
     Mar      1620.50Euro                 1700.00Euro
     Apr       750.00Euro                  900.00Euro

     Total    4270.90Euro                 4350.00Euro

     Difference: -79.10Euro

   Since the Euro symbol is attached to all values, `numdiff' cannot
compare them in the proper way if it is run with the default field
delimiters, as the output of the command `numdiff ledger1 ledger2'
shows:
     ----------------
     ##3       #:2   <== 1200.00Euro
     ##3       #:2   ==> 1100.00Euro
                                                          @
     ----------------
     ##4       #:3   <== 650.00Euro
     ##4       #:3   ==> 750.00Euro
                                                          @
     ----------------
     ##5       #:3   <== 1500.00Euro
     ##5       #:3   ==> 1700.00Euro
                                                          @
     ----------------
     ##6       #:2   <== 760.00Euro
     ##6       #:2   ==> 750.00Euro
                                                          @
     ----------------
     ##8       #:2   <== 4380.90Euro
     ##8       #:2   ==> 4270.90Euro
                                                          @
     ##8       #:3   <== 4050.00Euro
     ##8       #:3   ==> 4350.00Euro
                                                          @
     ----------------
     ##10      #:2   <== +330.90Euro
     ##10      #:2   ==> -79.10Euro
                                                          @

     +++  File "ledger1" differs from file "ledger2"

The trick to perform the comparison in the proper way consists in
specifying the Euro  symbol as field delimiter, in addition to blank,
horizontal tabulation and newline.  If `ledger1' and `ledger2' are
encoded in UTF-8, this can be done by using the option `-D' with the
argument `\xE2\x82\xAC \s \t \n', since the hexadecimal representation
of Euro  in UTF8 is given by the byte sequence 0xE2 0x82 0xAC.  On my
PC the output of the command `numdiff -D '\xE2\x82\xAC \s \t \n'
ledger1 ledger2' shows that in this case `numdiff' performs indeed a
numerical comparison of the values contained in the two files:
     ----------------
     ##3       #:2   <== 1200.00
     ##3       #:2   ==> 1100.00
      Absolute error = 1.0000000000e+2, Relative error = 9.0909090909e-2
     ----------------
     ##4       #:3   <== 650.00
     ##4       #:3   ==> 750.00
      Absolute error = 1.0000000000e+2, Relative error = 1.5384615385e-1
     ----------------
     ##5       #:3   <== 1500.00
     ##5       #:3   ==> 1700.00
      Absolute error = 2.0000000000e+2, Relative error = 1.3333333333e-1
     ----------------
     ##6       #:2   <== 760.00
     ##6       #:2   ==> 750.00
      Absolute error = 1.0000000000e+1, Relative error = 1.3333333333e-2
     ----------------
     ##8       #:2   <== 4380.90
     ##8       #:2   ==> 4270.90
      Absolute error = 1.1000000000e+2, Relative error = 2.5755695521e-2
     ##8       #:3   <== 4050.00
     ##8       #:3   ==> 4350.00
      Absolute error = 3.0000000000e+2, Relative error = 7.4074074074e-2
     ----------------
     ##10      #:2   <== +330.90
     ##10      #:2   ==> -79.10
      Absolute error = 4.1000000000e+2, Relative error = 5.1833122630e+0

     +++  File "ledger1" differs from file "ledger2"

   If `ledger1' and `ledger2' had been saved using a multi-byte
encoding different from UTF-8, then the sequence of bytes which
corresponds to Euro  in this other encoding should have been passed to
`-D'.

   As for `-s', with `-D' you can specify different delimiters for the
two files to compare by means of the prefixes `1:' and `2:', like in
`numdiff -D '1:\t \n' -D '2: -- \s \n' first_file second_file'.  The
recommendations about quoting the set of delimiters are valid also in
presence of a prefix.  Mind that, if you provide an explicit set of
delimiters for just one of the files to compare, `numdiff' uses the
default field delimiters *blank*, *tab* and *newline* for the other
file.

   If you run Numdiff with the option `-B' (`--binary') on files
created under MSDog/MSWindoze, you should always include the character
`\r' in the set of field delimiters.

   The option `-s' and `-D' can appear more than once on the command
line. In case of conflicts, `numdiff' assumes as set of delimiters for
a given file the one specified last on the command line.

   By means of the option `-#' the user can set the number of digits in
the significands used in multiple precision arithmetic.  The default
value is 35, the largest admissible value is 180.  If `numdiff' has
been linked against the GNU Multiple Precision Arithmetic Library (also
called GNU MP), then the precision it uses is typically higher than the
specified one. On my machine the actual value of the precision is 20 if
the user gives a value between 0 and 20, 30 if the user specifies a
precision between 21 and 30, 40 for a user-specified value between 31
and 40, and so on.  Anyway, the actual precision is never less than the
one required by the user.

Take into account that an higher precision makes the execution of
`numdiff' slower. This is particularly true if `numdiff' is not using
the computational routines from the GNU MP library and the files to
compare contain a lot of numerical fields.  In addition, you have to
care that `numdiff' truncates the value of a numerical field if it has
too much digits with respect to the current precision. To be precise,
denoted by P the current value of the precision, the following rules
apply.
   * If `numdiff' has been built with its own internal support for
     multiple precision arithmetic, then
        * if a number is written in ordinary decimal notation, `numdiff'
          will consider, in addition to all digits of the integer part,
          only the first P digits of the fractional part;

        * if a value is written in scientific notation, then `numdiff'
          will only consider the first P digits of the fractional part
          of the mantissa.

   * If `numdiff' uses the routines from the GNU MP library to perform
     its computations, the value of a numerical field is first
     translated into scientific notation and then only the first P
     digits of the fractional part of the mantissa are considered.

You can find out whether your local version of `numdiff' is relying on
GNU MP or not by executing the command `numdiff -v'.  If `numdiff' uses
GNU MP, then this command will display the following message or similar
(possibly translated into your mother language) among other information:
     The software has been linked against
     the GNU Multiple Precision Arithmetic Library,
     version number 4.2.4.

If `numdiff' does not rely on GNU MP, then the displayed message will
be (up to translation into your mother language)
     The software has been built with
     its own internal support for multiple precision arithmetic.

By means of the option `-c' the user can qualify a string as a symbol
or name for a currency. The string passed as argument to this option is
ignored by `numdiff' whenever it appears immediately before the first
digit of a number.  In particular, the presence of this string does not
prevent a field from being considered of numeric type. By prefixing the
argument of `-c' with `1:' or `2:' it is possible to set the currency
name/symbol only for one of the compared files, or to specify different
currency names for the two files. As example we consider the files
`money1':
	Profits			Expenses

	+$430.10		-$300.50
	+$750.20		-$550.02
	+$876.24		-$720.00

Totals  $2056.54               -$1570.52

and `money2':
	Profits			Expenses

	USD430.10		-USD300.50
	USD750.20		-USD550.02
	USD876.24		-USD720.15

Totals    2056.54                 -1570.67
 To properly compare them we have to tell `numdiff' that `$' and `USD'
are the currency symbols for `money1' and `money2' respectively.  This
can be achieved by `-c 1:$' and `-c 2:USD'.  The output of the command
`numdiff -c 1:$ -c 2:USD money1 money2' is
     ----------------
     ##5       #:2   <== -$720.00
     ##5       #:2   ==> -USD720.15
      Absolute error = 1.5000000000e-1, Relative error = 2.0833333333e-4
     ----------------
     ##7       #:3   <== -$1570.52
     ##7       #:3   ==> -1570.67
      Absolute error = 1.5000000000e-1, Relative error = 9.5509767466e-5

     +++  File "money1" differs from file "money2"

as it should be.

   The argument of `-c' may also be a multi-byte string, in particular
a multi-byte string encoded in UTF-8.  If your locale uses UTF-8 as
encoding, you can write the argument directly in this form. For
instance, you can write `-c Euro ' to specify as currency name the Euro
symbol.  If your locale does not use UTF-8 as encoding, or UTF-8 is not
supported by your terminal, you may still write an UTF-8 encoded string
as a multi-byte string by specifying each single byte of every
(multi-byte) character.  To this purpose you can use the same octal and
hexadecimal escape sequences recognized by the options `-s' and `-D'.

   For example, if the files to compare are encoded in UTF-8, you can
set Euro  as currency name by adding `-c '\xE2\x82\xAC'' to the command
line of `numdiff', since the hexadecimal representation of Euro  in
UTF-8 is given by the sequence of bytes 0xE2 0x82 0xAC.  Mind that in
this case the argument of `-c' has to be quoted to avoid the
interpretation of the hexadecimal escape sequences by the shell.

   To see this in practice, if `euro1' contains the text
     	Profits			       Expenses

     	+Euro 430.10		      -Euro 300.50
     	+Euro 750.20		      -Euro 550.02
     	+Euro 876.24		      -Euro 720.00

and `euro2' the text
     	Profits			       Expenses

     	+Euro 430.10		      -Euro 300.00
     	+Euro 750.20		      -Euro 550.02
     	+Euro 876.00		      -Euro 720.00

then the report of `numdiff -c '\xE2\x82\xAC' euro1 euro2' is
     ----------------
     ##3       #:2   <== -Euro 300.50
     ##3       #:2   ==> -Euro 300.00
      Absolute error = 5.0000000000e-1, Relative error = 1.6666666667e-3
     ----------------
     ##5       #:1   <== +Euro 876.24
     ##5       #:1   ==> +Euro 876.00
      Absolute error = 2.4000000000e-1, Relative error = 2.7397260274e-4

     +++  File "euro1" differs from file "euro2"

   Please consider that `-c' is only provided to let `numdiff' regard a
field as numeric also in presence of a currency name immediately before
its first digit: `numdiff' does not know anything about currencies and
can not perform any kind of conversion between them.  In addition, mind
that the number after the currency name can be written in any format,
not only in financial notation.  `numdiff' can even cope with the
currency name when it appears in a complex number. For example, with
`-c EUR' `numdiff' considers +EUR12-EUR0.24i and +12-0.24i as equal.

   The options `-d', `-t', `-g', `-p', `-n', `-e' and `-i' can be used
to instruct `numdiff' about the numeric formats used in the files which
it is going to compare.  The two files to compare do not have to adopt
the same numeric format and then `numdiff' allows to specify different
numeric formats for them.  Each of the options `-d', `-t', `-g', `-p',
`-n', `-e', and `-i' can have as argument one or two (single-byte)
characters, in particular one or two digits if the option is `-g'. In
the first case the argument refers to both files to compare, in the
second case the first character is for the file specified first on the
command line, the second character for the file specified last.  For
instance, the option `-d' can be used to tell `numdiff' which
character(s) is(are) used to indicate the decimal point in the two
files to compare.  If you give the command `numdiff -d_ FILE1 FILE2',
then `numdiff' will understand that both in FILE1 and in FILE2 the
character *underscore* (`_') is used in place of the default one (`.')
to indicate the position of the decimal point in the numerical values.
But if the command is `numdiff -d_: FILE1 FILE2', then `numdiff' will
understand that the decimal point is indicated by the character
*underscore* in FILE1, and by *colon* (`:') in FILE2.

   If you omit to use one of the options `-d', `-t', `-g', `-p', `-n',
`-e', and `-i', then the corresponding attribute will take its default
value, *note Default Numeric Format::.

   You should be careful whenever you use one or more of these options.
First, not all characters can be passed to them as arguments.  The
arguments of the option `-g' must be digits, the arguments of the
options `-d' and `-t' must be punctuation marks (punctuation marks are
all the characters of the ASCII set for which the standard C function
`ispunct' returns a non zero value), those ones of the options `-p',
`-n', `-e' and `-i' must be graphical characters but digits (graphical
characters are all the characters of the ASCII set for which the
standard C function `isgraph' returns a non zero value).

   It is not possible to set the decimal point, the thousands separator,
the positive sign, the negative sign, the prefix for decimal exponent
or the symbol of the imaginary unit in such a way that, for a same file,
two or more of these characters come out to be equal.  This rule also
applies if you miss/omit to explicitly select a symbol through the
appropriate option.  For instance, the command `numdiff -d,. FILE1
FILE2' will make `numdiff' abnormally terminate after printing the
error message:

     The numeric format specified for the first file is illegal,
     the following symbols should be all different
     while two or more of them are actually equal:

     Decimal point = `,'
     Thousands separator = `,'
     Leading positive sign = `+'
     Leading negative sign = `-'
     Prefix for decimal exponent = `e'
     Symbol used to denote the imaginary unit = `i'

With the option `-d' we have told `numdiff' that in the first file the
decimal point is indicated by the character *comma*, but at the same
time we have not modified the character in use to separate the groups
of thousands, which has remained the default one, i.e. *comma*, for
both files to compare.  In this way we have implicitly told that in
FILE1 the character *comma* represents both decimal point and thousands
separator. Since this is not reasonable, `numdiff' refuses to work.  To
avoid this problem it would be sufficient to set explicitly the
thousands separator by means of the option `-t': `numdiff -d,. -t.,
FILE1 FILE2'.  Of course, we assume here that the decimal point and the
thousands separator are represented in FILE1 by *comma* and *dot*
respectively, in FILE2 by *dot* and *comma*.

   I strongly suggest you, whenever you write a file, to avoid using
the same symbol to mean two different things (like would be using
*comma* for both decimal point and thousands separator), it is nonsense.

   At last, it is possible (but stupid) to use as argument for the
options `-d', `-t', `-g', `-p', `-n', `-e', and `-i' one of the
characters used as delimiters in the files to compare.  In such a case
`numdiff' does not complain, but you have to consider that it first
uses the set of field delimiters to split the files into fields and
then, when it has to distinguish between numerical and non-numerical
fields, it takes into account the numeric formats specified for the two
files.  However, it should never happen to specify as argument for one
of the options `-d', `-t', `-g', `-p', `-n', `-e', and `-i' a character
which is also used as field delimiter: in writing a file you should
avoid (and people usually avoid it) to use the same symbol to mean two
different things.  What we have said also explains why the argument of
the option `-c' should never contain one or more field delimiters.

   The option `-X' can be used to restrict the comparison between files
to a certain group of fields.  This option requires as argument a range
of positive integer values or eventually just one positive integer
number. The argument specifies the position(s) of the fields that
`numdiff' has to ignore.  Remember that the fields of a line are
numerated starting from the left hand of the line and proceeding
towards the right hand.

   The argument passed to `-X' can start with a prefix, which must be
either `1:' or `2:'.  `1:' refers to the file passed as first on the
command line, `2:' to the file specified as second.  With the prefix
`1:' only the fields of the first file corresponding to the given
position(s) are ignored.  Similarly, if you want to ignore only fields
from the second file you have to use the prefix `2:'.

   The option `-X' can appear more times on the command line, in which
case `numdiff' will ignore all fields located in the positions so
specified.  Some examples can clarify the use of ranges and prefixes.
If the file `List1' contains the data
* a  1    1     1     1
* b  2    2     2     2
* c  3    3     3     3
* d  4    4     4     4
* e  5    5     5     5

and `List2' the data
     1    1.1  1.01   A  1.001  1.0001
     2    2.2  2.02   B  2.002  2.0002
     3    3.3  3.03   C  3.003  3.0003
     4    4.4  4.04   D  4.004  4.0004
     5    5.5  5.05   E  5.005  5.0005

then the output of `numdiff -X 1:1-2 -X 2:4 -X 1:6 -X 2:5-6 List1
List2' is
     ----------------
     ##1       #:4   <== 1
     ##1       #:2   ==> 1.1
     @ Absolute error = 1.0000000000e-1, Relative error = 1.0000000000e-1
     ##1       #:5   <== 1
     ##1       #:3   ==> 1.01
     @ Absolute error = 1.0000000000e-2, Relative error = 1.0000000000e-2
     ----------------
     ##2       #:4   <== 2
     ##2       #:2   ==> 2.2
     @ Absolute error = 2.0000000000e-1, Relative error = 1.0000000000e-1
     ##2       #:5   <== 2
     ##2       #:3   ==> 2.02
     @ Absolute error = 2.0000000000e-2, Relative error = 1.0000000000e-2
     ----------------
     ##3       #:4   <== 3
     ##3       #:2   ==> 3.3
     @ Absolute error = 3.0000000000e-1, Relative error = 1.0000000000e-1
     ##3       #:5   <== 3
     ##3       #:3   ==> 3.03
     @ Absolute error = 3.0000000000e-2, Relative error = 1.0000000000e-2
     ----------------
     ##4       #:4   <== 4
     ##4       #:2   ==> 4.4
     @ Absolute error = 4.0000000000e-1, Relative error = 1.0000000000e-1
     ##4       #:5   <== 4
     ##4       #:3   ==> 4.04
     @ Absolute error = 4.0000000000e-2, Relative error = 1.0000000000e-2
     ----------------
     ##5       #:4   <== 5
     ##5       #:2   ==> 5.5
     @ Absolute error = 5.0000000000e-1, Relative error = 1.0000000000e-1
     ##5       #:5   <== 5
     ##5       #:3   ==> 5.05
     @ Absolute error = 5.0000000000e-2, Relative error = 1.0000000000e-2

     +++  File "List1" differs from file "List2"

Numdiff cuts off from `List1' the fields in the positions 1, 2 and 6
and from `List2' the fields in the positions 4, 5 and 6.  In this way
it compares the third, fourth and fifth field of every line of `List1'
with the first, second and third field respectively of the
corresponding line of `List2'.  An equivalent form of the command
`numdiff -X 1:1-2 -X 2:4 -X 1:6 -X 2:5-6 List1 List2' is given by
`numdiff -X 1:1-2 -X 2:4-5 -X 6 List1 List2' : since the sixth field is
cut off from both files we can refer to it without a prefix.

   As you can see, you can specify a range of fields by using the
notation `M-N', where M and N are the field numbers of the first and of
the last field in the range.  It is even possible to use range
expressions like `M-' or `-N'. The first expression corresponds to all
fields starting from the Mth one (inclusive) till to the end of line,
the second selects all fields from the first one till to the Nth one
(inclusive).  Therefore the command `numdiff -X 1:1-2 -X 2:4 -X 1:6 -X
2:5-6 List1 List2' is equivalent to `numdiff -X 1:-2 -X 2:4 -X 1:6 -X
2:5- List1 List2' or to `numdiff -X 1:-2 -X 1:6 -X 2:4- List1 List2'.

   Mind that the largest field number you can use while writing a
specification for the option `-X' is 32768.

   If you use the option `-X' the exit status of `numdiff' reflects the
outcome of the restricted comparison.  For instance, the exit status of
`numdiff -X 8- FILE1 FILE2' is 1 only if `numdiff' has found a
difference in the first seven fields of FILE1 and FILE2.  If the two
files differ only in the fields after the seventh one, then `numdiff'
ends with a zero exit status.

   Going back to the example with `List1' and `List2', the output of
`numdiff -X 1:1-2 -X 1:4- -X 2:2- List1 List2' is

     +++  Files "List1" and "List2" are equal

since every field of `List1' at position 3 is equal to the first field
in the corresponding line of `List2'.  The exit code returned by the
program to the shell is zero.

   The options `-z', `-Z', `-m', `-H', `-f', and `-T' influence the
action of the filter and their use is then described later, *note
Filtering::.  Care that `-z' and `-Z' need both an argument in the same
form required by `-X'.

   Since version 5 Numdiff accepts also long options to conform to the
GNU standards. Now it is then possible, e.g., to use
`--separators='\n\t %'' or `--separators '\n\t %'' instead of using `-s
'\n\t %''.

   The long options, which start all with two dashes, are listed at the
beginning of this chapter, each one near to the corresponding short
option.

   The argument of a long option may or may not preceded by the = sign.
The only exceptions are the options `--test-filter' and `--overview',
for which the presence of the = before the argument is mandatory.  Then
`--test-filter=60' is correct while `--test-filter 60' is not accepted.

6 Selecting lines and fields for the comparison
***********************************************

Together with the version 5.x of Numdiff is shipped the program
`ndselect'.  Originally, I decided to create this utility in order to
deal with a situation that comes out often in Numerical Analysis. Here
I present a very simple example of such a situation. Let us suppose
that file `list1' contains the values of the square root, rounded to
the 20th decimal digit, for all integer numbers between 12 and 24:
12       3.46410161513775458705
13       3.60555127546398929312
14       3.74165738677394138558
15       3.87298334620741688518
16       4
17       4.12310562561766054982
18       4.24264068711928514641
19       4.35889894354067355224
20       4.47213595499957939282
21       4.58257569495584000659
22       4.69041575982342955457
23       4.7958315233127195416
24       4.89897948556635619639

and LIST2 contains _suitable_ approximations of the square root only
for the numbers between 12 and 21 which are multiple of 3:

12       3.46410162002945508100
15       3.87298387096774193548
18       4.24264705882352941176
21       4.58260869565217391304

These approximations could have been obtained by using the famous
Heron's algorithm, which, given an approximation `a' for the square
root of a number `x', computes a better approximation by the formula `a
:= 0.5 * (x/a + a)'.  What we want now is to understand by using
`numdiff' how good the approximations contained in file LIST2 are.
Unfortunately, we cannot execute directly the command `numdiff list1
list2', since in this way we would compare the approximations provided
for the square roots of 15, 18, and 21 with the square roots of 13, 14,
and 15 respectively. To make the comparison in the right way, one could
open `list1' in a text editor and remove from this file all lines but
the ones related to the numbers 12, 15, 18, and 21.  This approach is
practicable since we have to remove only a few lines: one can easily
figure out how boring and inefficient would be to manually remove
hundreds or thousands of lines from a file.

   An expert GNU user would suggest that it is possible to automate
this removal by using the well known utilities `head' and `sed', in
this particular case `head -n 10 list1 | sed -n -e '1~3 p' > List1'.  A
quick explanation for the ones who do not know how to use `head' and
`sed': the previous command extracts from `list1' the first 10 lines,
namely the lines containing the square roots of the numbers from 12 to
21, then picks every third line starting from the first one to select
only the lines related to 12, 15, 18, and 21.  Finally, these lines are
printed on the file `List1', which then looks like:
12       3.46410161513775458705
15       3.87298334620741688518
18       4.24264068711928514641
21       4.58257569495584000659

   Once obtained `List1', we can perform the comparison between the
values we are interested in by means of `numdiff List1 list2' .
Unfortunately, this trick only works if you have installed the GNU
version of `sed', which, as far as I know, is the only one to provide
the extension FIRST~STEP to specify line addresses.  That is way I
decided to implement `ndselect', which allows to obtain the same result
as above with the simpler command `ndselect -b 1 -e 10 -s 3 list1 >
List1'

   The meaning of the arguments passed to the options `-b', `-e', and
`-s' is the following: we tell `ndselect' to print every third line of
file `list1' (the option `-s' specifies the step) starting from the
first one (the option `-b' specifies the beginning) and ending within
the tenth one possibly inclusive (the option `-e' specifies the end).
Because of the presence of the redirection operator `>', the previous
command sends to the file `List1' what `ndselect' would print on the
screen (standard output).

   Since version 5.6 `ndselect' can also be used to select particular
fields of a file.  Instead of printing all fields of every line, you
may want to print indeed only the fields at particular positions.  To
do this you can employ the option `-F' to indicate the position of the
first field to print, the option `-L' to indicate the position of the
last field that can be printed, the option `-I' to set the increment
when selecting the fields. In addition, the option `-S' can be used to
specify a set of field delimiters different from the default one (which
consists of *blank*, *tab* and *newline*). As for `numdiff', the field
delimiters are used to split the input lines into fields.

   The option `-S' of `ndselect' recognizes and accepts the same escape
sequences of `numdiff' options `-s', `-D', and `-c'.

   As example consider the selection of the even fields between the
second and the sixth one inclusive from the file `many_many_columns',
whose contents are shown here:
A |   I | 1.1 |   1.08  |   1.01  |  0.1  |   11.011    |    -1.0e-1
B |  II | 2.2 |   2.16  |   4.04  |  0.4  |   24.024    |    -1.0e-2
C | III | 3.3 |   3.24  |   9.09  |  0.9  |   39.039    |    -1.0e-3
D |  IV | 4.4 |   4.32  |  16.16  |  1.6  |  416.039    |    -1.0e-4
E |   V | 5.5 |   5.40  |  25.25  |  2.5  |  525.416    |    -1.0e-5
F |   # |  #  |    #    |    #    |   #   |     #       |      #

This selection can be accomplished by means of the command `ndselect -S
'| \t\n' -F 2 -L 6 -I 2 many_many_columns', whose output shows only the
selected fields:
     I | 1.08  |   0.1
     II | 2.16  |   0.4
     III | 3.24  |   0.9
     IV | 4.32  |  1.6
     V | 5.40  |  2.5
     # |  #    |    #

Of course, you can also select particular fields of particular lines,
as shown by the output of the command `ndselect -S '| \t\n' -b 1 -e 5
-s 3 -F 2 -L 6 -I 2 many_many_columns':
     I | 1.08  |   0.1
     IV | 4.32  |  1.6

   By default, `ndselect' reuses the delimiters found in the input
lines while writing the selected fields to the standard output.  You
can specify a custom separator by means of the option `-O'.  This one
recognizes and accepts the same escape sequences of `numdiff' options
`-s', `-D', and `-c'.  For example, `ndselect -S '| \t\n' -b 1 -e 5 -s
3 -F 2 -L 6 -I 2 -O '\t\t' many_many_columns' puts two horizontal
tabulations after every printed field:
I		1.08		0.1
IV		4.32		1.6

   Even if the implementation of a filter in `numdiff' and the addition
of the option `-X' have made `ndselect' much less useful than in the
past, this tool can still be used to handle some special cases.  In
addition, it can be used as a filter for other programs than `numdiff'.
The complete synopsis of `ndselect' can be found in the next chapter.

7 Invoking ndselect
*******************

*SYNOPSIS*

        ndselect -h|--help|-v|--version
   or
        ndselect [-b N][-e N][-s N][-F N][-L N][-I N][-S IFS][-D DELIMS]
        [-O OSEP][-x][-l PATH][-o PATH][FILE]

where FILE is the name of the file to read from.

   In the first case `ndselect' prints a short help or/and version
number, Copyright, License notice and NO-Warranty disclaimer.  In the
second case `ndselect' prints on the standard output a subset of lines
and fields from FILE.  The complete path of FILE should be given, a
directory name is not accepted.  If no input file is specified, the
program reads from the standard input.

*OPTIONS*

`-b, --beginning, --start=N'
     Set to N the number of the first line to print (The default
     behavior is to start with line number 1)

`-e, --end=N'
     Set to N the number of the last line that can be printed (The
     default behavior is to arrive till to the end of the file)

`-s, --step=N'
     Set to N the increment to use when selecting the lines to print
     (The default value for the increment is 1)

`-F, --first-field=N'
     Set to N the number of the first field to print (The default
     behavior is to start with field number 1)

`-L, --last-field=N'
     Set to N the number of the last field that can be printed (The
     default behavior is to arrive till to the end of every line)

`-I, --increment=N'
     Set to N the increment to use when selecting the fields to print
     (The default value for the increment is 1)

`-S, --separators=IFS'
     Specify the set of characters to use as delimiters while splitting
     the input lines into fields (The default set of characters is
     space, tab and newline)

`-D, --delimiters=DELIMS'
     Specify the set of strings to use as delimiters while splitting
     the input lines into fields (The default set of delimiters is
     space, tab and newline)

`-O, --output-separator=OSEP'
     Specify the string to use as separator while writing the selected
     fields to the standard output (The default behavior consists in
     reusing the delimiters found in the input lines)

`-x, --omit-empty-lines'
     Do not print empty lines

`-l, --warnings-to=PATH'
     Redirect warning and error messages from stderr to the indicated
     file

`-o, --output=PATH'
     Redirect output from stdout to the indicated file

`-h, --help'
     Show this help message

`-v, --version'
     Show version number, Copyright, Distribution Terms and NO-Warranty

Passing 0 as argument to the option `-L' or to `-e' is equivalent to
omit this option and leave enabled the default behavior (which consists
in scanning till to the end of the line and of the file, respectively).

*DIAGNOSTICS*

   The exit status is 0 in case of normal termination, -1 (255) in case
of error.

As `numdiff' does, since version 5 also `ndselect' accepts long
options. Thus, instead of `ndselect -b 1 -e 10 -s 3 list1 > List1' you
can write `ndselect --start=1 --end=10 --step=3 list1 > List1'.

   The usage of the option `-D' is the same as for `numdiff'.  The
option `-S' corresponds to the option `-s' of `numdiff'.

8 Using the filter of numdiff
*****************************

Since version 5 it is possible to activate a filter when calling
`numdiff', so that the program performs automatically the comparison in
the desired way.  Recalling the example of chapter 6, if you run the
command `numdiff -z 2- -V list1 list2' you obtain the following result:

     ----------------
     ##1       <== 12       3.46410161513775458705
     ##1       ==> 12       3.46410162002945508100

     ##1       #:2   <== 3.46410161513775458705
     ##1       #:2   ==> 3.46410162002945508100
      Absolute error = 4.8917004940e-9, Relative error = 1.4121122985e-9
     ----------------
     ##2       <== 13       3.60555127546398929312
               ==>

     ----------------
     ##3       <== 14       3.74165738677394138558
               ==>

     ----------------
     ##4       <== 15       3.87298334620741688518
     ##2       ==> 15       3.87298387096774193548

     ##4       #:2   <== 3.87298334620741688518
     ##2       #:2   ==> 3.87298387096774193548
      Absolute error = 5.2476032505e-7, Relative error = 1.3549253331e-7
     ----------------
     ##5       <== 16       4
               ==>

     ----------------
     ##6       <== 17       4.12310562561766054982
               ==>

     ----------------
     ##7       <== 18       4.24264068711928514641
     ##3       ==> 18       4.24264705882352941176

     ##7       #:2   <== 4.24264068711928514641
     ##3       #:2   ==> 4.24264705882352941176
      Absolute error = 6.3717042443e-6, Relative error = 1.5018250929e-6
     ----------------
     ##8       <== 19       4.35889894354067355224
               ==>

     ----------------
     ##9       <== 20       4.47213595499957939282
               ==>

     ----------------
     ##10      <== 21       4.58257569495584000659
     ##4       ==> 21       4.58260869565217391304

     ##10      #:2   <== 4.58257569495584000659
     ##4       #:2   ==> 4.58260869565217391304
      Absolute error = 3.3000696334e-5, Relative error = 7.2013423303e-6
     ----------------
     ##11      <== 22       4.69041575982342955457
               ==>

     ----------------
     ##12      <== 23       4.7958315233127195416
               ==>

     ----------------
     ##13      <== 24       4.89897948556635619639
               ==>


     +++  File "list1" differs from file "list2"

   Numdiff has recognized that the lines of `list1' with the square
roots for the numbers 13, 14, 16, 17, 19, 20, 22, 23 and 24 have been
deleted from `list2'.  The numerical comparison has been done by
likening each line of `list2' to the line of `list1' which displays the
square root for the same integer value.  The output obtained running
the filter of Numdiff by `numdiff -f -z 2- list1 list2' confirms this:
     12       3.46410161513775458705         12       3.46410162002945508100
     13       3.60555127546398929312      <
     14       3.74165738677394138558      <
     15       3.87298334620741688518         15       3.87298387096774193548
     16       4                           <
     17       4.12310562561766054982      <
     18       4.24264068711928514641         18       4.24264705882352941176
     19       4.35889894354067355224      <
     20       4.47213595499957939282      <
     21       4.58257569495584000659         21       4.58260869565217391304
     22       4.69041575982342955457      <
     23       4.7958315233127195416       <
     24       4.89897948556635619639      <


     +++  File "list1" differs from file "list2"

   If you compare the command `numdiff -z 2- -V list1 list2' with the
one used for the files `bill1' and `bill2', *note command::,  you
surely notice that the filter has been invoked in different ways, first
with `-z @' and then with `-z 2-'.

   The synchronization procedure used by the filter is based on
blurring and byte-by-byte comparison.  The options `-z' and `-Z' are
used to select which fields from which file have to be blurred. They
take both an argument in the same form requested by `-X', *note Use of
the option -X::, but accept additionally the special value `@' as
abbreviation for the range of fields `1-'.  Then the specifications
`1:@', `2:@' and `@' are used to mean all fields of the first file, of
the second one or of both, respectively.

   Employing `-z' and `-Z' in the right way is extremely important to
let the filter work as desired.  For instance, `numdiff -f -z @ list1
list2' matches the lines of `list1' and `list2' in the same wrong way
     12       3.46410161513775458705         12       3.46410162002945508100
     13       3.60555127546398929312         15       3.87298387096774193548
     14       3.74165738677394138558         18       4.24264705882352941176
     15       3.87298334620741688518         21       4.58260869565217391304
     16       4                           <
     17       4.12310562561766054982      <
     18       4.24264068711928514641      <
     19       4.35889894354067355224      <
     20       4.47213595499957939282      <
     21       4.58257569495584000659      <
     22       4.69041575982342955457      <
     23       4.7958315233127195416       <
     24       4.89897948556635619639      <


     +++  File "list1" differs from file "list2"

as Numdiff would do without employing the filter.

   It is essential then to understand what blurring a field means and
how the filter uses blurring to match the lines of the files to compare.

   After reading the files the filter removes from each of them (from
their images in the memory, actually) all the fields selected by the
option `-X', then it replaces each of the fields that have to be
blurred by a special character.  This special character is the same for
both files and it is so chosen that it cannot appear in the text.
Blurring a field means to replace it by this sort of place card.

   After doing this, the filter converts all remaining numerical fields
to a standard format and compares the files byte by byte neglecting the
field delimiters.  This comparison is just used to establish which
lines of the first file are not present in the second, which lines of
the second file are missing in the first one and how to match the
remaining lines to create a one-to-one correspondence.

   Only at this point `numdiff' inspects each couple of corresponding
lines, splits the two lines into the constituent fields, and neglecting
those ones eventually specified through the option `-X' compares
corresponding fields as it is supposed to do, performing a numerical
comparison whenever the fields are both legal numerical values.

   Blurring the right fields is essential to match the lines from the
two files appropriately before doing any numerical comparison.  Without
blurring, the numerical fields could prevent `numdiff' from an
appropriate matching of the lines, in case some of these are present in
only one file, by creating confusion with their (maybe small) numeric
differences.

   Blurring can be of two types, conditional or unconditional.  The
blurring is conditional if it has to be performed only for fields which
turn out to be legal numerical values.  The arguments of the option
`-z' indicate which fields of which file have to be blurred *under the
condition that they are recognized as numeric fields*. Non-numeric
fields are left by `-z' untouched (no blurring occurs for them).  Then
`-z 1:5-7' makes the filter blur the 5th, 6th and 7th field of each
line of the first file whenever they are recognized as numeric.

   By the option `-Z' you can specify which fields have to be
unconditionally blurred, i.e.  independently of their type, numeric or
not.  For example, `-Z 2:3-4' activates the blurring of the 3th and 4th
field of each line of the second file.

   Going back to the comparison of the files `list1' and `list2', the
option `-z 2-' of the command `numdiff -z 2- -V list1 list2' makes the
filter transform the (memory copies of the) two files as
     12       *
     13       *
     14       *
     15       *
     16       *
     17       *
     18       *
     19       *
     20       *
     21       *
     22       *
     23       *
     24       *

and

     12       *
     15       *
     18       *
     21       *

respectively. Here * denotes the special symbol used by the filter in
the blurring procedure, even if this symbol is not actually a bullet.
Since in this example space, tab and newline are used as field
delimiters, the byte-by-byte comparison between the transformed files
produces the same result displayed by the command `sdiff -W' when
applied to them:

     12       *                        12       *
     13       *                     <
     14       *                     <
     15       *                        15       *
     16       *                     <
     17       *                     <
     18       *                        18       *
     19       *                     <
     20       *                     <
     21       *                        21       *
     22       *                     <
     23       *                     <
     24       *                     <

If you put the blurred fields back you obtain exactly the output of
`numdiff -f -z 2- list1 list2':

     12       3.46410161513775458705           12       3.46410162002945508100
     13       3.60555127546398929312        <
     14       3.74165738677394138558        <
     15       3.87298334620741688518           15       3.87298387096774193548
     16       4                             <
     17       4.12310562561766054982        <
     18       4.24264068711928514641           18       4.24264705882352941176
     19       4.35889894354067355224        <
     20       4.47213595499957939282        <
     21       4.58257569495584000659           21       4.58260869565217391304
     22       4.69041575982342955457        <
     23       4.7958315233127195416         <
     24       4.89897948556635619639        <


     +++  File "list1" differs from file "list2"

Since the second field is a numerical value in all the lines of `list1'
and `list2', to use the option `-Z' instead of `-z' makes no difference
in this case.  The output of `numdiff -f -Z 2- list1 list2' is then the
same of `numdiff -f -z 2- list1 list2'.

   After this explanation you can also understand why `numdiff -f -z @
list1 list2' gives a wrong result.  Since also the first field is
always a numerical value, the option `-z @' makes the filter transform
the two given files as
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *
     *       *

and

     *       *
     *       *
     *       *
     *       *

respectively, so that it is not anymore possible to match the lines in
a reasonable way.

   We consider now a typical situation where it is better to use `-Z'
in place of `-z'.  If the file `Table1' contains
-6 	 2.449490
-5 	 2.236068
-4 	 2.000000
-3 	 1.732051
-2 	 1.414214
-1       1.000000
0        0
- - - - - - - - -
1 	 1.000000
2 	 1.414214
3 	 1.732051
4 	 2.000000
- - - - - - - - -
5 	 2.236068
6 	 2.449490
7 	 2.645751
- - - - - - - - -
8 	 2.828427
9 	 3.000000
10 	 3.162278
11 	 3.316625
12 	 3.464102
- - - - - - - - -
- - - - - - - - -
13 	 3.605551
14 	 3.741657

and `Table2' contains

-6       Not_defined
-4       Not_defined
-2       Not_defined
 0       0.000000
 2 	 1.414216
 4 	 2.000000
 6 	 2.449494
 8 	 2.828469
 10 	 3.162278
 12 	 3.464102
 14 	 3.741658
********************END

then the output of `numdiff -z 1:2 -Z 2:2 -f Table1 Table2' is
     -6       2.449490               -6       Not_defined
     -5       2.236068            <
     -4       2.000000               -4       Not_defined
     -3       1.732051            <
     -2       1.414214               -2       Not_defined
     -1       1.000000            <
     0        0                       0       0.000000
     - - - - - - - - -            <
     1        1.000000            <
     2        1.414214                2       1.414216
     3        1.732051            <
     4        2.000000                4       2.000000
     - - - - - - - - -            <
     5        2.236068            <
     6        2.449490                6       2.449494
     7        2.645751            <
     - - - - - - - - -            <
     8        2.828427                8       2.828469
     9        3.000000            <
     10       3.162278                10      3.162278
     11       3.316625            <
     12       3.464102                12      3.464102
     - - - - - - - - -            <
     - - - - - - - - -            <
     13       3.605551            <
     14       3.741657                14      3.741658
                                  >  ********************END


     +++  File "Table1" differs from file "Table2"

which is exactly what is expected.  On the other hand the command
`numdiff -z 2 -f Table1 Table2' displays
     -6       2.449490            |  -6       Not_defined
     -5       2.236068            |  -4       Not_defined
     -4       2.000000            |  -2       Not_defined
     -3       1.732051            <
     -2       1.414214            <
     -1       1.000000            <
     0        0                       0       0.000000
     - - - - - - - - -            <
     1        1.000000            <
     2        1.414214                2       1.414216
     3        1.732051            <
     4        2.000000                4       2.000000
     - - - - - - - - -            <
     5        2.236068            <
     6        2.449490                6       2.449494
     7        2.645751            <
     - - - - - - - - -            <
     8        2.828427                8       2.828469
     9        3.000000            <
     10       3.162278                10      3.162278
     11       3.316625            <
     12       3.464102                12      3.464102
     - - - - - - - - -            <
     - - - - - - - - -            <
     13       3.605551            <
     14       3.741657                14      3.741658
                                  >  ********************END


     +++  File "Table1" differs from file "Table2"

which is partially wrong.

   Notice that in `Table1' you find (truncated to the sixth decimal
digit) the square roots of the absolute values of the integer numbers
between -6 and 20, plus some randomly added lines.  The file `Table2'
contains some approximations (obtained by Newton's method) for the
square roots of the even numbers between -6 and 20. Since the (real)
square root is not defined for negative numbers, the values
corresponding to -6, -4 and -2 are replaced by Not_defined.

   Since Not_defined is not a numeric value, during the execution of
the last command the filter transforms `Table1' in this way
     -6       *
     -5       *
     -4       *
     -3       *
     -2       *
     -1       *
     0        *
     - - - - - - - - -
     1        *
     2        *
     3        *
     4        *
     - - - - - - - - -
     5        *
     6        *
     7        *
     - - - - - - - - -
     8        *
     9        *
     10       *
     11       *
     12       *
     - - - - - - - - -
     - - - - - - - - -
     13       *
     14       *

and `Table2' in this other way

     -6       Not_defined
     -4       Not_defined
     -2       Not_defined
      0       *
      2       *
      4       *
      6       *
      8       *
      10      *
      12      *
      14      *
     ********************END

Unfortunately the first three lines are enough to confuse the
synchronization procedure, which is based on a byte-by-byte comparison
with exclusion of the field delimiters, as we explained before.

   You can check that this is definitely the case by looking at the
output of the command `sdiff -W' on the transformed files, which is
     -6       *           |  -6       Not_defined
     -5       *           |  -4       Not_defined
     -4       *           |  -2       Not_defined
     -3       *           <
     -2       *           <
     -1       *           <
     0        *               0       *
     - - - - - - - - -    <
     1        *           <
     2        *               2       *
     3        *           <
     4        *               4       *
     - - - - - - - - -    <
     5        *           <
     6        *               6       *
     7        *           <
     - - - - - - - - -    <
     8        *               8       *
     9        *           <
     10       *               10      *
     11       *           <
     12       *               12      *
     - - - - - - - - -    <
     - - - - - - - - -    <
     13       *           <
     14       *               14      *
                          >  ********************END

   If we give the command `numdiff -z 1:2 -Z 2:2 -f Table1 Table2'
instead of `numdiff -z 2 -f Table1 Table2', the second field of the
lines of `Table2' is always blurred. The filter transforms then
`Table2' into
     -6       *
     -4       *
     -2       *
      0       *
      2       *
      4       *
      6       *
      8       *
      10      *
      12      *
      14      *
     ********************END

and re-synchronizes the files `Table1' and `Table2' in the right way.

   Using the unconditional blurring is suggested in all cases when a
certain field, which you want to include in the comparison (use `-X' to
completely neglect one or more fields), is of numeric type in almost
all lines of (one of) the given files. This can be the case e.g. when
in some lines the content of the field is given by a special numeric
value, like Infinity, Inf, +Inf or -Inf, or by NaN, abbreviation for
Not a Number.

   Concerning the numeric fields which are not blurred, one has to
remark that the filter is not confused by differences in the numeric
format. Before the byte-by-byte comparison, numeric values are
converted indeed to a standard format. To offer an example of this, let
us suppose that `short1' contains a list of numbers with their
logarithms
0.001          -3
0.01           -2
0.1            -1
1               0
1000            3
1000000         6
1000000000      9

and `short2' the same list of numbers and logarithms, but with
differences in the numeric format:
******************
0.0010000       -3
.0100           -2
0000.10         -1
1.               0
1,000.000        3
1,000,000.       6
1,000,000,000    9

Then `numdiff -f -z 2- short1 short2' displays
                                  >  ******************
     0.001          -3               0.0010000       -3
     0.01           -2               .0100           -2
     0.1            -1               0000.10         -1
     1               0               1.               0
     1000            3               1,000.000        3
     1000000         6               1,000,000.       6
     1000000000      9               1,000,000,000    9


     +++  File "short1" differs from file "short2"

showing that the filter has matched the lines in the right way.

   The filter can even handle the case when the same numerical value is
written in decimal notation in one file and in scientific notation in
the other one.  If the files `decimal' and `scientific' contain
.001           -3
.01            -2
.1             -1
* * * * * * * * *
1               0
1000            3
1000000         6
1000000000      9

and
*****************
1.0e-3         -3
1.0e-2         -2
1.0e-1         -1
1.0e0           0
1.0e3           3
1.0e6           6
1.0e9           9
*****************

respectively, then `numdiff -f -z 2- decimal scientific' shows
                                  >  *****************
     .001           -3               1.0e-3         -3
     .01            -2               1.0e-2         -2
     .1             -1               1.0e-1         -1
     * * * * * * * * *            <
     1               0               1.0e0           0
     1000            3               1.0e3           3
     1000000         6               1.0e6           6
     1000000000      9               1.0e9           9
                                  >  *****************


     +++  File "decimal" differs from file "scientific"

proving that the filter does not get confused.

   No problems come out also in the case when for the same not blurred
field the scientific notation is used in both files.  If the files
`sc1' and `sc2' contain
  1.E-3          -3
  1.00E-2        -2
  1.0E-1         -1
  1.0000E0        0
001.0E3           3
+01.000E6         6
  1.0E+09         9
  1.0E+10        10
* * * * * * * * * *

and
*****************
 1.0e-003      -3
 1.0e-2        -2
 1.0e-1        -1
 1.0e0          0
+1.0e3          3
 1.0e+6         6
 1.0e9          9

respectively, then `numdiff -f -z 2- sc1 sc2' correctly displays
                                  >  *****************
       1.E-3          -3              1.0e-003      -3
       1.00E-2        -2              1.0e-2        -2
       1.0E-1         -1              1.0e-1        -1
       1.0000E0        0              1.0e0          0
     001.0E3           3             +1.0e3          3
     +01.000E6         6              1.0e+6         6
       1.0E+09         9              1.0e9          9
       1.0E+10        10          <
     * * * * * * * * * *          <


     +++  File "sc1" differs from file "sc2"

The filter can even handle an improper use of the scientific notation,
meaning for example that it can recognize `123.456E+2' and `1.23456E+4'
as equal.

   We can see this in the case of the files `Scnot1':
-------------------------
1.2E0             *     1
2.45E-1           *     2
-3.678E-2         *     3

and `Scnot2':
12E-1             *     1
245E-3            *     2
-0.003678E+1      *     3

`numdiff -f -z 3- Scnot1 Scnot2' displays the report:
     -------------------------    <
     1.2E0             *     1       12E-1             *     1
     2.45E-1           *     2       245E-3            *     2
     -3.678E-2         *     3       -0.003678E+1      *     3


     +++  File "Scnot1" differs from file "Scnot2"

which is exactly what you would expect in such a case.  Also pretty
hard cases do not confuse the filter.  If `Scnot1' is given by
 1.2000e0         *     1
02.4500e-1        *     2
-003.678E-2       *     3

and `Scnot2' is the same file as before, the output of the command
`numdiff -f -z 3- Scnot1 Scnot2' is still right:
      1.2000e0         *     1       12E-1             *     1
     02.4500e-1        *     2       245E-3            *     2
     -003.678E-2       *     3       -0.003678E+1      *     3


     +++  Files "Scnot1" and "Scnot2" have the same structure

   Till now we have always used the option `-f' with no argument.  But
`-f' accepts an optional argument, which can be used to control how
`-f' displays its output.  If you provide an argument, care not to
leave any space between the option and the argument: `-f60' is correct
while `-f 60' makes Numdiff terminate after printing an error message.

   If the argument is a positive number NUM, then the side-by-side
output produced by `-f' will be NUM columns wide. The default value for
the width of the output is 130, which can fit onto a traditional
printer line, and is the one used when `-f' has no argument, or the
supplied argument is zero.  In other words, `-f' and `-f0' are just
easier to remind versions of `-f130'.

   A negative argument has the same effect as the positive number with
the same absolute value, but it causes in addition the removal of
common lines from the output.  For example, the command `numdiff -z 1:2
-Z 2:2 -f-130 Table1 Table2' displays the following text
     -5       2.236068            <
     -3       1.732051            <
     -1       1.000000            <
     - - - - - - - - -            <
     1        1.000000            <
     3        1.732051            <
     - - - - - - - - -            <
     5        2.236068            <
     7        2.645751            <
     - - - - - - - - -            <
     9        3.000000            <
     11       3.316625            <
     - - - - - - - - -            <
     - - - - - - - - -            <
     13       3.605551            <
                                  >  ********************END


     +++  File "Table1" differs from file "Table2"

   In conjunction with the option `-f' or `-O' you can use `-T' to
expand tabs to spaces in the output produced by `-f' / `-O'. This is
useful to preserve the alignment of tabs in the input files, if it is
thrown off by the presence of the gutter.

   The options `-H' and `-m' affect the performance of the filter of
Numdiff.  But performance has more than one dimension and these options
improve one aspect of performance at the cost of another, or they
improve performance in some cases while hurting it in others.

   The way that the filter re-synchronizes two files to compare always
comes up with a near-minimal set of deletions/insertions of lines.
Usually it is good enough for practical purposes.  If the filter
displays a large set of line deletions/insertions, you might want it to
use a modified algorithm that sometimes produces a smaller set of
differences.  The `-m' option does this; however, it can also cause the
filter to run more slowly than usual, so it is not the default behavior.

   If the files you are comparing are large and have small groups of
changes scattered throughout them, you can use the `-H' option to make
a different modification to the algorithm that the filter uses.  If the
input files have a constant small density of changes, where change
means deletion/insertion of lines, this option speeds up the
comparisons without changing the output or in the worst case
introducing minor modifications.

9 Warnings
**********

   * Bug reports have to be sent to the address
     <ivprimi(at)libero(dot)it> . Please, put Numdiff in the subject
     and indicate the version of the operating system you are running
     (in particular, do not forget to specify if it is a 32- or a
     64-bit system), and, if you know it, the version of the compiler
     used to build Numdiff.  Please write also whether your version of
     Numdiff uses the GNU MP library or not.  Before writing an email
     be sure to run the latest stable version of Numdiff, I do not
     provide support for older versions.

   * Numdiff does not accept numbers in scientific notation whose
     exponents lie outside the range -1073741824, ..., +1073741824.  If
     such a number is found in any of the files to compare, the
     execution of the program is stopped after printing a suitable
     error message on stderr.  Under the assumption that the numeric
     format in use is the default one, with "1.0001e-2147483640" the
     displayed error messages is
          numdiff: A number with a too small exponent has been found,
          namely "1.0001e-2147483640".
          Exponents smaller than -1073741824 are not accepted,
          the execution of the program ends now

   * If Numdiff has been built with its own internal support for
     multiple precision arithmetic instead of being linked against the
     GNU MP Library, then performance degradation and memory exhaustion
     can already make impossible to handle exponents of magnitude 10^6
     = 1000000.  This is what I obtained on my laptop, equipped with a
     dual core processor @1.50 GHz and with 1GB of RAM, when I tried to
     compare the numbers 1.101e1000000000 and 1.0e1000000000:
          numdiff: Insufficient memory for new allocation,
          the execution of the program ends now
     In addition, you can overload the processor with numbers whose
     exponents lie outside the range -1000000, ..., 1000000.  But at
     least on my machine, everything works fine and quick enough as
     long as exponent and size of the mantissa of the numbers are in
     the range -1000, ..., 1000.  Be careful and remember that Numdiff
     is distributed WITHOUT ANY WARRANTY; without even the implied
     warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     Consider also that, if you have numeric data with exponents
     outside the range -300, ..., 300, probably there is something
     wrong with your data: either you are using the wrong scale, or you
     should replace very small numbers, like 1e-100, by zero, or it is
     quite likely that the machine/program/algorithm which produced
     these data is not working right.

   * If Numdiff has been linked against the GNU Multiple Precision
     Arithmetic Library (also called GNU MP), then the precision it
     uses is typically higher than the specified one. On my machine the
     actual value of the precision is 20 if the user gives a value
     between 0 and 20, 30 if the user specifies a precision between 21
     and 30, 40 for a user-specified value between 31 and 40, and so on.
     Anyway, the actual precision is never less than the one required
     by the user.

   * After reading a numeric field, Numdiff truncates its value if this
     number has too much digits with respect to the current precision.
     To be precise, denoted by P the current value of the precision,
     the following rules apply.

     If `numdiff' has been built with its own internal support for
     multiple precision arithmetic, then
        * if the number is written in ordinary decimal notation,
          `numdiff' will consider, in addition to all digits of the
          integer part, only the first P digits of the fractional part;

        * if the value is written in scientific notation, then `numdiff'
          will only consider the first P digits of the fractional part
          of the mantissa.

     If `numdiff' uses the GNU MP library to perform its computations,
     the value of a numeric field is first translated into scientific
     notation and then only the first P digits of the fractional part
     of the mantissa are considered.

     By current value of the precision I mean the integer value
     specified by the option `-#', or the default one (35) when this
     option is not in use.

   * You can find out whether your local version of `numdiff' is
     relying on GNU MP or not by executing the command `numdiff -v'.
     If `numdiff' uses GNU MP, then this command will display the
     following message or similar (possibly translated into your mother
     language) among other information:
          The software has been linked against
          the GNU Multiple Precision Arithmetic Library,
          version number 4.2.4.

     If `numdiff' does not rely on GNU MP, then the displayed message
     will be (up to translation into your mother language)
          The software has been built with
          its own internal support for multiple precision arithmetic.

   * Numdiff can only be used on text files: the program terminates
     after printing a suitable error message if one of the files to
     compare turns out to be a binary file.  To detect if a file is
     binary or not, `numdiff' checks for the presence of null bytes
     (0x00) in the file.

   * If you are not including the so called white-space characters
     (usually ` ', `\t', `\f', `\v' and `\r') in the set of field
     delimiters, then a real and an imaginary number which are
     separated just by white-spaces can be coupled together and
     considered as a whole complex number.  For example, if you are
     using only *colon* (`:') and *newline* as field delimiters and
     Numdiff finds a line like that ::::3.0-5.6e-356i::::-12.9   +4.34i::::-12.9    4.34i::::New York::::

     then it will consider this line as formed by four fields, the
     first two are numeric and given by the complex numbers
     3.0-5.6e-356i and -12.9+4.34i, the last two ones are the strings
     New York and -12.9    4.34i.  I still do not know if I will modify
     this in the next version of Numdiff, so that the program
     recognizes only 3.0-5.6e-356i as numeric field and treats -12.9
     +4.34i as non-numeric due to the presence of spaces in the middle.
     -12.9    4.34i is already considered as non-numeric due to the
     absence of a leading sign in the imaginary value.

   * We have seen that one of the two files passed to `numdiff' can be
     -, which refers to stdin (standard input). In this way one of the
     two files to compare can be the output produced by another
     command, like in `cat file2 | numdiff -a 1.0e-3 file1 -'.
     However, if you activate the filter by means of the options `-z'
     or/and `-Z', Numdiff can not work with the standard input unless
     you use also the option `-f'. Therefore, the command `cat file2 |
     numdiff -a 1.0e-3 -z @ file1 -' displays only the error message numdiff: -: Illegal seek
      (or maybe the translation of this message in the language  you
     are using on your computer) but `cat file2 | numdiff -a 1.0e-3 -z
     @ -f file1 -' works as expected.

   * This manual describes the version 5.8 of Numdiff.  Prior 5.x
     versions did not recognize all the options that are currently
     accepted, versions 4.0.0 and 3.x used even a different format for
     the output.

Appendix A GNU Free Documentation License
*****************************************

                     Version 1.3, 3 November 2008

     Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
     `http://fsf.org/'

     Everyone is permitted to copy and distribute verbatim copies
     of this license document, but changing it is not allowed.

  0. PREAMBLE

     The purpose of this License is to make a manual, textbook, or other
     functional and useful document "free" in the sense of freedom: to
     assure everyone the effective freedom to copy and redistribute it,
     with or without modifying it, either commercially or
     noncommercially.  Secondarily, this License preserves for the
     author and publisher a way to get credit for their work, while not
     being considered responsible for modifications made by others.

     This License is a kind of "copyleft", which means that derivative
     works of the document must themselves be free in the same sense.
     It complements the GNU General Public License, which is a copyleft
     license designed for free software.

     We have designed this License in order to use it for manuals for
     free software, because free software needs free documentation: a
     free program should come with manuals providing the same freedoms
     that the software does.  But this License is not limited to
     software manuals; it can be used for any textual work, regardless
     of subject matter or whether it is published as a printed book.
     We recommend this License principally for works whose purpose is
     instruction or reference.

  1. APPLICABILITY AND DEFINITIONS

     This License applies to any manual or other work, in any medium,
     that contains a notice placed by the copyright holder saying it
     can be distributed under the terms of this License.  Such a notice
     grants a world-wide, royalty-free license, unlimited in duration,
     to use that work under the conditions stated herein.  The
     "Document", below, refers to any such manual or work.  Any member
     of the public is a licensee, and is addressed as "you".  You
     accept the license if you copy, modify or distribute the work in a
     way requiring permission under copyright law.

     A "Modified Version" of the Document means any work containing the
     Document or a portion of it, either copied verbatim, or with
     modifications and/or translated into another language.

     A "Secondary Section" is a named appendix or a front-matter section
     of the Document that deals exclusively with the relationship of the
     publishers or authors of the Document to the Document's overall
     subject (or to related matters) and contains nothing that could
     fall directly within that overall subject.  (Thus, if the Document
     is in part a textbook of mathematics, a Secondary Section may not
     explain any mathematics.)  The relationship could be a matter of
     historical connection with the subject or with related matters, or
     of legal, commercial, philosophical, ethical or political position
     regarding them.

     The "Invariant Sections" are certain Secondary Sections whose
     titles are designated, as being those of Invariant Sections, in
     the notice that says that the Document is released under this
     License.  If a section does not fit the above definition of
     Secondary then it is not allowed to be designated as Invariant.
     The Document may contain zero Invariant Sections.  If the Document
     does not identify any Invariant Sections then there are none.

     The "Cover Texts" are certain short passages of text that are
     listed, as Front-Cover Texts or Back-Cover Texts, in the notice
     that says that the Document is released under this License.  A
     Front-Cover Text may be at most 5 words, and a Back-Cover Text may
     be at most 25 words.

     A "Transparent" copy of the Document means a machine-readable copy,
     represented in a format whose specification is available to the
     general public, that is suitable for revising the document
     straightforwardly with generic text editors or (for images
     composed of pixels) generic paint programs or (for drawings) some
     widely available drawing editor, and that is suitable for input to
     text formatters or for automatic translation to a variety of
     formats suitable for input to text formatters.  A copy made in an
     otherwise Transparent file format whose markup, or absence of
     markup, has been arranged to thwart or discourage subsequent
     modification by readers is not Transparent.  An image format is
     not Transparent if used for any substantial amount of text.  A
     copy that is not "Transparent" is called "Opaque".

     Examples of suitable formats for Transparent copies include plain
     ASCII without markup, Texinfo input format, LaTeX input format,
     SGML or XML using a publicly available DTD, and
     standard-conforming simple HTML, PostScript or PDF designed for
     human modification.  Examples of transparent image formats include
     PNG, XCF and JPG.  Opaque formats include proprietary formats that
     can be read and edited only by proprietary word processors, SGML or
     XML for which the DTD and/or processing tools are not generally
     available, and the machine-generated HTML, PostScript or PDF
     produced by some word processors for output purposes only.

     The "Title Page" means, for a printed book, the title page itself,
     plus such following pages as are needed to hold, legibly, the
     material this License requires to appear in the title page.  For
     works in formats which do not have any title page as such, "Title
     Page" means the text near the most prominent appearance of the
     work's title, preceding the beginning of the body of the text.

     The "publisher" means any person or entity that distributes copies
     of the Document to the public.

     A section "Entitled XYZ" means a named subunit of the Document
     whose title either is precisely XYZ or contains XYZ in parentheses
     following text that translates XYZ in another language.  (Here XYZ
     stands for a specific section name mentioned below, such as
     "Acknowledgements", "Dedications", "Endorsements", or "History".)
     To "Preserve the Title" of such a section when you modify the
     Document means that it remains a section "Entitled XYZ" according
     to this definition.

     The Document may include Warranty Disclaimers next to the notice
     which states that this License applies to the Document.  These
     Warranty Disclaimers are considered to be included by reference in
     this License, but only as regards disclaiming warranties: any other
     implication that these Warranty Disclaimers may have is void and
     has no effect on the meaning of this License.

  2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either
     commercially or noncommercially, provided that this License, the
     copyright notices, and the license notice saying this License
     applies to the Document are reproduced in all copies, and that you
     add no other conditions whatsoever to those of this License.  You
     may not use technical measures to obstruct or control the reading
     or further copying of the copies you make or distribute.  However,
     you may accept compensation in exchange for copies.  If you
     distribute a large enough number of copies you must also follow
     the conditions in section 3.

     You may also lend copies, under the same conditions stated above,
     and you may publicly display copies.

  3. COPYING IN QUANTITY

     If you publish printed copies (or copies in media that commonly
     have printed covers) of the Document, numbering more than 100, and
     the Document's license notice requires Cover Texts, you must
     enclose the copies in covers that carry, clearly and legibly, all
     these Cover Texts: Front-Cover Texts on the front cover, and
     Back-Cover Texts on the back cover.  Both covers must also clearly
     and legibly identify you as the publisher of these copies.  The
     front cover must present the full title with all words of the
     title equally prominent and visible.  You may add other material
     on the covers in addition.  Copying with changes limited to the
     covers, as long as they preserve the title of the Document and
     satisfy these conditions, can be treated as verbatim copying in
     other respects.

     If the required texts for either cover are too voluminous to fit
     legibly, you should put the first ones listed (as many as fit
     reasonably) on the actual cover, and continue the rest onto
     adjacent pages.

     If you publish or distribute Opaque copies of the Document
     numbering more than 100, you must either include a
     machine-readable Transparent copy along with each Opaque copy, or
     state in or with each Opaque copy a computer-network location from
     which the general network-using public has access to download
     using public-standard network protocols a complete Transparent
     copy of the Document, free of added material.  If you use the
     latter option, you must take reasonably prudent steps, when you
     begin distribution of Opaque copies in quantity, to ensure that
     this Transparent copy will remain thus accessible at the stated
     location until at least one year after the last time you
     distribute an Opaque copy (directly or through your agents or
     retailers) of that edition to the public.

     It is requested, but not required, that you contact the authors of
     the Document well before redistributing any large number of
     copies, to give them a chance to provide you with an updated
     version of the Document.

  4. MODIFICATIONS

     You may copy and distribute a Modified Version of the Document
     under the conditions of sections 2 and 3 above, provided that you
     release the Modified Version under precisely this License, with
     the Modified Version filling the role of the Document, thus
     licensing distribution and modification of the Modified Version to
     whoever possesses a copy of it.  In addition, you must do these
     things in the Modified Version:

       A. Use in the Title Page (and on the covers, if any) a title
          distinct from that of the Document, and from those of
          previous versions (which should, if there were any, be listed
          in the History section of the Document).  You may use the
          same title as a previous version if the original publisher of
          that version gives permission.

       B. List on the Title Page, as authors, one or more persons or
          entities responsible for authorship of the modifications in
          the Modified Version, together with at least five of the
          principal authors of the Document (all of its principal
          authors, if it has fewer than five), unless they release you
          from this requirement.

       C. State on the Title page the name of the publisher of the
          Modified Version, as the publisher.

       D. Preserve all the copyright notices of the Document.

       E. Add an appropriate copyright notice for your modifications
          adjacent to the other copyright notices.

       F. Include, immediately after the copyright notices, a license
          notice giving the public permission to use the Modified
          Version under the terms of this License, in the form shown in
          the Addendum below.

       G. Preserve in that license notice the full lists of Invariant
          Sections and required Cover Texts given in the Document's
          license notice.

       H. Include an unaltered copy of this License.

       I. Preserve the section Entitled "History", Preserve its Title,
          and add to it an item stating at least the title, year, new
          authors, and publisher of the Modified Version as given on
          the Title Page.  If there is no section Entitled "History" in
          the Document, create one stating the title, year, authors,
          and publisher of the Document as given on its Title Page,
          then add an item describing the Modified Version as stated in
          the previous sentence.

       J. Preserve the network location, if any, given in the Document
          for public access to a Transparent copy of the Document, and
          likewise the network locations given in the Document for
          previous versions it was based on.  These may be placed in
          the "History" section.  You may omit a network location for a
          work that was published at least four years before the
          Document itself, or if the original publisher of the version
          it refers to gives permission.

       K. For any section Entitled "Acknowledgements" or "Dedications",
          Preserve the Title of the section, and preserve in the
          section all the substance and tone of each of the contributor
          acknowledgements and/or dedications given therein.

       L. Preserve all the Invariant Sections of the Document,
          unaltered in their text and in their titles.  Section numbers
          or the equivalent are not considered part of the section
          titles.

       M. Delete any section Entitled "Endorsements".  Such a section
          may not be included in the Modified Version.

       N. Do not retitle any existing section to be Entitled
          "Endorsements" or to conflict in title with any Invariant
          Section.

       O. Preserve any Warranty Disclaimers.

     If the Modified Version includes new front-matter sections or
     appendices that qualify as Secondary Sections and contain no
     material copied from the Document, you may at your option
     designate some or all of these sections as invariant.  To do this,
     add their titles to the list of Invariant Sections in the Modified
     Version's license notice.  These titles must be distinct from any
     other section titles.

     You may add a section Entitled "Endorsements", provided it contains
     nothing but endorsements of your Modified Version by various
     parties--for example, statements of peer review or that the text
     has been approved by an organization as the authoritative
     definition of a standard.

     You may add a passage of up to five words as a Front-Cover Text,
     and a passage of up to 25 words as a Back-Cover Text, to the end
     of the list of Cover Texts in the Modified Version.  Only one
     passage of Front-Cover Text and one of Back-Cover Text may be
     added by (or through arrangements made by) any one entity.  If the
     Document already includes a cover text for the same cover,
     previously added by you or by arrangement made by the same entity
     you are acting on behalf of, you may not add another; but you may
     replace the old one, on explicit permission from the previous
     publisher that added the old one.

     The author(s) and publisher(s) of the Document do not by this
     License give permission to use their names for publicity for or to
     assert or imply endorsement of any Modified Version.

  5. COMBINING DOCUMENTS

     You may combine the Document with other documents released under
     this License, under the terms defined in section 4 above for
     modified versions, provided that you include in the combination
     all of the Invariant Sections of all of the original documents,
     unmodified, and list them all as Invariant Sections of your
     combined work in its license notice, and that you preserve all
     their Warranty Disclaimers.

     The combined work need only contain one copy of this License, and
     multiple identical Invariant Sections may be replaced with a single
     copy.  If there are multiple Invariant Sections with the same name
     but different contents, make the title of each such section unique
     by adding at the end of it, in parentheses, the name of the
     original author or publisher of that section if known, or else a
     unique number.  Make the same adjustment to the section titles in
     the list of Invariant Sections in the license notice of the
     combined work.

     In the combination, you must combine any sections Entitled
     "History" in the various original documents, forming one section
     Entitled "History"; likewise combine any sections Entitled
     "Acknowledgements", and any sections Entitled "Dedications".  You
     must delete all sections Entitled "Endorsements."

  6. COLLECTIONS OF DOCUMENTS

     You may make a collection consisting of the Document and other
     documents released under this License, and replace the individual
     copies of this License in the various documents with a single copy
     that is included in the collection, provided that you follow the
     rules of this License for verbatim copying of each of the
     documents in all other respects.

     You may extract a single document from such a collection, and
     distribute it individually under this License, provided you insert
     a copy of this License into the extracted document, and follow
     this License in all other respects regarding verbatim copying of
     that document.

  7. AGGREGATION WITH INDEPENDENT WORKS

     A compilation of the Document or its derivatives with other
     separate and independent documents or works, in or on a volume of
     a storage or distribution medium, is called an "aggregate" if the
     copyright resulting from the compilation is not used to limit the
     legal rights of the compilation's users beyond what the individual
     works permit.  When the Document is included in an aggregate, this
     License does not apply to the other works in the aggregate which
     are not themselves derivative works of the Document.

     If the Cover Text requirement of section 3 is applicable to these
     copies of the Document, then if the Document is less than one half
     of the entire aggregate, the Document's Cover Texts may be placed
     on covers that bracket the Document within the aggregate, or the
     electronic equivalent of covers if the Document is in electronic
     form.  Otherwise they must appear on printed covers that bracket
     the whole aggregate.

  8. TRANSLATION

     Translation is considered a kind of modification, so you may
     distribute translations of the Document under the terms of section
     4.  Replacing Invariant Sections with translations requires special
     permission from their copyright holders, but you may include
     translations of some or all Invariant Sections in addition to the
     original versions of these Invariant Sections.  You may include a
     translation of this License, and all the license notices in the
     Document, and any Warranty Disclaimers, provided that you also
     include the original English version of this License and the
     original versions of those notices and disclaimers.  In case of a
     disagreement between the translation and the original version of
     this License or a notice or disclaimer, the original version will
     prevail.

     If a section in the Document is Entitled "Acknowledgements",
     "Dedications", or "History", the requirement (section 4) to
     Preserve its Title (section 1) will typically require changing the
     actual title.

  9. TERMINATION

     You may not copy, modify, sublicense, or distribute the Document
     except as expressly provided under this License.  Any attempt
     otherwise to copy, modify, sublicense, or distribute it is void,
     and will automatically terminate your rights under this License.

     However, if you cease all violation of this License, then your
     license from a particular copyright holder is reinstated (a)
     provisionally, unless and until the copyright holder explicitly
     and finally terminates your license, and (b) permanently, if the
     copyright holder fails to notify you of the violation by some
     reasonable means prior to 60 days after the cessation.

     Moreover, your license from a particular copyright holder is
     reinstated permanently if the copyright holder notifies you of the
     violation by some reasonable means, this is the first time you have
     received notice of violation of this License (for any work) from
     that copyright holder, and you cure the violation prior to 30 days
     after your receipt of the notice.

     Termination of your rights under this section does not terminate
     the licenses of parties who have received copies or rights from
     you under this License.  If your rights have been terminated and
     not permanently reinstated, receipt of a copy of some or all of
     the same material does not give you any rights to use it.

 10. FUTURE REVISIONS OF THIS LICENSE

     The Free Software Foundation may publish new, revised versions of
     the GNU Free Documentation License from time to time.  Such new
     versions will be similar in spirit to the present version, but may
     differ in detail to address new problems or concerns.  See
     `http://www.gnu.org/copyleft/'.

     Each version of the License is given a distinguishing version
     number.  If the Document specifies that a particular numbered
     version of this License "or any later version" applies to it, you
     have the option of following the terms and conditions either of
     that specified version or of any later version that has been
     published (not as a draft) by the Free Software Foundation.  If
     the Document does not specify a version number of this License,
     you may choose any version ever published (not as a draft) by the
     Free Software Foundation.  If the Document specifies that a proxy
     can decide which future versions of this License can be used, that
     proxy's public statement of acceptance of a version permanently
     authorizes you to choose that version for the Document.

 11. RELICENSING

     "Massive Multiauthor Collaboration Site" (or "MMC Site") means any
     World Wide Web server that publishes copyrightable works and also
     provides prominent facilities for anybody to edit those works.  A
     public wiki that anybody can edit is an example of such a server.
     A "Massive Multiauthor Collaboration" (or "MMC") contained in the
     site means any set of copyrightable works thus published on the MMC
     site.

     "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
     license published by Creative Commons Corporation, a not-for-profit
     corporation with a principal place of business in San Francisco,
     California, as well as future copyleft versions of that license
     published by that same organization.

     "Incorporate" means to publish or republish a Document, in whole or
     in part, as part of another Document.

     An MMC is "eligible for relicensing" if it is licensed under this
     License, and if all works that were first published under this
     License somewhere other than this MMC, and subsequently
     incorporated in whole or in part into the MMC, (1) had no cover
     texts or invariant sections, and (2) were thus incorporated prior
     to November 1, 2008.

     The operator of an MMC Site may republish an MMC contained in the
     site under CC-BY-SA on the same site at any time before August 1,
     2009, provided the MMC is eligible for relicensing.


ADDENDUM: How to use this License for your documents
====================================================

To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and license
notices just after the title page:

       Copyright (C)  YEAR  YOUR NAME.
       Permission is granted to copy, distribute and/or modify this document
       under the terms of the GNU Free Documentation License, Version 1.3
       or any later version published by the Free Software Foundation;
       with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
       Texts.  A copy of the license is included in the section entitled ``GNU
       Free Documentation License''.

   If you have Invariant Sections, Front-Cover Texts and Back-Cover
Texts, replace the "with...Texts." line with this:

         with the Invariant Sections being LIST THEIR TITLES, with
         the Front-Cover Texts being LIST, and with the Back-Cover Texts
         being LIST.

   If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.

   If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License, to
permit their use in free software.

Index
*****

Acknowledgments:                               See 2.        (line   60)
Build:                                         See 4.        (line 1025)
Caveats:                                       See 9.        (line 3705)
Command line options for ndselect:             See 7.        (line 2893)
Command line options for numdiff:              See 5.        (line 1142)
Compile:                                       See 4.        (line 1025)
Copying Conditions:                            See 1.        (line   44)
Diagnostics (ndselect):                        See 7.        (line 2893)
Diagnostics (numdiff):                         See 5.        (line 1142)
FDL:                                           See Appendix A.
                                                             (line 3842)
Filter:                                        See 8.        (line 2985)
Filter output (numdiff):                       See 3.3.      (line  977)
Format of the reports:                         See 3.1.      (line  592)
GNU FDL:                                       See Appendix A.
                                                             (line 3842)
GNU Free Documentation License:                See Appendix A.
                                                             (line 3842)
GNU General Public License:                    See 1.        (line   44)
GPL:                                           See 1.        (line   44)
How to use numdiff:                            See 3.        (line   76)
Install:                                       See 4.        (line 1025)
Introduction:                                  See 3.        (line   76)
Invoking ndselect:                             See 7.        (line 2893)
Invoking numdiff:                              See 5.        (line 1142)
License:                                       See 1.        (line   44)
ndselect (introduction to its use):            See 6.        (line 2759)
Notes:                                         See 9.        (line 3705)
Options, command line (ndselect):              See 7.        (line 2893)
Options, command line (numdiff):               See 5.        (line 1142)
Output format (numdiff):                       See 3.1.      (line  592)
Overview mode of numdiff:                      See 3.2.      (line  754)
Predefined settings of numdiff:                See 5.        (line 1142)
Purposes:                                      See 3.        (line   76)
Side-by-side report (numdiff):                 See 3.2.      (line  754)
Synopsis (ndselect):                           See 7.        (line 2893)
Synopsis (numdiff):                            See 5.        (line 1142)
Thanks:                                        See 2.        (line   60)
Tools:                                         See 6.        (line 2759)
Usage of numdiff:                              See 3.        (line   76)
Warnings:                                      See 9.        (line 3705)