README

This is a slightly patched version of the Core Library.
The most notable difference is the fact that everything is accessible under the CORE_TWO namespace.
Please see the git commit log for all changes.
Do NOT file any bugs against the original Core source code if you are using this repository!
We hope that this form of distribution is in the spirit of the QPL.
The original Readme follows.

************************************************************
*                        Core Library
*
*            A C/C++ Library for Robust Computation
*
*                Department of Computer Science
*          Courant Institute of Mathematical Sciences
*                     New York University
*                      251 Mercer Street
*                      New York, NY 10012
*                            USA
*
*                    http://cs.nyu.edu/exact/
*        
*  Copyright (c) 1998-2016 by Exact Computation Project, NYU
*
*  $Id: README,v 1.12 2010/07/12 13:48:47 exact Exp $
*************************************************************

README FILE: TABLE OF CONTENTS

    1) ORIENTATION
    2) DIRECTORIES AND FILES
    3) DOWNLOAD AND INSTALLATION
    4) NEWS, PLANS AND BUGS
    5) ACKNOWLEDGMENT AND HISTORY
    6) LICENSE INFORMATION

*************************************************************
1) ORIENTATION

   This is Core Library Version 2.1, released on July 1, 2010.

   The Core Library is a collection of C/C++ classes for exact
   computation with real algebraic numbers.
   It embodies our precision-driven approach and is 
   useful for robust numerical (especially geometric) algorithms.
   The library supports the Exact Geometric Computation (EGC) philosophy
   through its novel and easy-to-use notion of accuracy levels.
   We define four "Core Accuracy Levels":

     Level I:  Machine Accuracy
                    This is the IEEE 754 Standard.
     Level II: Arbitrary Accuracy
                    If a user specify "1000 bits" of accuracy,
                    this means no overflow/underflow occurs as long
                    as long as number sizes do not exceed 1000 bits. 
     Level III: Guaranteed Accuracy
                    The user can specify "10 bits" (relative or
                    absolute) and any computed number will have
                    at least 10 bits of accuracy.
     Level IV: Mixed Accuracy
                    The above three levels are intermixed.

   Level III is the most interesting level. It is the default level,
   Ideally, a single C++ program can be compiled to run at any chosen
   level.  Level IV is not fully defined at the present time.
   The current library focuses on Level III.  Most programs should
   be able to run both Levels I and III.   Such multi-level capability
   in a single program is useful for debugging and program development.

   A key design goal is to allow any standard C/C++ program to access
   critical CORE facilities with minimal changes to the program.  Most
   basic standalone C++ programs can be ``CORE-ized'' with little fuss.
   In the ideal case, one only has to insert the following statement

                #include "CORE/CORE.h"

   following all other standard include statements.  The default accuracy
   is Level III.  At this level, a variable with the machine type
   "double" or "long" is redefined as an instance of the class "Expr"
   (a C++ class).  All comparisons involving Expr are error-free,
   provided your expressions involve only the four arithmetic operations
   and square-roots.  Example:

                double x, y, a, b;
                x = 2; y = 3;
                a = sqrt(x) + sqrt(y);
                b = sqrt(x + y + 2*sqrt(x)*sqrt(y));
                if (a == b)  cout << "Equal   (CORRECT!)\n";
                    else     cout << "Not Equal (ERROR!)\n";

   When run in Level III, the expressions for a and b are always equal
   regardless of the values (x=2 and y=3) assigned to x and y.
   
   In the framework of EGC, error-free comparisons amount to
   computing the ''exact geometry'' in your programs.  Nonrobustness
   issues arising from round-off errors are thereby abolished.

   A tutorial in this "doc" subdirectory, plus the many
   sample programs in the "progs" subdirectory, should allow you
   to start writing CORE programs rather quickly.   In particular,
   you may use the examples in progs/generic as template
   to develop your own CORE programs.  For extra functionality, 
   we define domain specific "CORE eXtensions" (COREX for short). 
   These include a linear algebra COREX and a geometry
   COREX.  They are still rudimentary.
   
   The Core Library is relatively small.  For convenience, we offer
   three distributions: "base distribution" has the basics, including
   source code and examples.  The "full distribution" is the 
   base distribution, plus documentation, gmp and mpfr.
   The library has been tested on MacOS, linux, cygwin and Windows. 
   It is also fully compatible with CGAL, and distributed with CGAL.

   NOTE ON NAMES.
   "Core" is not an abbreviation; we chose this name to suggest its
   role as the "numerical core" for robust geometric computations.
   It is also to remind ourselves (the designers) that "cores" are
   usually small.  However, we use the capitalized sequence "CORE"
   as a shorthand for "Core Library" (e.g., a CORE program).

*************************************************************
2) DIRECTORIES AND FILES

   The Core Directory is denoted ${CORE_PATH} in these notes.
   The base distribution has the following files and subdirectories
   under ${CORE_PATH}:

       README        This file
       ANNOUNCEMENT  Release announcement
       AUTHORS       Authors of Core Library
       Makefile      Makefile for the whole library
       Make.config   Compiler flags (other Makefiles include this)
       FAQs          Frequently asked questions
       LICENSE.QPL   The Q Public License 

       ext/          CORE eXtensions (linear algebra, geometry, etc)
       inc/          The header files.
       lib/          The compiled library is found here.
       progs/        Sample programs that use Core Library.
       src/          Source code for the Core Library.
       tmp/          Temporary files (e.g., installation diagnostics)

       python/       Python interface to Core Library.
       win32/        Windows related files (can be deleted for unix-base)
       bin/          executable programs (mostly for development currently)
       cmake/        directory for CMake builds and modules
       builds/       directory for building gmp and mpfr (it is OK to
                         delete this directory if you don't need it later)
       gmp/          gmp installation directory (may be a link).  This
                         directory may be missing if you have your own gmp.
                         From the SVN repository, you can link to ../gmp.
       mpfr/         mpfr installation directory (may be a link).  Omit
                         this directory if you have your own mpfr.
                         From the SVN repository, you can link to ../gmp.

   The full distribution has these additional files and directories:

       doc/            Core Library Documentation  
       builds/gmp-xxx.tar.gz  GNU's gmp distribution (version xxx)
       builds/gmp-xxx/        GNU's gmp directory to unpack & compile
                       (can be deleted after installing gmp)
       builds/mpfr-xxx.tar.gz  mpfr distribution (version xxx)
       builds/mpfr-xxx/        mpfr directory for unpacking and compiling

*************************************************************
3) DOWNLOAD AND INSTALLATION

   3.1) PREREQUISITE
        You must have "make" (GNU's gmake is recommended), and "svn"
        to download Core Library.  You will need the "g++" compiler
        and gmp/mpfr libraries to compile Core Library.

        For Windows: you could replace g++ with Visual C++.
        Here, we highly recommend the free "Cygwin".  This is a free
        and easy-to-install Unix-OS that sits on top of all 
        versions of Windows.   All the needed tools (g++, make, svn,
        etc) are available to download (you must choose them).
        Advantages of Cygwin:
        (I) There is no need for dual booting;
        (II) There is no need to partition your disc;
        (III) You can share files between Windows and Cygwin; 
        (IV) There is a convenient setup.exe utility that allows you to
                install / update / re-install / uninstall 
        any component in a large suite of Unix tools & utilities, directly 
        from the web.  You can download Cygwin at

                http://www.cygwin.com.  

   3.2) DOWNLOAD 
        Please follow the download instructions in

              https://cs.nyu.edu/exact/core_pages/downloads.html 

        Let the variable "CORE_PATH" refer to the folder where you
        downloaded the Core Library.  Go to this folder:

            > cd $(CORE_PATH)
        
        (">" denotes the command window prompt)
        There is a README file (in fact, you are currently
        reading this file:) with instructions for installing Core Library.

            =========================================================
            | Before 2014, we distributed Core Library in           |
            | tar-gzipped files. These files may be downloaded at   |
            |    https://cs.nyu.edu/exact/core_pages/downloads.html |
            | The pre-2014 versions may still be useful.            |
            | For example, Core1 implemented a variety of root      |
            | bounds but Core2 only implemented one such bound.     |
            | There are situations where the other bounds may speed |
            | up difficult zero computation.                        |
            | NOTE:   Core1 was last tested using                   |
            |      gcc 5.4.0,    gmp-4.2.4,     mpfr-3.1.5.         |
            =========================================================

   3.3) There are four stages of installation:
                STAGE 1 -- gmp installation
                STAGE 2 -- mpfr installation
                STAGE 3 -- create the core library, extensions, samples
                STAGE 4 -- run tests and timings
        The screen output from the individual steps in these stages are
        stored in the files ${CORE_PATH}/tmp/DIAG_* for your diagnostics.

        But before any of these steps, you MUST choose your PLATFORM.
        PLATFORM is a combination of OS plus compiler.
        This variable is set in the Make.config file:

            > cd $(CORE_PATH)
            > vi Make.config -- use "vi" or your favorite editor to edit
                                the file ``Make.config''.   
                             -- Follow the instructions in the file.
                             -- For most users, you only need to set one
                                variable, the PLATFORM variable.
  
        To see what your current (default) options are, do:
            > make options   --this will show the various global settings
                             --e.g. PLATFORM=gnu, LINKAGE=static, VAR=,etc.

   3.4) STAGE 1: INSTALLING GMP 
        a) PRELIMINARY STEP:   For Unix-like OS (e.g., Solaris, Linux),
        gmp will automatically build BOTH a static and a dynamic library.  
        In this case, most compilers will link to the dynamic gmp library.
        Therefore, you need to add the path of your dynamic gmp library to
        the environmental variable LD_LIBRARY_PATH.
        
        E.g., in csh/tcsh,
        
         > setenv LD_LIBRARY_PATH ${LD_LIBRARY_PATH}:${your_gmp_dir}/lib
               
        E.g., in bash,
        
         > export LD_LIBRARY_PATH=${LD_LIBRARY_PATH}:${your_gmp_dir}/lib

        For instance, ${your_gmp_dir} might be /usr/local.
        If you are installing gmp using option d) below, we suggest
        letting ${your_gmp_dir}=${CORE_PATH}/gmp.

        NOW we are ready to proceed with installing gmp.
        Choose one of the following three situations that apply to you:

        b) YOU HAVE AN INSTALLED GMP ALL READY TO GO:
        You already have GMP header files and library files on 
        your system directory, AND they can be found by your
        C++ compiler.  In this case, there is nothing to do.

        In any case, if you want to quickly check if you are in this
        best case situation, do the following:

                > cd ${CORE_PATH} 
                > make testgmp

        This test simply computes 12345 * 67890 = 838102050,
        and 111111111 * 111111111 = 12,345,678,987,654,321.

        If you pass the test, skip this stage and go to STAGE 2.

        c) YOU HAVE AN INSTALLED GMP BUT CORE COULD NOT FIND IT:
        Assuming you have a properly installed GMP in some directory, say
                ${your_gmp_dir}
        which was not found by the above "testgmp".  
        For instance, for the MacOS, you might have your gmp (and mpfr)
        under /opt/local/lib/libgmp*  and /opt/local/include/gmp*.
        In that case, your_gmp_dir=/opt/local.

        Typically, you will find under ${your_gmp_dir} several
        subdirectories, including the following:
                lib    include     share
        Then you can make a symbolic link from ${CORE_PATH}/gmp
        using the following command:

            > cd ${CORE_PATH}
            > ln -s ${your_gmp_dir} gmp
            > make testgmp

        If you pass this test, you can go to STAGE 2.

        E.g., The default gmp installation will install the
        header file and library files in /usr/local/include
        and /usr/local/lib (respectively).  Hence, in this case,
        ${your_gmp_dir}=/usr/local.

        E.g., You had successfully installed an older version of Corelib
        in ${OLD_CORE_PATH}.  Say gmp and mpfr are in these
        directories.  Then you can simply link from ${CORE_PATH} to
        both these directories as above.

        d) YOU NEED TO INSTALL GMP:
        If you did not download the full distribution of CORE, see 3.10).
        Here we assume you have downloaded our full distribution,
        which comes with gmp:

            > cd ${CORE_PATH}
            > make first     -- this is equivalent to "make config-gmp"
                             -- it uses the configuration program in gmp
                             -- it searches for system information
                                (compiler, CPU, files, etc).
                             -- screen output from this process is placed 
                                in the file ${CORE_PATH}/tmp/DIAG_CONFIG_GMP.
                                (E.g., if you do not have m4, the error will
                                be detected here.)
            > make second    -- this is equivalent to "make build-gmp"
                             -- this compiles gmp (takes a while!)
                             -- screen output from this process is placed
                                in the file ${CORE_PATH}/tmp/DIAG_BUILD_GMP
            > make third     -- installs gmp (same as "make install-gmp")
                             -- screen output from this process is placed
                                in the file ${CORE_PATH}/tmp/DIAG_INSTALL_GMP
                                (at this point, you can go to the gmp directory
                                and do a "make check" for extra assurance)
            > make testgmp   -- you must pass this test to continue.
                                output in file ${CORE_PATH}/tmp/DIAG_TEST_GMP

        IMPORTANT: Our top level make file is written for GNU's gmake.
        The remaining make files in subdirectories should work with other
        make programs.  The first three steps takes 15-25 minutes.

   3.5) STAGE 2: INSTALLING MPFR
        Stage 2 is very similar to Stage 1.
        As for Stage 1, we have three possible scenarios:

        a) If you have already installed MPFR, you can verify this by

                > cd ${CORE_PATH} 
                > make testmpfr

        If your screen shows the answer 8.3810205000e8 and 3.1415929208,
        you have passed the test -- then skip this stage and go to STAGE 3.

        b) You have installed MPFR, but it could not be found 
        by the compiler in a).  Suppose your MPFR is installed in
        the directory called ${your_mpfr_dir}.  
        This might be the same directory as ${your_gmp_dir}.
        For instance, you expect to find the file mpfr.h in 
        ${your_mpfr_dir}/include.  In this case, we could make a 
        symbolic link from ${CORE_PATH}/mpfr using the following command:

            > cd ${CORE_PATH}
            > ln -s ${your_mpfr_dir} mpfr
            > make testmpfr

        If you pass this test, you can go to STAGE 3.

        c) If you do not have MPFR, we will install MPFR as follows:

            > make fourth    -- configures mpfr (or "make config-mpfr")
                             -- screen output is placed in the file
                                ${CORE_PATH}/tmp/DIAG_CONFIG_MPFR
            > make fifth     -- builds mpfr library (or "make build-mpfr")
                             -- screen output is placed in the
                                file ${CORE_PATH}/tmp/DIAG_BUILD_MPFR
            > make sixth     -- installs mpfr (same as "make install-mpfr")
                             -- screen output is placed in the
                                file ${CORE_PATH}/tmp/DIAG_INSTALL_MPFR
            > make testmpfr  -- you must pass this test to continue.
                                Output in ${CORE_PATH}/tmp/DIAG_TEST_MPFR

   3.6) STAGE 3: BUILD CORE LIBRARY.
        Now that gmp and mpfr are properly installed, you are ready to 
        install CORE.  IMPORTANT: the compilation settings for STAGE 3 
        are found in ${CORE_PATH}/Make.config.  They are described 
        in Step 3.3) above, and also Step 3.10) below.  Do the following:

            > cd ${CORE_PATH}
            > make seventh    -- this is equivalent to a plain "make", or
                                "make core", or "make corelib; make corex; 
                                make demo".
                             -- screen output is placed in the files
                                ${CORE_PATH}/tmp/DIAG_X where X = CORELIB,
                                COREX or BUILD_DEMO.

        This creates the library files "libcore++.a", "libcorex++_level1.a"
        and "libcorex++_level3.a", and store them under ${CORE_PATH}/lib.

        The source files for the Core Library are found in
        ${CORE_PATH}/src.  If you only want to make "libcore++.a", type
        "make" while in ${CORE_PATH}/src.

   3.7) STAGE 4: TESTING AND TIMING.
        If you had chosen LINKAGE to be "shared" above, then make sure
        that your LD_LIBRARY_PATH contains the directory ${CORE_PATH}/lib
        (see 3.10(4) below).

            > cd ${CORE_PATH}
            > make eighth     -- or, "make test".  Screen output is
                                 placed in the file 
                                    ${CORE_PATH}/tmp/DIAG_TEST_DEMO

        This tests all the previously compiled programs 
        in ${CORE_PATH}/progs.  Many of the programs are self-validating:
        if an unexpected value is computed, it will output "ERROR!"
        instead of "CORRECT!".  But an "INCORRECT!" output
        is not considered an error (e.g., with Level I accuracy, this
        is expected).  You can browse the output in the DIAG_TEST_DEMO file
        (search for the word "ERROR" to catch obvious errors).
        You can also time it:

            > make ninth       -- or "make time"; it uses the "time"
                                  utility to measure the time "make test" 
                               -- screen output is placed in 
                                  the file ${CORE_PATH}/tmp/DIAG_TIME

   3.8) USING CORE LIBRARY.
        Assume that you have a stand alone C++ program "foo.cpp".
        In the ideal case, you only have to insert the preamble:

             #ifndef CORE_LEVEL
             #   define CORE_LEVEL N   // N=1,2,3.  Default is N=3 
             #endif
             #include "CORE/CORE.h"  

        This should be placed after your standard include
        files (e.g., <iostream.h>).  You can now compile it as usual:

             g++   -I${CORE_PATH}/inc -I${CORE_PATH}/gmp/include \ 
                   foo.cpp -o foo \                       
                   -L${CORE_PATH}/lib -L${CORE_PATH}/gmp/lib \
                   -lcore${VAR} -lgmp -lm                      

        For more information, please go to ${CORE_PATH}/progs/generic for
        samples, and also read the tutorial in ${CORE_PATH}/doc.

   3.9) PLATFORMS.
        Version 2.1 has been tested on

        sun solaris 5.8         --  g++-2.95.3, g++-3.2, g++-3.3, g++-3.4
                                --  (gmp 4.1.3, gmp 4.1.4)
                                --  Sun's WorkShop 6 (C++ 5.3)
        Debian Linux 2.4.9      --  gcc version 2.95
        cygwin                  --  g++-3.2, g++-3.3, g++-3.4
        mingw                   --  g++-3.2, g++-3.3, g++-3.4
        Windows                 --  Visual C++, version 6.0, 7.0, 7.1 
        MacOS                   --  g++-4.0.1 (Apple build 5493)

        NOTE : for users who prefer a unix/linux type environment that is
        embedded within the Windows' world, we highly recommend the Cygwin
        platform as an easy-to-use environment.  Cygwin and Windows
        can freely access each other's files (a big bonus).

   3.10) VARIANT INSTALLATIONS
        (0) INSTALLING GMP AND/OR MPFR:
        Perhaps you only downloaded the base or standard version of CORE,
        but the above steps for installing GMP and/or MPFR did not work.
        The easiest is to go to our website and download GMP and MPFR
        separately (our site has the latest versions of GMP and MPFR that
        has been tested with CORE).  Suppose the downloaded GMP tar file is
        "gmp-4.2.4.tar.gz".  Put this into $(CORE_PATH) and do:

                > gzip -cd gmp-4.2.4.tar.gz | tar -xvf -
                > ln -s gmp-4.2.4 gmp
                > make gmp

        That is it!  The screen output at the end will include a test of
        gmp, showing some correctly computed digits.
        Do the same for your MPFR downloaded file:

                > gzip -cd mpfr-2.4.0.tar.gz | tar -xvf -
                > ln -s gmp-2.4.0 mpfr
                > make mpfr

        Again, the screen output at the end include a test of mpfr.

        (1) INSTALLING BASE DISTRIBUTION:
        This assumes you already have an installed gmp and mpfr.

          -- Download the file core-X.Y.Z.tgz (for version X.Y.Z) from our
             website, http://cs.nyu.edu/exact/core.   This version has no
             gmp, mpfr and no documentation.
          -- Do the UNPACK STEP (Step 3.2) above.  This creates the files
             under ${CORE_PATH} as described above.

        At this point there are two possibilities.
        You may not have to do anything.  To test this, try the following:

                > cd ${CORE_PATH}
                > make testgmp
                > make testmpfr

        If these tests print encouraging messages,
        you may continue from STAGE 3 (Step 3.6) above.  
        Otherwise you need to do STAGES 1 and 2 first.

        (2) INSTALLATION FOR WINDOWS PLATFORM
        If you use cygwin on your windows platform, then the installation
        is same as on Unix platform as described above. 
        But if you use Visual C++ instead, then follow these steps.

          -- Download file core-X.Y.Z_full.tgz (for version X.Y.Z) from our
             website. (We recommend the Full Distribution since it contains
             gmp already and you cannot easily link to pre-installed gmp)
          -- Unpack it into some directory ${CORE_PATH}. 
          -- Unpack ${CORE_PATH}\gmp-xxx.tar.gz into ${CORE_PATH}\win32 to
             create a subdirectory ${CORE_PATH}\win32\gmp-xxx. Rename this
             to plain ${CORE_PATH}\win32\gmp.
          -- Install gmp patches
                
                c:\>cd ${CORE_PATH}\win32\patches
                c:\>patch xxx      // where "xxx" is the gmp version;
                                   // (xxx = "3.1.1", "4.0.1","4.1-static",
                                   //  and  "4.1-dynamic" available)
                                   // for GMP 4.1.2, you still can choose
                                   // "4.1-static" or "4.1-dynamic".

          -- Open a "Command Prompt" window and run "vcvars32.bat" to setup
             Visual C++ command line environment. File vcvars32.bat is
             automatically created by your Visual C++.  E.g., it may be
             found in C:\"Microsoft Visual Studio"\vc98\Bin.

          -- Compile gmp and the Core Library: 
                
                c:\>cd ${CORE_PATH}\win32
                c:\>nmake
          
             This nmake also creates the Core Extension libraries and 
             and all the demo programs.

          -- Optional: Testing (for the demo programs)

                c:\>cd ${CORE_PATH}\win32
                c:\>set PATH=%PATH%;${CORE_PATH}\win32\lib
                c:\>nmake test
      
          -- Alternatively, use Visual Studio IDE to compile Core Library
             and Core Extensions.  E.g., to compile Core Library, open the 
             project file "core.dsw" under ${CORE_PATH}\win32\corelib, and 
             compile.  To compile Core Extensions, open the project file "
             corex.dsw" under ${CORE_PATH}\win32\ext, and compile.

        (3) CORE USAGE FOR WINDOWS PLATFORM
        If you want to create your own Visual C++ project file, do the
        following steps:
 
          --add "${CORE_PATH}\inc", "${CORE_PATH}\win32\gmp" to IncludePath
          --add "${CORE_PATH}\win32\lib" in "Lib Path".
          --Enable "Run-Time Type Information" in your project settings. 

        (4) SHARED OR STATIC LIBRARY LINKAGE:
        In Make.config, you can set the LINKAGE variable to "shared" or
        "static".  The default is "static" for simplicity; but the
        executables will be fairly large (over 1MB each).
        To run programs using the static library there is nothing
        special to do;  to run the dynamic version of the Core Library,
        you could move your compiled CORE, gmp, etc, libraries 
        into the standard library paths (e.g., /lib or /usr/lib).
        Alternatively, you can set the environment
        variable LD_LIBRARY_PATH as follows:

        for "csh/tcsh",  
            > setenv LD_LIBRARY_PATH \
                ${CORE_PATH}/lib:${CORE_PATH}/gmp/lib:${LD_LIBRARY_PATH}.
        
        for "bash",
            $ export LD_LIBRARY_PATH= \
                ${CORE_PATH}/lib:${CORE_PATH}/gmp/lib:${LD_LIBRARY_PATH}.

        If gmp is not installed in ${CORE_PATH}, adjust accordingly.

        (5) RECOMPILATION OF THE CORE LIBRARY:
        Sometimes you may want to recompile the Core Library (perhaps after
        changing a file in ${CORE_PATH}/src or $(CORE_PATH)/inc).  
        The simplest is to go into ${CORE_PATH}/src and type
        "make clean; make".  This will automatically update
        the files libcore++.a and/or libcore++.so in ${CORE_PATH}/lib. 
        But for experimental purposes, we may want to keep two or more
        versions of libcore++.a around.  In this case, we suggest using the
        VAR variable described next.

        (6) DEBUGGING AND OTHER VERSIONS OF CORE LIBRARY: 
        It may be useful to have different compiled versions of the Core 
        Library around.  For this, edit file ${CORE_PATH}/Make.config.
        This file defines the compilation flags, depending on the platform.
        There are two main variables to set: VAR and PLATFORM.

        VAR indicates the "variant" of the library you want to compile.
        The Makefile creates the library file "libcore++$(VAR).a" to be
        placed in ${CORE_PATH}/lib/.  Since the default value of VAR is
        the empty string, the default library is the plain "libcore++.a".

        Here are two useful variants, Debug variant and NoOpt variant:

        (a) VAR=Debug: you need this variant to run the GNU debugger, gdb.
           If you set VAR=Debug in Make.config, and re-do STAGE 3 above
           (STAGES 1 and 2 are still fine), you will create the debugging
           variant of the library, namely "libcore++Debug.a".  Other stages
           will similarly create the Debug variants, e.g., Debug versions
           Core Extension libraries, sample programs, etc.
        (b) VAR=NoOpt: this version turns off all compiler optimizations.
           Try this if you find mysterious bugs, and to rule out possible
           errors caused by aggressive compiler optimizations.

        PLATFORM is used to define various variables that are platform-
        and compiler-specific.  It defaults to "gnu".  
        Other possible values are "sun", "cyg", "mingw", "sgi", "mac".
        The "sgi" option has not been tested in the latest library.

  3.11) CORE EXTENSIONS.
        Under ${CORE_PATH}/ext, you will find extensions of Core Library
        to encode knowledge of algebraic and geometric domains.  These
        "CORE Extensions" (COREX) are quite rudimentary at present, but 
        feel free to contribute.   

        For instance, there is a geometry2d and geometry3d.  We compile a
        Level 1 and Level 3 versions of these libraries, so that you can
        compile your application in either of these two accuracy levels.

  3.12) HOUSEKEEPING AND UNINSTALL
        To uninstall the library, simple delete all the files under
        ${CORE_PATH} (i.e., we do not put files anywhere).  If gmp and
        mpfr are installed under ${CORE_PATH} you may remove them first.
        
        In every directory, we have a Makefile with two targets called
        "clean" and "veryclean".  If you want to save space, you can type
        "make clean" in any directory to remove temporary files.
        E.g., *.o files.  If you type "make veryclean", this will
        remove more files (in particular, all executable files).
        NOTE: both targets are recursively propagated into subdirectories.

*************************************************************
4) NEWS, PLANS AND BUGS
   NEW in Version 2.1 (July 2010)
   -- introduction of anary nodes (operators with no fixed arity,
      such as sum and product)
   -- introduced a DoubleWrapper as a thinwrapper over
      machine double in Level 1: now Level 2 and Level 3 programs
      can also compile at Level 1.
   -- templated linearAlgebra extension
   -- templated interval class 
   -- templated complex number class 
   -- much improved curve subdivision algorithm (Cxy), and related
      subdivision code for complex root isolation and interval Newton.
   -- various bug fixes, including root bound overflow bug and
      printout of garbage digits but.
   -- Willi Mann provided a patch to allow the default constructor for
      Expr to use a static object.  If you have functions that have local
      Expr variables, this can greatly improve speed (30% speedup in Mann's
      application in the FIST software of Martin Held for triangulation.)
   -- Acknowledgments:  Thanks for feedback and bug reports from
          Sven Kohler, Michael Haag, Volmar Klatt, Martin Held,
          Willi Mann, Narayan Kamath.   Nararan contributed the interval
		  class, complex number class, and improvements in subdivision
		  algorithms.

   NEW in Version 2.0:
   -- Complete redesign of Core Library, based on Zilin's PhD thesis.
      However, the original simple interfaces are retained.
   -- Expr class is now templated to receive three parameter classes:
      Filter Class, RootBound Class and BigFloat Class.  
   -- Complete redesign of BigFloat (a new class called BigFloat2, for
      interval arithmetic, is split off the original BigFloat).
      The new BigFloat classes are based on MPFR, which in turn is
      based on GMP.
   -- Complete redesign of the basic Expr evaluation algorithms to improve
      efficiency, avoiding redundant computations.
   -- Introduction of transcendental functions into Expr.
      All the elementary functions are available in Expr,
      with EscapeBound used in lieu of root bounds.  CutOffBounds are also
      introduce to limit approximation near zero.
   -- Redesigned hypergeometric function classes.
   -- New facilities for user-defined nodes in Expr.
   -- Use of Descartes method for real root isolation,
      in place of Sturm.  This gives general speedup improvements.
   -- Improved InCore, the interactive python version of CORE.
   -- Bug Fixes
          * Newton iteration fixed to really have quadratic convergence
            (previous implementation only has linear convergence)
          * Bug fix in Sturm root counting functions (off by one)
   -- Miscellaneous:
          * Versioning of Core Library is now Core-X.Y.Z (starting with
            X.Y.Z = 2.0.0).  Previously it was "Core_v1.7.0").
          * Upgraded Core Library to be compatible with latest gmp 4.1.4.
   -- Acknowledgments:  Thanks for feedback and bug reports from
            Daniel Russell, Ron Wein, Andreas Fabri, Sylvain Pion.
            
=============================
   NEW in Version 1.7:
   -- Introduction of plane algebraic curves and bi-variate polynomials 
         The main capability is to plot curves and do basic
         intersection tests.  See CORE_PATH/progs/curves/.
   -- Introduced "InCore", an interactive version of Core Library 
         based on Python (this has to be downloaded separately)
   -- Enhancement of the univariate polynomial and real algebraic
        number facilities. New methods such as
        polynomial GCD, resultant, square free part, primitive part, etc.
        * Polynomial<NT> can now work with all choices of NT
                NT = BigInt, int, BigRat, Expr, BigFloat
          But not all functionality are fully available for
          NT=BigFloat and NT=Expr (e.g., rootbounds).
        * Sturm<NT> can now work with NT=BigFloat
        * To support the above, various new methods are added to the
          BigInt, BigRat, BigFloat, Real classes (isDivisible, gcd, etc).
        * Polynomials now accept string inputs. 
            E.g., Polynomial<BigInt> p = "x^3 - 2x^2 + 17x - 4";
   -- Compatibility with gmp 4.1, and gcc 3.3
   -- Introduced a common Reference Counting facility for all
         Core number types, encoded in the two templated classes:
                RCRepImpl<class N>
      to create Reps of the class N.  The basic functions provided by
      this class is reference counting, and gives us the "Reps" of
      each number type (e.g., BigIntRep is derived from RCRepImpl).  
      The other class is
                RCImpl<class T>
      which is the actual number type (e.g., BigInt is derived
      from RCImpl<BigIntRep>).  As a result former Rep Files
      such as RealRep.h can be removed, and the code size is reduced.
      Also, BigInt and BigRat now have reference counting in
      their Rep classes (none before).
   -- Speedup from reorganization of Core Library number classes.
      We wrote wrappers around gmp's C function library.
      We also tested the possibility of using gmp's C++ classes (since
      gmp 4.1) which are template based.  CORE's interface are not fully
      compatible with gmp since we have exact algebraic representation
      and precision bounds.
      We compared three versions of Core Library:
           (A) -- old code: version 1.6
           (B) -- new code: some optimization
           (C) -- new code: optimization + using gmp C++ class
      Here are the running times on two machines:
   
      Test              Pentium III (1G Memory)        Jinai (Solaris)
      =======================================================
      > A               64.12s                         1m:07s
      > B               56.52s                         52.6s
      > C               50.71s                         50.4s
      > Speedup         20%                            25%
    
      Although the use of gmp C++ classes (which use expression template
      to eliminate temporary variables) is faster, it is
      incompatible with visual C++ (not sure about Sun CC).
      So we added as an option for compiling the Core Library:
      to turn on gmp C++ classes, un-comment macro CORE_USE_GMPXX
      in CoreImpl.h and recompile Core Library; there is no need to compile
      GMP C++ library since all necessary code are in Core Library already.
   -- Updated Core Library Tutorial
   -- Simple openGL display for curves: see CORE_PATH/ext/graphics/
   -- Bug fixes
          * rootOf(P,i) now works properly when P have multiple roots
            (reason: we assumed the endpoints of isolating intervals
            have distinct signs)
          * Fixed bugs in Newton and Sturm methods
          * Expr::doubleValue() is now correctly implemented.
            E.g., suppose you compute
                double s = sqrt(n);
                double ss = sqrt(Expr(n)).doubleValue();
            Then we guarantee that |s-ss| has relative error at
            most 4*CORE_EPS = 4.44089e-16 (CORE_EPS is machine epsilon). 
            This factor of 4 is essentially the best possible (see
            CORE_PATH/progs/testIO/testSqrt.cpp).
          * fixed Expr::degreeBound().  Previously it always returned 1 at
            leaves.  With algebraic numbers, it must return d_e()).  This
            has dramatic improvement in speed. Ron Wein's program for
            intersecting two ellipses used to take overnight, but now take
            0.4 sec.
          * Fixed an output bug that has been around since Core 1.4.
            BigFloat can print an output whose exponent is off by 1.
                E.g., sqrt(100) = 9.999e0 but BigFloatRep::round(...),
                an internal function, returns the string "1.0000e0"
                instead of "1.0000e1".  Thanks to Blazi for noticing this.
   -- Miscellaneous:
          * Previously, the library version numbers have two numbers
            (e.g., "1" and "6" in Version 1.6).  Between official releases,
            we call it Version 1.6x.  Thus "Version 1.6x" refers to any
            number of unofficial releases between 1.6 and 1.7.
            We now have a third number.  E.g., this official release
            is Version 1.7.0.
          * CORE::core_error() is enhanced to write its results
            into a file "Core_Diagnostics" but also writes errors to
            std::cerr (as before).  
          * We introduce new versions of the BigFloat::makeExact(), namely,
            makeCeilExact() and makeFloorExact().
          * Upgraded Core Library to be compatible with the latest gmp
            4.1.3 (released 4/28/2004).
          * Use the program "astyle" to beautify all our code, so the
            files are more consistent and readable.  Tab are converted
            to spaces, use k&r style, etc):
                       ./astyle --style=kr -s2 filename
          * More use of BigFloats to replace Expr, when possible.  E.g.
            CauchyLowerBound() and CauchyUpperBound() rewritten using
            BigFloat instead of Expr (30% improvement here).
          * In CORE_PATH, you can type "make options" 
            to see all your currently selected CORE ENVIRONMENT VARIABLES.
            If you type "make alloptions", this will also show
            all possible alternatives that you could have chosen.
          * Simple timing facility (see src/Timer.h)
   -- Acknowledgments:  Thanks for feedback and bug reports from
            Janos Blazi, Arno Eigenwillig, Ovidiu Daescu,
            Andreas Fabri, Michael Hemmer, Athanasios Kakargias,
            Daniel Russell, Ron Wein.

   PLANS:
       -- improved precision-sensitive algorithm
       -- improved bounds (measure of Sekigawa, etc)
       -- better floating point filters
       -- optimized determinant primitive and filters
       -- compilation and optimization of expressions
       -- Expressions should have the ability to output exact values
                in case of integers or rational numbers.  Currently,
                only bigFloat values are output.
       -- complex numbers
       -- better implementation of relative precision bounds
       -- Newton-based algorithms for elementary functions
       -- development of Core Extensions (geometry, algebra, meshes, etc)
       -- graphical support and interface
       -- File I/O of Core Objects (e.g., expressions)
       -- CORE versions of "printf" and "scanf" (this reduces the fuss to
                CORE-ize a standard C program).
       -- the exponent of a bigFloat number is represented by 
             a machine long.  This should not be a problem in practice.  An
             improvement is to use machine double for the exponent,
             yielding 53 bits of precision.
   
   BUGS:
       -- To report a bug, please send email to exact@cs.nyu.edu, with as
             much details as possible (including your platform/compiler).
       -- Let r = 54.  Suppose you convert a rational p/q to a BigFloat bf 
			 using r bits of relative precision.  Next convert bf to a
			 machine double md,
                       double md = (double)bf;
			 This md should be equal to rounding "p/q" to the nearest
			 machine double.  Turns out (Core 1.4) this may not be true!  
			 But if r=59, our tests indicated that md does equal the
			 machine rounding of p/q.  This bug is not too serious since,
             for any particular inputs, it can be removed by increasing r.
             Tests suggest that r=59 is sufficient for
             all p/q.  See ${CORE_PATH}/src/test/ for details.
       -- we should allow defOutputDigits to be CORE_INFTY, and when
             a rational number is printed in this case, we ought to print
             it with no errors.  Currently, we let defBigFloatOutputDigits
             control this output; since this value is never infinite,
             the printed value may have error.
       -- level II is not fully supported
       -- level IV is not fully defined 

*************************************************************
5) ACKNOWLEDGMENT and BRIEF HISTORY

   This work has been supported by a National Science Foundation
        Grant \#CCR 9402464 and \#CCR 0082056 (an ITR grant).
   The Real/Expr Package (1994-96) was developed by
        Chee Yap and Tom Dube.  
   Koji Ouchi and Chee Yap further improved the Real/Expr Package with
        its concept of composite precision bounds.
        The new algorithms for BigFloat with automatic error bounds
        is documented in Koji's masters thesis.
   A Numerical Accuracy API (the 4 levels of accuracy) was 
        proposed by Chee Yap in Oct 1998.
   Core Library Version 1.1 (Jan 1999) was adapted from Real/Expr by
        Chee Yap, Vijay Karamcheti, Igor Pechtchanski and Chen Li to
        implement the Numerical Accuracy API.
        Compiler-based optimizations techniques were investigated.
   Version 1.2 (Sep 1999) is a debugged and improved version.
        The BFMS root bound was incorporated. 
   Version 1.3 (Sep 2000) is significantly
        faster than its predecessor because of new improved root 
        bounds and adoption of LiDIA/CLN's bignumber as default kernel.
   Version 1.4 (Aug 2001) moved from LiDIA/CLN to gmp as the main kernel.
		Incremental square roots, improved precision-sensitivity
		algorithms, simple floating point filters, hypergeometry package.  
   Version 1.5 (Aug 2002) improvements in speed and root bounds (k-ary
		bounds), CGAL compatibility changes, file I/O for large
		mathematical constants (BigInt, BigFloat, BigRat),
		improved hypergeometric package,
   Version 1.6 (June 2003) introduced arbitrary
        real algebraic numbers in Expr's, incorporated Polynomial and Sturm
        classes into Core Library.  CORE is now distributed with CGAL,
        and issued under the QPL agreement.
   Version 1.7 (Nov 2004) introduced algebraic curves and bivariate 
       polynomials.  An interactive version of Core Library called "InCore"
       is available.  Beginning graphic capability for display of curves.
   Version 2.0 (Jun 2006) complete redesign of Core Library (templated
        Expr class taking parameters for Filter class, Rootbound class
        and BigFloat class), incorporation of transcendental functions into
		Expr, Redesigned BigFloat (split off another class called
		BigFloat2), general speed improvement 2-10 times.
   Version 2.1 (Jul 2010) 
        -- introduce anary nodes for sum and product
        -- DoubleWrapper for machine double in Level 1
                so that Levels 2 and 3 programs can compile in Level 1.
        -- templated classes for linearAlgebra, intervals, complex numbers

*************************************************************
6)  LICENSE INFORMATION

   Core Library is now under the terms of the Q Public License
   version 1.0. See the file LICENSE.QPL distributed with CORE.
*************************************************************

*************************************************************
*  CONTACT AND FURTHER INFORMATION:
*       For comments and bug report, send email to:
*               exact@cs.nyu.edu.
*       Core Library Homepage:
*               http://cs.nyu.edu/exact/
*       CGAL Homepage:
*               http://www.cgal.org/
*       GEOMETRY FACTORY Homepage:
*               http://www.geometryfactory.com/
*       GNU/gmp Homepage:
*               http://www.gnu.org/software/gmp/gmp.html
*               http://www.gnu.org/home.html
*************************************************************