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************************************************************ * 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) 1998-2010 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) ACKNOWLEDGEMENT 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 TERMINOLOGY. "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 (this file is inherited by other Makefiles) 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 diagnostic messages) python/ Python interface to Core Library. win32/ Windows related files (can be deleted for unix-base) bin/ executable programs (mostly for development currently) builds/ directory for building gmp and mpfr (it is possible 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. mpfr/ mpfr installation directory (may be a link). This directory may be missing if you have your own mpfr. 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 for unpacking and compiling (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 3.0) PREREQUISITE: You must have "make" (GNU's gmake is recommended), "tar" and "gzip". For compilers, "g++" is also recommended but not required. In turn, "g++" will need "m4" (the macro preprocessor). We DO NOT assume that you already have "gmp" (GNU's multiprecision number package). If you do not download the full version, we will use "wget" to automatically download gmp from the web). For Windows: you could replace g++ with Visual C++. However we highly recommend "Cygwin" from Red Hat. This is a free and easy-to-install Unix-like environment that sits on top of all versions of Windows. All the needed tools (g++, m4, make, tar, gzip, wget, etc) are available (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 and utilities, directly from the web. You can download Cygwin at http://www.cygwin.com. 3.1) DOWNLOAD. First download the file core-X.Y.Z_full.tgz (for version X.Y.Z). The current distribution is X.Y.Z=2.1.0. The website for download is http://cs.nyu.edu/exact/core. There are three distributions, a "base distribution", a "standard distribution" and a "full distribution". The standard distribution includes the base distribution and documentation. The full distribution includes the standard distribution, plus gmp and mpfr. The main steps here describes an installation from a full distribution in a unix-like environment: it is the simplest to install. But see 3.10) below for *variant* installations: these include installing only the base distribution, using dynamic libraries, Windows platform, re-installing the Core Library, building Debugging Versions of Core Library, etc. (dated!) The overall installation time is 25-40 minutes on a typical machine. If you already have gmp, the time is halved (on Pentium III 800 MHz, 256 MB RAM, it takes 15 minutes). 3.2) UNPACK. In your computer, go to the directory where you wish to install the Core Library. You don't have to be a superuser. If this location is the value of the environment variable CORE_INSTALL (in unix), you would do: % cd $(CORE_INSTALL) (% denotes the unix prompt in these instructions) Move the file core-X.Y.Z_full.tgz there. Now unzip and untar the file by typing: % gzip -cd core-X.Y.Z_full.tgz | tar xvf - This creates a directory called core-X.Y.Z. All the files and directories mentioned in 2) above, including this README file, should now be in this directory. In the following, the variable CORE_PATH denotes the full path name of this newly created directory. Hence, $(CORE_PATH) is really $(CORE_INSTALL)/core-X.Y.Z 3.3) There are three 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 the PLATFORM variable. To see what your chosen 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 most 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_install_dir}/lib E.g., in bash, % export LD_LIBRARY_PATH=${LD_LIBRARY_PATH}:${your_gmp_install_dir}/lib For instance, ${your_gmp_install_dir} might be /usr/local. If you are installing gmp using option d) below, we suggest letting ${your_gmp_install_dir}=${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, you may 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 which cannot be found by the above test, then make a symbolic link from ${CORE_PATH}/gmp using the following command: % cd ${CORE_PATH} % ln -s ${your_gmp_install_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_install_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 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 and may go to STAGE 3. b) You have installed MPFR, but it could not be found by the compiler in a). In this case, we make a symbolic link from ${CORE_PATH}/gmp using the following command: % cd ${CORE_PATH} % ln -s ${your_mpfr_install_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 (same as "make config-mpfr") -- screen output is placed in the file ${CORE_PATH}/tmp/DIAG_CONFIG_MPFR % make fifth -- builds mpfr library (same as "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 file ${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 -- equivalent to "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!". Sometimes, the output says "INCORRECT!" but this 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 -- equivalent to "make time"; it uses the "time" utility to measure the time "make test" -- screen output is placed in the file ${CORE_PATH}/tmp/DIAG_TIME 3.8) USING CORE LIBRARY. Assume that you have a stand alone C++ program "foo.cpp". In the ideal case, you only have to insert the preamble: #ifndef CORE_LEVEL # define CORE_LEVEL N // N=1,2,3. Defaults to N=3 if omitted #endif #include "CORE/CORE.h" This should be placed after your standard include files (e.g., <iostream.h>). You can now compile it as usual: g++ -I${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 the 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 preinstalled gmp) -- Unpack it into some directory ${CORE_PATH}. -- Unpack ${CORE_PATH}\gmp-xxx.tar.gz into ${CORE_PATH}\win32: this creates 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., sometimes it is 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 the 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", "${CORE_PATH}\win32\gmp" to "Include Path". -- add "${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, you must 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", which will 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 then 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". The 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 want 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. 3.10) 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. 3.11) 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 want to move 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% improvement in Mann's application in the FIST software of Martin Held for triangulation.) -- Acknowledgements: 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 the latest gmp 4.1.4. -- Acknowledgements: 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 and 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 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: optimzation + 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 incompatiable 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, uncomment macro CORE_USE_GMPXX in CoreImpl.h and recompile Core Library; there is no need to compile GMP C++ library since all necessory 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. But 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 Wersion 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) -- Acknowledgements: 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 will reduce 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) ACKNOWLEDGEMENT 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. ************************************************************* 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 *************************************************************
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