Html Documentation MPFRCX 0.4.2

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This manual documents how to install and use the Multiple Precision Real and Complex Polynomial Library, version 0.4.2 as of May 2013.

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MPFRCX Copying Conditions

The MPFRCX Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version, see the file COPYING.LIB.

The MPFRCX Library 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 Lesser General Public License for more details.

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1 Introduction to MPFRCX

MPFRCX is a portable library written in C for arithmetic of polynomials with arbitrary precision real or complex floating point coefficients. It is based on the GNU MP, the MPFR and the MPC libraries.

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2 Installing MPFRCX

To build MPFRCX, you first have to install GNU MP, MPFR and MPC on your computer. For MPC, the minimal supported version is 1.0; the minimally required versions of GNU MP and MPFR depend on the MPC version used. You need a C compiler, preferably GCC, but any reasonable compiler should work. And you need a standard Unix ‘make’ program, plus some other standard Unix utility programs.

Here are the steps needed to install the library on Unix systems:

  1. tar xzf mpfrcx-0.4.2.tar.gz
  2. cd mpfrcx-0.4.2
  3. ./configure

    if GMP, MPFR and MPC are installed into standard directories, that is, directories that are searched by default by the compiler and the linking tools.

    ./configure --with-gmp=<gmp_install_dir>

    is used to indicate a different location where GMP is installed.

    ./configure --with-mpfr=<mpfr_install_dir>

    is used to indicate a different location where MPFR is installed.

    ./configure --with-mpc=<mpc_install_dir>

    is used to indicate a different location where MPC is installed.

    ./configure --with-gmp=<gmp_install_dir> --with-mpfr=<mpfr_install_dir> --with-mpc=<mpc_install_dir>

    is used to indicate different locations of GMP, MPFR and MPC.

    Warning! Do not use these options if you have CPPFLAGS and/or LDFLAGS containing a -I or -L option with a directory that contains a GMP header or library file, as these options just add -I and -L options to CPPFLAGS and LDFLAGS after the ones that are currently declared, so that DIR will have a lower precedence. Also, this may not work if DIR is a system directory.

  4. make

    This compiles MPFRCX in the working directory.

  5. make check

    This will make sure MPFRCX was built correctly.

    If you get error messages, please report them to ‘’ (See Reporting Bugs, for information on what to include in useful bug reports).

  6. make install

    This will copy the file mpfrcx.h to the directory /usr/local/include, the file libmpfrcx.a to the directory /usr/local/lib, and the file to the directory /usr/local/share/info (or if you passed the ‘--prefix’ option to configure, using the prefix directory given as argument to ‘--prefix’ instead of /usr/local). Note: you need write permissions on these directories.

2.1 Other `make' Targets

There are some other useful make targets:

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3 Reporting Bugs

If you think you have found a bug in the MPFRCX library, please investigate and report it.

There are a few things you should take into account when you compose your bug report. Please send us a minimal test case that makes it possible for us to reproduce the bug. Include instructions on how to run the test case, and why the result differs from what you would expect.

Please include compiler version information in your bug report. This can be extracted using ‘cc -V’ on some machines, or, if you are using gcc, ‘gcc -v’. Also, include the output from ‘uname -a’.

Send your bug report to: ‘’.

Beware that the MPFRCX library is in a very early development stage, and some functions, while working correctly, may be terribly inefficient. You might want to send an e-mail to the above address if you are interested in one of the more exotic functions to enquire about its status.

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4 MPFRCX Basics

MPFRCX provides types and functions for working with univariate polynomials, taking as coefficients either real or complex floating point numbers of arbitrary precision. The functions are collected in the library libmpfrcx.a and declared in the header file mpfrcx.h.

4.1 Nomenclature and Types

A real polynomial is a polynomial whose coefficients are of type mpfr_t. The C data type for such objects is mpfrx_t. All coefficients are supposed to have the same floating point precision. Besides its list of coefficients, a variable of type mpfrx_t contains the degree of the polynomial as an int and the precision of its coefficients as an mp_prec_t. If the degree of a polynomial increases, its list of coefficients is lengthened accordingly; on the other hand, if the degree decreases, the memory allocated to the now superfluous coefficients is not freed, unless explicitly requested by a call to mpfrx_realloc, see Initialisation Functions.

A complex polynomial is a polynomial whose coefficients are of type mpc_t. The C data type for such objects is mpcx_t. All coefficients are supposed to have the same floating point precision, and this both for their real and their imaginary parts. Otherwise, complex polynomials behave like real ones.

When calling the functions described in the following, arguments of type mpfrx_ptr or mpfrx_srcptr stand for arbitrary variables of type mpfrx_t; the former may be modified by the function, the latter not. The same holds for mpcx_ptr and mpcx_srcptr.

Notice that unlike for operations with real numbers of type mpfr_t and complex numbers of type mpc_t, there are no rounding modes for operations with polynomials and no precise semantics; polynomial arithmetic is realised by calls to functions on the coefficients, which may accumulate rounding errors.

4.2 Function Classes

Functions and macros working with real polynomials begin with mpfrx_, those treating complex polynomials begin with mpcx_. For the time being, there are no mixed operations.

4.3 MPFRCX Variable Conventions

As a general rule, all MPFRCX functions expect output arguments before input arguments, in analogy with GMP, MPFR and MPC.

MPFRCX allows you to use the same variable for both input and output in the same expression. For example, the main function for multiplication of real polynomials, mpfrx_mul, can be used like this: mpfrx_mul (f, f, f) to replace f by its square.

Before you can assign to an MPFRCX variable, you need to initialise it by calling one of the special initialisation functions. When you are done with a variable, you need to clear it out, using one of the functions for that purpose.

A variable should be initialised only once; after a variable has been initialised, values may be assigned to it any number of times.

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5 Functions

All publicly visible functions exist with the same behaviour for real or complex polynomials; their names start with mpfrx_ resp. mpcx_. For the time being, the only specific functions concern the fast Fourier transform and are internal to the library.

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5.1 Initialisation Functions

An mpfrx_t or mpcx_t object must be initialised before storing the first value in it, using the function mpfrx_init or mpcx_init.

— Function: void mpfrx_init (mpfrx_ptr f, const int size, const mp_prec_t prec)
— Function: void mpcx_init (mpcx_ptr f, const int size, const mp_prec_t prec)

Initialise f with initial space for size coefficients of precision prec, and set it to zero. It is possible to assign a polynomial with more than size coefficients to f later on; the size of f is then increased automatically. Beware that size=d+1 coefficients are needed to store a polynomial of degree d.

— Function: void mpfrx_clear (mpfrx_ptr z)
— Function: void mpcx_clear (mpcx_ptr z)

Free the space currently occupied by z. Make sure to call this function on each variable precisely once.

— Function: void mpfrx_realloc (mpfrx_ptr f, const int size)
— Function: void mpcx_realloc (mpcx_ptr f, const int size)

Changes the number of coefficients stored in f to size, which may be more or less than (or equal to) the previous size, while preserving the precision of the coefficients. If f still fits (that is, its degree is at most size-1), its value is preserved, otherwise, it is replaced by 0.

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5.2 Assignment Functions

These functions assign new values to already initialised polynomials (see Initialisation Functions).

— Function: void mpfrx_set (mpfrx_ptr h, mpfrx_srcptr f)
— Function: void mpcx_set (mpcx_ptr h, mpcx_srcptr f)

Set the value of h from f. The precision of h is preserved, and the coefficients of f are rounded if the target precision is lower.

— Function: void mpfrx_swap (mpfrx_ptr f, mpfrx_ptr g)
— Function: void mpcx_swap (mpcx_ptr f, mpcx_ptr g)

Swap the contents of the variables f and g. If their coefficients do not have the same precision, precisions are swapped as well. The effect is thus not the same as obtained by several calls to mpfrx_set or mpcx_set, respectively, with an additional temporary variable, which would keep the respective precisions of f and g unchanged.

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5.3 Combined Initialisation and Assignment Functions

— Function: void mpfrx_init_set (mpfrx_ptr h, mpfrx_srcptr f)
— Function: void mpcx_init_set (mpcx_ptr h, mpcx_srcptr f)

Initialise h with the same precision as f and set its value from f.

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5.4 Access Functions

— Macro: int mpfrx_get_prec (mpfrx_srcptr f)
— Macro: int mpcx_get_prec (mpcx_srcptr f)

Return the common precision of the coefficients of f.

— Function: void mpfrx_set_prec (mpfrx_ptr f, const mp_prec_t prec)
— Function: void mpcx_set_prec (mpcx_ptr f, const mp_prc_t prec)

Set the precision of the coefficients of f to prec and replace f by the zero polynomial. The effect is similar to a call to mpfrx_clear resp. mpcx_clear followed by a call to mpfrx_init resp. mpcx_init, but the number of coefficients in the polynomial is kept.

— Macro: int mpfrx_get_deg (mpfrx_srcptr f)
— Macro: int mpcx_get_deg (mpcx_srcptr f)

Return the degree of f, which is -1 for the zero polynomial.

— Function: void mpfrx_set_deg (mpfrx_ptr f, const int deg)
— Function: void mpcx_set_deg (mpcx_ptr f, const int deg)

Set the degree of f to deg while preserving the coefficients. If the degree increases, the new coefficients are set to NaN and need to be set manually before computing with the variable, see mpfrx_set_coeff and mpcx_set_coeff. If necessary, new coefficients are allocated.

— Function: mpfr_ptr mpfrx_get_coeff (mpfrx_srcptr f, const unsigned int i)
— Function: mpc_ptr mpcx_get_coeff (mpcx_srcptr f, const unsigned int i)

Return a pointer to coefficient i of f, or NULL if the degree of f is less than i.

— Function: void mpfrx_set_coeff (mpfrx_ptr f, const unsigned int i, mpfr_srcptr coeff)
— Function: void mpcx_set_coeff (mpcx_ptr f, const unsigned int i, mpc_scrptr coeff)

Set the coefficient i of f to coeff. If the current degree of f is smaller than i, then the degree of f is set to i; intermediate coefficients are set to NaN.

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5.5 Comparison and Miscellaneous Functions

— Function: int mpfrx_cmp (mpfrx_srcptr f, mpfrx_srcptr g)
— Function: int mpcx_cmp (mpcx_srcptr f, mpcx_srcptr g)

Return 0 if f equals g and -1 if not. The coefficients of f and g are compared one by one; so even if the two polynomials have different precisions, they may be recognised as equal.

— Function: void mpfrx_urandom (mpfrx_ptr f, int deg, gmp_rand_state_t state)
— Function: void mpcx_urandom (mpcx_ptr f, int deg, gmp_rand_state_t state)

If deg<0, set f to be the 0 polynomial. Otherwise, generate a uniformly distributed random degree between 0 and deg (inclusive), and a random polynomial of this degree. Each coefficient is chosen uniformly at random in the unit interval [0, 1] resp. the unit square [0, 1] x [0, 1]; except for the leading coefficient, which cannot be 0.

state is a gmp_randstate_t structure which should be created using the GMP rand_init function, see the GMP manual.

— Macro: const int MPFRCX_VERSION_MAJOR
— Macro: const int MPFRCX_VERSION_MINOR
— Macro: const char* MPFRCX_VERSION_STRING

The major, minor and patchlevel version number of the library. These are concatenated and separated by '.' to form the version string; '-dev' is added to the version string of development snapshots.

— Function: const char * mpfrcx_get_version (void)

Return the MPFRCX version as a null-terminated string.

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5.6 Input and Output Functions

The following function writes a polynomial to an output stream. When using it, you need to include stdio.h before mpfrcx.h.

— Function: size_t mpfrx_out_str (FILE* stream, int base, size_t n_digits, mpfrx_srcptr f)
— Function: size_t mpcx_out_str (FILE* stream, int base, size_t n_digits, mpcx_srcptr f)

Output f to stream using the given base and the given number of digits n_digits for the coefficients. If n_digits is 0, then a suitable number of digits is chosen so that reading the polynomial into a variable of the same precision as f yields the same polynomial again (this input function needs yet to be written...).

The output starts with an opening bracket '(', followed by the degree and a list of coefficients in decreasing order of degree (separated by spaces) and a closing bracket ')'.

Return the number of written characters.

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5.7 Basic Polynomial Arithmetic

— Function: void mpfrx_add (mpfrx_ptr h, mpfrx_srcptr f, mpfrx_srcptr g)
— Function: void mpcx_add (mpcx_ptr h, mpcx_srcptr f, mpcx_srcptr g)

Set h to f + g.

— Function: void mpfrx_sub (mpfrx_ptr h, mpfrx_srcptr f, mpfrx_srcptr g)
— Function: void mpfrx_si_sub (mpfrx_ptr h, const long int f, mpfrx_srcptr g)
— Function: void mpcx_sub (mpcx_ptr h, mpcx_srcptr f, mpcx_srcptr g)
— Function: void mpcx_si_sub (mpcx_ptr h, const long int f, mpcx_srcptr g)

Set h to f - g.

— Function: void mpfrx_neg (mpfrx_ptr h, mpfrx_srcptr f)
— Function: void mpcx_neg (mpcx_ptr h, mpcx_srcptr f)

Set h to −f.

— Function: void mpfrx_mul (mpfrx_ptr h, mpfrx_srcptr f, mpfrx_srcptr g)
— Function: void mpcx_mul (mpcx_ptr h, mpcx_srcptr f, mpcx_srcptr g)

Set h to f * g.

— Function: void mpfrx_rem (mpfrx_ptr r, mpfrx_srcptr f, mpfrx_srcptr g)
— Function: void mpcx_rem (mpcx_ptr r, mpcx_srcptr f, mpcx_srcptr g)

Set r to the remainder of f divided by g.

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5.8 Higher Level Functions

— Function: void mpfrx_root (mpfr_ptr root, mpfrx_srcptr f)
— Function: void mpcx_root (mpc_ptr root, mpcx_srcptr f)

Computes a root of f. The variable root is supposed to contain an initial approximation, that is refined via Newton iterations until it does not change any more. No special care is taken to avoid infinite loops.

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5.9 Tree Based Functions

The following functions implement asymptotically fast operations on arrays of polynomials, usually through the use of subproduct trees. Such a tree is a binary tree constructed from an array of polynomials by storing these polynomials in the leaves of the tree. Each parent node contains the product of the child nodes, so that the root of the tree contains the product of all the leaves. Internally, subproduct trees are variables of types mpfrx_tree_t and mpcx_tree_t. (Analogously to the situation with polynomials, in the following, mpfrx_tree_ptr and mpfcx_tree_ptr are used for variables of type mpfrx_tree_t and mpcx_tree_t that may be modified by the function, and mpfrx_tree_srcptr and mpfcx_tree_srcptr for variables that are not modified.)

— Function: void mpfrx_tree_init (mpfrx_tree_ptr t, int no, mpfr_prec_t prec)
— Function: void mpcx_tree_init (mpcx_tree_ptr t, int no, mpfr_prec_t prec)

Initialises t as a subproduct tree with no leaves in which all polynomials stored in the nodes will have precision prec. All nodes are initialised by a call to mpfrx_init or mpcx_init, respectively.

— Function: void mpfrx_tree_clear (mpfrx_tree_ptr t)
— Function: void mpcx_tree_clear (mpcx_tree_ptr t)

Frees the subproduct tree referenced by t, and all the polynomials stored in its nodes by calls to mpfrx_clear or mpcx_clear, respectively.

— Function: void mpfrx_subproducttree (mpfrx_tree_ptr t, mpfrx_t *factors)
— Function: void mpcx_subproducttree (mpcx_tree_ptr t, mpcx_t *factors)

Computes the subproduct tree t whose leaves contains the polynomials in the array factors. The variable t needs to have been initialised by a call to mpfrx_tree_init or mpcx_tree_init, respectively, and factors needs to contain at least as many elements as there are leaves in t. (If there are more elements in factors, the last ones are ignored.)

So a typical usage is a call to mpfrx_tree_init, followed by a call to mpfrx_subproducttree, and finally a call to mpfrcx_tree_clear.

factors is not modified by the function.

— Function: void mpfrx_tree_get_root (mpfrx_ptr f, mpfrx_tree_srcptr t)
— Function: void mpcx_tree_get_root (mpcx_ptr f, mpcx_tree_srcptr t)

Assigns the root of the tree t to f.

For instance, if t has been obtained by a call to mpfrx_subproducttree or mpcx_subproducttree, this retrieves the product of all polynomials on the leaves.

— Function: void mpfrx_reconstruct (mpfrx_ptr h, mpfrx_t* factors, int no)
— Function: void mpcx_reconstruct (mpcx_ptr h, mpcx_t* factors, int no)

Computes the product of the first no polynomials in the array factors and stores it in h (which may be one of the elements of factors; apart from that, factors is not modified).

The same effect could be obtained by a call to mpfrx_subproducttree or mpcx_subproducttree, respectively, followed by mpfrx_tree_get_root or mpcx_tree_get_root, respectively. But to save space by a factor of O(log(no)), this function uses a separate implementation.

— Function: void mpfrx_hecke (mpfrx_ptr rop, mpfrx_tree_srcptr subproducts, mpfrx_t *vals)
— Function: void mpcx_hecke (mpcx_ptr rop, mpcx_tree_srcptr subproducts, mpcx_t *vals)

Assume that t has been created (by a call to mpfrx_subproducttree or mpcx_subproducttree, respectively) with the elements of the array factors on its leaves, and let f be the product of all the elements in factors (or, equivalently, the polynomial at the root of subproducttree). Then the function computes Σ vals[i]*f/factors[i] and returns the result in rop. It can be used to compute the Hecke representation of algebraic numbers, whence its name.

— Function: void mpfrx_product_and_hecke (mpfrx_t *rop, mpfrx_t **vals, int no_pols, int no_factors)
— Function: void mpcx_product_and_hecke (mpcx_t *rop, mpcx_t **vals, int no_pols, int no_factors)

Combines one call to mpfrx_subproducttree (resp. mpcx_subproducttree) and one or more calls to mpfrx_hecke (resp. mpcx_hecke) into one function, which allows to not store the subproduct tree and thus to conserve memory without computational overhead. The function behaves as if a subproduct tree were created from vals[0], which needs to contain no_factors elements; the root of the tree is returned in rop[0]. For i from 1 to no_pols-1, it then behaves as if it called mpfrx_hecke (resp. mpcx_hecke) with this subproduct tree and vals[i], which needs to also contain no_factors values; the result of the operation is stored in rop[i].

— Function: void mpfrx_multieval (mpfr_t* values, mpfr_t* args, int no, mpfrx_t f)
— Function: void mpcx_multieval (mpc_t* values, mpc_t* args, int no, mpcx_t f)

Evaluates the polynomial f in the first no elements of the array args, and store the values in the first no entries of values (that must exist and already be initalised).

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The main developer of the MPFRCX library is Andreas Enge.

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Concept Index

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Function Index

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Appendix A GNU Free Documentation License

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    4. Preserve all the copyright notices of the Document.
    5. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
    6. 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.
    7. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice.
    8. Include an unaltered copy of this License.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. Delete any section Entitled “Endorsements”. Such a section may not be included in the Modified Version.
    14. Do not retitle any existing section to be Entitled “Endorsements” or to conflict in title with any Invariant Section.
    15. 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.


    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.”


    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.


    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.


    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.


    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.


    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

    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.


    “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.