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WebIn addition, XSD clarifies and updates many aspects of the definitions of the existing datatypes: for example, it extends the value space of xs:double to allow both positive and negative zero, and extends the lexical space to allow +INF; it modifies the value space of xs:Name to permit additional Unicode characters; it allows year zero and Web1 day ago · Unadorned integer literals (including hex, octal and binary numbers) yield integers. Numeric literals containing a decimal point or an exponent sign yield floating point numbers. Appending 'j' or 'J' to a numeric literal yields an imaginary number (a complex number with a zero real part) which you can add to an integer or float to get a complex Web原创 Python量化交易实战教程汇总. B站配套视频教程观看设计适合自己并能适应市场的交易策略，才是量化交易的灵魂课程亲手带你设计并实现两种交易策略，快速培养你的策略思维能力择时策略：通过这个策略学会如何利用均线，创建择时策略，优化股票买入卖出的时间点。 WebIt is inadvisable to use a dependence on R with patchlevel (the third digit) other than zero. Doing so with packages which others depend on will cause the other packages to become unusable under earlier versions in the series, and e.g. versions 4.x.1 are widely used throughout the Northern Hemisphere academic year WebRésidence officielle des rois de France, le château de Versailles et ses jardins comptent parmi les plus illustres monuments du patrimoine mondial et constituent la plus complète réalisation de l’art français du XVIIe siècle ... read more

The Sūnzĭ Suànjīng , of unknown date but estimated to be dated from the 1st to 5th centuries AD , and Japanese records dated from the 18th century, describe how the c. As noted in Xiahou Yang's Suanjing — AD that states that to multiply or divide a number by 10, , , or , all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places, [27] According to A History of Mathematics , the rods "gave the decimal representation of a number, with an empty space denoting zero".

In AD , Empress Wǔ promulgated Zetian characters , one of which was "〇"; originally meaning 'star', it subsequently [ when? Zero was not treated as a number at that time, but as a "vacant position".

Pingala c. The concept of zero as a written digit in the decimal place value notation was developed in India. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together. The Lokavibhāga , a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD Saka era , uses a decimal place-value system , including a zero.

In this text, śūnya "void, empty" is also used to refer to zero. The Aryabhatiya c. Rules governing the use of zero appeared in Brahmagupta 's Brahmasputha Siddhanta 7th century , which states the sum of zero with itself as zero, and incorrectly division by zero as: [44] [45].

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.

Zero divided by zero is zero. A black dot is used as a decimal placeholder in the Bakhshali manuscript , portions of which date from AD — There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt. A stone tablet found in the ruins of a temple near Sambor on the Mekong , Kratié Province , Cambodia , includes the inscription of "" in Khmer numerals a set of numeral glyphs for the Hindu—Arabic numeral system.

The number is the year of the inscription in the Saka era , corresponding to a date of AD The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior , in India, dated The Arabic -language inheritance of science was largely Greek , [50] followed by Hindu influences.

In AD , astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī , using Hindu numerals; [51] and about , he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.

This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started to acquire a meaning of any arithmetic based on decimals.

Muhammad ibn Ahmad al-Khwarizmi , in , stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr. The Hindu—Arabic numeral system base 10 reached Western Europe in the 11th century, via Al-Andalus , through Spanish Muslims , the Moors , together with knowledge of classical astronomy and instruments like the astrolabe ; Gerbert of Aurillac is credited with reintroducing the lost teachings into Catholic Europe.

For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa was instrumental in bringing the system into European mathematics in , stating:.

After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business.

I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus Modus Indorum. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art.

I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now.

If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign any number may be written. Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication.

From the 13th century, manuals on calculation adding, multiplying, extracting roots, etc. became common in Europe where they were called algorismus after the Persian mathematician al-Khwārizmī.

The most popular was written by Johannes de Sacrobosco , about and was one of the earliest scientific books to be printed in Until the late 15th century, Hindu—Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.

Zero is an even number [57] because it is divisible by 2 with no remainder. Zero is a number which quantifies a count or an amount of null size. In most cultures , 0 was identified before the idea of negative things i. As a value or a number , zero is not the same as the digit zero, used in numeral systems with positional notation. Successive positions of digits have higher weights, so the digit zero is used inside a numeral to skip a position and give appropriate weights to the preceding and following digits.

A zero digit is not always necessary in a positional number system e. In some instances, a leading zero may be used to distinguish a number.

The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number , but it is an integer, and hence a rational number and a real number as well as an algebraic number and a complex number. The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number.

It cannot be prime because it has an infinite number of factors , and cannot be composite because it cannot be expressed as a product of prime numbers as 0 must always be one of the factors. a multiple of 2, as well as being a multiple of any other integer, rational, or real number. The following are some basic elementary rules for dealing with the number 0.

These rules apply for any real or complex number x , unless otherwise stated. The sum of 0 numbers the empty sum is 0, and the product of 0 numbers the empty product is 1.

The factorial 0! evaluates to 1, as a special case of the empty product. The value zero plays a special role for many physical quantities.

For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an absolute temperature as measured in kelvins , zero is the lowest possible value negative temperatures are defined, but negative-temperature systems are not actually colder.

This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons , the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics , the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.

Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right. This would create an element with no protons and no charge on its nucleus. As early as , Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table.

It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements. The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer programming languages such as Fortran and COBOL.

However, in the late s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts allowing any positive, negative, or zero integer as base for array subscripts , and most subsequent programming languages adopted one or other of these positions.

This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based languages precalculate the array's base address to be the position one element before the first. There can be confusion between 0- and 1-based indexing; for example, Java's JDBC indexes parameters from 1 although Java itself uses 0-based indexing.

In databases, it is possible for a field not to have a value. It is then said to have a null value. For text fields this is not blank nor the empty string. The presence of null values leads to three-valued logic. No longer is a condition either true or false , but it can be undetermined. Any computation including a null value delivers a null result.

A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern possibly different values for different data types. However, in some computer hardware signed number representations , zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero , with the latter form sometimes called negative zero.

In binary, 0 represents the value for "off", which means no electricity flow. The Unix epoch the date and time associated with a zero timestamp begins the midnight before the first of January The Classic Mac OS epoch and Palm OS epoch the date and time associated with a zero timestamp begins the midnight before the first of January Many APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and non-zero values to indicate specific error or warning conditions.

Programmers often use a slashed zero to avoid confusion with the letter " O ". The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0. The distinction came into prominence on modern character displays. The digit 0 with a dot in the center seems to have originated as an option on IBM displays and has continued with some modern computer typefaces such as Andalé Mono , and in some airline reservation systems.

One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made one of the capital-O—digit-0 pair more rounded and the other more angular closer to a rectangle. A further distinction is made in falsification-hindering typeface as used on German car number plates by slitting open the digit 0 on the upper right side. In some systems either the letter O or the numeral 0, or both, are excluded from use, to avoid confusion. In the BC calendar era , the year 1 BC is the first year before AD 1; there is not a year zero.

Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "0" — news · newspapers · books · scholar · JSTOR July Learn how and when to remove this template message. Natural number. List of numbers Integers. Main articles: Names for the number 0 and Names for the number 0 in English. See also: History of the Hindu—Arabic numeral system. See also: Null mathematics. Main article: Symbols for zero.

Main article: Year zero. Number words and number symbols: a cultural history of numbers. Courier Dover Publications. ISBN Archived from the original on 25 April Retrieved 5 January Oxford University Press. December Archived from the original on 7 March Retrieved 4 March Arabic sifr in the meaning of zero is a translation of the corresponding India sunya. The Universal History of Numbers: From Prehistory to the Invention of the Computer.

com — Retrieved April Archived from the original on 15 November Retrieved 21 December The Crest of the Peacock: Non-European Roots of Mathematics Third ed. Princeton UP. The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press. Maths History. Archived from the original on 21 September Retrieved 7 September Archived from the original on 7 September The Guardian. Archived from the original on 28 November Retrieved 10 October YaleGlobal online.

The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived from the original on 25 August Retrieved 1 September Zero: The Biography of a Dangerous Idea. OCLC Archived from the original on 30 April Retrieved 30 April A Brain for Numbers: The Biology of the Number Instinct.

MIT Press. The Nothing that Is: A Natural History of Zero. Archived from the original on 24 June In Zalta, Edward N. The Stanford Encyclopedia of Philosophy Winter ed.

Metaphysics Research Lab, Stanford University. Archived from the original on 10 January Retrieved 9 August The Exact Sciences in Antiquity. Acta Historica Scientiarum Naturalium et Medicinalium. Dover Publications. PMID Home of Kairos. Archived PDF from the original on 5 November Retrieved 28 March Ptolemy's Almagest.

csv is not the standard 22 CSV format , or save images. RData or. The files should not be hidden have names starting with a dot. Images extensions. RData 23 or. rda can contain references to the namespaces of packages that were used to create them. Preferably there should be no such references in data files, and in any case they should only be to packages listed in the Depends and Imports fields, as otherwise it may be impossible to install the package.

To check for such references, load all the images into a vanilla R session, run str on all the datasets, and look at the output of loadedNamespaces. Particular care is needed where a dataset or one of its components is of an S4 class, especially if the class is defined in a different package. First, the package containing the class definition has to be available to do useful things with the dataset, so that package must be listed in Imports or Depends even if this gives a check warning about unused imports.

Second, the definition of an S4 class can change, and often is unnoticed when in a package with a different author. So it may be wiser to use the. R scripts to produce your data, loading your namespace, you can speed up installation by providing a file datalist in the data subdirectory. csv files can be compressed by gzip , bzip2 or xz , optionally with additional extension. bz2 or. If your package is to be distributed, do consider the resource implications of large datasets for your users: they can make packages very slow to download and use up unwelcome amounts of storage space, as well as taking many seconds to load.

It is normally best to distribute large datasets as. Using bzip2 or xz compression will usually reduce the size of both the package tarball and the installed package, in some cases by a factor of two or more. Package tools has a couple of functions to help with data images: checkRdaFiles reports on the way the image was saved, and resaveRdaFiles will re-save with a different type of compression, including choosing the best type for that particular image.

Useful values are bzip2 , xz and the default, gzip : value none is also accepted. rdb file. A function to do that quoting sizes in KB is. If you see that, run CheckLazyDataCompression and set the field — to gzip in the unlikely event 24 that is the best choice. The analogue for sysdata. Lazy-loading is not supported for very large datasets those which when serialized exceed 2GB, the limit for the format on bit platforms.

Next: Specifying URLs , Previous: Data in packages , Up: Package structure [ Contents ][ Index ]. Subdirectory exec could be used for scripts for interpreters such as the shell, BUGS, JavaScript, Matlab, Perl, php amap , Python or Tcl Simile , or even R.

Java code is a special case: except for very small programs,. java files should be byte-compiled to a. class file and distributed as part of a. jar file: the conventional location for the. It is desirable and required under an Open Source license to make the Java source files available: this is best done in a top-level java directory in the package—the source files should not be installed. This is fairly easy to do: first find the Tcl search path:. If no location on that search path is writeable, you will need to add one each time BWidget is to be used with tcltk::addTclPath.

gz , but needed patching for current Tk 8. Previous: Non-R scripts in packages , Up: Package structure [ Contents ][ Index ]. URLs in many places in the package documentation will be converted to clickable hyperlinks in at least some of their renderings.

So care is needed that their forms are correct and portable. Spaces in URLs are not portable and how they are handled does vary by HTTP server and by client. Next: Checking and building packages , Previous: Package structure , Up: Creating R packages [ Contents ][ Index ].

Note that most of this section is specific to Unix-alikes: see the comments later on about the Windows port of R. If your package needs some system-dependent configuration before installation you can include an executable Bourne 26 shell script configure in your package which if present is executed by R CMD INSTALL before any other action is performed.

This can be a script created by the Autoconf mechanism, but may also be a script written by yourself. Use this to detect if any nonstandard libraries are present such that corresponding code in the package can be disabled at install time rather than giving error messages when the package is compiled or used. To summarize, the full power of Autoconf is available for your extension package including variable substitution, searching for libraries, etc. Under a Unix-alike only, an executable Bourne shell script cleanup is executed as the last thing by R CMD INSTALL if option --clean was given, and by R CMD build when preparing the package for building from its source.

As an example consider we want to use functionality provided by a C or Fortran library foo. For example, if a function named bar is to be made available by linking against library foo i. From this file configure creates the actual R source file foo.

R looking like. if library foo was not found with the desired functionality. In this case, the above R code effectively disables the function. One could also use different file fragments for available and missing functionality, respectively. You will very likely need to ensure that the same C compiler and compiler flags are used in the configure tests as when compiling R or your package. Under a Unix-alike, you can achieve this by including the following fragment early in configure.

If your code does load checks for example, to check for an entry point in a library or to run code then you will also need. You can use R CMD config to get the value of the basic configuration variables, and also the header and library flags necessary for linking a front-end executable program against R, see R CMD config --help for details.

Note that FLIBS as determined by R must be used to ensure that Fortran code works on all R platforms. Otherwise R CMD build may ship the files that are created. For example, package RODBC has. As this example shows, configure often creates working files such as config. If your configure script needs auxiliary files, it is recommended that you ship them in a tools directory as R itself does. You should bear in mind that the configure script will not be used on Windows systems. If your package is to be made publicly available, please give enough information for a user on a non-Unix-alike platform to configure it manually, or provide a configure.

win script or configure. ucrt to be used on that platform. Optionally, there can be a cleanup. win script or cleanup. Both should be shell scripts to be executed by ash , which is a minimal version of Bourne-style sh.

When configure. win or configure. In some rare circumstances, the configuration and cleanup scripts need to know the location into which the package is being installed. Usually, the object that is dynamically loaded by R is linked against the second, dependent, object.

On some systems, we can add the location of this dependent object to the object that is dynamically loaded by R. Another example is when a package installs support files that are required at run time, and their location is substituted into an R data structure at installation time. The names of the top-level library directory i. Additionally, the name of the package e.

Its main use is in configure. One of the more tricky tasks can be to find the headers and libraries of external software. One tool which is increasingly available on Unix-alikes but not by default 28 on macOS to do this is pkg-config.

Note that pkg-config --libs gives the information required to link against the default version 30 of that library usually the dynamic one , and pkg-config --static --libs may be needed if the static library is to be used.

Static libraries are commonly used on macOS and Windows to facilitate bundling external software with binary distributions of packages.

This means that portable source packages need to allow for this. so which would be needed to use -ljbig sometimes included in pkg-config --static --libs libtiff Another issue is that pkg-config --exists may not be reliable. XQuartz 2. x only distributed dynamic libraries and not some of the. pc files needed for --exists. Sometimes the name by which the software is known to pkg-config is not what one might expect e.

To get a complete list use. Some external software provides a -config command to do a similar job to pkg-config , including. curl-config is for libcurl not curl. nc-config is for netcdf. Most have an option to use static libraries. These commands indicate what header paths and libraries are needed, but they do not obviate the need to check that the recipes they give actually work.

This is especially necessary for platforms which use static linking. If using Autoconf it is good practice to include all the Autoconf sources in the package and required for an Open Source package and tested by R CMD check --as-cran. This will include the file configure. ac 31 in the top-level directory of the package. If extensions written in m4 are needed, these should be included under the directory tools and included from configure. ac via e. Alternatively, Autoconf can be asked to search all.

m4 files in a directory by including something like Next: Configure example , Previous: Configure and cleanup , Up: Configure and cleanup [ Contents ][ Index ]. Sometimes writing your own configure script can be avoided by supplying a file Makevars : also one of the most common uses of a configure script is to make Makevars from Makevars.

A Makevars file is a makefile and is used as one of several makefiles by R CMD SHLIB which is called by R CMD INSTALL to compile code in the src directory. It should be written if at all possible in a portable style, in particular except for Makevars.

win and Makevars. ucrt without the use of GNU extensions. When writing a Makevars file for a package you intend to distribute, take care to ensure that it is not specific to your compiler: flags such as -O2 -Wall -pedantic and all other -W flags: for the Oracle compilers these are used to pass arguments to compiler phases are all specific to GCC and compilers such as clang which aim to be options-compatible with it.

Also, do not set variables such as CPPFLAGS , CFLAGS etc. See Customizing package compilation in R Installation and Administration ,. That makefile is included as a Makefile after Makevars[. win] , and the macros it defines can be used in macro assignments and make command lines in the latter. These include. A macro containing the set of libraries need to link Fortran code.

A macro containing the BLAS libraries used when building R. Beware that if it is empty then the R executable will contain all the double-precision and double-complex BLAS routines, but no single-precision nor complex routines. A macro containing the LAPACK libraries and paths where appropriate used when building R. It may point to a dynamic library libRlapack which contains the main double-precision LAPACK routines as well as those double-complex LAPACK routines needed to build R, or it may point to an external LAPACK library, or may be empty if an external BLAS library also contains LAPACK.

See the example later in this section. This can be useful in conjunction with implicit rules to allow other types of source code to be compiled and included in the shared object. It can also be used to control the set of files which are compiled, either by excluding some files in src or including some files in subdirectories.

Note that Makevars should not normally contain targets, as it is included before the default makefile and make will call the first target, intended to be all in the default makefile. If you really need to circumvent that, use a suitable phony target all before any actual targets in Makevars. needed to ensure that the LAPACK routines find some constants without infinite looping.

The Windows equivalent was. Note that the first target in Makevars will be called, but for back-compatibility it is best named all. If you want to create and then link to a library, say using code in a subdirectory, use something like. Be careful to create all the necessary dependencies, as there is no guarantee that the dependencies of all will be run in a particular order and some of the CRAN build machines use multiple CPUs and parallel makes.

In particular,. but that is not portable. dmake and pmake allow the similar. Note that on Windows it is required that Makevars[. ucrt] does create a DLL: this is needed as it is the only reliable way to ensure that building a DLL succeeded. If you want to use the src directory for some purpose other than building a DLL, use a Makefile. win or Makefile. ucrt file. win or Makevars. ucrt : this will be used by R CMD build to clean up a copy of the package sources. which asks make to clean in parallel with compiling the code.

Not only does this lead to hard-to-debug installation errors, it wipes out all the evidence of any error from a parallel make or not. It is much better to leave cleaning to the end user using the facilities in the previous paragraph.

If you want to run R code in Makevars , e. to find configuration information, please do ensure that you use the correct copy of R or Rscript : there might not be one in the path at all, or it might be the wrong version or architecture. The correct way to do this is via. Environment or make variables can be used to select different macros for and bit code, for example GNU make syntax, allowed on Windows. On Windows there is normally a choice between linking to an import library or directly to a DLL.

Where possible, the latter is much more reliable: import libraries are tied to a specific toolchain, and in particular on bit Windows two different conventions have been commonly used.

So for example instead of. where the first and second are conventionally import libraries, the third and fourth often static libraries with. html WIN The fly in the ointment is that the DLL might not be named libxxx.

dll , and in fact on bit Windows there is a libxml2. dll whereas on one build for bit Windows the DLL is called libxml Using import libraries can cover over these differences but can cause equal difficulties. If static libraries are available they can save a lot of problems with run-time finding of DLLs, especially when binary packages are to be distributed and even more when these support both architectures.

Where using DLLs is unavoidable we normally arrange via configure. ucrt to ship them in the same directory as the package DLL.

Next: Using pthreads , Previous: Using Makevars , Up: Using Makevars [ Contents ][ Index ]. There is some support for packages which wish to use OpenMP The make macros. If you do use your own checks, make sure that OpenMP support is complete by compiling and linking an OpenMP-using program: on some platforms the runtime library is optional and on others that library depends on other optional libraries.

Some care is needed when compilers are from different families which may use different OpenMP runtimes e. For a package with Fortran code using OpenMP the appropriate lines are. as the C compiler will be used to link the package code. There are platforms on which this does not work for some OpenMP-using code and installation will fail.

So cross-check that e. h header: some compilers but not all include it when OpenMP mode is switched on e. via flag -fopenmp. There is nothing 37 to say what version of OpenMP is supported: version 4. Apple clang on macOS has no OpenMP support. Rarely, using OpenMP with clang on Linux generates calls in libatomic , resulting in loading messages like. The performance of OpenMP varies substantially between platforms. The Windows implementation has substantial overheads, so is only beneficial if quite substantial tasks are run in parallel.

Also, on Windows new threads are started with the default 38 FPU control word, so computations done on OpenMP threads will not make use of extended-precision arithmetic which is the default for the main process. Many functions in the R API modify internal R data structures and might corrupt these data structures if called simultaneously from multiple threads.

Most R API functions can signal errors, which must only happen on the R main thread. Also, external libraries e. LAPACK may not be thread-safe. Packages are not standard-alone programs, and an R process could contain more than one OpenMP-enabled package as well as other components for example, an optimized BLAS making use of OpenMP.

So careful consideration needs to be given to resource usage. Parallel regions can be nested, although it is common to use only a single thread below the first level.

R uses Note that setting environment variables to control OpenMP is implementation-dependent and may need to be done outside the R process or before any use of OpenMP which might be by another process or R itself. Next: Compiling in sub-directories , Previous: OpenMP support , Up: Using Makevars [ Contents ][ Index ].

There is no direct support for the POSIX threads more commonly known as pthreads : by the time we considered adding it several packages were using it unconditionally so it seems that nowadays it is universally available on POSIX operating systems hence not Windows. For reasonably recent versions of gcc and clang the correct specification is. For other platforms the specification is.

and note that the library name is singular. This is what -pthread does on all known current platforms although earlier versions of OpenBSD used a different library name. POSIX threads are not normally used on Windows, which has its own native concepts of threads. However, there are two projects implementing pthreads on top of Windows, pthreads-w32 and winpthreads part of the MinGW-w64 project.

Whether Windows toolchains implement pthreads is up to the toolchain provider. See also the comments on thread-safety and performance under OpenMP: on all known R platforms OpenMP is implemented via pthreads and the known performance issues are in the latter. Previous: Using pthreads , Up: Using Makevars [ Contents ][ Index ]. Package authors fairly often want to organize code in sub-directories of src , for example if they are including a separate piece of external software to which this is an R interface.

One simple way is simply to set OBJECTS to be all the objects that need to be compiled, including in sub-directories. For example, CRAN package RSiena has. One problem with that approach is that unless GNU make extensions are used, the source files need to be listed and kept up-to-date. As in the following from CRAN package lossDev :. Where the subdirectory is self-contained code with a suitable makefile, the best approach is something like.

Note the quotes: the macros can contain spaces, e. Others forget the need 42 for position-independent code. Next: Using F9x code , Previous: Using Makevars , Up: Configure and cleanup [ Contents ][ Index ]. The configure. ac file follows: configure is created from this by running autoconf in the top-level package directory containing configure.

A user can then be advised to specify the location of the ODBC driver manager files by options like lines broken for easier reading. R assumes that source files with extension.

f95 , but those are not used by R itself so this is not required. The same compiler is used 43 for both fixed-form and free-form Fortran code with different file extensions and possibly different flags.

However, Fortran 95 is widely supported. Intel Fortran has full Fortran support from version Modern versions of Fortran support modules, whereby compiling one source file creates a module file which is then included in others. Module files typically have a. mod extension: they do depend on the compiler used and so should never be included in a package. This creates a dependence which make will not know about and often causes installation with a parallel make to fail.

For example, if file iface. f90 and dmi. Note that it is not portable although some platforms do accept it to define a module of the same name in multiple source files. Next: Using cmake , Previous: Using F9x code , Up: Configure and cleanup [ Contents ][ Index ]. As from R 4. ucrt on Windows should include the line.

Such a check could be. Note that Windows builds prior to R 4. html to see if the features you want to use are widely implemented. Packages often wish to include the sources of other software and compile that for inclusion in their.

so or. dll , which is normally done by including or unpacking the sources in a subdirectory of src , as considered above. Further issues arise when the external software uses another build system such as cmake , principally to ensure that all the settings for compilers, include and load paths etc are made.

This section has already mentioned the need to set at least some of. html manual:cmake-env-variables 7 , but it may be desirable to translate these into native settings such as. Two approaches have been used. It is often most convenient to build the external software in a directory other than its sources particularly during development when the build directory can be removed between builds rather than attempting to clean the sources — this is illustrated in the first approach. Next: Writing package vignettes , Previous: Configure and cleanup , Up: Creating R packages [ Contents ][ Index ].

Before using these tools, please check that your package can be installed. R CMD check will inter alia do this, but you may get more detailed error messages doing the install directly.

Renviron are used to set environment variables when using these utilities. win , cleanup. win , configure. ucrt or cleanup. ucrt scripts or a src directory and e. need vignettes built. You may need to set the environment variable TMPDIR to point to a suitable writable directory with a path not containing spaces — use forward slashes for the separators.

Also, the directory needs to be on a case-honouring file system some network-mounted file systems are not. Next: Building package tarballs , Previous: Checking and building packages , Up: Checking and building packages [ Contents ][ Index ]. Using R CMD check , the R package checker, one can test whether source R packages work correctly. It can be run on one or more directories, or compressed package tar archives with extension.

tgz ,. It is strongly recommended that the final checks are run on a tar archive prepared by R CMD build. A warning is given for directory names that look like R package check directories — many packages have been submitted to CRAN containing these.

Note that the latter might give false positives in that the symbols might be pulled in with external libraries and could never be called. Of course, released packages should be able to run at least their own examples. If there is an error 51 in executing the R code in vignette foo. ext , a log file foo. log is created in the check directory.

Use R CMD check --help to obtain more information about the usage of the R package checker. A subset of the checking steps can be selected by adding command-line options. Rprofile , by e. which reports unused local assignments. Not only does this point out computations which are unnecessary because their results are unused, it also can uncover errors. This can give false positives, most commonly because of non-standard evaluation for formulae and because the intention is to return objects in the environment of a function for later use.

Complete checking of a package which contains a file README. You do need to ensure that the package is checked in a suitable locale if it contains non- ASCII characters. Such packages are likely to fail some of the checks in a C locale, and R CMD check will warn if it spots the problem. You should be able to check any package in a UTF-8 locale if one is available. Beware that although a C locale is rarely used at a console, it may be the default if logging in remotely or for batch jobs.

Often R CMD check will need to consult a CRAN repository to check details of uninstalled packages. Next: Building binary packages , Previous: Checking packages , Up: Checking and building packages [ Contents ][ Index ]. gz files or in binary form. The source form can be installed on all platforms with suitable tools and is the usual form for Unix-like systems; the binary form is platform-specific, and is the more common distribution form for the Windows and macOS platforms.

Using R CMD build , the R package builder, one can build R package tarballs from their sources for example, for subsequent release. Prior to actually building the package in the standard gzipped tar file format, a few diagnostic checks and cleanups are performed.

Run-time checks whether the package works correctly should be performed using R CMD check prior to invoking the final build procedure. To exclude files from being put into the package, one can specify a list of exclude patterns in file. Rbuildignore in the top-level source directory. These patterns should be Perl-like regular expressions see the help for regexp in R for the precise details , one per line, to be matched case-insensitively against the file and directory names relative to the top-level package source directory.

In addition, directories from source control systems 54 or from eclipse 55 , directories with names check , chm , or ending. In addition, same-package tarballs from previous builds and their binary forms will be excluded from the top-level directory, as well as those files in the R , demo and man directories which are flagged by R CMD check as having invalid names.

Use R CMD build --help to obtain more information about the usage of the R package builder. To do so it installs the current package into a temporary library tree, but any dependent packages need to be installed in an available library tree see the Note: at the top of this section. Similarly, if the. If there are any install-time or render-time macros, a. pdf version of the package manual will be built and installed in the build subdirectory. This allows CRAN or other repositories to display the manual even if they are unable to install the package.

One of the checks that R CMD build runs is for empty source directories. The --resave-data option allows saved images. rda and. RData files in the data directory to be optimized for size. It will also compress tabular files and convert. R files to saved images. Where a non-POSIX file system is in use which does not utilize execute permissions, some care is needed with permissions. This applies on Windows and to e. FAT-formatted drives and SMB-mounted file systems on other OSes.

info returns. A particular issue is packages being built on Windows which are intended to contain executable scripts such as configure and cleanup : R CMD build ensures those two are recorded with execute permission. Directory build of the package sources is reserved for use by R CMD build : it contains information which may not easily be created when the package is installed, including index information on the vignettes and, rarely, information on the help pages and perhaps a copy of the PDF reference manual see above.

Previous: Building package tarballs , Up: Checking and building packages [ Contents ][ Index ]. Binary packages are compressed copies of installed versions of packages. The format and filename are platform-specific; for example, a binary package for Windows is usually supplied as a. zip file, and for the macOS platform the default binary package file extension is.

R CMD INSTALL --build pkg where pkg is either the name of a source tarball in the usual. gz format or the location of the directory of the package source to be built. This operates by first installing the package and then packing the installed binaries into the appropriate binary package file for the particular platform.

By default, R CMD INSTALL --build will attempt to install the package into the default library tree for the local installation of R. This has two implications:. To prevent changes to the present working installation or to provide an install location with write access, create a suitably located directory with write access and use the -l option to build the package in the chosen location. The usage is then.

where location is the chosen directory with write access. The package will be installed as a subdirectory of location , and the package binary will be created in the current directory.

Other options for R CMD INSTALL can be found using R CMD INSTALL --help , and platform-specific details for special cases are discussed in the platform-specific FAQs. Note that this is intended for developers on other platforms who do not have access to Windows but wish to provide binaries for the Windows platform.

Next: Package namespaces , Previous: Checking and building packages , Up: Creating R packages [ Contents ][ Index ]. In addition to the help files in Rd format, R packages allow the inclusion of documents in arbitrary other formats. Pointers from package help indices to the installed documents are automatically created.

To ensure that they can be accessed from a browser as an HTML index is provided , the file names should start with an ASCII letter and be comprised entirely of ASCII letters or digits or hyphen or underscore. A special case is package vignettes. Sweave, provided by the R distribution, is the default engine. Other vignette engines besides Sweave are supported; see Non-Sweave vignettes. Package vignettes have their sources in subdirectory vignettes of the package sources. Sweave vignette sources are normally given the file extension.

Rnw or. Rtex , but for historical reasons extensions Snw and. Stex are also recognized. Sweave allows the integration of LaTeX documents: see the Sweave help page in R and the Sweave vignette in package utils for details on the source document format.

Package vignettes are tested by R CMD check by executing all R code chunks they contain except those marked for non-evaluation, e. The R working directory for all vignette tests in R CMD check is a copy of the vignette source directory. All other files needed to re-make the vignettes such as LaTeX style files, BibTeX input files and files for any figures not created by running the code in the vignette must be in the vignette source directory.

By including the vignette outputs in the package sources it is not necessary that these can be re-built at install time, i. By default R CMD build will run Sweave on all Sweave vignette source files in vignettes. If Makefile is found in the vignette source directory, then R CMD build will try to run make after the Sweave runs, otherwise texi2pdf is run on each. tex file produced. All the usual caveats about including a Makefile apply. It must be portable no GNU extensions , use LF line endings and must work correctly with a parallel make : too many authors have written things like.

Metadata lines can be placed in the source file, preferably in LaTeX comments in the preamble. This index is linked from the HTML help index for the package. Do watch that PDFs are not too large — one in a CRAN package was 72MB! This is usually caused by the inclusion of overly detailed figures, which will not render well in PDF viewers. Sometimes it is much better to generate fairly high resolution bitmap PNG, JPEG figures and include those in the PDF document.

See the description of the. Rinstignore file for full details. Next: Non-Sweave vignettes , Previous: Writing package vignettes , Up: Writing package vignettes [ Contents ][ Index ]. Vignettes will in general include descriptive text, R input, R output and figures, LaTeX include files and bibliographic references. As any of these may contain non- ASCII characters, the handling of encodings can become very complicated. The vignette source file should be written in ASCII or contain a declaration of the encoding see below.

This applies even to comments within the source file, since vignette engines process comments to look for options and metadata lines. Sweave will produce a. tex file in the current encoding, or in UTF-8 if that is declared. Non- ASCII encodings need to be declared to LaTeX via a line like. For files where this line is not needed e. chapters included within the body of a larger document, or non-Sweave vignettes , the encoding may be declared using a comment like. If no declaration is given in the vignette, it will be assumed to be in the encoding declared for the package.

If there is no encoding declared in either place, then it is an error to use non- ASCII characters in the vignette. Sweave will also parse and evaluate the R code in each chunk. One thing people often forget is that the R output may not be ASCII even for ASCII R sources, for many possible reasons.

The final issue is the encoding of figures — this applies only to PDF figures and not PNG etc. The PDF figures will contain declarations for their encoding, but the Sweave option pdf. encoding may need to be set appropriately: see the help for the pdf graphics device.

That package did not have a declared encoding, and its vignette was in ASCII. However, the data it displays are read from a UTF-8 CSV file and will be assumed to be in the current encoding, so fortunes.

tex will be in UTF-8 in any locale. Had read. table been told the data were UTF-8, fortunes. Previous: Encodings and vignettes , Up: Writing package vignettes [ Contents ][ Index ]. For example knitr version 1. tex files from a variation on Sweave format, and. These engines replace the Sweave function with other functions to convert vignette source files into LaTeX files for processing into. pdf , or directly into. pdf or.

html files. The Stangle function is replaced with a function that extracts the R source from a vignette. R recognizes non-Sweave vignettes using filename extensions specified by the engine. For example, the knitr package supports the extension. This specifies the name of a package and an engine to use in place of Sweave in processing the vignette.

If more than one package is specified as a builder, they will be searched in the order given there. The utils package is always implicitly appended to the list of builder packages, but may be included earlier to change the search order. The vignette engine can produce. tex ,. pdf , or. html files as output. If it produces.

tex files, R will call texi2pdf to convert them to. pdf for display to the user unless there is a Makefile in the vignettes directory. Package writers who would like to supply vignette engines need to register those engines in the package.

onLoad function. For example, that function could make the call. The actual registration in knitr is more complicated, because it supports other input formats. See the? tools::vignetteEngine help topic for details on engine registration. Next: Writing portable packages , Previous: Writing package vignettes , Up: Creating R packages [ Contents ][ Index ]. R has a namespace management system for code in packages.

This system allows the package writer to specify which variables in the package should be exported to make them available to package users, and which variables should be imported from other packages. The namespace for a package is specified by the NAMESPACE file in the top level package directory. This file contains namespace directives describing the imports and exports of the namespace.

Additional directives register any shared objects to be loaded and any S3-style methods that are provided. Note that although the file looks like R code and often has R-style comments it is not processed as R code. Only very simple conditional processing of if statements is implemented. Packages are loaded and attached to the search path by calling library or require. Only the exported variables are placed in the attached frame. Loading a package that imports variables from other packages will cause these other packages to be loaded as well unless they have already been loaded , but they will not be placed on the search path by these implicit loads.

Thus code in the package can only depend on objects in its own namespace and its imports including the base namespace being visible Namespaces are sealed once they are loaded. Sealing means that imports and exports cannot be changed and that internal variable bindings cannot be changed.

Sealing allows a simpler implementation strategy for the namespace mechanism and allows code analysis and compilation tools to accurately identify the definition corresponding to a global variable reference in a function body. The namespace controls the search strategy for variables used by functions in the package.

If not found locally, R searches the package namespace first, then the imports, then the base namespace and then the normal search path so the base namespace precedes the normal search rather than being at the end of it. Next: Registering S3 methods , Previous: Package namespaces , Up: Package namespaces [ Contents ][ Index ].

Exports are specified using the export directive in the NAMESPACE file. A directive of the form. specifies that the variables f and g are to be exported. fractions must be. For packages with many variables to export it may be more convenient to specify the names to export with a regular expression using exportPattern.

The directive. exports all variables that do not start with a period. However, such broad patterns are not recommended for production code: it is better to list all exports or use narrowly-defined groups. This pattern applies to S4 classes. Beware of patterns which include names starting with a period: some of these are internal-only variables and should never be exported, e.

Packages implicitly import the base namespace. Variables exported from other packages with namespaces need to be imported explicitly using the directives import and importFrom.

The import directive imports all exported variables from the specified package s. Thus the directives. specifies that all exported variables in the packages foo and bar are to be imported. If only some of the exported variables from a package are needed, then they can be imported using importFrom. specifies that the exported variables f and g of the package foo are to be imported. Using importFrom selectively rather than import is good practice and recommended notably when importing from packages with more than a dozen exports and especially from those written by others so what they export can change in future.

To import every symbol from a package but for a few exceptions, pass the except argument to import. imports every symbol from foo except bar and baz.

The value of except should evaluate to something coercible to a character vector, after substituting each symbol for its corresponding string. It is possible to export variables from a namespace which it has imported from other namespaces: this has to be done explicitly and not via exportPattern. If a package only needs a few objects from another package it can use a fully qualified variable reference in the code instead of a formal import.

A fully-qualified reference to the function f in package foo is of the form foo::f. Evaluating foo::f will cause package foo to be loaded, but not attached, if it was not loaded already—this can be an advantage in delaying the loading of a rarely used package. Using the foo::f form will be necessary when a package needs to use a function of the same name from more than one namespace. Using foof instead of foo::f allows access to unexported objects.

This is generally not recommended, as the existence or semantics of unexported objects may be changed by the package author in routine maintenance. Next: Load hooks , Previous: Specifying imports and exports , Up: Package namespaces [ Contents ][ Index ]. The standard method for S3-style UseMethod dispatching might fail to locate methods defined in a package that is imported but not attached to the search path. To ensure that these methods are available the packages defining the methods should ensure that the generics are imported and register the methods using S3method directives.

If a package defines a function print. foo intended to be used as a print method for class foo , then the directive. ensures that the method is registered and available for UseMethod dispatch, and the function print. foo does not need to be exported. Since the generic print is defined in base it does not need to be imported explicitly.

It is possible to specify a third argument to S3method, the function to be used as the method, for example. As from R 3. function gen. cls will get registered as an S3 method for class cls and generic gen from package pkg only when the namespace of pkg is loaded. Next: useDynLib , Previous: Registering S3 methods , Up: Package namespaces [ Contents ][ Index ]. There are a number of hooks called as packages are loaded, attached, detached, and unloaded.

See help ". onLoad" for more details. Since loading and attaching are distinct operations, separate hooks are provided for each. These hook functions are called.

onLoad and. They both take arguments 63 libname and pkgname ; they should be defined in the namespace but not exported. Packages can use a. onDetach or. lib function provided the latter is exported from the namespace when detach is called on the package.

It is called with a single argument, the full path to the installed package. There is also a hook. onUnload and. onDetach should be defined in the namespace and not exported, but. lib does need to be exported. Packages are not likely to need. onAttach except perhaps for a start-up banner ; code to set options and load shared objects should be placed in a. onLoad function, or use made of the useDynLib directive described next. These hooks are often used incorrectly.

People forget to export. Compiled code should be loaded in. onLoad or via a useDynLb directive: see below and unloaded in. by pkgname::fun and that a package can be detached and re-attached whilst its namespace remains loaded.

It is good practice for these functions to be quiet. Any messages should use packageStartupMessage so users include check scripts can suppress them if desired.

The useDynLib directive also accepts the names of the native routines that are to be used in R via the. Call ,. Fortran and. External interface functions. These are given as additional arguments to the directive, for example,. These can be used in the. External calls in place of the name of the routine and the PACKAGE argument. For instance, we can call the routine myRoutine from R with the code. There are at least two benefits to this approach. Firstly, the symbol lookup is done just once for each symbol rather than each time the routine is invoked.

Secondly, this removes any ambiguity in resolving symbols that might be present in more than one DLL. However, this approach is nowadays deprecated in favour of supplying registration information see below. In some circumstances, there will already be an R variable in the package with the same name as a native symbol. For example, we may have an R function in the package named myRoutine. In this case, it is necessary to map the native symbol to a different R variable name.

This can be done in the useDynLib directive by using named arguments. It may be too costly to compute these for many routines when the package is loaded if many of these routines are not likely to be used. In this case, one can still perform the symbol resolution correctly using the DLL, but do this each time the routine is called.

Given a reference to the DLL as an R variable, say dll , we can call the routine myRoutine using the expression. This is the same computation as above where we resolve the symbol when the package is loaded. In order to use this dynamic approach e.

Built-in Constants. Built-in Exceptions. The principal built-in types are numerics, sequences, mappings, classes, instances and exceptions. Some collection classes are mutable. Some operations are supported by several object types; in particular, practically all objects can be compared for equality, tested for truth value, and converted to a string with the repr function or the slightly different str function. The latter function is implicitly used when an object is written by the print function.

Any object can be tested for truth value, for use in an if or while condition or as operand of the Boolean operations below. constants defined to be false: None and False. zero of any numeric type: 0 , 0. empty sequences and collections: '' , , [] , {} , set , range 0.

Operations and built-in functions that have a Boolean result always return 0 or False for false and 1 or True for true, unless otherwise stated. Important exception: the Boolean operations or and and always return one of their operands.

x and y. not x. if x is false, then True , else False. This is a short-circuit operator, so it only evaluates the second argument if the first one is false. This is a short-circuit operator, so it only evaluates the second argument if the first one is true.

There are eight comparison operations in Python. They all have the same priority which is higher than that of the Boolean operations. is not. Objects of different types, except different numeric types, never compare equal. The behavior of the is and is not operators cannot be customized; also they can be applied to any two objects and never raise an exception. There are three distinct numeric types: integers , floating point numbers , and complex numbers.

In addition, Booleans are a subtype of integers. Integers have unlimited precision. Floating point numbers are usually implemented using double in C; information about the precision and internal representation of floating point numbers for the machine on which your program is running is available in sys.

Complex numbers have a real and imaginary part, which are each a floating point number. To extract these parts from a complex number z , use z. real and z. The standard library includes the additional numeric types fractions. Fraction , for rationals, and decimal.

Decimal , for floating-point numbers with user-definable precision. Numbers are created by numeric literals or as the result of built-in functions and operators. Unadorned integer literals including hex, octal and binary numbers yield integers. Numeric literals containing a decimal point or an exponent sign yield floating point numbers. Appending 'j' or 'J' to a numeric literal yields an imaginary number a complex number with a zero real part which you can add to an integer or float to get a complex number with real and imaginary parts.

A comparison between numbers of different types behaves as though the exact values of those numbers were being compared. The constructors int , float , and complex can be used to produce numbers of a specific type. All numeric types except complex support the following operations for priorities of the operations, see Operator precedence :. complex re, im. a complex number with real part re , imaginary part im. im defaults to zero. divmod x, y.

pow x, y. Also referred to as integer division. Not for complex numbers. Instead convert to floats using abs if appropriate. Conversion from floating point to integer may round or truncate as in C; see functions math. floor and math. ceil for well-defined conversions. The numeric literals accepted include the digits 0 to 9 or any Unicode equivalent code points with the Nd property. txt for a complete list of code points with the Nd property. All numbers. Real types int and float also include the following operations:.

trunc x. x truncated to Integral. round x[, n]. x rounded to n digits, rounding half to even. If n is omitted, it defaults to 0. floor x. ceil x. For additional numeric operations see the math and cmath modules. Bitwise operations only make sense for integers. Negative shift counts are illegal and cause a ValueError to be raised.

A left shift by n bits is equivalent to multiplication by pow 2, n. A right shift by n bits is equivalent to floor division by pow 2, n. The int type implements the numbers. Integral abstract base class. In addition, it provides a few more methods:. Return the number of bits necessary to represent an integer in binary, excluding the sign and leading zeros:.

More precisely, if x is nonzero, then x. If x is zero, then x. Return the number of ones in the binary representation of the absolute value of the integer. This is also known as the population count. The integer is represented using length bytes, and defaults to 1. An OverflowError is raised if the integer is not representable with the given number of bytes. The byteorder argument determines the byte order used to represent the integer, and defaults to "big".

If byteorder is "big" , the most significant byte is at the beginning of the byte array. If byteorder is "little" , the most significant byte is at the end of the byte array. If signed is False and a negative integer is given, an OverflowError is raised. The default value for signed is False. The default values can be used to conveniently turn an integer into a single byte object. Changed in version 3. The argument bytes must either be a bytes-like object or an iterable producing bytes.

To request the native byte order of the host system, use sys. byteorder as the byte order value. Return a pair of integers whose ratio is exactly equal to the original integer and with a positive denominator.

The integer ratio of integers whole numbers is always the integer as the numerator and 1 as the denominator. The float type implements the numbers. Real abstract base class. float also has the following additional methods. Return a pair of integers whose ratio is exactly equal to the original float and with a positive denominator.

Raises OverflowError on infinities and a ValueError on NaNs. Return True if the float instance is finite with integral value, and False otherwise:.

Two methods support conversion to and from hexadecimal strings. In contrast, hexadecimal strings allow exact representation and specification of floating-point numbers. This can be useful when debugging, and in numerical work. Return a representation of a floating-point number as a hexadecimal string. For finite floating-point numbers, this representation will always include a leading 0x and a trailing p and exponent.