OpenVMS VAX RTL Mathematics (MTH$) Manual

Document revision date: 19 July 1999

Order Number: AA--PVXJC--TE

January 1999

This manual documents the mathematics routines contained in the MTH$ facility of the OpenVMS Run-Time Library.

Revision/Update Information: This manual supersedes the OpenVMS VAX RTL Mathematics (MTH$) Manual, Version 5.5.

Software Version: OpenVMS VAX Version 7.1 The content of this document has not changed since OpenVMS Version 7.1.

Compaq Computer Corporation
Houston, Texas

Reprinted January 1999

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Contents Index

Preface

This manual provides users of the OpenVMS operating system with detailed usage and reference information on mathematics routines supplied in the MTH$ facility of the Run-Time Library.

Run-Time Library routines can be used only in programs written in languages that produce native code for the VAX hardware. At present, these languages include VAX MACRO and the following compiled high-level languages:

DEC Pascal for OpenVMS VAX Systems
Digital Fortran for OpenVMS VAX Systems
VAX Ada
VAX BASIC
VAX BLISS-32
VAX C
VAX COBOL
VAX CORAL
VAX DIBOL
VAX PL/I
VAX RPG
VAX SCAN

Interpreted languages that can also access Run-Time Library routines include VAX DSM and VAX DATATRIEVE.

Intended Audience

This manual is intended for system and application programmers who write programs that call MTH$ Run-Time Library routines.

Document Structure

This manual contains two tutorial chapters, two reference sections, and two appendixes:

Chapter 1 is an introductory chapter that provides guidelines on using the MTH$ scalar routines.
Chapter 2 provides guidelines on using the MTH$ vector routines.
The Scalar MTH$ Reference Section provides detailed reference information on each scalar mathematics routine contained in the MTH$ facility of the Run-Time Library.
The Vector MTH$ Reference Section provides detailed reference information on the BLAS Level 1 (Basic Linear Algebra Subroutines) and FOLR (First Order Linear Recurrence) routines.
Reference information is presented using the documentation format described in the OpenVMS Programming Interfaces: Calling a System Routine. Routine descriptions are in alphabetical order by routine name.
Appendix A lists supported MTH$ routines not included with the routines in the Scalar MTH$ Reference Section, because they are rarely used.
Appendix B contains a table of the vector MTH$ routines that you can call from VAX MACRO.

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Conventions

The following conventions are used in this manual:

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Part 1
MTH$ Tutorial Section

This part of the OpenVMS VAX RTL Mathematics (MTH$) Manual contains tutorial information about the OpenVMS RTL MTH$ Facility, and is structured as follows:

Chapter 1 is an introductory chapter that provides guidelines on using the MTH$ scalar routines.
Chapter 2 provides guidelines on using the MTH$ vector routines.

Chapter 1
OpenVMS Run-Time Library Mathematics (MTH$) Facility

The OpenVMS Run-Time Library Mathematics (MTH$) facility contains routines to perform a wide variety of computations including the following:

Floating-point trigonometric function evaluation
Exponentiation
Complex function evaluation
Complex exponentiation
Miscellaneous function evaluation
Vector operations (VAX only)

The OTS$ facility provides additional language-independent arithmetic support routines (see the OpenVMS RTL General Purpose (OTS$) Manual).

This chapter contains an introduction to the MTH$ facility and includes examples of how to call mathematics routines from BASIC, COBOL, Fortran, MACRO, Pascal, PL/I, and Ada.

Chapter 2 contains an overview of the vector routines available on VAX processors.

The Scalar MTH$ Reference Section describes the MTH$ scalar routines.

The Vector MTH$ Reference Section describes the MTH$ vector routines.

1.1 Entry Point Names

The names of the mathematics routines are formed by adding the MTH$ prefix to the function names.

When function arguments and returned values are of the same data type, the first letter of the name indicates this data type. When function arguments and returned values are of different data types, the first letter indicates the data type of the returned value, and the second letter indicates the data type of the arguments.

The letters used as data type prefixes are listed below.

Letter Data Type

I Word

J Longword

D D_floating

G G_floating

H H_floating

C F_floating complex

CD D_floating complex

CG G_floating complex

Letter	Data Type
I	Word
J	Longword
D	D_floating
G	G_floating
H	H_floating
C	F_floating complex
CD	D_floating complex
CG	G_floating complex

Generally, F-floating data types have no letter designation. For example, MTH$SIN returns an F-floating value of the sine of an F-floating argument and MTH$DSIN returns a D-floating value of the sine of a D-floating argument. However, in some of the miscellaneous functions, F-floating data types are referenced by the letter designation A.

1.2 Calling Conventions

For calling conventions specific to the MTH$ vector routines, refer to Chapter 2.

All calls to mathematics routines, as described in the Format section of each routine, accept arguments passed by reference. JSB entry points accept arguments passed by value.

All mathematics routines return values in R0 or R0/R1 except those routines for which the values cannot fit in 64 bits. D-floating complex, G-floating complex, and H-floating values are data structures which are larger than 64 bits. Routines returning values that cannot fit in registers R0/R1 return their function values into the first argument in the argument list.

The notation JSB MTH$NAME_Rn, where n is the highest register number referenced, indicates that an equivalent JSB entry point is available. Registers R0:Rn are not preserved.

Routines with JSB entry points accept a single argument in R0:Rm, where m, which is defined in the following table, is dependent on the data type.

Data Type m

F_floating 0

D_floating 1

G_floating 1

H_floating 3

Data Type	m
F_floating	0
D_floating	1
G_floating	1
H_floating	3

A routine returning one value returns it to registers R0:Rm.

When a routine returns two values (for example, MTH$SINCOS), the first value is returned in R0:Rm and the second value is returned in (R<m+1>:R<2*m+1>).

Note that for routines returning a single value, n>=m. For routines returning two values, n>=2*m + 1.

In general, CALL entry points for mathematics routines do the following:

Disable floating-point underflow
Enable integer overflow
Cause no floating-point overflow or other arithmetic traps or faults
Preserve all other enabled operations across the CALL

JSB entry points execute in the context of the caller with the enable operations as set by the caller. Since the routines do not cause arithmetic traps or faults, their operation is not affected by the setting of the arithmetic trap enables, except as noted.

For more detailed information on CALL and JSB entry points, refer to the OpenVMS Programming Interfaces: Calling a System Routine.

1.3 Algorithms

For those mathematics routines having corresponding algorithms, the complete algorithm can be found in the Description section of the routine description appearing in the Scalar MTH$ Reference Section of this manual.

1.4 Condition Handling

Error conditions are indicated by using the VAX signaling mechanism. The VAX signaling mechanism signals all conditions in mathematics routines as SEVERE by calling LIB$SIGNAL. When a SEVERE error is signaled, the default handler causes the image to exit after printing an error message. A user-established condition handler can be written to cause execution to continue at the point of the error by returning SS$_CONTINUE. A mathematics routine returns to its caller after the contents of R0/R1 have been restored from the mechanism argument vector CHF$L_MCH_SAVR0/R1. Thus, the user-established handler should correct CHF$L_MCH_SAVR0/R1 to the desired function value to be returned to the caller of the mathematics routine.

D-floating complex, G-floating complex, and H-floating values cannot be corrected with a user-established condition handler, because R2/R3 is not available in the mechanism argument vector.

Note that it is more reliable to correct R0 and R1 to resemble R0 and R1 of a double-precision floating-point value. A double-precision floating-point value correction works for both single- and double-precision values.

If the correction is not performed, the floating-point reserved operand --0.0 is returned. A floating-point reserved operand is a floating-point datum with a sign bit of 1 and a biased exponent of 0. Accessing the floating-point reserved operand will cause a reserved operand fault. See the OpenVMS RTL Library (LIB$) Manual for a complete description of how to write user condition handlers for SEVERE errors.

A few mathematics routines signal floating underflow if the calling program (JSB or CALL) has enabled floating underflow faults or traps.

All mathematics routines access input arguments and the real and imaginary parts of complex numbers using floating-point instructions. Therefore, a reserved operand fault can occur in any mathematics routine.

1.5 Complex Numbers

A complex number y is defined as an ordered pair of real numbers r and i, where r is the real part and i is the imaginary part of the complex number.

y=(r,i)

OpenVMS supports three floating-point complex types: F-floating complex, D-floating complex, and G-floating complex. There is no H-floating complex data type.

Run-Time Library mathematics routines that use complex arguments require a pointer to a structure containing two x-floating values to be passed by reference for each argument. The first x-floating value contains r, the real part of the complex number. The second x-floating value contains i, the imaginary part of the complex number. Similarly, Run-Time Library mathematics routines that return complex function values return two x-floating values. Some Language Independent Support (OTS$) routines also calculate complex functions.

Note that complex functions have no JSB entry points.

1.6 Mathematics Routines Not Documented in the MTH$ Reference Section

The mathematics routines in Table 1-1 are not found in the reference section of this manual. Instead, their entry points and argument information are listed in Appendix A of this manual.

A reserved operand fault can occur for any floating-point input argument in any mathematics routine. Other condition values signaled by each mathematics routine are indicated in the footnotes.

Table 1-1 Additional Mathematics Routines
Entry Point Function

Absolute Value Routines

MTH$ABS F-floating absolute value

MTH$DABS D-floating absolute value

MTH$GABS G-floating absolute value

MTH$HABS H-floating absolute value ¹

MTH$IIABS Word absolute value ²

MTH$JIABS Longword absolute value ²

Bitwise AND Operator Routines

MTH$IIAND Bitwise AND of two word parameters

MTH$JIAND Bitwise AND of two longword parameters

F-Floating Conversion Routines

MTH$DBLE Convert F-floating to D-floating (exact)

MTH$GDBLE Convert F-floating to G-floating (exact)

MTH$IIFIX Convert F-floating to word (truncated) ²

MTH$JIFIX Convert F-floating to longword (truncated) ²

Floating-Point Positive Difference Routines

MTH$DIM Positive difference of two F-floating parameters ³

MTH$DDIM Positive difference of two D-floating parameters ³

MTH$GDIM Positive difference of two G-floating parameters ³

MTH$HDIM Positive difference of two H-floating parameters ^1,3

MTH$IIDIM Positive difference of two word parameters ²

MTH$JIDIM Positive difference of two longword parameters ²

Bitwise Exclusive OR Operator Routines

MTH$IIEOR Bitwise exclusive OR of two word parameters

MTH$JIEOR Bitwise exclusive OR of two longword parameters

Integer to Floating-Point Conversion Routines

MTH$FLOATI Convert word to F-floating (exact)

MTH$DFLOTI Convert word to D-floating (exact)

MTH$GFLOTI Convert word to G-floating (exact)

MTH$FLOATJ Convert longword to F-floating (rounded)

MTH$DFLOTJ Convert longword to D-floating (exact)

MTH$GFLOTJ Convert longword to G-floating (exact)

Conversion to Greatest Floating-Point Integer Routines

MTH$FLOOR Convert F-floating to greatest F-floating integer

MTH$DFLOOR Convert D-floating to greatest D-floating integer

MTH$GFLOOR Convert G-floating to greatest G-floating integer

MTH$HFLOOR Convert H-floating to greatest H-floating integer ¹

Floating-Point Truncation Routines

MTH$AINT Convert F-floating to truncated F-floating

MTH$IINT Convert F-floating to truncated word ²

MTH$JINT Convert F-floating to truncated longword ²

MTH$DINT Convert D-floating to truncated D-floating

MTH$IIDINT Convert D-floating to truncated word ²

MTH$JIDINT Convert D-floating to truncated longword ²

MTH$GINT Convert G-floating to truncated G-floating

MTH$IIGINT Convert G-floating to truncated word ²

MTH$JIGINT Convert G-floating to truncated longword ²

MTH$HINT Convert H-floating to truncated H-floating ¹

MTH$IIHINT Convert H-floating to truncated word ²

MTH$JIHINT Convert H-floating to truncated longword ²

Bitwise Inclusive OR Operator Routines

MTH$IIOR Bitwise inclusive OR of two word parameters

MTH$JIOR Bitwise inclusive OR of two longword parameters

Maximum Value Routines

MTH$AIMAX0 F-floating maximum of n word parameters

MTH$AJMAX0 F-floating maximum of n longword parameters

MTH$IMAX0 Word maximum of n word parameters

MTH$JMAX0 Longword maximum of n longword parameters

MTH$AMAX1 F-floating maximum of n F-floating parameters

MTH$DMAX1 D-floating maximum of n D-floating parameters

MTH$GMAX1 G-floating maximum of n G-floating parameters

MTH$HMAX1 H-floating maximum of n H-floating parameters ¹

MTH$IMAX1 Word maximum of n F-floating parameters ²

MTH$JMAX1 Longword maximum of n F-floating parameters ²

Minimum Value Routines

MTH$AIMIN0 F-floating minimum of n word parameters

MTH$AJMIN0 F-floating minimum of n longword parameters

MTH$IMIN0 Word minimum of n word parameters

MTH$JMIN0 Longword minimum of n longword parameters

MTH$AMIN1 F-floating minimum of n F-floating parameters

MTH$DMIN1 D-floating minimum of n D-floating parameters

MTH$GMIN1 G-floating minimum of n G-floating parameters

MTH$HMIN1 H-floating minimum of n H-floating parameters ¹

MTH$IMIN1 Word minimum of n F-floating parameters ²

MTH$JMIN1 Longword minimum of n F-floating parameters ²

Remainder Routines

MTH$AMOD Remainder of two F-floating parameters, arg1/arg2 ^3,6

MTH$DMOD Remainder of two D-floating parameters, arg1/arg2 ^3,6

MTH$GMOD Remainder of two G-floating parameters, arg1/arg2 ³

MTH$HMOD Remainder of two H-floating parameters, arg1/arg2 ^1,3

MTH$IMOD Remainder of two word parameters, arg1/arg2 ⁵

MTH$JMOD Remainder of two longword parameters, arg1/arg2 ⁵

Floating-Point Conversion to Nearest Value Routines

MTH$ANINT Convert F-floating to nearest F-floating integer

MTH$ININT Convert F-floating to nearest word integer ²

MTH$JNINT Convert F-floating to nearest longword integer ²

MTH$DNINT Convert D-floating to nearest D-floating integer

MTH$IIDNNT Convert D-floating to nearest word integer ²

MTH$JIDNNT Convert D-floating to nearest longword integer ²

MTH$GNINT Convert G-floating to nearest G-floating integer

MTH$IIGNNT Convert G-floating to nearest word integer ²

MTH$JIGNNT Convert G-floating to nearest longword integer ²

MTH$HNINT Convert H-floating to nearest H-floating integer ¹

MTH$IIHNNT Convert H-floating to nearest word integer ²

MTH$JIHNNT Convert H-floating to nearest longword integer ²

Bitwise Complement Operator Routines

MTH$INOT Bitwise complement of word parameter

MTH$JNOT Bitwise complement of longword parameter

Floating-Point Multiplication Routines

MTH$DPROD D-floating product of two F-floating parameters ³

MTH$GPROD G-floating product of two F-floating parameters

Bitwise Shift Operator Routines

MTH$IISHFT Bitwise shift of word

MTH$JISHFT Bitwise shift of longword

Floating-Point Sign Function Routines

MTH$SGN F- or D-floating sign function

MTH$SIGN F-floating transfer of sign of y to sign of x

MTH$DSIGN D-floating transfer of sign of y to sign of x

MTH$GSIGN G-floating transfer of sign of y to sign of x

MTH$HSIGN H-floating transfer of sign of y to sign of x ¹

MTH$IISIGN Word transfer of sign of y to sign of x

MTH$JISIGN Longword transfer of sign of y to sign of x

Conversion of Double to Single Floating-Point Routines

MTH$SNGL Convert D-floating to F-floating (rounded) ³

MTH$SNGLG Convert G-floating to F-floating (rounded) ^3,4

**Table 1-1 Additional Mathematics Routines**
Entry Point	Function
Absolute Value Routines
MTH$ABS	F-floating absolute value
MTH$DABS	D-floating absolute value
MTH$GABS	G-floating absolute value
MTH$HABS	H-floating absolute value ¹
MTH$IIABS	Word absolute value ²
MTH$JIABS	Longword absolute value ²
Bitwise AND Operator Routines
MTH$IIAND	Bitwise AND of two word parameters
MTH$JIAND	Bitwise AND of two longword parameters
F-Floating Conversion Routines
MTH$DBLE	Convert F-floating to D-floating (exact)
MTH$GDBLE	Convert F-floating to G-floating (exact)
MTH$IIFIX	Convert F-floating to word (truncated) ²
MTH$JIFIX	Convert F-floating to longword (truncated) ²
Floating-Point Positive Difference Routines
MTH$DIM	Positive difference of two F-floating parameters ³
MTH$DDIM	Positive difference of two D-floating parameters ³
MTH$GDIM	Positive difference of two G-floating parameters ³
MTH$HDIM	Positive difference of two H-floating parameters ^1,3
MTH$IIDIM	Positive difference of two word parameters ²
MTH$JIDIM	Positive difference of two longword parameters ²
Bitwise Exclusive OR Operator Routines
MTH$IIEOR	Bitwise exclusive OR of two word parameters
MTH$JIEOR	Bitwise exclusive OR of two longword parameters
Integer to Floating-Point Conversion Routines
MTH$FLOATI	Convert word to F-floating (exact)
MTH$DFLOTI	Convert word to D-floating (exact)
MTH$GFLOTI	Convert word to G-floating (exact)
MTH$FLOATJ	Convert longword to F-floating (rounded)
MTH$DFLOTJ	Convert longword to D-floating (exact)
MTH$GFLOTJ	Convert longword to G-floating (exact)
Conversion to Greatest Floating-Point Integer Routines
MTH$FLOOR	Convert F-floating to greatest F-floating integer
MTH$DFLOOR	Convert D-floating to greatest D-floating integer
MTH$GFLOOR	Convert G-floating to greatest G-floating integer
MTH$HFLOOR	Convert H-floating to greatest H-floating integer ¹
Floating-Point Truncation Routines
MTH$AINT	Convert F-floating to truncated F-floating
MTH$IINT	Convert F-floating to truncated word ²
MTH$JINT	Convert F-floating to truncated longword ²
MTH$DINT	Convert D-floating to truncated D-floating
MTH$IIDINT	Convert D-floating to truncated word ²
MTH$JIDINT	Convert D-floating to truncated longword ²
MTH$GINT	Convert G-floating to truncated G-floating
MTH$IIGINT	Convert G-floating to truncated word ²
MTH$JIGINT	Convert G-floating to truncated longword ²
MTH$HINT	Convert H-floating to truncated H-floating ¹
MTH$IIHINT	Convert H-floating to truncated word ²
MTH$JIHINT	Convert H-floating to truncated longword ²
Bitwise Inclusive OR Operator Routines
MTH$IIOR	Bitwise inclusive OR of two word parameters
MTH$JIOR	Bitwise inclusive OR of two longword parameters
Maximum Value Routines
MTH$AIMAX0	F-floating maximum of n word parameters
MTH$AJMAX0	F-floating maximum of n longword parameters
MTH$IMAX0	Word maximum of n word parameters
MTH$JMAX0	Longword maximum of n longword parameters
MTH$AMAX1	F-floating maximum of n F-floating parameters
MTH$DMAX1	D-floating maximum of n D-floating parameters
MTH$GMAX1	G-floating maximum of n G-floating parameters
MTH$HMAX1	H-floating maximum of n H-floating parameters ¹
MTH$IMAX1	Word maximum of n F-floating parameters ²
MTH$JMAX1	Longword maximum of n F-floating parameters ²
Minimum Value Routines
MTH$AIMIN0	F-floating minimum of n word parameters
MTH$AJMIN0	F-floating minimum of n longword parameters
MTH$IMIN0	Word minimum of n word parameters
MTH$JMIN0	Longword minimum of n longword parameters
MTH$AMIN1	F-floating minimum of n F-floating parameters
MTH$DMIN1	D-floating minimum of n D-floating parameters
MTH$GMIN1	G-floating minimum of n G-floating parameters
MTH$HMIN1	H-floating minimum of n H-floating parameters ¹
MTH$IMIN1	Word minimum of n F-floating parameters ²
MTH$JMIN1	Longword minimum of n F-floating parameters ²
Remainder Routines
MTH$AMOD	Remainder of two F-floating parameters, arg1/arg2 ^3,6
MTH$DMOD	Remainder of two D-floating parameters, arg1/arg2 ^3,6
MTH$GMOD	Remainder of two G-floating parameters, arg1/arg2 ³
MTH$HMOD	Remainder of two H-floating parameters, arg1/arg2 ^1,3
MTH$IMOD	Remainder of two word parameters, arg1/arg2 ⁵
MTH$JMOD	Remainder of two longword parameters, arg1/arg2 ⁵
Floating-Point Conversion to Nearest Value Routines
MTH$ANINT	Convert F-floating to nearest F-floating integer
MTH$ININT	Convert F-floating to nearest word integer ²
MTH$JNINT	Convert F-floating to nearest longword integer ²
MTH$DNINT	Convert D-floating to nearest D-floating integer
MTH$IIDNNT	Convert D-floating to nearest word integer ²
MTH$JIDNNT	Convert D-floating to nearest longword integer ²
MTH$GNINT	Convert G-floating to nearest G-floating integer
MTH$IIGNNT	Convert G-floating to nearest word integer ²
MTH$JIGNNT	Convert G-floating to nearest longword integer ²
MTH$HNINT	Convert H-floating to nearest H-floating integer ¹
MTH$IIHNNT	Convert H-floating to nearest word integer ²
MTH$JIHNNT	Convert H-floating to nearest longword integer ²
Bitwise Complement Operator Routines
MTH$INOT	Bitwise complement of word parameter
MTH$JNOT	Bitwise complement of longword parameter
Floating-Point Multiplication Routines
MTH$DPROD	D-floating product of two F-floating parameters ³
MTH$GPROD	G-floating product of two F-floating parameters
Bitwise Shift Operator Routines
MTH$IISHFT	Bitwise shift of word
MTH$JISHFT	Bitwise shift of longword
Floating-Point Sign Function Routines
MTH$SGN	F- or D-floating sign function
MTH$SIGN	F-floating transfer of sign of y to sign of x
MTH$DSIGN	D-floating transfer of sign of y to sign of x
MTH$GSIGN	G-floating transfer of sign of y to sign of x
MTH$HSIGN	H-floating transfer of sign of y to sign of x ¹
MTH$IISIGN	Word transfer of sign of y to sign of x
MTH$JISIGN	Longword transfer of sign of y to sign of x
Conversion of Double to Single Floating-Point Routines
MTH$SNGL	Convert D-floating to F-floating (rounded) ³
MTH$SNGLG	Convert G-floating to F-floating (rounded) ^3,4

¹Returns value to the first argument; value exceeds 64 bits.
²Integer overflow exceptions can occur.
³Floating-point overflow exceptions can occur.
⁴Floating-point underflow exceptions can occur.
⁵Divide-by-zero exceptions can occur.
⁶Floating-point underflow exceptions are signaled.

Contents

Index

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PF1 x	A sequence such as PF1 x indicates that you must first press and release the key labeled PF1 and then press and release another key or a pointing device button.
`[Return]`	In examples, a key name enclosed in a box indicates that you press a key on the keyboard. (In text, a key name is not enclosed in a box.)
...	A horizontal ellipsis in examples indicates one of the following possibilities: Additional optional arguments in a statement have been omitted. The preceding item or items can be repeated one or more times. Additional parameters, values, or other information can be entered.
. . .	A vertical ellipsis indicates the omission of items from a code example or command format; the items are omitted because they are not important to the topic being discussed.
( )	In command format descriptions, parentheses indicate that, if you choose more than one option, you must enclose the choices in parentheses.
[ ]	In command format descriptions, brackets indicate optional elements. You can choose one, none, or all of the options. (Brackets are not optional, however, in the syntax of a directory name in an OpenVMS file specification or in the syntax of a substring specification in an assignment statement.)
{ }	In command format descriptions, braces indicate a required choice of options; you must choose one of the options listed.
bold text	This text style represents the introduction of a new term or the name of an argument, an attribute, or a reason.
italic text	Italic text indicates important information, complete titles of manuals, or variables. Variables include information that varies in system output (Internal error number), in command lines (/PRODUCER= name), and in command parameters in text (where device-name contains up to five alphanumeric characters).
UPPERCASE TEXT	Uppercase text indicates a command, the name of a routine, the name of a file, or the abbreviation for a system privilege.
Monospace text	Monospace type indicates code examples and interactive screen displays. In the C programming language, monospace type in text identifies the following elements: keywords, the names of independently compiled external functions and files, syntax summaries, and references to variables or identifiers introduced in an example.
-	A hyphen at the end of a command format description, command line, or code line indicates that the command or statement continues on the following line.
numbers	All numbers in text are assumed to be decimal unless otherwise noted. Nondecimal radixes---binary, octal, or hexadecimal---are explicitly indicated.