Ubuntu Manpage: eqn - format mathematics (equations) for groff or MathML

Name

       eqn - format mathematics (equations) for groff or MathML

Synopsis

       eqn [-CNrR] [-d xy] [-f F] [-m n] [-M dir] [-p n] [-s n] [-T dev] [file ...]

       eqn --help

       eqn -v
       eqn --version

Description

The GNU implementation of eqn is part of the groff(7) document formatting system. eqn is
a troff(1) preprocessor that translates expressions in its own language, embedded in
roff(7) input files, into mathematical notation typeset by troff(1). It copies each
file's contents to the standard output stream, translating each equation between lines
starting with .EQ and .EN, or within a pair of user-specified delimiters. Normally, eqn
is not executed directly by the user, but invoked by specifying the -e option to groff(1).
While GNU eqn's input syntax is highly compatible with AT&T eqn, the output eqn produces
cannot be processed by AT&T troff; GNU troff (or a troff implementing relevant GNU
extensions) must be used. If no file operands are given on the command line, or if file
is “-”, eqn reads the standard input stream.

Unless the -R option is used, eqn searches for the file eqnrc in the directories given
with the -M option first, then in /usr/share/groff/site-tmac, and finally in the standard
macro directory /usr/share/groff/1.23.0/tmac. If it exists and is readable, eqn processes
it before any input files.

This man page primarily discusses the differences between GNU eqn and AT&T eqn. Most of
the new features of the GNU eqn input language are based on TeX. There are some
references to the differences between TeX and GNU eqn below; these may safely be ignored
if you do not know TeX.

Three points are worth special note.

• GNU eqn emits Presentation MathML output when invoked with the “-T MathML” option.

• GNU eqn does not support terminal devices well, though it may suffice for simple inputs.

• GNU eqn sets the input token “...” as an ellipsis on the text baseline, not the three
centered dots of AT&T eqn. Set an ellipsis on the math axis with the GNU extension
macro cdots.

Anatomy of an equation
eqn input consists of tokens. Consider a form of Newton's second law of motion. The
input

.EQ
F =
m a
.EN

becomes F=ma. Each of F, =, m, and a is a token. Spaces and newlines are
interchangeable; they separate tokens but do not break lines or produce space in the
output.

The following input characters not only separate tokens, but manage their grouping and
spacing as well.

{ } Braces perform grouping. Whereas “e sup a b” expresses “(e to the a) times b”, “e
sup { a b }” means “e to the (a times b)”. When immediately preceded by a “left”
or “right” primitive, a brace loses its special meaning.

^ ~ are the half space and full space, respectively. Use them to tune the appearance
of the output.

Tab and leader characters separate tokens as well as advancing the drawing position to the
next tab stop, but are seldom used in eqn input. When they occur, they must appear at the
outermost lexical scope. This roughly means that they can't appear within braces that are
necessary to disambiguate the input; eqn will diagnose an error in this event. (See
subsection “Macros” below for additional token separation rules.)

Other tokens are primitives, macros, an argument to either of the foregoing, or components
of an equation.

Primitives are fundamental keywords of the eqn language. They can configure an aspect of
the preprocessor's state, as when setting a “global” font selection or type size (gfont
and gsize), or declaring or deleting macros (“define” and undef); these are termed
commands. Other primitives perform formatting operations on the tokens after them (as
with fat, over, sqrt, or up).

Equation components include mathematical variables, constants, numeric literals, and
operators. eqn remaps some input character sequences to groff special character escape
sequences for economy in equation entry and to ensure that glyphs from an unstyled font
are used; see groff_char(7).

+ \[pl] ' \[fm]
- \[mi] <= \[<=]
= \[eq] >= \[>=]

Macros permit primitives, components, and other macros to be collected and referred to by
a single token. Predefined macros make convenient the preparation of eqn input in a form
resembling its spoken expression; for example, consider cos, hat, inf, and lim.

Spacing and typeface
GNU eqn imputes types to the components of an equation, adjusting the spacing between them
accordingly. Recognized types are as follows; most affect spacing only, whereas the
“letter” subtype of “ordinary” also assigns a style.

ordinary character such as “1”, “a”, or “!”
letter character to be italicized by default
digit n/a
operator large operator such as “Σ”
binary binary operator such as “+”
relation relational operator such as “=”
opening opening bracket such as “(”
closing closing bracket such as “)”
punctuation punctuation character such as “,”
inner sub-formula contained within brackets
suppress component to which automatic spacing is not applied

Two primitives apply types to equation components.

type t e
Apply type t to expression e.

chartype t text
Assign each character in (unquoted) text type t, persistently.

eqn sets up spacings and styles as if by the following commands.

chartype "letter" abcdefghiklmnopqrstuvwxyz
chartype "letter" ABCDEFGHIKLMNOPQRSTUVWXYZ
chartype "letter" \[*a]\[*b]\[*g]\[*d]\[*e]\[*z]
chartype "letter" \[*y]\[*h]\[*i]\[*k]\[*l]\[*m]
chartype "letter" \[*n]\[*c]\[*o]\[*p]\[*r]\[*s]
chartype "letter" \[*t]\[*u]\[*f]\[*x]\[*q]\[*w]

chartype "binary" *\[pl]\[mi]
chartype "relation" <>\[eq]\[<=]\[>=]
chartype "opening" {([
chartype "closing" })]
chartype "punctuation" ,;:.
chartype "suppress" ^~

eqn assigns all other ordinary and special roff characters, including numerals 0–9, the
“ordinary” type. (The “digit” type is not used, but is available for customization.) In
keeping with common practice in mathematical typesetting, lowercase, but not uppercase,
Greek letters are assigned the “letter” type to style them in italics. The macros for
producing ellipses, “...”, cdots, and ldots, use the “inner” type.

Primitives
eqn supports without alteration the AT&T eqn primitives above, back, bar, bold, define,
down, fat, font, from, fwd, gfont, gsize, italic, left, lineup, mark, matrix, ndefine,
over, right, roman, size, sqrt, sub, sup, tdefine, to, under, and up.

New primitives
The GNU extension primitives “type” and chartype are discussed in subsection “Spacing and
typeface” above; “set” in subsection “Customization” below; and grfont and gbfont in
subsection “Fonts” below. In the following synopses, X can be any character not appearing
in the parameter thus bracketed.

e1 accent e2
Set e2 as an accent over e1. e2 is assumed to be at the appropriate height for a
lowercase letter without an ascender; eqn vertically shifts it depending on e1's
height. For example, hat is defined as follows.

accent { "^" }

dotdot, dot, tilde, vec, and dyad are also defined using the accent primitive.

big e Enlarge the expression e; semantics like those of CSS “large” are intended. In
troff output, the type size is increased by 5 scaled points. MathML output emits
the following.

copy file
include file
Interpolate the contents of file, omitting lines beginning with .EQ or .EN. If a
relative path name, file is sought relative to the current working directory.

ifdef name X anything X
If name is defined as a primitive or macro, interpret anything.

nosplit text
As "text", but since text is not quoted it is subject to macro expansion; it is not
split up and the spacing between characters not adjusted per subsection “Spacing
and typeface” above.

e opprime
As prime, but set the prime symbol as an operator on e. In the input “A opprime
sub 1”, the “1” is tucked under the prime as a subscript to the “A” (as is
conventional in mathematical typesetting), whereas when prime is used, the “1” is a
subscript to the prime character. The precedence of opprime is the same as that of
bar and “under”, and higher than that of other primitives except accent and
uaccent. In unquoted text, a neutral apostrophe (') that is not the first
character on the input line is treated like opprime.

sdefine name X anything X
As “define”, but name is not recognized as a macro if called with arguments.

e1 smallover e2
As over, but reduces the type size of e1 and e2, and puts less vertical space
between e1 and e2 and the fraction bar. The over primitive corresponds to the TeX
\over primitive in displayed equation styles; smallover corresponds to \over in
non-display (“inline”) styles.

space n
Set extra vertical spacing around the equation, replacing the default values, where
n is an integer in hundredths of an em. If positive, n increases vertical spacing
before the equation; if negative, it does so after the equation. This primitive
provides an interface to groff's \x escape sequence, but with the opposite sign
convention. It has no effect if the equation is part of a pic(1) picture.

special troff-macro e
Construct an object by calling troff-macro on e. The troff string 0s contains the
eqn output for e, and the registers 0w, 0h, 0d, 0skern, and 0skew the width,
height, depth, subscript kern, and skew of e, respectively. (The subscript kern of
an object indicates how much a subscript on that object should be “tucked in”, or
placed to the left relative to a non-subscripted glyph of the same size. The skew
of an object is how far to the right of the center of the object an accent over it
should be placed.) The macro must modify 0s so that it outputs the desired result,
returns the drawing position to the text baseline at the beginning of e, and
updates the foregoing registers to correspond to the new dimensions of the result.

Suppose you want a construct that “cancels” an expression by drawing a diagonal
line through it.

.de Ca
. ds 0s \
\Z'\\*(0s'\
\v'\\n(0du'\
\D'l \\n(0wu -\\n(0hu-\\n(0du'\
\v'\\n(0hu'
..
.EQ
special Ca "x \[mi] 3 \[pl] x" ~ 3
.EN

We use the \[mi] and \[pl] special characters instead of + and - because they are
part of the argument to a troff macro, so eqn does not transform them to
mathematical glyphs for us. Here's a more complicated construct that draws a box
around an expression; the bottom of the box rests on the text baseline. We define
the eqn macro box to wrap the call of the troff macro Bx.

.de Bx
.ds 0s \
\Z'\\h'1n'\\*[0s]'\
\v'\\n(0du+1n'\
\D'l \\n(0wu+2n 0'\
\D'l 0 -\\n(0hu-\\n(0du-2n'\
\D'l -\\n(0wu-2n 0'\
\D'l 0 \\n(0hu+\\n(0du+2n'\
\h'\\n(0wu+2n'
.nr 0w +2n
.nr 0d +1n
.nr 0h +1n
..
.EQ
define box ' special Bx $1 '
box(foo) ~ "bar"
.EN

split "text"
As text, but since text is quoted, it is not subject to macro expansion; it is
split up and the spacing between characters adjusted per subsection “Spacing and
typeface” above.

e1 uaccent e2
Set e2 as an accent under e1. e2 is assumed to be at the appropriate height for a
letter without a descender; eqn vertically shifts it depending on whether e1 has a
descender. utilde is predefined using uaccent as a tilde accent below the
baseline.

undef name
Remove definition of macro or primitive name, making it undefined.

vcenter e
Vertically center e about the math axis, a horizontal line upon which fraction bars
and characters such as “+” and “−” are aligned. MathML already behaves this way,
so eqn ignores this primitive when producing that output format. The built-in sum
macro is defined as if by the following.

define sum ! { type "operator" vcenter size +5 \(*S } !

Extended primitives
GNU eqn extends the syntax of some AT&T eqn primitives, introducing one deliberate
incompatibility.

delim on
eqn recognizes an “on” argument to the delim primitive specially, restoring any
delimiters previously disabled with “delim off”. If delimiters haven't been
specified, neither command has effect. Few eqn documents are expected to use “o”
and “n” as left and right delimiters, respectively. If yours does, consider
swapping them, or select others.

col n { ... }
ccol n { ... }
lcol n { ... }
rcol n { ... }
pile n { ... }
cpile n { ... }
lpile n { ... }
rpile n { ... }
The integer value n (in hundredths of an em) increases the vertical spacing between
rows, using groff's \x escape sequence (the value has no effect in MathML mode).
Negative values are accepted but have no effect. If more than one n occurs in a
matrix or pile, the largest is used.

Customization
When eqn generates troff input, the appearance of equations is controlled by a large
number of parameters. They have no effect when generating MathML, which delegates
typesetting to a MathML rendering engine. Configure these parameters with the set
primitive.

set p n
assigns parameter p the integer value n; n is interpreted in units of hundredths of
an em unless otherwise stated. For example,

set x_height 45

says that eqn should assume that the font's x-height is 0.45 ems.

Available parameters are as follows; defaults are shown in parentheses. We intend
these descriptions to be expository rather than rigorous.

minimum_size sets a floor for the type size (in scaled points) at which
equations are set (5).

fat_offset The fat primitive emboldens an equation by overprinting two copies
of the equation horizontally offset by this amount (4). In MathML
mode, components to which fat_offset applies instead use the
following.
<mstyle mathvariant='double-struck'>

over_hang A fraction bar is longer by twice this amount than the maximum of
the widths of the numerator and denominator; in other words, it
overhangs the numerator and denominator by at least this amount
(0).

accent_width When bar or under is applied to a single character, the line is
this long (31). Normally, bar or under produces a line whose
length is the width of the object to which it applies; in the case
of a single character, this tends to produce a line that looks too
long.

delimiter_factor Extensible delimiters produced with the left and right primitives
have a combined height and depth of at least this many thousandths
of twice the maximum amount by which the sub-equation that the
delimiters enclose extends away from the axis (900).

delimiter_shortfall
Extensible delimiters produced with the left and right primitives
have a combined height and depth not less than the difference of
twice the maximum amount by which the sub-equation that the
delimiters enclose extends away from the axis and this amount
(50).

null_delimiter_space
This much horizontal space is inserted on each side of a fraction
(12).

script_space The width of subscripts and superscripts is increased by this
amount (5).

thin_space This amount of space is automatically inserted after punctuation
characters. It also configures the width of the space produced by
the ^ token (17).

medium_space This amount of space is automatically inserted on either side of
binary operators (22).

thick_space This amount of space is automatically inserted on either side of
relations. It also configures the width of the space produced by
the ~ token (28).

x_height The height of lowercase letters without ascenders such as “x”
(45).

axis_height The height above the baseline of the center of characters such as
“+” and “−” (26). It is important that this value is correct for
the font you are using.

default_rule_thickness
This should be set to the thickness of the \[ru] character, or the
thickness of horizontal lines produced with the \D escape sequence
(4).

num1 The over primitive shifts up the numerator by at least this amount
(70).

num2 The smallover primitive shifts up the numerator by at least this
amount (36).

denom1 The over primitive shifts down the denominator by at least this
amount (70).

denom2 The smallover primitive shifts down the denominator by at least
this amount (36).

sup1 Normally superscripts are shifted up by at least this amount (42).

sup2 Superscripts within superscripts or upper limits or numerators of
smallover fractions are shifted up by at least this amount (37).
Conventionally, this is less than sup1.

sup3 Superscripts within denominators or square roots or subscripts or
lower limits are shifted up by at least this amount (28).
Conventionally, this is less than sup2.

sub1 Subscripts are normally shifted down by at least this amount (20).

sub2 When there is both a subscript and a superscript, the subscript is
shifted down by at least this amount (23).

sup_drop The baseline of a superscript is no more than this much below the
top of the object on which the superscript is set (38).

sub_drop The baseline of a subscript is at least this much below the bottom
of the object on which the subscript is set (5).

big_op_spacing1 The baseline of an upper limit is at least this much above the top
of the object on which the limit is set (11).

big_op_spacing2 The baseline of a lower limit is at least this much below the
bottom of the object on which the limit is set (17).

big_op_spacing3 The bottom of an upper limit is at least this much above the top
of the object on which the limit is set (20).

big_op_spacing4 The top of a lower limit is at least this much below the bottom of
the object on which the limit is set (60).

big_op_spacing5 This much vertical space is added above and below limits (10).

baseline_sep The baselines of the rows in a pile or matrix are normally this
far apart (140). Usually equal to the sum of num1 and denom1.

shift_down The midpoint between the top baseline and the bottom baseline in a
matrix or pile is shifted down by this much from the axis (26).
Usually equal to axis_height.

column_sep This much space is added between columns in a matrix (100).

matrix_side_sep This much space is added at each side of a matrix (17).

draw_lines If non-zero, eqn draws lines using the troff \D escape sequence,
rather than the \l escape sequence and the \[ru] special
character. The eqnrc file sets the default: 1 on ps, html, and
the X11 devices, otherwise 0.

body_height is the presumed height of an equation above the text baseline; eqn
adds any excess as extra pre-vertical line spacing with troff's \x
escape sequence (85).

body_depth is the presumed depth of an equation below the text baseline; eqn
adds any excess as extra post-vertical line spacing with troff's
\x escape sequence (35).

nroff If non-zero, then ndefine behaves like define and tdefine is
ignored, otherwise tdefine behaves like define and ndefine is
ignored. The eqnrc file sets the default: 1 on ascii, latin1,
utf8, and cp1047 devices, otherwise 0.

Macros
In GNU eqn, macros can take arguments. A word defined by any of the define, ndefine, or
tdefine primitives followed immediately by a left parenthesis is treated as a
parameterized macro call: subsequent tokens up to a matching right parenthesis are treated
as comma-separated arguments. In this context only, commas and parentheses also serve as
token separators. A macro argument is not terminated by a comma inside parentheses nested
within it. In a macro definition, $n, where n is between 1 and 9 inclusive, is replaced
by the nth argument; if there are fewer than n arguments, it is replaced by nothing.

Predefined macros
GNU eqn supports the predefined macros offered by AT&T eqn: and, approx, arc, cos, cosh,
del, det, dot, dotdot, dyad, exp, for, grad, half, hat, if, inter, Im, inf, int, lim, ln,
log, max, min, nothing, partial, prime, prod, Re, sin, sinh, sum, tan, tanh, tilde, times,
union, vec, ==, !=, +=, ->, <-, <<, >>, and “...”. The lowercase classical Greek letters
are available as alpha, beta, chi, delta, epsilon, eta, gamma, iota, kappa, lambda, mu,
nu, omega, omicron, phi, pi, psi, rho, sigma, tau, theta, upsilon, xi, and zeta. Spell
them with an initial capital letter (Alpha) or in full capitals (ALPHA) to obtain
uppercase forms.

GNU eqn further defines the macros cdot, cdots, and utilde (all discussed above), dollar,
which sets a dollar sign, and ldots, which sets an ellipsis on the text baseline.

Fonts
eqn uses up to three typefaces to set an equation: italic (oblique), roman (upright), and
bold. Assign each a groff typeface with the primitives gfont, grfont, and gbfont. The
defaults are the styles I, R, and B (applied to the current font family). The chartype
primitive (see above) sets a character's type, which determines the face used to set it.
The “letter” type is set in italics; others are set in roman. Use the bold primitive to
select an (upright) bold style.

gbfont f
Select f as the bold font. This is a GNU extension.

gfont f
Select f as the italic font.

grfont f
Select f as the roman font. This is a GNU extension.

Options

--help displays a usage message, while -v and --version show version information; all exit
afterward.

-C Recognize .EQ and .EN even when followed by a character other than space or
newline.

-d xy Specify delimiters x for left and y for right ends of equations not bracketed by
.EQ/.EN. x and y need not be distinct. Any “delim xy” statements in the source
file override this option.

-f F is equivalent to “gfont F”.

-m n is equivalent to “set minimum_size n”.

-M dir Search dir for eqnrc before those listed in section “Description” above.

-N Prohibit newlines within delimiters. This option allows eqn to recover better from
missing closing delimiters.

-p n Set sub- and superscripts n points smaller than the surrounding text. This option
is deprecated. eqn normally sets sub- and superscripts at 70% of the type size of
the surrounding text.

-r Reduce the type size of subscripts at most once relative to the base type size for
the equation.

-R Don't load eqnrc.

-s n is equivalent to “gsize n”. This option is deprecated.

-T dev Prepare output for the device dev. In most cases, the effect of this is to define
a macro dev with a value of 1; eqnrc uses this to provide definitions appropriate
for the device. However, if the specified driver is “MathML”, the output is MathML
markup rather than troff input, and eqnrc is not loaded at all. The default output
device is ps.

Files

       /usr/share/groff/1.23.0/tmac/eqnrc
              Initialization file.

MathML mode limitations

       MathML   is   designed   on  the  assumption  that  it  cannot  know  the  exact  physical
       characteristics of the media and devices on which  it  will  be  rendered.   It  does  not
       support control of motions and sizes to the same degree troff does.

       • eqn customization parameters have no effect on generated MathML.

       • The  special,  up,  down,  fwd,  and  back primitives cannot be implemented, and yield a
         MathML “<merror>” message instead.

       • The vcenter primitive is silently ignored, as centering on the math axis is  the  MathML
         default.

       • Characters  that eqn sets extra large in troff mode—notably the integral sign—may appear
         too small and need to have their “<mstyle>” wrappers adjusted by hand.

       As in its troff mode, eqn in MathML mode leaves the .EQ and .EN tokens in place, but emits
       nothing  corresponding to delim delimiters.  They can, however, be recognized as character
       sequences that begin with “<math>”, end with “</math>”, and do not cross line boundaries.

Caveats

       Tokens must be double-quoted in eqn input if they are not to be  recognized  as  names  of
       macros  or  primitives,  or  if they are to be interpreted by troff.  In particular, short
       ones, like “pi” and “PI”, can collide with  troff  identifiers.   For  instance,  the  eqn
       command  “gfont  PI”  does  not  select groff's Palatino italic font for the global italic
       face; you must use “gfont "PI"” instead.

       Delimited equations are set at the type size current at the beginning of the  input  line,
       not necessarily that immediately preceding the opening delimiter.

       Unlike  TeX, eqn does not inherently distinguish displayed and inline equation styles; see
       the smallover primitive above.  However, macro packages frequently define EQ and EN macros
       such  that the equation within is displayed.  These macros may accept arguments permitting
       the equation to be labeled or captioned; see the package's documentation.

Bugs

       eqn  abuses  terminology—its  “equations”  can  be  inequalities,  bare  expressions,   or
       unintelligible gibberish.  But there's no changing it now.

       In nroff mode, lowercase Greek letters are rendered in roman instead of italic style.

       In  MathML  mode,  the  mark  and  lineup features don't work.  These could, in theory, be
       implemented with “<maligngroup>” elements.

       In MathML mode, each digit of a numeric literal gets  a  separate  “<mn></mn>”  pair,  and
       decimal  points  are  tagged  with “<mo></mo>”.  This is allowed by the specification, but
       inefficient.

Examples

       We first illustrate eqn usage with a trigonometric identity.

              .EQ
              sin ( alpha + beta ) = sin alpha cos beta + cos alpha sin beta
              .EN

       It can be convenient to set up delimiters if mathematical content will  appear  frequently
       in running text.

              .EQ
              delim $$
              .EN
              Having cached a table of logarithms,
              the property $ln ( x y ) = ln x + ln y$ sped calculations.

       The  quadratic  formula  illustrates  use  of  fractions  and  radicals,  and  affords  an
       opportunity to use the full space token ~.

              .EQ
              x = { - b ~ \[+-] ~ sqrt { b sup 2 - 4 a c } } over { 2 a }
              .EN

       Alternatively, we could define the  plus-minus  sign  as  a  binary  operator.   Automatic
       spacing  puts  0.06 em less space on either side of the plus-minus than ~ does, this being
       the difference between the widths of the medium_space parameter used by  binary  operators
       and  that  of  the  full  space.   Independently, we can define a macro “frac” for setting
       fractions.

              .EQ
              chartype "binary" \[+-]
              define frac ! { $1 } over { $2 } !
              x = frac(- b \[+-] sqrt { b sup 2 - 4 a c }, 2 a)
              .EN