@Section
   @Title { Mathematical functions, loops, and tests }
   @Tag { functions }
@Begin
@PP
@Code "@Graph" offers quite a large selection of mathematical functions,
graphs. @RawIndex { graphs (statistical) }
graphs.mathematical @SubIndex { mathematical functions }
mathematical.functions @Index { mathematical functions in graphs }
available everywhere that x and y coordinates are required:  within
the @Code xticks and @Code yticks options, within the points within
the @Code "objects" option, and within the right parameter of the
@Code "@Data" symbol.  For example,
@ID @OneRow @Code {
"@Data"
"    pairs { solid }"
"{"
"    0 0  pi sin { pi/2 }"
"}"
}
draws a solid line from @Eq {(0, 0)} to @Eq {(pi, sin(pi "/" 2))}.  Section
{@NumberOf grsummary} lists all the functions; they include the four
arithmetical operators @Eq { non + }, @Eq { non - }, @Eq { non * }, and
@Eq { "/" }, as well as {@Code "sin"}, {@Code "cos"}, {@Code "sqrt"}, and
many others.  Braces are used for grouping, never parentheses.
@PP
For plotting functions there are three looping symbols, {@Code "xloop"},
{@Code "yloop"}, and {@Code "zloop"}.  For example, the following plots
the two functions @Eq { y = 2 } and @Eq { y = sqrt { pi x "/" 4 } + 1 }
for @Eq { x } from 10 to 500:
@ID -1px @Break @OneRow @Code @Verbatim {
-2p @Font @Graph
    style { axes }
    xorigin { 0 }
    yorigin { 0 }
    width { 8c }
    xticks { 10@ 50@ 100@ 200@ 500@ }
    objects {  @NE at { 300 2 } @I { Exponential }
               @SE at { 300 sqrt { pi*300/4 } + 1 } @I { Uniform } }
    belowcaption { @I n }
    belowgap { 0c }
    leftcaption { Right shell nodes }
{
    @Data points { filledcircle }
    { 10 1.97  50 2.01  100 2.00  200 2.0  500 2.00 }

    @Data points { filledcircle }
    { 10 3.53  50 7.45  100 9.32  200 13.41  500 21.63 }

    @Data pairs { dashed }
    { 10 2  500 2 }

    @Data pairs { dashed }
    {
        xloop from { 10 } to { 500 } by { 20 } do
        {
            x  sqrt { pi*x / 4 } + 1
        }
    }
}
}
The @Code "do" option of @Code xloop is replicated repeatedly with each
occurrence of @Code x replaced by 10, 30, 50, ... up to 490.  The
result is
@FootNote { Source:  Jeffrey H. Kingston, Analysis of tree algorithms
for the simulation event list.  @I { Acta Informatica } {@B 22},
pp. 15--33 (1985). }
@CD -2p @Font @Graph
    style { axes }
    xorigin { 0 }
    yorigin { 0 }
    width { 8c }
    xticks { 10@ 50@ 100@ 200@ 500@ }
    objects {
       @NE at { 300 2 } @I { Exponential }
       @SE at { 300 sqrt { pi*300/4 } + 1 } @I { Uniform }
    }
    belowcaption { @I n }
    belowgap { 0c }
    leftcaption { Right shell nodes }
{
    @Data points { filledcircle }
    { 10 1.97  50 2.01  100 2.00  200 2.0  500 2.00 }

    @Data points { filledcircle }
    { 10 3.53  50 7.45  100 9.32  200 13.41  500 21.63 }

    @Data pairs { dashed }
    { 10 2  500 2 }

    @Data pairs { dashed }
    {
        xloop from { 10 } to { 500 } by { 20 } do
        {
            x  sqrt { pi*x / 4 } + 1
        }
    }
}
The points are connected by straight line segments as usual, but a
smallish @Code "by" option of about one-twentieth of the range creates
the illusion of a smooth curve quite well.
@PP
There is also an @Code "if" symbol which produces alternative results,
depending on whether a condition evaluates to @Code "true" or
{@Code"false"}:
@ID @OneRow @Code @Verbatim {
xloop from { -5 } to { +5 } by { 0.2 } do
{
    if cond { abs { x } > 0.1 } then { x 1/x } else {}
}
}
This plots the function @Eq { y = 1 "/" x }, skipping points near
zero.  Actually the @Code "else" part could be omitted since its default
value is empty.
@PP
Adventurous users might enjoy nesting a @Code "yloop" or @Code "zloop"
within an {@Code "xloop"}, or using loops to generate ticks, like this:
@ID @OneRow @Code @Verbatim {
xticks {
    xloop from { 0 } to { 20 } do
    {
       x if cond { x mod 5 = 0 } then { @ }
    }
}
}
The missing @Code "by" option defaults to 1, so this produces x ticks at
0, 1, 2, ..., 20, with labels at 0, 5, 10, 15, and 20.  It is quite all
right to mix @Code "@" and even labels in with numbers, as long as the
final result obeys the rules of Section {@NumberOf ticks}.
@PP
You can define your own functions using Lout definitions, placed in your
@Code "mydefs" file as explained in Section {@NumberOf definitions}.  Here
is an example of a function definition:
@ID @OneRow @Code @Verbatim {
import @Graph @Data
def @Tan
    precedence 40
    right x
{
    sin x / cos x
}
}
This defines a function called @Code "@Tan" which implements the
trigonometric tangent function.  It may then be used in expressions
just like any other function:
@ID @OneRow @Code @Verbatim {
@Data {
    yloop from { 0 } to { 0.95 } by { 0.05 } do
    {
        y   @Tan { y / pi }
    }
}
}
Following is a detailed explanation.
@PP
The first line, {@Code "import @Graph @Data"}, is the import clause.  Its
function is to grant the definition access to the three previously defined
functions (symbols) that it uses, namely {@Code "sin"}, {@Code "cos"},
and {@Code "/"}.  These are found within the @Code "@Data" symbol within
{@Code "@Graph"}.
@PP
After the import clause comes the @Code "def" keyword, meaning
`define,' and then the name of the symbol being defined, in this case
@Code "@Tan".  We have chosen @Code "@Tan" rather than @Code "tan"
because symbols defined by the user in this way are visible throughout
the document, and we do not want the literal word @Code "tan" to be
taken as a symbol.
@PP
Next comes the symbol's precedence, in this case the same as @Code "sin" and
@Code "cos" (see Section {@NumberOf dia_summ} for the precedence of
each symbol).  Next is a list of the formal parameters, in this case
just one, called {@Code "x"}, that is to be passed on the right.
@PP
Finally comes the body of the definition enclosed in braces.  When
@Code "@Tan" is invoked, its value will be this body with each occurrence
of the formal parameter @Code "x" replaced by the object following the
@Code "@Tan" symbol.  For example, the @Code "do" option of the @Code
"yloop" above becomes
@ID @Code "y   sin { y / pi } / cos { y / pi }"
as you would expect.
@End @Section
