User-Defined Compositions
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The `Z C' (`calc-user-define-composition') command lets you define the
display format for any algebraic function.  You provide a formula
containing a certain number of argument variables on the stack.  Any
time Calc formats a call to the specified function in the current
language mode and with that number of arguments, Calc effectively
replaces the function call with that formula with the arguments
replaced.

Calc builds the default argument list by sorting all the variable
names that appear in the formula into alphabetical order.  You can
edit this argument list before pressing RET if you wish.  Any
variables in the formula that do not appear in the argument list will
be displayed literally; any arguments that do not appear in the
formula will not affect the display at all.

You can define formats for built-in functions, for functions you have
defined with `Z F' (See Algebraic Definitions), or for functions
which have no definitions but are being used as purely syntactic
objects.  You can define different formats for each language mode, and
for each number of arguments, using a succession of `Z C' commands.
When Calc formats a function call, it first searches for a format
defined for the current language mode (and number of arguments); if
there is none, it uses the format defined for the Normal language
mode.  If neither format exists, Calc uses its built-in standard
format for that function (usually just `FUNC(ARGS)').

If you execute `Z C' with the number 0 on the stack instead of a
formula, any defined formats for the function in the current language
mode will be removed.  The function will revert to its standard
format.

For example, the default format for the binomial coefficient function
`choose(n, m)' in the Big language mode is

      n
     ( )
      m

You might prefer the notation,

      C
     n m

To define this notation, first make sure you are in Big mode, then put
the formula

     choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])

on the stack and type `Z C'.  Answer the first prompt with `choose'.
The second prompt will be the default argument list of `(C m n)'.
Edit this list to be `(n m)' and press RET.  Now, try it out: For
example, turn simplification off with `m O' and enter `choose(a,b) +
choose(7,3)' as an algebraic entry.

      C  +  C
     a b   7 3

As another example, let's define the usual notation for Stirling
numbers of the first kind, `stir1(n, m)'.  This is just like the
regular format for binomial coefficients but with square brackets
instead of parentheses.

     choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])

Now type `Z C stir1 RET', edit the argument list to `(n m)', and type
RET.

The formula provided to `Z C' usually will involve composition
functions, but it doesn't have to.  Putting the formula `a + b + c'
onto the stack and typing `Z C foo RET RET' would define the function
`foo(x,y,z)' to display like `x + y + z'.  This "sum" will act exactly
like a real sum for all formatting purposes (it will be parenthesized
the same, and so on).  However it will be computationally unrelated to
a sum.  For example, the formula `2 * foo(1, 2, 3)' will display as `2
(1 + 2 + 3)'.  Operator precedences have caused the "sum" to be
written in parentheses, but the arguments have not actually been
summed.  (Generally a display format like this would be undesirable,
since it can easily be confused with a real sum.)

The special function `eval' can be used inside a `Z C' composition
formula to cause all or part of the formula to be evaluated at display
time.  For example, if the formula is `a + eval(b + c)', then `foo(1,
2, 3)' will be displayed as `1 + 5'.  Evaluation will use the default
simplifications, regardless of the current simplification mode.  There
are also `evalsimp' and `evalextsimp' which simplify as if by `a s'
and `a e' (respectively).  Note that these "functions" operate only in
the context of composition formulas (and also in rewrite rules, where
they serve a similar purpose; See Rewrite Rules).  On the stack, a
call to `eval' will be left in symbolic form.

It is not a good idea to use `eval' except as a last resort.  It can
cause the display of formulas to be extremely slow.  For example,
while `eval(a + b)' might seem quite fast and simple, there are
several situations where it could be slow.  For example, `a' and/or
`b' could be polar complex numbers, in which case doing the sum
requires trigonometry.  Or, `a' could be the factorial `fact(100)'
which is unevaluated because you have typed `m O'; `eval' will
evaluate it anyway to produce a large, unwieldy integer.

You can save your display formats permanently using the `Z P' command
(See Creating User Keys).