====================================================================
Make Function
====================================================================

It is sometimes useful to be able to define a function given by
the result of a calculation.

Suppose that you have obtained the following expression after several
computations and that you now want to tabulate the numerical values of
f for x between -1 and +1 with increment 0.1. ::

  expr := (x - exp x + 1)^2 * (sin(x^2) * x + 1)^3
       3   x 2        4     3   x    5     4    3      2 3
     (x (%e )  + (- 2x  - 2x )%e  + x  + 2x  + x )sin(x )
   + 
        2   x 2        3     2   x     4     3     2      2 2
     (3x (%e )  + (- 6x  - 6x )%e  + 3x  + 6x  + 3x )sin(x )
   + 
            x 2        2        x     3     2           2       x 2
     (3x (%e )  + (- 6x  - 6x)%e  + 3x  + 6x  + 3x)sin(x ) + (%e )
   + 
                 x    2
     (- 2x - 2)%e  + x  + 2x + 1
                               Type: Expression Integer

You could, of course, use the function eval within a loop and evaluate
expr twenty-one times, but this would be quite slow.  A better way is
to create a numerical function f such that f(x) is defined by the
expression expr above, but without retyping expr!  The package
MakeFunction provides the operation function which does exactly this.

Issue this to create the function f(x) given by expr. ::

  function(expr, f, x)
    f
                               Type: Symbol

To tabulate expr, we can now quickly evaluate f 21 times. ::

  tbl := [f(0.1 * i - 1) for i in 0..20];
                               Type: List Float

Use the list [x1,...,xn] as the third argument to function to create a
multivariate function f(x1,...,xn). ::

  e := (x - y + 1)^2 * (x^2 * y + 1)^2 
      4 4        5     4     2  3     6     5    4     3     2      2
     x y  + (- 2x  - 2x  + 2x )y  + (x  + 2x  + x  - 4x  - 4x  + 1)y
   + 
        4     3     2               2
     (2x  + 4x  + 2x  - 2x - 2)y + x  + 2x + 1
                                Type: Polynomial Integer

  function(e, g, [x, y])
    g
                               Type: Symbol

In the case of just two variables, they can be given as arguments
without making them into a list. ::

  function(e, h, x, y)
    h
                               Type: Symbol

Note that the functions created by function are not limited to
floating point numbers, but can be applied to any type for which they
are defined. ::

  m1 := squareMatrix [ [1, 2], [3, 4] ]
    +1  2+
    |    |
    +3  4+
                               Type: SquareMatrix(2,Integer)

  m2 := squareMatrix [ [1, 0], [-1, 1] ]
    + 1   0+
    |      |
    +- 1  1+
                               Type: SquareMatrix(2,Integer)

  h(m1, m2)
    +- 7836   8960 +
    |              |
    +- 17132  19588+
                               Type: SquareMatrix(2,Integer)

See Also:

* )show MakeFunction