map list

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How about a serious 'variables' function?

Map List

The Derive in-built function MAP_LIST( ) is very useful. As a simple example, suppose we want to create, say, a vector of twenty variables for use within a function:

vars(z, n) := MAP_LIST(APPEND(STRING(z), r), r, 1, n)

vars(q, 20)

simplifies to

[q1, q2, q3, q4, q5, q6, q7, q8, q9, q10, q11, q12, q13, q14, q15, q16, q17, q18, q19, q20]

Or to achieve a similar effect, how about:

vars(z, n) := VECTOR(APPEND(STRING(z), r), r, 1, n)

As a simpler example of map_list, and to avoid the function nesting above, here is another one. Take a list (vector): [3, 4, 5], and square all the elements:

MAP_LIST(x² , x, [3, 4, 5])

[9, 16, 25]

In this example, we are mapping the function x², for variable x, to the vector [3, 4, 5]. Similarly,

MAP_LIST(x²y, x, [3, 4, 5])

[9·y, 16·y, 25·y]

MAP_LIST(xy, y, [3, 4, 5])

[ 3·x , 4·x , 5·x ]

If we now consider the Fibonacci sequence, as described in the append section. The solution vector is given by [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]. So, we now need to take natural logs of each element:

MAP_LIST(LN(x), x, [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144])

[0, 0, 0.6931, 1.098, 1.609, 2.079, 2.564, 3.044, 3.526, 4.007, 4.488, 4.969]

At this point, we can introduce Derive's very useful FIT() function. Taking logs of the Fibonacci sequence may look linear, but is it? A least-squares fit of the data points should help. To allow plotting of these points (y co-ordinates), we need to create a matrix containing corresponding x co-ordinates.

The function to create a Fibonacci sequence referred to here is:

fib(z, x:=[1], n:=1) :=
    Prog
       Loop
        If n > z
        RETURN REVERSE( x)
       x := ADJOIN(n, x)
       n :+ FIRST(REST(x))

or in 1-D

fib(z, x := [1], n := 1) := LOOP(IF(n > z, RETURN REVERSE(x)), x := ADJOIN(n, x), n :+ FIRST(REST(x)))

And the function which takes the necessary natural logs is:

fibona(q, a) :=
    Prog
    a := fib(q)
    [[0, ..., DIM(a) - 1], MAP_LIST(LN(x), x, a)]`

Again, in the Entry bar in 1-D this would be:

fibona(q, a) := PROG(a := fib(q), [[0, ..., DIM(a) - 1], MAP_LIST(LN(x), x, a)]`)

So,

fibona(200)

This returns the following pairs of data points, which can be plotted, as well as tested for a linear fit:

As an example of using Derives' FIT() command, the above pairs of data points can be used to find a least squares fit. For a linear fit (that is, a constant plus some co-efficient of x-only):

the general form is

FIT([h,a+bh], [x,y])

where [x,y] is the above 12 x 2 matrix of co-ordinates. Explicitly,

and the solution on approximating this is

0.4727h-0.2597

which, when plotted, is a straight line very close to the above data points. To extend the fit() function to non-linear a fit, the format is

and again approximating to give

0.00049h⁴-0.0121h³+0.1009h²+0.175h-0.0728