functional programming

Question

Suppose I have this Mathematica code, whose output, a real number, depends on the input, say, x,y,z. How do I make a real-valued function in x,y,z based on the code?

If the code describes a simple relationship among x,y,z, I could define this function directly. The point here is that the given code is a very complicated block (or module).

For example, if the code simply sums x,y,z, I would simply define

f[x_,y_,z_]=x+y+z

What if I have a very complex example, like the one below:

s0[a_, b_, x_] :=
{1, 0, (a + b) x + (1 - a - b)}

s1[a_, b_, c_, d_, p_, q_, n_, x_] :=

Which[0 <= x <= c, {2, n - 1, x/c*q + p},
c <= x <= c + d, {2, n, (x - c)/d*p},
c + d <= x <= 1, {1, n + 1, (x - (c + d))/(1 - c - d)*(1 - a - b)}]

s2[s_, t_, c_, d_, p_, q_, n_, x_] :=

Which[0 <= x <= 1 - s - t, {2, n - 1, 
x/(1 - s - t)*(1 - p - q) + p + q},
1 - s - t <= x <= 1 - s, {3, 
n - 1, (x - (1 - s - t))/t*(1 - c - d) + c + d},
1 - s <= x <= 1, {3, n, (x - (1 - s))/s*d + c}]

s3[c_, a_, b_, s_, t_, n_, x_] :=

Which[0 <= x <= 1 - a - b, {4, n - 1, x/(1 - a - b)*t + 1 - s - t},
1 - a - b <= x <= 1 - a, {4, n, (x - (1 - a - b))/b*(1 - s - t)},
1 - a <= x <= 1, {3, n + 1, (x - (1 - a))/a*c}]

s4[p_, q_, s_, a_, b_, n_, x_] :=

Which[0 <= x <= p, {4, n - 1, x/p*s + 1 - s},
p <= x <= p + q, {5, n - 1, (x - p)/q*a/(a + b) + b/(a + b)},
p + q <= x <= 1, {5, n, (x - (p + q))/(1 - p - q)*b/(a + b)}]

F[{k_, n_, x_}] :=
Which[k == 0, s0[a, b, x],
k == 1, s1[a, b, c, d, p, q, n, x],
k == 2, s2[s, t, c, d, p, q, n, x],
k == 3, s3[c, a, b, s, t, n, x],
k == 4, s4[p, q, s, a, b, n, x]]

G[x_] := NestWhile[F, {0, 0, x}, Function[e, Extract[e, {1}] != 5]]
H[x_] := Extract[G[x], {2}] + Extract[G[x], {3}]
H[0]

For the above code to run, one needs to specify the list

{a,b,c,d,p,q,s,t}

And the output are real numbers. How does one define a function in a,b,c,d,p,q,s,t that spits out these real numbers?

good question becuase the trick in mathematica is to use the pattern sign "_" to do functional programming with pattern. to split the numbers use Partition[] function in mathematica or the Split[] function. remember to use "." for numbers to make them real becuase mathematica likes symbols and do not know it is like real numbers without the "." there — Robert H
– Robert H, Commented Aug 7, 2012 at 4:04
Check out Piecewise and Switch in the online help. You can use Switch for your F definition at least. You haven't defined e, so you need to do that. But otherwise, it's not clear what your problem is. Yes, you need to define auxiliary functions like s1 to keep the code readable, but there's nothing wrong with that. To pass the other parameters, you need to redefine each function to take all parameters in your list, or to have options where you can set them globally. Look up SetOptions in the help. — Verbeia
– Verbeia, Commented Aug 7, 2012 at 6:05
@Verbeia My code seems to be running fine without defining e. I thought e would just refer to the function to be iterated, which in my case would be F. Is this the correct understanding? — Michael C
– Michael C, Commented Aug 15, 2012 at 22:42
@MichaelC you are quite right. Your code is a bit confusing and I missed that it was just the equivalent of Slot. — Verbeia
– Verbeia, Commented Aug 15, 2012 at 22:43

Verbeia · Accepted Answer · 2012-08-07 06:40:45Z

6

Your essential problem is that you have a large number of parameters in your auxiliary functions, but your big-letter functions (F, G and H and by the way single-capital-letter function names in Mathematica are a bad idea) only take three parameters and your auxiliary functions (s0 etc) only return three values in the returned list.

You have two possible ways to fix this.

You can either redefine everything to require all the parameters required in the whole system - I'm assuming that common parameter names across the auxiliary functions really are common values - like this:

 G[x_, a_, b_, c_, d_, p_, q_, s_, t_] := 
 NestWhile[F, {0, 0, x, a, b, c, d, p, q, s, t}, 
  Function[e, Extract[e, {1}] != 5]]

or

You can set some options that set these parameters globally for the whole system. Look up Options and OptionsPattern. You would do something like this:

First, define default options:

Options[mySystem] = {aa -> 0.2, bb -> 1., cc -> 2., dd -> 4., 
     pp -> 0.2, qq -> 0.1, ss -> 10., tt -> 20.}
SetOptions[mySystem, {aa->0.2, bb->1., cc->2., dd->4., pp->0.2, 
     qq->0.1, ss->10., tt->20.}]

Then write your functions like this:

F[{k_, n_, x_}, OptionsPattern[mySystem]] :=
 With[{a = OptionValue[aa], b = OptionValue[bb], c = OptionValue[cc], 
   d = OptionValue[dd], p = OptionValue[pp], q = OptionValue[qq], 
   s = OptionValue[ss], t = OptionValue[tt]}, 
  Which[k == 0, s0[a, b, x], k == 1, s1[a, b, c, d, p, q, n, x], 
   k == 2, s2[s, t, c, d, p, q, n, x], k == 3, 
   s3[c, a, b, s, t, n, x], k == 4, s4[p, q, s, a, b, n, x]] ]

There is also something quite wrong with your use of Extract (you are assuming there are more parts in your list than are actually there in the first few iterations), but this answers your main issue.

answered Aug 7, 2012 at 6:40

Verbeia

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2 Comments

Michael C Over a year ago

the first option worked. For the second one, I'll probably have to spend more time in order to understand it. I don't see what is wrong with the use of Extract. My code seems to be working. Could you explain a bit?

Verbeia Over a year ago

@MichaelC it is just that when I ran your code, it was wanting to Extract the nth part of something when there was, say n-2 parts in the list at that point. So it couldn't find the nth part and threw an error.

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