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using Symata # load Symata and enter Symata mode
There are (more or less) three ways to define functions in Symata.
Function
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f(x_) := x^2
f(2)
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Note the trailing underscore in x_. This function is implemented via a rule for rewriting expressions.
The argument to f can be any expression:
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[f(x+y), f("dog"), f(2.0)]
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ClearAll(f) # remove the definition for `f`
This computes the nth Fibonacci number.
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fib(1) := 1
fib(2) := 1
fib(n_) := fib(n-1) + fib(n-2)
fib(10)
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A compound statement can be written like this.
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addone(x_) := (a = 1, x + a)
addone(y)
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But, the variable a is not local to the function. The global value of a is set.
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a
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To make variables local to a function, use Module
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g(x_) := Module([c,d],(c=1,d=3,a+d+x)) # local variables are lexically scoped.
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g(z)
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The global values of c and d are not affected.
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[c,d]
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The argument to the function f in In(2) can be any expression. You can also make multiple definitions that apply only when the argument satisfies some requirements. For example, the following makes a definition for g that only applies for floating point arguments greater than 5.
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gt5(x_) := x > 5
g(x_Float:?(gt5)) := 1
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g(10.0) # The conditions in the new definition are satisfied by 10.0
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[g(4.0), g(10), g(q + r)] # The definition in In(13) applies here
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You can do structural matching on the argument. This matches anything that has the form of a square.
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h(x_^2) := x
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[h(r^2), h(r^3), h((r+q)^2), h(Expand((r+q)^2)), 4 , 2]
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Although 4 is the square of 2, 2 is the square of Sqrt(2), and q^2 +2*q*r + r^2 is the square of r+q, none of these match. Only expressions of the form Power(expr,2) match. We see why h(r^2) matches by looking at the full form of r^2.
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FullForm(r^2)
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ClearAll(f,h,a)
Use two trailing underscores to match one or more elements
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g(x_Symbol, p__Integer) := Apply(Plus, x^[p])
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g(y,1,4,7,10)
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Apply(ClearAll, UserSyms())
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f(x_) := x^2
g(x_) = x^2
x = 3
h(x_) = x^2
j(x_) := x^2
[f(a), g(a), h(a), j(a)]
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In the definition of h, x is not a dummy variable. If you are beginning with Symata it is enough to know that := should be preferred and you can skip the following.
What does f(x_) := x^2 do ? When Symata evaluates this expression, it does not evaluate the right hand side, but rather holds it in its unevaluated form. When the rule is applied, say by evaluating f(a+b), the value that matches x_, in this case a+b, is substituted for x in the right hand side and the result is evaluated with the result (a+b)^2. Note that the value of x at the time the definition is first evaluated has no affect on this result: The right hand side is never evaluated before subsituting for x.
The definition g(x_) = x^2 differs in that the right hand side is evaluated when the definition is made. After this, everything is the same as for :=. The value that matches x_ is substituted in the right hand side and the result is evaluated. If x was unbound when the definition was made, the results are identical, because the substituion will be made before evaluating the right hand side again, so assignments to x in the mean time will have no effect. However if x is bound at the time of the definition, then the value of x^2 at the time of defnition is stored and the match of x_ is substituted into this expression. If x had a value of 3, then a+b is substituted into 3, which just results in 3.
These "function" definitions actually define DownValues. We can see that the definitions of f,g, and j are the same, no matter what the future values of x.
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Map(DownValues, [f,g,h,j])
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ClearAll(f,g,h,j,x) # clear the symbols used in this example
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1;
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ClearAll(g,h,a,b)
# J(expr) interprets expr as Julia code
f = J((x) -> x^2 ) ;
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f(3)
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f("dog") # This calls the Julia function, so we get what "dog"^2 means in Julia
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Note that we did not use := in the definition of f. That is, f := :((x) -> x^2 ). This would create a new anonymous function everytime f is called, which would be very slow.
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f(a+b)
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The last example works because ^ has been overloaded in Julia to operate on Symata expressions. (Symata expressions are objects of type Symata.Mxpr in Julia.)
Anonymous or pure functions are used like this:
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Function(x, x^2)(a)
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The infix operator -> is equivalent to Function
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(x -> x^2)(a)
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A function of two variables is defined like this:
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([x,y] -> x^2 + y^2)(3,4)
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