Named Function

def <functionName> ([parameters]) {statements}

or

def <functionName> ([parameters]): statement (can only have one statement here)

We will discuss user-defined aggregate functions at the end of this section. Their syntax is the same as the named functions except that their definitions start with "defg" instead of "def".

• Function parameters are always passed by references.

• An input parameter can be modified within function body if and only if it is qualified by "mutable".

Define a named function:

def f(a){return a+1};
f(2);
// output: 3

def f(a):a+1;
f(3);
// output: 4

def f(a=1, b=2){return a + b}
f(,3)
// output: 4

Assign a function or an array of functions to variables:

g=sin;
g(3.1415926);
// output: 5.358979e-008

g=[sin, cos, log];
g(1 2);

If a function and a variable have the same name, we can use the address operator (&) to indicate a function.

sum=15;
g=sum;
g;
// output: 15

g=∑
g;
// output: sum

g(1..4);
// output: 10

Immutable parameters cannot be modified within the function:

def f(a){a+=1; return a};
// output: Syntax Error: [line #1] Constant variable [a] can't be modified.

Mutable parameters can be modified within the function:

def f(mutable a){a+=1; return a};
x=1;
f(x);
// output: 2

f(x);
// output: 3

x;
// output: 3

x=1..6$3:2;
x;

def f(mutable a){if(a.rows()>=2 && a.cols()>=2){a[1,1]=0}; return a};
f(x);

Declare a function first then define it later:

def f2(a,b)
def f1(a,b){return f2(a,b)}
def f2(a,b){a pow b};
// ";" should not be put in-between the lines otherwise the system cannot tell f2 is a declaration without the follow up definition.
f1(2,3);
// output: 8

Return multiple values:

def summary(a){return sum(a), avg(a), std(a)};
x=1..11;
summary(x);
// output: [66, 6, 3.316625]

Let's write a function to calculate the covariance between input vectors of equal length.

(1) If neither a nor b has null elements, calculate the covariance between them.

(2) If either a or b contains null elements, we first retrieve the sub vectors that do not contain any null elements and then calculate the covariance.

def calcovar(a, b){
    aNull=hasNull a;                                                  // return true if input vector a contains null values; otherwise, false
    bNull=hasNull b;
    if(!aNull && !bNull){                                             // if neither a or b contains null values
            am=avg a;                                                 // calculate the mean of input vector a using function avg
            bm=avg b;                                                 // calculate the mean of input vector b using function avg
            ab=a ** b;                                               // calculate the inner product of vector a and b
            n=count a;                                               // get the number of non-null values
            return (ab-n*am*bm) \ (n-1);                             // return a covariance value
   }
        else{                                                         // get all positions in which values are not null in both a and b
               if(!aNull)                                              // if a does not contain any null values
                       index=!isNull b;                               // get the indices of non-null values in b
                   else {
                           if(!bNull)                                   // if b does not contain any null values
                               index=!isNull a;                          // get the indices of non-null values in a
                           else
                               index=!((isNull a) || (isNull b));       // get the positions that are not nulls in both a and b
                           }
                    c=a[index];
                    d=b[index];
                    am=avg c;
        bm=avg d;
        ab=c ** d;
        n=count c;
        return (ab-n*am*bm) \ (n-1);
      }
}

A user-defined aggregate function is a function that restricts the data form of the output to be scalar. There are times when we would like to make sure a function returns a scalar. Aggregate functions serve this purpose.

User-defined aggregate functions have the same syntax as the named functions except that their definitions start with "defg" instead of "def".

defg f(x, y){
    a = sum(abs(x+y))
    b=sum(abs(x))+sum(abs(y))
    return a\b
};
x=-5..5; y=0..10;
f(x,y);
// output: 0.858824