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2.6 Broadcasting and conformability

Two arrays need not have identical shape in order for a binary operation between them to make perfect sense:

y = a*x^3 + b*x^2 + c*x + d;

The obvious intent of this assignment statement is that y should have the same shape as x, each value of y being the value of the polynomial at the corresponding y.

Alternatively, array valued coefficients a, b, ..., represent an array of several polynomials -- perhaps Legendre polynomials. Then, if x is a scalar value, the meaning is again obvious; y should have the same shape as the coefficient arrays, each y being the value of the corresponding polynomial.

A binary operation is performed once for each element, so that a+b, say, means

a(i,j,k,l) + b(i,j,k,l)

for every i, j, k, l (taking a and b to be four dimensional). The lengths of corresponding indices must match in order for this procedure to make sense; Yorick signals a "conformability error" when the shapes of binary operands do not match.

However, if a had only three dimensions, a+b still makes sense as:

a(i,j,k) + b(i,j,k,l)

This sense extends to two dimensional, one dimensional, and finally to scalar a:

a + b(i,j,k,l)

which is how Yorick interpreted the monomial x^3 in the first example of this section. The shapes of a and b are conformable as long as the dimensions which they have in common all have the same lengths; the shorter operand simply repeats its values for every index of a dimension it doesn't have. This repitition is called "broadcasting".

Broadcasting is the key to Yorick's array syntax. In practical situations, it is just as likely for the a array to be missing the second (j) dimension of the b array as its last (l) dimension. To handle this case, Yorick will broadcast any unit-length dimension in addition to a missing final dimension. Hence, if the a array has a second dimension of length one, a+b means:

a(i,1,k,l) + b(i,j,k,l)

for every i, j, k, l. The pseudo-index can be used to generate such unit length indices when necessary (see section 2.3.5 Creating a pseudo-index).