FreeMat
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Section: Mathematical Operators
There are two distinct syntaxes for the colon :
operator - the two argument form
y = a : c
and the three argument form
y = a : b : c
The two argument form is exactly equivalent to a:1:c
. The output y
is the vector
where a+nb <= c
. There is a third form of the colon operator, the no-argument form used in indexing (see indexing
for more details).
The colon operator turns out to be trickier to implement than one might believe at first, primarily because the floating point versions should do the right thing, which is not the obvious behavior. For example, suppose the user issues a three point colon command
y = a : b : c
The first question that one might need to answer is: how many points in this vector? If you answered
then you would be doing the straighforward, but not correct thing. because a, b, and c are all floating point values, there are errors associated with each of the quantities that can lead to n not being an integer. A better way (and the way FreeMat currently does the calculation) is to compute the bounding values (for b positive)
where
means we replace x by the floating point number that is closest to it in the direction of y. Once we have determined the number of points we have to compute the intermediate values
but one can readily verify for themselves that this may not be the same as the vector
even for the case where
for some n. The reason is that the roundoff in the calculations may be different depending on the nature of the sum. FreeMat uses the following strategy to compute the double-colon vector:
n
is computed by taking the floor of the larger value in the interval defined above. n
falls inside the interval defined above, then it is assumed that the user intended c = a + n*b
, and the symmetric algorithm is used. Otherwise, the nonsymmetric algorithm is used.
a:b:c
and c:-b:a
(which basically means that instead of moving towards infinity, we move towards the signed infinity where the sign is inherited from b
). If you think this is all very obscure, it is. But without it, you will be confronted by mysterious vectors where the last entry is dropped, or where the values show progressively larger amounts of accumulated roundoff error.
Some simple examples of index generation.
--> y = 1:4 y = 1 2 3 4
Now by half-steps:
--> y = 1:.5:4 y = 1.0000 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000
Now going backwards (negative steps)
--> y = 4:-.5:1 y = 4.0000 3.5000 3.0000 2.5000 2.0000 1.5000 1.0000
If the endpoints are the same, one point is generated, regardless of the step size (middle argument)
--> y = 4:1:4 y = 4
If the endpoints define an empty interval, the output is an empty matrix:
--> y = 5:4 y = Empty array 1x0