Array¶

Array Displays¶

Array values can be constructed by enclosing a comma-separated lists of component expressions (all of the same type) in brackets, as in [1,-4,3*6], which is called a row display (the given one will be printed as [1,-4,18]). They are also implicitly constructed by loops.

Array Selection¶

If v is an identifier, or more generally an expression, whose value is an array, then the individual components of the array can be accessed by subscripting, as in v[i]. Interpretation of the index always starts from 0: the first element of v is v[0], and the last is v[#v-1] (for meaning of #, see Basic Operators). However, one can also index counting from the end by prefixing the [ by ~, so the last element is also v~[0], and the before-last v~[1] and so on.

atlas> set v = [1,2,3,4]
Variable v: [int]
atlas> v[0]
Value: 1
atlas> v[#v-1]
Value: 4
atlas> v~[0]
Value: 4
atlas> v~[1]
Value: 3

Array Slicing¶

One can also select multiple elements at once (combined into a new array value) using a “slice”, specifying the a lower and an upper bound separated by :. Each should be thought of as indicating a separation between two elements, namely between the one at the given index and at an index one less; the slice then selects all elements between the lower separator and the upper separator. This means that in a v[a:b] includes the element v[a] but excludes v[b], and therefore selects b-a elements in all.

atlas> set v = [0,1,2,3,4,5]
Variable v: [int]
atlas> v[1:3]
Value: [1,2]

In case a>=b the slice returns an empty array of the same type as v (having a=b is normal, having a>b is allowed but rarely useful). It is an error if the lower bound is negative, or if the upper bound exceeds the size of the array.

atlas> v[3:1]
Value: []

Individual bounds in a range can also be counted from the end by appending a ~, in which case they indicate the separation after (that is, closer to the end) then the element the index would reversely select. As a convenience a completely omitted lower bound means 0 (from the beginning) and an omitted upper bound means 0~ (up to the end). Finally one can (like for subscription) select from the reverses array by prefixing the ‘[‘ by ‘~’.

atlas> v~[1:3]
Value: [4,3]

Here are some examples of various uses of slices

Expression Effect
v[0:3] or v[:3] the first 3 elements of v
v[3:0~] or v[3:] everything except the first 3 elements of v
v[2~:0~] or v[2~:] the last 2 elements of v
v[0:2~] or v[:2~] everything except the last 2 elements of v
v[i:k] the elements [v[i],v[i+1],...,v[k-1]
v[i:k~] all of v except its first i and its last k elements
v~[0:0~] or v~[:] all of v in reverse order
v~[4:0~] or v~[4:] all of v except the last 4 elements, in reverse order
v~[0:5~] or v~[:5~] all of v except the first 5 elements, in reverse order
v~[i:k] the k-i elements [v[n-1-i],...,v[n-k-1],v[n-k]] (n=#v)
v~[k~:i~] the k-i elements [v[k-1],...,v[i+1],v[i]]

Other Array-Like Types¶

While vectors, rational vectors, matrices and strings are not axis arrays (they are instead members of primitive types), they do behave like arrays for the purpose of selection; the result is respectively an integer, rational number, vector (column of the matrix) or a one-character string (the selection can also cause an error, if the index is out of bounds). They can also be used in slicing, in which case they return a value of the same type as what was selected from. For instance "reverse me!"~[:] gives "!em esrever" and

atlas> id_mat(6)
Value:
| 1, 0, 0, 0, 0, 0 |
| 0, 1, 0, 0, 0, 0 |
| 0, 0, 1, 0, 0, 0 |
| 0, 0, 0, 1, 0, 0 |
| 0, 0, 0, 0, 1, 0 |
| 0, 0, 0, 0, 0, 1 |

atlas> id_mat(6)[1:2~]
Value:
| 0, 0, 0 |
| 1, 0, 0 |
| 0, 1, 0 |
| 0, 0, 1 |
| 0, 0, 0 |
| 0, 0, 0 |

For individual entries of a matrix, one can provide row and column index at once (in that order), separated by a comma as in A[i,j] (this actually indexes A by a tuple of type (int,int), as in A[(i,j)]. One can apply reverse indexing provided it applies to both indexes, as in A~[0,0] for the bottom right entry. Slicing applies somewhat differently in case of matrices (unless it involves a single integer range, in which case a range of columns is selected). The general form is A[i:k,j:l] where each of the bounds i,k,j,l may be followed by ~ (but not A; this syntax allows no global reversal). This selects a block of A with the interpretation above for each of the ranges. For more general slicing of matrices (with optional reversal of row and/or columns, transposition, and negation of individual entries, atlas provides a built-in function swiss_matix_knife (the A[i:k,j:l] syntax in fact uses it).

Finally there is selection of ParamPol values, which can be seen as associative arrays mapping certain Param values to Split coefficients, by Param values used as index: for a ParamPol: P and a Pram value q, the subscription P[q] returns the coefficient of q in P (or Split:0 if no term q is present). There is a technical restriction to this use due to the fact that terms in ParamPol can only involve so-called standard parameters, and these are automatically expanded to nonzero final parameters; therefore P[q] gives an error unless q is standard, nonzero and final. There is no such thing as reversal in associative arrays, so the syntax P~[q] is not allowed here.