Nested Lists
| {list1,list2,...} | list of lists |
| Table[expr,{i,m},{j,n},...] | m×n×... table of values of expr |
| Array[f,{m,n,...}] | m×n×... array of values  |
| Normal[SparseArray[{{i1,j1,...}->v1,...},{m,n,...}]] |
| m×n×... array with element  being  |
| Outer[f,list1,list2,...] | generalized outer product with elements combined using f |
| Tuples[list,{m,n,...}] | all possible m×n×... arrays of elements from list |
Ways to construct nested lists.
This generates a table corresponding to a 2×3 nested list.
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This generates an array corresponding to the same nested list.
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Elements not explicitly specified in the sparse array are taken to be
.
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Each element in the final list contains one element from each input list.
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Functions like
Array,
SparseArray, and
Outer always generate
full arrays, in which all sublists at a particular level are the same length.
| Dimensions[list] | the dimensions of a full array |
| ArrayQ[list] | test whether all sublists at a given level are the same length |
| ArrayDepth[list] | the depth to which all sublists are the same length |
Functions for full arrays.
Mathematica can handle arbitrary nested lists. There is no need for the lists to form a full array. You can easily generate ragged arrays using
Table.
This generates a triangular array.
| Out[5]= |  |
| Flatten[list] | flatten out all levels of list |
| Flatten[list,n] | flatten out the top n levels |
| ArrayFlatten[list,rank] | create a flattened array from an array of arrays |
Flattening out sublists and subarrays.
This generates a 2×3 array.
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Flatten in effect puts elements in lexicographic order of their indices.
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This creates a matrix from a block matrix.
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| Transpose[list] | transpose the top two levels of list |
| Transpose[list,{n1,n2,...}] | put the k level in list at level  |
Transposing levels in nested lists.
This generates a 2×2×2 array.
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This permutes levels so that level 3 appears at level 1.
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This restores the original array.
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| Map[f,list,{n}] | map f across elements at level n |
| Apply[f,list,{n}] | apply f to the elements at level n |
| MapIndexed[f,list,{n}] | map f onto parts at level n and their indices |
Applying functions in nested lists.
This maps a function
at level 2.
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This applies the function at level 2.
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This applies
to both parts and their indices.
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| Partition[list,{n1,n2,...}] | partition into  × ×... blocks |
| PadLeft[list,{n1,n2,...}] | pad on the left to make an  × ×... array |
| PadRight[list,{n1,n2,...}] | pad on the right to make an  × ×... array |
| RotateLeft[list,{n1,n2,...}] | rotate  places to the left at level k |
| RotateRight[list,{n1,n2,...}] | rotate  places to the right at level k |
Operations on nested lists.
This rotates different amounts at each level.
| Out[17]= |  |
This pads with zeros to make a 2×3×3 array.
| Out[18]= |  |
