7.14.4 Tensor Component Array Filling

Tensor Component Array Filling is the systematic process of completing every cell of a tensor's component array through a coordinated sequence of assignment operations, ensuring that the entire multi-dimensional array, rather than only a handful of scattered entries, is populated with definite numerical values.

The Goal of Filling

Completing the Full Array

While a single assignment operation places a value at one index target, filling refers to the broader task of carrying out enough assignment operations, in a coordinated way, to leave no cell of the component array without a definite value, producing a fully specified tensor relative to the chosen basis.

Distinguishing Filling From a Single Assignment

A single assignment operation is a local, isolated act affecting one cell, whereas filling describes the cumulative, global outcome of applying such operations systematically until the entire array reaches a fully determined state.

Strategies for Filling an Array

Exhaustive Enumeration

The most direct filling strategy visits every index tuple in a systematic order, such as lexicographic order, and performs one assignment at each, guaranteeing that no cell is skipped and providing a straightforward way to verify completeness once the enumeration finishes.

Formula-Driven Filling

When the tensor's entries follow a known formula in terms of the index values, filling can be automated by evaluating that formula at every tuple in turn, filling the array far more efficiently than manually specifying each entry individually.

Symmetry-Assisted Filling

When the tensor is known to be symmetric or antisymmetric across some indices, filling can proceed by directly assigning only the independent entries and then automatically propagating those values, or their negatives, to every related cell dictated by the symmetry, reducing the total number of independent assignment decisions required.

Verifying That Filling Is Complete

Checking for Unfilled Cells

A completed filling process should leave no cell without an assigned value, and a careful check for any remaining unfilled cells, whether by direct inspection for small arrays or by systematic scanning for larger ones, confirms that the array truly represents a fully specified tensor.

Checking for Consistency With Declared Structure

Beyond mere completeness, a filled array should also be checked against any declared structural properties, such as symmetry, to confirm that the filling process has not inadvertently produced entries that contradict those properties.

Partial and Default Filling

Filling With an Initial Default

A common practice initializes every cell of the array with a default value, typically zero, before beginning the substantive filling process, ensuring that any cell not explicitly addressed by a later assignment still holds a well-defined, if trivial, value rather than being left undefined.

Sparse Filling for Mostly Trivial Tensors

For tensors expected to have only a small number of nonzero entries, filling may deliberately remain partial by design, explicitly assigning only the known nonzero cells and relying on the default value to correctly represent every other cell, a practical approach for tensors with substantial built-in sparsity.

Diagrammatic Illustration

An array progressively filled cell by cell, moving from an entirely empty grid toward full completion.

Practical Importance of Array Filling

Prerequisite for Reliable Computation

A tensor cannot be reliably used in further arithmetic, such as contraction or the tensor product, until its component array has been fully and correctly filled, since any missing entry introduces an undefined gap that later computations cannot resolve on their own.

Efficiency Considerations at Scale

For high-rank tensors in high-dimensional spaces, where the total number of cells grows exponentially, filling strategies that exploit formulas or symmetry rather than relying on manual, individual assignment become increasingly important for completing the array within a practical amount of effort.