= Solution
Within the sparse-vector implementation, use:
``
class Node(var value: Double, val index: Int,
var prev: Node, var next: Node)
``
The header's `index` is the fixed vector dimension, so it need not change after construction. For a data node it is the coordinate index. The <representation invariant> requires a finite simple cycle through the header, reciprocal predecessor/successor links, strictly increasing indices in the forward data-node traversal, indices in $[0,dimension)$, and a nonzero value at every data node. The header's value is ignored. The <abstraction function> assigns stored values to their indices and zero to all missing coordinates.
The numerical code below takes entries to be finite `Double` values. Excluding NaN and infinities makes the relationship to a real <sparse vector> explicit; arithmetic results still follow ordinary <floating-point arithmetic>.
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