Choosing a Fibonacci implementation in APL for zero-indexed n

The decision

You need nth Fibonacci with zero-indexed n in APL without pulling in external libraries. The choice is not about correctness alone, it is about growth, numeric stability, and dialect portability.

Constraints that matter in practice: zero-indexed definition where fib 0 = 0, fib 1 = 1; predictable time for repeated calls; integer exactness for large n; and readability for team members who maintain APL.

Options compared

Trade-offs

Scan builds the whole prefix sequence with cumulative sum. It is easy to read, stays integer, and matches APL’s array style. Cost is linear in n and memory proportional to n if you keep the full vector. For a single nth value you can take the last element.

Matrix exponentiation gives logarithmic time via fast power of the 2×2 Fibonacci matrix. It is exact for integers in APL’s arbitrary precision. The cost is conceptual: you must know the correct matrix, power operator, and how to extract the element. Dialects differ in power syntax and matrix shape notation.

Recursive definition is the textbook definition and useful for teaching. Without memoization it is exponential and will exhaust stack for modest n. Do not use in production.

Binet is constant time but uses floating point. Rounding with ⌊ or ⎕ must be applied carefully. Errors appear for n beyond about 70 in double precision, and dialect differences in sqrt and power affect results.

Concrete implementation: scan for a prefix

Run in an interactive APL session, e.g., Dyalog APL or GNU APL. No special permissions are required.

fibs←{+\\1,⍵⍴0}
nth←{⍵[⍵]}
fib←nth∘fibs

fibs n returns the first n Fibonacci numbers with zero-indexing: 0 1 1 2 3... The scan +\\ applies + cumulatively to the seed vector 1,0 repeated n times.

Validation steps you can perform in the session:

• Evaluate fibs 10 and check the sequence starts 0 1 1 2 3 5 8 13 21 34.
• Evaluate fib 10 and compare to 55. This is the standard check for zero-indexed definition.
• Compare fib n for n=0 to 20 against a matrix power version if both are available in your dialect.

Limitations. Scan is O(n) time and O(n) intermediate storage. For very large n, memory use dominates before CPU. Matrix power avoids building the prefix but requires correct power operator. In some dialects power is ⋆ not *. Floor is ⌊ in most dialects but verify.

When to choose what

Pick scan when you need the whole prefix, want integer exactness, and n is in the thousands to low millions. Pick matrix power when you need a single large nth value and logarithmic time is required. Avoid recursion for n > 30. Avoid Binet unless n is small and you accept floating point rounding risk.

Practical check after implementation: run the three validation steps above for a small range and for one larger value you can cross-check with the matrix method. If results diverge, inspect dialect differences in power and indexing before changing code.