Does ⎕CT tolerance apply to complex numbers in nested arrays?

Answer

In Dyalog APL (version 14.0 and later) the comparison tolerance ⎕CT does apply to complex numbers, and when primitives such as match (≡) or find (⍳) compare nested arrays the tolerance is used for each leaf complex element. In many other APL implementations the behavior is either unspecified or treats complex equality as exact, so tolerance may not affect complex numbers.

Confirmed facts (Dyalog APL)

• ⎕CT is defined for all numeric scalars; for complex numbers the tolerance test compares the magnitude of the difference to ⎕CT × max(|a|,|b|).
• Match (≡) and find (⍳) recursively compare leaf scalars using this tolerant equality, so tolerance propagates to complex leaves inside nested arrays.

Likely explanation for other APLs

Because the APL2 standard does not explicitly mention complex numbers in the definition of ⎕CT, some interpreters either leave the behavior undefined or implement complex equality as an exact test (or apply ⎕CT separately to real and imaginary parts). This leads to variation across vendors.

Verification steps

1. Set a known tolerance, e.g., ⎕CT←1E-14.
2. Create two complex scalars that differ by less than the tolerance, e.g., a←1+1E-15j and b←1.
3. Evaluate a≡b; a result of 1 indicates tolerance is active for complex numbers.
4. Build nested arrays containing those scalars, e.g., X←(a 2) (3 b) and Y←(1 2) (3 1).
5. Evaluate X≡Y; a result of 1 shows tolerance propagated to the leaf complex elements.
6. Repeat the test with ⎕CT←0 (exact comparison) to confirm the same expressions return 0.

If you are using an APL other than Dyalog, consult its language reference for the "Comparison tolerance" section to see whether complex numbers are mentioned.

Missing diagnostic detail

To give a definitive recommendation, please specify which APL interpreter you are using (e.g., Dyalog, GNU APL, NARS2000, APLX, etc.).