Replace the sole recursive call by updates to the current segment's parameters:
PROCEDURE rotate(i, n, k: INTEGER);
VAR a, b: INTEGER;
BEGIN
  WHILE (k > 0) AND (k < n) DO
    a := n-k; b := k;
    IF a = b THEN
      blockswap(i, i+a, a);
      RETURN
    ELSIF a < b THEN
      blockswap(i, i+b, a);
      n := b; k := b-a
    ELSE
      blockswap(i, i+a, b);
      i := i+b; n := a; k := b
    END
  END
END rotate;
The loop invariant is that every position outside the current unresolved segment already has its final value, while rotating the current segment by the current k completes the original array rotation. The block decompositions in part (ii) prove preservation. The positive unresolved length decreases in each continuing branch; the equal-block branch finishes immediately. The same rotation now uses auxiliary space, including control storage, and leaves the rest of the array untouched.

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