trace: TraceProc := NIL;
        If "trace" is not NIL, the minimizer calls "trace(s, x, f, g, d)"
        at every "valid" point along the descent path; where
        |
        |   "x" is the current argument vector, 
        |   "f" is the corresponding function value "F(x)",
        |   "g" is the magnitude of the gradient at "x",
        |   "d" is the direction of steepest descent, and
        |   "s" is the length of the path from the starting point
        |       to the current "x".
        |
        The "valid" points are 
        
        
PROCEDURE PerpDev(READONLY e, a, b: Point): Length =
  (*
    The length of the component of "e" that is orthogonal
    to "b - a".
  *)
  VAR ee, ed, dd: LONGREAL := 0.0d0;
  BEGIN
    FOR i := 0 TO LAST(e) DO 
      WITH
        ei = e[i],  di = b[i] - a[i]
      DO
        ee := ee + ei*ei;
        ed := ed + ei*di;
        dd := dd + di*di;
      END
    END;
    RETURN Math.sqrt(ABS(ee - ed*ed/dd))
  END PerpDev;