Understanding differences between Maple and Mathematica in examples picked by Maplesoft

Question

I am reading the document How Maple Compares to Mathematica. On page 15 there is an example where Mathematica produces wrong results. Does anybody know why?

MAPLE:

MATHEMATICA:

Also on page 17 the given results are extreme:

Is this really true and if so, why?

Answer 1

The first example seems to intentionally set Mathematica up to "fail" by specifying insufficient input accuracy. With additional precision:

ClearAll[s]
s[i_] := s[i] = 2*s[i - 1] - 3*s[i - 1]^2
s[0] = 0.3`30;
s[40]

0.333333

And Mathematica is capable of far greater precision if necessary:

ClearAll[s]
$RecursionLimit = ∞
s[i_] := s[i] = 2*s[i - 1] - 3*s[i - 1]^2
s[0] = 0.3`5000;
s[8280]

0.333333333333333

By the way this kind of iteration can be nicely written with Nest:

Nest[2 # - 3 #^2 &, 0.3`5000, 8280]

0.333333333333333

---

I read the section of the linked PDF from which this example comes. I think Maple is simply using machine precision here, e.g.:

Nest[2 # - 3 #^2 &, .3, 40]

0.333333

To imply that this is superior to Mathematica's result while specifically triggering the Mathematica arbitrary precision engine seems disingenuous. Further the paper makes the claim:

The last term in the output says that s40=0.×1062 , which is not a good approximation of 3.

There is nothing in the computation to warn the user that the results may not be reliable at every step.

This is false. Hovering over the pink error box tells you exactly what is going on:

No significant digits are available to display.

I think this is an example of attempting to paint a weakness of Maple as a strength, though admittedly I haven't used Maple in many years so I don't know if it also has generalized precision tracking.

Answer 2

Without commenting on how much attention one should pay to marketing literature: the second example is somewhat relevant. Mathematica's polynomial factoring algorithm is known to be at least fifteen years behind the state of the art, and things that Maple will factor in seconds will go away (literally) forever in Mathematica. This is, of course, not too surprising: Maple is, at heart, a computer algebra system, while, as people on this forum know well, Wolfram Research has been prioritizing adding snazzy features to strengthening the base. I have nothing against snazzy features, though it annoys me to have to go to other providers if I need to factor a polynomial.

Answer 3

The critical issue in the first example is that Mathematica is using significance arithmetic to track precision. This is certainly billed as a feature by Wolfram Research. As we see in this example though, it can be portrayed as a weakness. In truth, you might need to know what you're doing to use it correctly. In this answer, I mentioned that significance arithmetic can be problematic in iterative dynamics in the neighborhood of a point a super-attractive fixed point where $f'(x_0)=0$, which is exactly the situation here. To see what's going on, consider the following:

Clear[s];
s[i_] := s[i] = 2*s[i - 1] - 3*s[i - 1]^2;
s[0] = SetPrecision[3/10, 20];
values = Table[s[i], {i, 0, 40, 5}];
precision = Precision /@ values;
Grid[Transpose[{values, precision}],
  Dividers -> All, Alignment -> Left]

As we examine every fifth iterate, we see that it decreases just about linearly.

Differences[precision]
(* Out: {-2.94007, -3.0103, -3.0103, -3.0103, -3.0103, -3.0103, -2.00843, 0.} *)

In fact, if you understand significance arithmetic as described in that answer and the references therein, we expect the significance of each iterate to decrease by about $2\log_{10}(2) \approx 0.60206$ and 5 times this yields the $-3.0103$ that we see above.

However, if we track the significance by examining $f(x)=2x-3x^2$ near $x=1/3$ using the derivative to obtain a first order approximation, we should expect the number of significant digits to increase! Thus, sometimes it is incumbent on the user to increase the significance manually.

By resetting the precision this way, we can emulate the Maple behavior.

Clear[s2];
s2[i_] := s2[i] = SetPrecision[2*s2[i - 1] - 3*s2[i - 1]^2, 20];
s2[0] = SetPrecision[3/10, 20];
values = Table[s2[i], {i, 0, 40, 5}];
precision = Precision /@ values;
Grid[Transpose[{values, precision}],
  Dividers -> All, Alignment -> Left]

To be absolutely clear, though, we are essentially using fixed precision by resetting the precision in this way, though the precision is greater than machine precision. The ability to track the precision can be quite nice and we'd certainly prefer to the conservative estimate that Mathematica gives, rather than an optimistic estimate that returns false precision.