Trefethen & Bau & MATLAB & Julia, Lectures 12-13: Conditioning and floating point

I’ve run into trouble managing gists with lots of files in them, so I’m back to doing one per lecture. Here are Lecture 12 and Lecture 13.

We’ve entered Part 3 of the book, which is on conditioning and stability matters. The lectures in this part are heavily theoretical and often abstract, so I find a little occasional computer time helps to clear the cobwebs.

Right off the top, in reproducing Figure 12.1, I ran right into the trap I worried about in my last post regarding polynomials in Julia. In MATLAB, the polynomial coefficients are just a plain vector. That makes perturbing them trivial:

p = poly([1,1,1,0.4,2.2]);    % polynomial with these roots
q = p + 1e-9*randn(size(p));    % perturb its coefficients

In Julia, you can use the Polynomials package and get polynomial objects. Behold:

using Polynomials
p = poly([1,1,1,0.4,2.2]);  # polynomial with these roots
q = p + Poly(1e-9*randn(6));    # perturb coefficients

Note that poly constructs a polynomial from a vector of roots, while Poly constructs one from a vector of coefficients. Sure enough, I used poly in both lines the first time around. It’s a pernicious mistake, because it produces no error—the polynomials can be added no matter what. The mistake was mine, but I think this is an unfortunate design.

The only other notable usage in Lecture 12 is my first use of a comprehension:

hilb(n) = [ 1.0/(i+j) for i=1:n, j=1:n ];

This is a pretty handy way to create a matrix.

In Lecture 13 I had some fun dissecting floating point numbers in both systems. There was only one area in which Julia didn’t go as smoothly as I would hope. MATLAB offers realmin and realmax , which give the smallest and largest normalized floating point numbers. While Julia has similar-sounding commands, they are interpreted differently:

julia> typemin(Float64), typemax(Float64)
(-Inf,Inf)

Eh, not so much. There is even one more layer of subtlety. Consider

julia> (prevfloat(Inf),nextfloat(0.0))
(1.7976931348623157e308,5.0e-324)

The first of these values is exactly the same as realmax, but the second is not realmin. IEEE 754 double precision has “denormalized” numbers that let you trade away bits of precision to get closer to zero in magnitude. Julia is reporting the smallest denormalized number, not the smallest full-precision number. Julia’s not wrong, but access to the extreme finite double precision values isn’t as straightforward as it could be.

One last observation. Trefethen & Bau refer to the value $2^{-53}$ as “machine epsilon.” This isn’t what MATLAB and Julia use, which is $2^{-52}$. Nick Higham’s Accuracy and Stability of Numerical Algorithms also has “machine epsilon” at $2^{-52}$ and calls $2^{-53}$ “unit roundoff.” Stoer and Bulirsch (2nd ed.) call $2^{-53}$ “machine precision.” Corless and Fillion seem to agree with Higham. Golub and Van Loan (3rd ed.) don’t use “machine epsilon” at all, and in the index one finds

Machine precision. See unit roundoff.

Sigh. The mathematical uses are, unsurprisingly, consistent. Frankly, I feel better about my personal inconsistencies at using those terms: at least I stood on the shoulders of giants. Toby Driscoll
Professor of Mathematical Sciences

My research interests are in scientific computation, mathematical software, and applications of mathematics in the life sciences.