4 ms·

I see it in statistics. It's not uncommon in a lot of statistical algorithms to need to take the product (or sum) of a lot of very small numbers and then divide

by gh02t 2y ago

I see it in statistics. It's not uncommon in a lot of statistical algorithms to need to take the product (or sum) of a lot of very small numbers and then divide by that product when computing likelihoods of a bunch of independent measurements. There are some algebra tricks you can use to get around loss of precision like working in the log domain or some tedious scaling tricks, but working in quad precision can help when those aren't enough. The article also gives an example from quantum physics. Some chaotic or highly numerically unstable problems just need quad precision to not blow up; primarily problems/algorithms (can be the problem itself or the algorithm to solve it that causes the issues) that tend to catastrophically amplify small errors. Matrix inversion is another example, to the point that most algorithms try to do almost anything to avoid explicitly inverting a matrix. But sometimes you just have to.

slashdave 2y ago
Matrix inversion precision can be fixed iteratively (at a CPU cost, of course). Nearly singular matrices should probably be factored in any case.
- gh02t 2y ago
  Probably, that falls into "doing almost anything to avoid inverting a matrix explicitly", but direct inversion is the classic example here.
  - kolbe 2y ago
    For numerical PDE solvers, you never do the actual inversion, but the discretization of the system (denoted A) can have significant sensitivity to numerical precision. Look up Condition Number and Preconditioning. In fact, resolving the problems with numerical precision for some A takes as much time as actually solving the system, and it's worth it.
    - gh02t 2y ago
      Oh I'm very aware, I'm lumping those all into "tricks used to avoid/work around the numerical issues associated with matrix inversion" for simplicity, because explicit computation of a matrix inverse is one of the classic examples of a numerically unstable task. Hence why a large subfield of applied math can be summarized as "coming up with ways to avoid matrix inversion." PDE solvers like you mention are one of the main applications for that. Tricks like clever factorization (which a lot of factorization algorithms have their own severe numerical issues e.g. some of the ways to compute QR or SVD), preconditioners, sparse and/or iterative algorithms like GMRES, randomized algorithms (my favorite) etc are all workarounds you wouldn't need if there was a fast and stable way to exactly invert any arbitrary non-singular matrix. Well, you would have less need for, there are other benefits to those methods but improving numerical stability by avoiding inverses is one of the main ones.
Dylan16807 2y ago
To put some specifics on "a lot of very small numbers", it's around a million numbers with part-per-billion precision, or a billion numbers with part-per-million precision, where you approach the limits of naive summation in double precision.
- gh02t 2y ago
  For sums, yes, but products and division by small numbers are the real killer and those come up a lot in stats. Hence why you try to avoid them by working with log-probabilities (where products become sums), but sometimes you can't. Quad precision is a bit of a bandaid that just pushes the issue off when you don't have a better algorithm, but it works sometimes.
  - belkala 2y ago
    How so? All primitive operations are backward stable, and unlike addition and subtraction, division and multiplication are well-conditioned.
    - gh02t 2y ago
      It's more about resolution. Products of small numbers get small a lot faster than sums. Likewise for dividing two small numbers. The smallest possible quad precision number is a lot smaller than the smallest possible double precision one.
      - Dylan16807 2y ago
        But the main feature of floating point is that you keep the same relative precision at all sizes. As long as you're not hitting infinity or denormals, multiplying doesn't lose information like adding a big number to a small number does. Do stats often deal with distict probabilities below 10^-300? And do they need to store them all over, or just in a couple variables that could do something clever?
        2y ago

by gh02t 2y ago

slashdave 2y ago

Matrix inversion precision can be fixed iteratively (at a CPU cost, of course). Nearly singular matrices should probably be factored in any case.

gh02t 2y ago
Probably, that falls into "doing almost anything to avoid inverting a matrix explicitly", but direct inversion is the classic example here.
- kolbe 2y ago
  For numerical PDE solvers, you never do the actual inversion, but the discretization of the system (denoted A) can have significant sensitivity to numerical precision. Look up Condition Number and Preconditioning. In fact, resolving the problems with numerical precision for some A takes as much time as actually solving the system, and it's worth it.
  - gh02t 2y ago
    Oh I'm very aware, I'm lumping those all into "tricks used to avoid/work around the numerical issues associated with matrix inversion" for simplicity, because explicit computation of a matrix inverse is one of the classic examples of a numerically unstable task. Hence why a large subfield of applied math can be summarized as "coming up with ways to avoid matrix inversion." PDE solvers like you mention are one of the main applications for that. Tricks like clever factorization (which a lot of factorization algorithms have their own severe numerical issues e.g. some of the ways to compute QR or SVD), preconditioners, sparse and/or iterative algorithms like GMRES, randomized algorithms (my favorite) etc are all workarounds you wouldn't need if there was a fast and stable way to exactly invert any arbitrary non-singular matrix. Well, you would have less need for, there are other benefits to those methods but improving numerical stability by avoiding inverses is one of the main ones.

Dylan16807 2y ago

To put some specifics on "a lot of very small numbers", it's around a million numbers with part-per-billion precision, or a billion numbers with part-per-million precision, where you approach the limits of naive summation in double precision.

gh02t 2y ago
For sums, yes, but products and division by small numbers are the real killer and those come up a lot in stats. Hence why you try to avoid them by working with log-probabilities (where products become sums), but sometimes you can't. Quad precision is a bit of a bandaid that just pushes the issue off when you don't have a better algorithm, but it works sometimes.
- belkala 2y ago
  How so? All primitive operations are backward stable, and unlike addition and subtraction, division and multiplication are well-conditioned.
  - gh02t 2y ago
    It's more about resolution. Products of small numbers get small a lot faster than sums. Likewise for dividing two small numbers. The smallest possible quad precision number is a lot smaller than the smallest possible double precision one.
    - Dylan16807 2y ago
      But the main feature of floating point is that you keep the same relative precision at all sizes. As long as you're not hitting infinity or denormals, multiplying doesn't lose information like adding a big number to a small number does. Do stats often deal with distict probabilities below 10^-300? And do they need to store them all over, or just in a couple variables that could do something clever?
      - 2y ago