4 ms·

I've always argued that if you are getting close to having to worry about underflow, overflow etc. then you have an ill-conditioned problem and just increasing

I've always argued that if you are getting close to having to worry about underflow, overflow etc. then you have an ill-conditioned problem and just increasing the size of your intermediate results won't help you a huge amount because you need more precision from your inputs. There are very few fields where you need more than the 7 decimal digits afforded by floats. Maybe the only exceptions are in astrophysics.

auggierose 6y ago
That argument would be wrong. Double can be a big help, for example when performing solid modelling operations on triangle meshes. Best would be actually exact arithmetic, but it being often too slow, Double is often good enough, while Float isn't.
evanb 6y ago
Lattice QCD, especially near the physical point, has poorly-conditioned matrices that one wishes to solve Mx=b for. The state-of-the-art is that the [sparse] matrices can be as large as 4×3×(128^3 × 192) ~ 5e9 on a side. It's not so rare to find legitimately difficult problems in hard sciences.
jabl 6y ago
Double precision has been the mainstay of scientific computing for decades, and no, it's not because all those scientists are dumb.
godelski 6y ago
Pretty much any ODE solver you want double. All of CFD (fluid sims) and FEM (mechanics) use double. It can also be a big help in graphics. Basically, if you're doing fine meshing, doubles will help you. There's a balance between your meshing size as well. Your life is a whole lot easier and you'll get better answers if you just use double. Ask literally any computational {physicist,scientist,engineer}. We've all tried without double. Trust me, I'd love to have lower memory constraints.
fluffything 6y ago
When solving linear systems of equations (which arise pretty much everywhere) Krylov subspace methods are usually quite effective because the Krylov subspace is orthonormal. If your floating-point precision isn't high enough, you'll end up instead with a "subspace" that isn't spanned by orthogonal vectors, and the consequences of this are pretty drastic (requiring n-times the number of iterations to solve the system, you'll never find a solution, etc.). These all happen even if your system of equations has a good condition number, you just need to make the precision low enough. This is why most people use double precision, and some people use quad precision. For many systems, e.g., if you are using CG as your solver, quad precision can cut the number of iterations by a large factor (2x-4x). These problems are still bandwidth bound, and using quad precision duplicates your memory bandwidth requirements, but if it reduces the number of iterations by 4x, you just halved your time to solution.

auggierose 6y ago

That argument would be wrong. Double can be a big help, for example when performing solid modelling operations on triangle meshes. Best would be actually exact arithmetic, but it being often too slow, Double is often good enough, while Float isn't.

evanb 6y ago

Lattice QCD, especially near the physical point, has poorly-conditioned matrices that one wishes to solve Mx=b for. The state-of-the-art is that the [sparse] matrices can be as large as 4×3×(128^3 × 192) ~ 5e9 on a side. It's not so rare to find legitimately difficult problems in hard sciences.

jabl 6y ago

Double precision has been the mainstay of scientific computing for decades, and no, it's not because all those scientists are dumb.

godelski 6y ago

Pretty much any ODE solver you want double. All of CFD (fluid sims) and FEM (mechanics) use double. It can also be a big help in graphics. Basically, if you're doing fine meshing, doubles will help you. There's a balance between your meshing size as well. Your life is a whole lot easier and you'll get better answers if you just use double. Ask literally any computational {physicist,scientist,engineer}. We've all tried without double. Trust me, I'd love to have lower memory constraints.

fluffything 6y ago

When solving linear systems of equations (which arise pretty much everywhere) Krylov subspace methods are usually quite effective because the Krylov subspace is orthonormal. If your floating-point precision isn't high enough, you'll end up instead with a "subspace" that isn't spanned by orthogonal vectors, and the consequences of this are pretty drastic (requiring n-times the number of iterations to solve the system, you'll never find a solution, etc.). These all happen even if your system of equations has a good condition number, you just need to make the precision low enough. This is why most people use double precision, and some people use quad precision. For many systems, e.g., if you are using CG as your solver, quad precision can cut the number of iterations by a large factor (2x-4x). These problems are still bandwidth bound, and using quad precision duplicates your memory bandwidth requirements, but if it reduces the number of iterations by 4x, you just halved your time to solution.