33 votes
Accepted

Is it possible for user written algorithms to outperform libraries' built-in optimized functions?

Although LAPACK has some incredibly optimized code, it can still be worth it to write your own version in a few cases. The most important reason (and the reason they make you do it in your course). ...
  • 1,266
20 votes
Accepted

The real myth of GPU (specifically CUDA) really speed up FEM/CFD

Here's the deal with GPUs. On a GPU, every single core is slow. Really slow. However, you have thousands of cores. If you can effectively use the thousands of cores at a time, then your algorithm will ...
19 votes

What is the preferred and efficient approach for interpolating multidimensional data?

For the first part of my question, I found this very useful comparison for performance of different linear interpolation methods using python libraries: http://nbviewer.ipython.org/github/pierre-...
14 votes
Accepted

How to compare runtimes of two algorithms in a reproducible way

To make more robust comparisons (on linux), you can : 1) On Intel CPUs the turbo overclocks your CPU. This is controlled by the temperature of the CPU, so it can behave differently from one run to ...
14 votes
Accepted

Beating typical BLAS libraries matrix multiplication performance

Consolidating the comments: No, you are very unlikely to beat a typical BLAS library such as Intel's MKL, AMD's Math Core Library, or OpenBLAS.1 These not only use vectorization, but also (at least ...
14 votes
Accepted

Increasing computational performance by using 16 bit numbers

There has been considerable recent interest in numerical linear algebra using mixed precision with some combination of 16, 32, 64, and 128 bit floating point arithmetic. For example, a low ...
13 votes
Accepted

Whittaker-Shannon interpolation: Accuracy dies with speedup; can it be fixed?

I was able to reproduce the behavior reported in the question, and traced the observed inaccuracies to the following line: ...
  • 1,320
13 votes
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Is there an algorithm or graph theory that allows me to not need to store an intermediate matrix when calculating AT*Y1*A + BT*Y2*B?

BLAS may not have a function to compute what you are asking for, but the product $$ Y_N = A^TY_AA + B^T Y_B B $$ means that the entries $(Y_N)_{ij}$ are defined by $$ (Y_N)_{ij} = \sum_{k,l} (A^T)...
12 votes
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Hardware performance, floating point functions

In the 1980's era Intel 80x86 architecture, there was a scalar floating point unit that had instructions like FSIN, FCOS, etc. for computing functions like sin and cos. These functions were ...
12 votes

Faster Logistic Function

Yes! There are nice approximations of the logistic. Plot of Approximating Functions As shown below, several functions approximate the logistic (shown as blue dots). This graph is available ...
  • 3,586
11 votes

Open source implementation of rational approximation to a function

Doing one-off best rational approximations can often be accomplished by "manual" iterations of the Remez algorithm: interpolate a rational approximation with (relative or absolute) alternating sign ...
  • 3,219
9 votes
Accepted

Loop optimization with f2py, Cython and Numba

I think that the problem is linked to the way in which f2py generates the fortran interface: the argument to fortranrun.f2py should be stored as a F_CONTIGUOUS ...
  • 3,809
8 votes
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What is the fastest opensource implementation of Bessel functions computation?

The appropriate and fastest library depends on several things. Which Bessel functions (only J, Y & Hankel or modified Bessel functions I & K too), for which types of arguments (real or complex,...
8 votes
Accepted

Integer operations vs floating point operations

There is a nice discussion on StackOverflow regarding floating point vs integer operations. In short, the performance of the operations depends a lot on processor architecture how the data is stored ...
  • 8,542
8 votes
Accepted

Fastest Way to Mutiply $10^4$ 2x2 Matrices

In general, I agree with Chris's comment that using a compiled language with the allocation of the matrices on the stack can help significantly. Several possibilities if we are limited to Python and ...
  • 8,542
8 votes

How to properly calculate CPU and GPU FLOPS performance?

You can calculate GFLOP rates this way, but the numbers are pretty meaningless on today's hardware: Floating point operations require a variable number of clock cycles. An addition is generally ...
8 votes

Is it possible for user written algorithms to outperform libraries' built-in optimized functions?

@ThiysSteel covers a lot, here is another perspective I find important: Even if you have available excellent implementations of any algorithm you might need, you still need to understand some of the ...
  • 181
7 votes
Accepted

How to evaluate a series of derivatives?

You can convert $\mathbf{b}-\mathbf{x}$ into polar coordinates, and do the dot product in this system. This changes $((\mathbf{b}-\mathbf{x})\cdot\nabla)^n\frac{1}{r}$ to $$\left((\mathbf{b}-\mathbf{...
7 votes

Performance differences between ATLAS and MKL?

Prior answers to this question have covered most of the salient points, but I want to add one comment with respect to this: does MKL have the upper hand for some tasks? The MKL team is in a unique ...
  • 2,096
7 votes

Is it possible for user written algorithms to outperform libraries' built-in optimized functions?

To clarify @ThiysSteel's good answer: The point is not to attempt to surpass the optimization of code written by very experienced people, who'd wrangled with it for decades. The point is to acquaint ...
7 votes

Is it possible for user written algorithms to outperform libraries' built-in optimized functions?

I used to try to optimize code via using assembly language (as opposed to C). I had some clear success, where a real-time microphone array worked with assembly language, but it would not work at all ...
6 votes
Accepted

What is the state of the art algorithm for diagonalizing real symmetric matrices?

The QR/Francis algorithm is the go-to choice for dense eigenproblems, but there are a few competitors around: The Jacobi algorithm (like QR, another algorithm with an unfortunate name, which can be ...
6 votes
Accepted

Why is the speed of the parts of the LU-decomposition so different?

First, don't forget to also time the LU decomposition in a loop! Otherwise it's not really a fair comparison. If I do that, I get the following timings: ...
6 votes
Accepted

Implementation of Jacobi iteration

If you really want to speed it up, I'd suggest changing your iteration to a SSOR iteration. This can converge much faster than jacobi, and if you are already OpenMP parallelized, there's not much you ...
  • 2,059
6 votes
Accepted

Is operation count a reliable predictor of performance when comparing two formulations?

No. There a number of aspects of modern computer architecture that make operation counts unreliable as a way to compare the performance of algorithms. These include memory caches, vector ...
6 votes

Is it possible for user written algorithms to outperform libraries' built-in optimized functions?

This might be tangent but i feel it worth adding. A common way for user code to beat libraries is to exploit structure the libraries are not aware of. Block diagonal matrices are easier to work in ...
5 votes

Why am I not seeing faster neural network training after upgrading to a vastly better GPU?

I suppose, you right and your network is not that big to 100%-utilize the GPU. The bottle-neck here seems to be not the GPU itself, but the transfer rate between RAM and VRAM and here the difference ...
  • 114
5 votes

Can other running processes harm the performance of my simulation?

If you just want the computational results and aren't running benchmarking tests then this isn't a serious problem. If you're trying to benchmark the performance of the code and get repeatable run ...
5 votes

The real myth of GPU (specifically CUDA) really speed up FEM/CFD

To extend Chris Rackauckas's exhaustive answer with a reference try to look pdf by Torres, Gonzalez-Escribano, Llanos. It is about the tuning of a gpu, that is an important aspect for performance. ...

Only top scored, non community-wiki answers of a minimum length are eligible