I need to calculate the function y=x/(1+x^2) on a small FPGA in fixed point, can you help me finding the best algorithm?
I thought of those possibilities:
as the FPGA is small I think I will use shift&add/subtract algebra a
multiplier and a divider will use about the same number of cycles,
right?
this function is similar to a digital filter, can I calculate
divisionless using a feedback loop?
I don't think I will have much memory so I'd prefer not using a LUT
Are there other options?
The time requirement isn't very strict so I thought of a simple shift&subtract but if there is something simpler, that might be better.
Related
I am trying to convolve a 16-bit input data stream with a Dirac Delta on a Xilinx Virtex 7.
More specifically, instead of multiplying my input stream by a cosine in the time domain, I would like to convolve it with the following expression in the frequency domain:
F(f) = 0.5 * (delta(f - f0) + delta(f + f0))
Does anybody have any idea about how to implement that ? Indeed, the only possibly interesting Xilinx IP core for my problem is the FIR Complier but I don't know how to represent my function F(f) as the 'coefficient' input of this IP core.
EDIT: mathematically, since the target convolution involves only Dirac Deltas, there may exist a shorter way to avoid the convolution by simply evaluating the input function at the point f0. But I have no idea about how to implement that neither ..
Thank you in advance
Xilinx has an IP to perform Fast Fourier Transform on the FPGA. Once in frequency domain, you are somewhat on your own to perform your operations. You could use the FIR ip core, but since your function is quite simple it would waste a lot of resources compared to a custom implementation. Finally, the Xilinx's core can do inverse FFT to go back to time-domain.
AFAIK, there is no core to help perform convolution in frequency domain. So don't forget to overlap-add your transforms to do the proper calculation. Matlab will be your friend there!
Finally, you may be interested in Number Theoretic Transform (NTT). The algorithm is more efficient than FFT for FPGA and can be used to perform convolution. The drawback is that there are limitations to the length of the transform you can have and that the "frequency-domain coefficient" are totally unrelated to frequency (they are somewhat random). If all you want if fast-convolution, NTT is for you, if you're looking for other uses for these fourier-coefficient, it's not. However, the NTT expression of the cosine would be much more complicated and would defeat the purpose of your work, but I thought you may be interested on an academic standpoint. As I stated in my comment, multiplying with a cosine is simpler in time-domain after all.
I am implementing a SPICE solver. I have the following problem: say I put two diodes and a current source in serial (standard diodes). I use MNA and Boost LU-decomposition. The problem is that the nodal matrix becomes very quickly near-singular. I think I have to scale the values but I don't know how and I couldn't find anything on the Internet. Any ideas how to do this scaling?
In the perspective of numerical, there is a scale technique for this kind of near-singular matrices. Basically, this technique is to divide each row of A by the sum (or maximum) of the absolute values in that row. You can find KLU which is a linear solver for circuit simulations for more details.
In perspective of SPICE simulation, it uses so-call Gmin stepping technique to iteratively compute and approach a real answer. You can find this in the documents of a SPICE project QUCS (Quite Universal Circuit Simulator).
Scaling does not help when the matrix has both very large and very small entries.
It is necessary to use some or all of the many tricks that were developed for circuit solver applications. A good start is clipping the range of the exponential and log function arguments to reasonable values -- in most circuits a diode forward voltage is never more than 1V and the diode reverse current not less than 1pA.
Actually, look at all library functions and wrap them in code that makes their arguments and results suitable for circuit-solving purposes. Simple clipping is sometimes good enough, but it is way better to make sure the functions stay (twice) differentiable and continuous.
For a university lecture I am looking for floating point algorithms with known asymptotic runtime, but potential for low-level (micro-)optimization. This means optimizations such as minimizing cache misses and register spillages, maximizing instruction level parallelism and taking advantage of SIMD (vector) instructions on new CPUs. The optimizations are going to be CPU-specific and will make use of applicable instruction set extensions.
The classic textbook example for this is matrix multiplication, where great speedups can be achieved by simply reordering the sequence of memory accesses (among other tricks). Another example is FFT. Unfortunately, I am not allowed to choose either of these.
Anyone have any ideas, or an algorithm/method that could use a boost?
I am only interested in algorithms where a per-thread speedup is conceivable. Parallelizing problems by multi-threading them is fine, but not the scope of this lecture.
Edit 1: I am taking the course, not teaching it. In the past years, there were quite a few projects that succeeded in surpassing the current best implementations in terms of performance.
Edit 2: This paper lists (from page 11 onwards) seven classes of important numerical methods and some associated algorithms that use them. At least some of the mentioned algorithms are candidates, it is however difficult to see which.
Edit 3: Thank you everyone for your great suggestions! We proposed to implement the exposure fusion algorithm (paper from 2007) and our proposal was accepted. The algorithm creates HDR-like images and consists mainly of small kernel convolutions followed by weighted multiresolution blending (on the Laplacian pyramid) of the source images. Interesting for us is the fact that the algorithm is already implemented in the widely used Enfuse tool, which is now at version 4.1. So we will be able to validate and compare our results with the original and also potentially contribute to the development of the tool itself. I will update this post in the future with the results if I can.
The simplest possible example:
accumulation of a sum. unrolling using multiple accumulators and vectorization allow a speedup of (ADD latency)*(SIMD vector width) on typical pipelined architectures (if the data is in cache; because there's no data reuse, it typically won't help if you're reading from memory), which can easily be an order of magnitude. Cute thing to note: this also decreases the average error of the result! The same techniques apply to any similar reduction operation.
A few classics from image/signal processing:
convolution with small kernels (especially small 2d convolves like a 3x3 or 5x5 kernel). In some sense this is cheating, because convolution is matrix multiplication, and is intimately related to the FFT, but in reality the nitty-gritty algorithmic techniques of high-performance small kernel convolutions are quite different from either.
erode and dilate.
what image people call a "gamma correction"; this is really evaluation of an exponential function (maybe with a piecewise linear segment near zero). Here you can take advantage of the fact that image data is often entirely in a nice bounded range like [0,1] and sub-ulp accuracy is rarely needed to use much cheaper function approximations (low-order piecewise minimax polynomials are common).
Stephen Canon's image processing examples would each make for instructive projects. Taking a different tack, though, you might look at certain amenable geometry problems:
Closest pair of points in moderately high dimension---say 50000 or so points in 16 or so dimensions. This may have too much in common with matrix multiplication for your purposes. (Take the dimension too much higher and dimensionality reduction silliness starts mattering; much lower and spatial data structures dominate. Brute force, or something simple using a brute-force kernel, is what I would want to use for this.)
Variation: For each point, find the closest neighbour.
Variation: Red points and blue points; find the closest red point to each blue point.
Welzl's smallest containing circle algorithm is fairly straightforward to implement, and the really costly step (check for points outside the current circle) is amenable to vectorisation. (I suspect you can kill it in two dimensions with just a little effort.)
Be warned that computational geometry stuff is usually more annoying to implement than it looks at first; don't just grab a random paper without understanding what degenerate cases exist and how careful your programming needs to be.
Have a look at other linear algebra problems, too. They're also hugely important. Dense Cholesky factorisation is a natural thing to look at here (much more so than LU factorisation) since you don't need to mess around with pivoting to make it work.
There is a free benchmark called c-ray.
It is a small ray-tracer for spheres designed to be a benchmark for floating-point performance.
A few random stackshots show that it spends nearly all its time in a function called ray_sphere that determines if a ray intersects a sphere and if so, where.
They also show some opportunities for larger speedup, such as:
It does a linear search through all the spheres in the scene to try to find the nearest intersection. That represents a possible area for speedup, by doing a quick test to see if a sphere is farther away than the best seen so far, before doing all the 3-d geometry math.
It does not try to exploit similarity from one pixel to the next. This could gain a huge speedup.
So if all you want to look at is chip-level performance, it could be a decent example.
However, it also shows how there can be much bigger opportunities.
I need to implement a exponential term "exp((-x²*0.008)-0.5)" in VHDL. Here 'x' is 18-bit fixed point signal ranging between (-90.0 to 90.0). When I went through internet, one solution I found is using taylor series expansion. Is there an alternative solution which better suits my need. If using taylor series is the only solution to evaluate the 'exp' function then what should be the summation limit ? Thank you.
The summation limit should be whatever is required to meet your required accuracy. It is usually best to find that out in other tools (even a spreadsheet!) before worrying about VHDL.
For sufficiently smooth curves, a quadratic interpolation using coefficients from a small lookup table may be accurate enough, (again : evaluate in MatLab, Octave or spreadsheet) and is fairly easy to implement.
What are some good tips and/or techniques for optimizing and improving the performance of calculation heavy programs. I'm talking about things like complication graphics calculations or mathematical and simulation types of programming where every second saved is useful, as opposed to IO heavy programs where only a certain amount of speedup is helpful.
While changing the algorithm is frequently mentioned as the most effective method here,I'm trying to find out how effective different algorithms are in the first place, so I want to create as much efficiency with each algorithm as is possible. The "problem" I'm solving isn't something thats well known, so there are few if any algorithms on the web, but I'm looking for any good advice on how to proceed and what to look for.
I am exploring the differences in effectiveness between evolutionary algorithms and more straightforward approaches for a particular group of related problems. I have written three evolutionary algorithms for the problem already and now I have written an brute force technique that I am trying to make as fast as possible.
Edit: To specify a bit more. I am using C# and my algorithms all revolve around calculating and solving constraint type problems for expressions (using expression trees). By expressions I mean things like x^2 + 4 or anything else like that which would be parsed into an expression tree. My algorithms all create and manipulate these trees to try to find better approximations. But I wanted to put the question out there in a general way in case it would help anyone else.
I am trying to find out if it is possible to write a useful evolutionary algorithm for finding expressions that are a good approximation for various properties. Both because I want to know what a good approximation would be and to see how the evolutionary stuff compares to traditional methods.
It's pretty much the same process as any other optimization: profile, experiment, benchmark, repeat.
First you have to figure out what sections of your code are taking up the time. Then try different methods to speed them up (trying methods based on merit would be a better idea than trying things at random). Benchmark to find out if you actually did speed them up. If you did, replace the old method with the new one. Profile again.
I would recommend against a brute force approach if it's at all possible to do it some other way. But, here are some guidelines that should help you speed your code up either way.
There are many, many different optimizations you could apply to your code, but before you do anything, you should profile to figure out where the bottleneck is. Here are some profilers that should give you a good idea about where the hot spots are in your code:
GProf
PerfMon2
OProfile
HPCToolkit
These all use sampling to get their data, so the overhead of running them with your code should be minimal. Only GProf requires that you recompile your code. Also, the last three let you do both time and hardware performance counter profiles, so once you do a time (or CPU cycle) profile, you can zoom in on the hotter regions and find out why they might be running slow (cache misses, FP instruction counts, etc.).
Beyond that, it's a matter of thinking about how best to restructure your code, and this depends on what the problem is. It may be that you've just got a loop that the compiler doesn't optimize well, and you can inline or move things in/out of the loop to help the compiler out. Or, if you're running as fast as you can with basic arithmetic ops, you may want to try to exploit vector instructions (SSE, etc.) If your code is parallel, you might have load balance problems, and you may need to restructure your code so that data is better distributed across cores.
These are just a few examples. Performance optimization is complex, and it might not help you nearly enough if you're doing a brute force approach to begin with.
For more information on ways people have optimized things, there were some pretty good examples in the recent Why do you program in assembly? question.
If your optimization problem is (quasi-)convex or can be transformed into such a form, there are far more efficient algorithms than evolutionary search.
If you have large matrices, pay attention to your linear algebra routines. The right algorithm can make shave an order of magnitude off the computation time, especially if your matrices are sparse.
Think about how data is loaded into memory. Even when you think you're spending most of your time on pure arithmetic, you're actually spending a lot of time moving things between levels of cache etc. Do as much as you can with the data while it's in the fastest memory.
Try to avoid unnecessary memory allocation and de-allocation. Here's where it can make sense to back away from a purely OO approach.
This is more of a tip to find holes in the algorithm itself...
To realize maximum performance, simplify everything inside the most inner loop at the expense of everything else.
One example of keeping things simple is the classic bouncing ball animation. You can implement gravity by looking up the definition in your physics book and plugging in the numbers, or you can do it like this and save precious clock cycles:
initialize:
float y = 0; // y coordinate
float yi = 0; // incremental variable
loop:
y += yi;
yi += 0.001;
if (y > 10)
yi = -yi;
But now let's say you're having to do this with nested loops in an N-body simulation where every particle is attracted to every other particle. This can be an enormously processor intensive task when you're dealing with thousands of particles.
You should of course take the same approach as to simplify everything inside the most inner loop. But more than that, at the very simplest level you should also use data types wisely. For example, math operations are faster when working with integers than floating point variables. Also, addition is faster than multiplication, and multiplication is faster than division.
So with all of that in mind, you should be able to simplify the most inner loop using primarily addition and multiplication of integers. And then any scaling down you might need to do can be done afterwards. To take the y and yi example, if yi is an integer that you modify inside the inner loop then you could scale it down after the loop like this:
y += yi * 0.01;
These are very basic low-level performance tips, but they're all things I try to keep in mind whenever I'm working with processor intensive algorithms. Of course, if you then take these ideas and apply them to parallel processing on a GPU then you can take your algorithm to a whole new level. =)
Well how you do this depends the most on which language
you are using. Still, the key in any language
in the profiler. Profile your code. See which
functions/operations are taking the most time and then determine
if you can make these costly operations more efficient.
Standard bottlenecks in numerical algorithms are memory
usage (do you access matrices in the order which the elements
are stored in memory); communication overhead, etc. They
can be little different than other non-numerical programs.
Moreover, many other factors such as preconditioning, etc.
can lead to drastically difference performance behavior
of the SAME algorithm on the same problem. Make sure
you determine optimal parameters for your implementations.
As for comparing different algorithms, I recommend
reading the paper
"Benchmarking optimization software with performance profiles,"
Jorge Moré and Elizabeth D. Dolan, Mathematical Programming 91 (2002), 201-213.
It provides a nice, uniform way to compare different algorithms being
applied to the same problem set. It really should be better known
outside of the optimization community (in my not so humble opinion
at least).
Good luck!