I am working on a n-body gravitational simulator that takes input and produces output in metric MKS units. This involves dealing with some very large numbers (like solar masses expressed in kilograms, semimajor axes of planetary orbits expressed in meters, and timescales of years expressed in seconds), which get multiplied by some very small numbers (notably, the gravitational constant, which is 6.67384e-11 in MKS units), and also the occasional very small number getting added to or subtracted from a very large number (mainly when summing up pairwise accelerations), which gets me concerned about the effects of rounding errors.
I've already taken the step of replacing all masses m by Gm (premultiplying by the gravitational constant), which significantly reduces the total number of multiplies, and makes the mass numbers much smaller, and that seems to have had a positive effect on both efficiency and accuracy, as judged by how well the simulator conserves energy.
I am wondering, however: is potentially it worth trying to do some internal re-scaling into different units to further minimize floating point errors? And if so, what kind of range (for double-precision floats) should I be trying to get my numbers centered on for maximum accuracy?
In general if you want precise results in physical based rendering you don't want to use floats or doubles since they have massive rounding problems and thus introduce errors in your simulation.
If you need or want to stick with floats/double you probably should rescale around zero. The reason is that often floating point representations have a higher "density" of values around this point and tend to have fewer on the min/max sides. Image example from google
I would suggest that you change all values to integer based number variables. This erases rounding errors (over/underflow can still happen!) and speeds up the calculation process by an order of magnitude because normal CPUs work faster with integer operations. In case of GPU its basically the same but thats another story all by its own...
But before you take such an effort to further improve your accuracy i would strongly advise an arbitrary precision number library. This may come with an performance loss but should be way easier and yield better results than a rescaling of your values.
Most of the numerical mathematicians come across this problem.
At first let me remind you that you can not deal with numbers (or phsycal values) smaller than the machine epsilon for each calculation. Unfortunately the epsilon depends around which number you are analyzing. You can try eps(a) for any value of a in MATLAB, as far as I remember eps(1.0)~=2.3e-16 and eps(0)~1e-298.
That's why in numerical methods you avoid calculations using very different scaled numbers. Because one is just an ignored (smaller than its epsilon) by the other value and rounding errors are inevitable.
But what else people do? If they encounter such physical problems, before coding, mathematicians analyse the problem theoritically, they make simplifications to use similarly scaled numbers.
Related
First, sorry if this question looks "silly", because I'm new to MPFR, LOL.
I have two mpfr_t variables with precision of 1024, and they have the value of 0.2 and 0.06 stored in them.
But when I add these variables, things goes wrong and the result (which is also a mpfr_t variable) has the value of 0.2599999...
This is strange because the MPFR library should maintain the precision (isn't it?).
Could you please help me with this? Thanks so much, so much in advance.
MPFR numbers are represented in binary (base 2). In this system, the only numbers that can be represented exactly have the form N·2k, where N and k are integers. Neither 0.2 = 1/5 nor 0.06 = 3/50 have this form, so that they are approximated with some small error. When you add these variables, you are seeing a consequence of this error (there may be also another error in the addition operation since in binary these numbers have many nonzero digits, unlike in decimal).
This is the same issue as the one described in: Is floating point math broken?
EDIT:
To answer the question in comment "Is there a way to avoid this situation?", no, there is no way to avoid this situation in practice, except in very specific cases. For instance, if all your numbers (inputs and results of each intermediate operations) are decimal numbers, representable with a small enough number of digits, you can use a decimal arithmetic (but MPFR can't do that). Computer algebra systems may help in some cases. There's also iRRAM... I'll come back to it later.
However, there are solutions to attempt to hide issues with numerical errors. You need to estimate the maximum possible error on a computed value. With an error analysis, you can obtain rigorous bounds, but this may be difficult or take time to do. Note that rigorous bounds are pessimistic in general, but if you use arbitrary precision (e.g. with MPFR), this is less an issue. The analysis can be done dynamically with interval arithmetic (still pessimistic, even worse). But perhaps a simple estimate is sufficient for you. Once you have an estimate of the maximum error:
For the output, choose the number of displayed digits so that the error is less than the weight of the last displayed digit.
For discontinuous functions (e.g. equality test, floor, ceil): if the distance between the computed value and a discontinuity point is less than the maximum error, assume that the actual value is equal to the discontinuity point. Note that this is just a heuristic, but if it fails (this may remain unnoticed and will probably invalidate your estimate), this means that you have not done your computations with enough precision.
Note: MPFR won't do that for you. But you can write code to take these rules into account.
The iRRAM package, which is based on MPFR, can track the error in a rigorous way (like with interval arithmetic) and automatically redo all the computations in a higher precision if it notices that the accuracy is too low. However, if some mathematical result is a discontinuity point, iRRAM won't help. In particular, it cannot provide a rigorous equality test.
Finally, I suggest that you have a look at Goldberg's paper What Every Computer Scientist Should Know About Floating-Point Arithmetic, in particular the notion of cancellation.
I was working with PIC18f4550 and the program is critical to speed. when I multiply two floating variables it tooks the PIC about 140 cycles to perform the multiplication. I am measuring it with PIC18f4550 timer1.
variable_1 = variable_2 * variable_3; // took 140 cycles to implement
On the the other hand when I add the same two variables the PIC tooks 280 cycles to perfom the addition.
variable_1 = variable_2 + variable_3; // took 280 cycles to implement
I have seen that the number of cycles vary if the variables changed depend on their exponents.
What is the reason of those more cycles? though I was thinking the addition is more simple than multiplication.
Is there any solution?
For floating point addition, the operands need to be adjusted so that they have the same exponent before the add, and that involves shifting one of the mantissas across byte boundaries, whereas a multiply is basically multiplying the mantissas and adding the exponents.
Since the PIC apparently has a small hardware multiplier, it may not be surprising that sometimes the multiply can be faster than doing a multi-byte shift (especially if the PIC only has single bit shift instructions).
Unless a processor has direct support for it, floating point is always slow, and you should certainly consider arranging your code to use fixed point if at all possible. Getting rid of the floating point library would probably free up a lot of code space as well.
I searched the site but did not find exactly what I was looking for... I wanted to generate a discrete random number from normal distribution.
For example, if I have a range from a minimum of 4 and a maximum of 10 and an average of 7. What code or function call ( Objective C preferred ) would I need to return a number in that range. Naturally, due to normal distribution more numbers returned would center round the average of 7.
As a second example, can the bell curve/distribution be skewed toward one end of the other? Lets say I need to generate a random number with a range of minimum of 4 and maximum of 10, and I want the majority of the numbers returned to center around the number 8 with a natural fall of based on a skewed bell curve.
Any help is greatly appreciated....
Anthony
What do you need this for? Can you do it the craps player's way?
Generate two random integers in the range of 2 to 5 (inclusive, of course) and add them together. Or flip a coin (0,1) six times and add 4 to the result.
Summing multiple dice produces a normal distribution (a "bell curve"), while eliminating high or low throws can be used to skew the distribution in various ways.
The key is you are going for discrete numbers (and I hope you mean integers by that). Multiple dice throws famously generate a normal distribution. In fact, I think that's how we were first introduced to the Gaussian curve in school.
Of course the more throws, the more closely you approximate the bell curve. Rolling a single die gives a flat line. Rolling two dice just creates a ramp up and down that isn't terribly close to a bell. Six coin flips gets you closer.
So consider this...
If I understand your question correctly, you only have seven possible outcomes--the integers (4,5,6,7,8,9,10). You can set up an array of seven probabilities to approximate any distribution you like.
Many frameworks and libraries have this built-in.
Also, just like TokenMacGuy said a normal distribution isn't characterized by the interval it's defined on, but rather by two parameters: Mean μ and standard deviation σ. With both these parameters you can confine a certain quantile of the distribution to a certain interval, so that 95 % of all points fall in that interval. But resticting it completely to any interval other than (−∞, ∞) is impossible.
There are several methods to generate normal-distributed values from uniform random values (which is what most random or pseudorandom number generators are generating:
The Box-Muller transform is probably the easiest although not exactly fast to compute. Depending on the number of numbers you need, it should be sufficient, though and definitely very easy to write.
Another option is Marsaglia's Polar method which is usually faster1.
A third method is the Ziggurat algorithm which is considerably faster to compute but much more complex to program. In applications that really use a lot of random numbers it may be the best choice, though.
As a general advice, though: Don't write it yourself if you have access to a library that generates normal-distributed random numbers for you already.
For skewing your distribution I'd just use a regular normal distribution, choosing μ and σ appropriately for one side of your curve and then determine on which side of your wanted mean a point fell, stretching it appropriately to fit your desired distribution.
For generating only integers I'd suggest you just round towards the nearest integer when the random number happens to fall within your desired interval and reject it if it doesn't (drawing a new random number then). This way you won't artificially skew the distribution (such as you would if you were clamping the values at 4 or 10, respectively).
1 In testing with deliberately bad random number generators (yes, worse than RANDU) I've noticed that the polar method results in an endless loop, rejecting every sample. Won't happen with random numbers that fulfill the usual statistic expectations to them, though.
Yes, there are sophisticated mathematical solutions, but for "simple but practical" I'd go with Nosredna's comment. For a simple Java solution:
Random random=new Random();
public int bell7()
{
int n=4;
for (int x=0;x<6;++x)
n+=random.nextInt(2);
return n;
}
If you're not a Java person, Random.nextInt(n) returns a random integer between 0 and n-1. I think the rest should be similar to what you'd see in any programming language.
If the range was large, then instead of nextInt(2)'s I'd use a bigger number in there so there would be fewer iterations through the loop, depending on frequency of call and performance requirements.
Dan Dyer and Jay are exactly right. What you really want is a binomial distribution, not a normal distribution. The shape of a binomial distribution looks a lot like a normal distribution, but it is discrete and bounded whereas a normal distribution is continuous and unbounded.
Jay's code generates a binomial distribution with 6 trials and a 50% probability of success on each trial. If you want to "skew" your distribution, simply change the line that decides whether to add 1 to n so that the probability is something other than 50%.
The normal distribution is not described by its endpoints. Normally it's described by it's mean (which you have given to be 7) and its standard deviation. An important feature of this is that it is possible to get a value far outside the expected range from this distribution, although that will be vanishingly rare, the further you get from the mean.
The usual means for getting a value from a distribution is to generate a random value from a uniform distribution, which is quite easily done with, for example, rand(), and then use that as an argument to a cumulative distribution function, which maps probabilities to upper bounds. For the standard distribution, this function is
F(x) = 0.5 - 0.5*erf( (x-μ)/(σ * sqrt(2.0)))
where erf() is the error function which may be described by a taylor series:
erf(z) = 2.0/sqrt(2.0) * Σ∞n=0 ((-1)nz2n + 1)/(n!(2n + 1))
I'll leave it as an excercise to translate this into C.
If you prefer not to engage in the exercise, you might consider using the Gnu Scientific Library, which among many other features, has a technique to generate random numbers in one of many common distributions, of which the Gaussian Distribution (hint) is one.
Obviously, all of these functions return floating point values. You will have to use some rounding strategy to convert to a discrete value. A useful (but naive) approach is to simply downcast to integer.
Have done fft (see earlier posting if you are interested!) and got a result, which helps me. Would like to analyse the noisiness / spikiness of an array (actually a vb.nre collection of single). Um, how to explain ...
When signal is good, fft power results is 512 data points (frequency buckets) with low values in all but maybe 2 or 3 array entries, and a decent range (i.e. the peak is high, relative to the noise value in the nearly empty buckets. So when graphed, we have a nice big spike in the values in those few buckets.
When signal is poor/noisy, data values spread (max to min) is low, and there's proportionally higher noise in many more buckets.
What's a good, computationally non-intensive was of analysing the noisiness of this data set? Would some kind of statistical method, standard deviations or something help ?
The key is defining what is noise and what is signal, for which modelling assumptions must be made. Often an assumption is made of white noise (constant power per frequency band) or noise of some other power spectrum, and that model is fitted to the data. The signal to noise ratio can then be used to measure the amount of noise.
Fitting a noise model depends on the nature of your data: if you know that the real signal will have no power in the high frequency components, you can look there for an indication of the noise level, and use the model to predict what the noise will be at the lower frequency components where there is both signal and noise. Alternatively, if your signal is constant in time, taking multiple FFTs at different points in time and comparing them to get a standard deviation for each frequency band can give the level of noise present.
I hope I'm not patronising you to mention the issues inherent with windowing functions when performing FFTs: these can have the effect of introducing spurious "noise" into the frequency spectrum which is in fact an artifact of the periodic nature of the FFT. There's a tradeoff between getting sharp peaks and 'sideband' noise - more here www.ee.iitm.ac.in/~nitin/_media/ee462/fftwindows.pdf
Calculate a standard deviation and then you decide the threshold that will indicate noise. In practice this is usually easy and allows you to easily tweak the "noise level" as needed.
There is a nice single pass stddev algorithm in Knuth. Here is link that describes an implementation.
Standard Deviation
calculate the signal to noise ratio
http://en.wikipedia.org/wiki/Signal-to-noise_ratio
you could also check the stdev for each point and if it's under some level you choose then the signal is good else it's not.
wouldn't the spike be
treated as a noise glitch in SNR, an
outlier to be discarded, as it were?
If it's clear from the time-domain data that there are such spikes, then they will certainly create a lot of noise in the frequency spectrum. Chosing to ignore them is a good idea, but unfortunately the FFT can't accept data with 'holes' in it where the spikes have been removed. There are two techniques to get around this. The 'dirty trick' method is to set the outlier sample to be the average of the two samples on either site, and compute the FFT with a full set of data.
The harder but more-correct method is to use a Lomb Normalised Periodogram (see the book 'Numerical Recipes' by W.H.Press et al.), which does a similar job to the FFT but can cope with missing data properly.
I need to find the frequency of a sample, stored (in vb) as an array of byte. Sample is a sine wave, known frequency, so I can check), but the numbers are a bit odd, and my maths-foo is weak.
Full range of values 0-255. 99% of numbers are in range 235 to 245, but there are some outliers down to 0 and 1, and up to 255 in the remaining 1%.
How do I normalise this to remove outliers, (calculating the 235-245 interval as it may change with different samples), and how do I then calculate zero-crossings to get the frequency?
Apologies if this description is rubbish!
The FFT is probably the best answer, but if you really want to do it by your method, try this:
To normalize, first make a histogram to count how many occurrances of each value from 0 to 255. Then throw out X percent of the values from each end with something like:
for (i=lower=0;i< N*(X/100); lower++)
i+=count[lower];
//repeat in other direction for upper
Now normalize with
A[i] = 255*(A[i]-lower)/(upper-lower)-128
Throw away results outside the -128..127 range.
Now you can count zero crossings. To make sure you are not fooled by noise, you might want to keep track of the slope over the last several points, and only count crossings when the average slope is going the right way.
The standard method to attack this problem is to consider one block of data, hopefully at least twice the actual frequency (taking more data isn't bad, so it's good to overestimate a bit), then take the FFT and guess that the frequency corresponds to the largest number in the resulting FFT spectrum.
By the way, very similar problems have been asked here before - you could search for those answers as well.
Use the Fourier transform, it's much more noise insensitive than counting zero crossings
Edit: #WaveyDavey
I found an F# library to do an FFT: From here
As it turns out, the best free
implementation that I've found for F#
users so far is still the fantastic
FFTW library. Their site has a
precompiled Windows DLL. I've written
minimal bindings that allow
thread-safe access to FFTW from F#,
with both guru and simple interfaces.
Performance is excellent, 32-bit
Windows XP Pro is only up to 35%
slower than 64-bit Linux.
Now I'm sure you can call F# lib from VB.net, C# etc, that should be in their docs
If I understood well from your description, what you have is a signal which is a combination of a sine plus a constant plus some random glitches. Say, like
x[n] = A*sin(f*n + phi) + B + N[n]
where N[n] is the "glitch" noise you want to get rid of.
If the glitches are one-sample long, you can remove them using a median filter which has to be bigger than the glitch length. On both sides of the glitch. Glitches of length 1, mean you will have enough with a median of 3 samples of length.
y[n] = median3(x[n])
The median is computed so: Take the samples of x you want to filter (x[n-1],x[n],x[n+1]), sort them, and your output is the middle one.
Now that the noise signal is away, get rid of the constant signal. I understand the buffer is of a limited and known length, so you can just compute the mean of the whole buffer. Substract it.
Now you have your single sinus signal. You can now compute the fundamental frequency by counting zero crossings. Count the amount of samples above 0 in which the former sample was below 0. The period is the total amount of samples of your buffer divided by this, and the frequency is the oposite (1/x) of the period.
Although I would go with the majority and say that it seems like what you want is an fft solution (fft algorithm is pretty quick), if fft is not the answer for whatever reason you may want to try fitting a sine curve to the data using a fitting program and reading off the fitted frequency.
Using Fityk, you can load the data, and fit to a*sin(b*x-c) where 2*pi/b will give you the frequency after fitting.
Fityk can be used from a gui, from a command-line for scripting and has a C++ API so could be included in your programs directly.
I googled for "basic fft". Visual Basic FFT Your question screams FFT, but be careful, using FFT without understanding even a little bit about DSP can lead results that you don't understand or don't know where they come from.
get the Frequency Analyzer at http://www.relisoft.com/Freeware/index.htm and run it and look at the code.