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What is time-complexity of math.sqrt implementation in Java ?
Java has time-complexity implemented in some technique whose, time-complexity I am trying to determine.
In most cases, Java attempts to use the "smart-power" algorithm, which results in a time-complexity of O(log n).
Smart power Algorithm
Also, it appears that in different cases, you could end up with different complexities; Why is multiplied many times faster than taking the square root?
It looks like it is implemented by delegating to the sqrt method StrictMath which is a native method.
Thus it seems the answer would be implementation specific.
Strictly speaking it is O(1). In theory (but obviously not practice), we could iterate over all doubles and find the maximum time.
In addition, the time complexity of Math.sqrt(n) does not directly depend on n but instead on the amount of space needed to represent n which for doubles should be constant.
I need to evaluate a logarithm of any base, it does not matter, to some precision. Is there an algorithm for this? I program in Java, so I'm fine with Java code.
How to find a binary logarithm very fast? (O(1) at best) might be able to answer my question, but I don't understand it. Can it be clarified?
Use this identity:
logb(n) = loge(n) / loge(b)
Where log can be a logarithm function in any base, n is the number and b is the base. For example, in Java this will find the base-2 logarithm of 256:
Math.log(256) / Math.log(2)
=> 8.0
Math.log() uses base e, by the way. And there's also Math.log10(), which uses base 10.
I know this is extremely late, but this may come to be useful for some since the matter here is precision. One way of doing this is essentially implementing a root-finding algorithm that uses, from its base, the high precision types you might want to be using, consisting of simple +-x/ operations.
I would recommend implementing Newton's ​method since it demands relatively few iterations and has great convergence. For this sort of application, specifically, I believe it's fair to say it will always provide the correct result provided good input validation is implemented.
Considering a simple constant "a" where
Where a is sought to be solved for such that it obeys, then
We can use the Newton method iteratively to find "a" within any specified tolerance, where each a-ith iteration can be computed by
and the denominator is
,
because that's the first derivative of the function, as necessary for the Newton method. Once this is solved for, "a" is the direct answer for the "a = log,b(x)" problem, obtainable by simple +-x/ operations, so you're already good to go. "Wait, but there's a power there?". Yes. If you can rely on your power function as being accurate enough, then there are no issues with going ahead and using it there. Otherwise, you can further break down the power operation into a series of other +-x/ operations by using these methods to simplify whatever decimal number that is on the power into two integer power operations that can be computed easily with a series of multiplication operations. This process will eventually leave you with nth-roots to solve for, which you can also find with the Newton method. If you do go down that road, you can use this for the newton method
which, as you can see, has to be solved for recursively until you reach b = 1.
Phew, but yeah, that's it. This is the way you can solve the problem by making sure you use high precision types along the whole way with only +-x/ operations. Below is a quick implementation I did in Excel to solve for log,2(3), compared with the solution given by the software's original function. As you can see, I can just keep refining "a" until I reach the tolerance I want by monitoring what the optimization function gives me. In this, I used a=2 as the initial guess, which you can use and should be fine for most cases.
I am working on probabilistic models, and when doing inference on those models, the estimated probabilities can become very small. In order to avoid underflow, I am currently working in the log domain (I store the log of the probabilities). Multiplying probabilities is equivalent to an addition, and summing is done by using the formula:
log(exp(a) + exp(b)) = log(exp(a - m) + exp(b - m)) + m
where m = max(a, b).
I use some very large matrices, and I have to take the element-wise exponential of those matrices to compute matrix-vector multiplications. This step is quite expensive, and I was wondering if there exist other methods to deal with underflow, when working with probabilities.
Edit: for efficiency reasons, I am looking for a solution using primitive types and not objects storing arbitrary-precision representation of real numbers.
Edit 2: I am looking for a faster solution than the log domain trick, not a more accurate solution. I am happy with the accuracy I currently get, but I need a faster method. Particularly, summations happen during matrix-vector multiplications, and I would like to be able to use efficient BLAS methods.
Solution: after a discussion with Jonathan Dursi, I decided to factorize each matrix and vector by its largest element, and to store that factor in the log domain. Multiplications are straightforward. Before additions, I have to factorize one of the added matrices/vectors by the ratio of the two factors. I update the factor every ten operations.
This issue has come up recently on the computational science stack exchange site as well, and although there the immediate worry there was overflow, the issues are more or less the same.
Transforming into log space is certainly one reasonable approach. Whatever space you're in, to do a large number of sums correctly, there's a couple of methods you can use to improve the accuracy of your summations. Compensated summation approaches, most famously Kahan summation, keep both a sum and what's effectively a "remainder"; it gives you some of the advantages of using higher precision arithmeitic without all of the cost (and only using primitive types). The remainder term also gives you some indication of how well you're doing.
In addition to improving the actual mechanics of your addition, changing the order of how you add your terms can make a big difference. Sorting your terms so that you're summing from smallest to largest can help, as then you're no longer adding terms as frequently that are very different (which can cause significant roundoff problems); in some cases, doing log2 N repeated pairwise sums can also be an improvement over just doing the straight linear sum, depending on what your terms look like.
The usefullness of all these approaches depend a lot on the properties of your data. The arbitrary precision math libraries, while enormously expensive in compute time (and possibly memory) to use, have the advantage of being a fairly general solution.
I ran into a similar problem years ago. The solution was to develop an approximation of log(1+exp(-x)). The range of the approximation does not need to be all that large (x from 0 to 40 will more than suffice), and at least in my case the accuracy didn't need to be particularly high, either.
In your case, it looks like you need to compute log(1+exp(-x1)+exp(-x2)+...). Throw out those large negative values. For example, suppose a, b, and c are three log probabilities, with 0>a>b>c. You can ignore c if a-c>38. It's not going to contribute to your joint log probability at all, at least not if you are working with doubles.
Option 1: Commons Math - The Apache Commons Mathematics Library
Commons Math is a library of lightweight, self-contained mathematics and statistics components addressing the most common problems not
available in the Java programming language or Commons Lang.
Note: The API protects the constructors to force a factory pattern while naming the factory DfpField (rather than the somewhat more intuitive DfpFac or DfpFactory). So you have to use
new DfpField(numberOfDigits).newDfp(myNormalNumber)
to instantiate a Dfp, then you can call .multiply or whatever on this. I thought I'd mention this because it's a bit confusing.
Option 2: GNU Scientific Library or Boost C++ Libraries.
In these cases you should use JNI in order to call these native libraries.
Option 3: If you are free to use other programs and/or languages, you could consider using programs/languages for numerical computations such as Octave, Scilab, and similar.
Option 4: BigDecimal of Java.
Rather than storing values in logarithmic form, I think you'd probably be better off using the same concept as doubles, namely, floating-point representation. For example, you might store each value as two longs, one for sign-and-mantissa and one for the exponent. (Real floating-point has a carefully tuned design to support lots of edge cases and avoid wasting a single bit; but you probably don't need to worry so much about any of those, and can focus on designing it in a way that's simple to implement.)
I don't understand why this works, but this formula seems to work and is simpler:
c = a + log(1 + exp(b - a))
Where c = log(exp(a)+exp(b))
What complexity are the methods multiply, divide and pow in BigInteger currently? There is no mention of the computational complexity in the documentation (nor anywhere else).
If you look at the code for BigInteger (provided with JDK), it appears to me that
multiply(..) has O(n^2) (actually the method is multiplyToLen(..)). The code for the other methods is a bit more complex, but you can see yourself.
Note: this is for Java 6. I assume it won't differ in Java 7.
As noted in the comments on #Bozho's answer, Java 8 and onwards use more efficient algorithms to implement multiplication and division than the naive O(N^2) algorithms in Java 7 and earlier.
Java 8 multiplication adaptively uses either the naive O(N^2) long multiplication algorithm, the Karatsuba algorithm or the 3 way Toom-Cook algorithm depending in the sizes of the numbers being multiplied. The latter are (respectively) O(N^1.58) and O(N^1.46).
Java 8 division adaptively uses either Knuth's O(N^2) long division algorithm or the Burnikel-Ziegler algorithm. (According to the research paper, the latter is 2K(N) + O(NlogN) for a division of a 2N digit number by an N digit number, where K(N) is the Karatsuba multiplication time for two N-digit numbers.)
Likewise some other operations have been optimized.
There is no mention of the computational complexity in the documentation (nor anywhere else).
Some details of the complexity are mentioned in the Java 8 source code. The reason that the javadocs do not mention complexity is that it is implementation specific, both in theory and in practice. (As illustrated by the fact that the complexity of some operations is significantly different between Java 7 and 8.)
There is a new "better" BigInteger class that is not being used by the sun jdk for conservateism and lack of useful regression tests (huge data sets). The guy that did the better algorithms might have discussed the old BigInteger in the comments.
Here you go http://futureboy.us/temp/BigInteger.java
Measure it. Do operations with linearly increasing operands and draw the times on a diagram.
Don't forget to warm up the JVM (several runs) to get valid benchmark results.
If operations are linear O(n), quadratic O(n^2), polynomial or exponential should be obvious.
EDIT: While you can give algorithms theoretical bounds, they may not be such useful in practice. First of all, the complexity does not give the factor. Some linear or subquadratic algorithms are simply not useful because they are eating so much time and resources that they are not adequate for the problem on hand (e.g. Coppersmith-Winograd matrix multiplication).
Then your computation may have all kludges you can only detect by experiment. There are preparing algorithms which do nothing to solve the problem but to speed up the real solver (matrix conditioning). There are suboptimal implementations. With longer lengths, your speed may drop dramatically (cache missing, memory moving etc.). So for practical purposes, I advise to do experimentation.
The best thing is to double each time the length of the input and compare the times.
And yes, you do find out if an algorithm has n^1.5 or n^1.8 complexity. Simply quadruple
the input length and you need only the half time for 1.5 instead of 2. You get again nearly half the time for 1.8 if you multiply the length 256 times.
I need to calculate Math.exp() from java very frequently, is it possible to get a native version to run faster than java's Math.exp()??
I tried just jni + C, but it's slower than just plain java.
This has already been requested several times (see e.g. here). Here is an approximation to Math.exp(), copied from this blog posting:
public static double exp(double val) {
final long tmp = (long) (1512775 * val + (1072693248 - 60801));
return Double.longBitsToDouble(tmp << 32);
}
It is basically the same as a lookup table with 2048 entries and linear interpolation between the entries, but all this with IEEE floating point tricks. Its 5 times faster than Math.exp() on my machine, but this can vary drastically if you compile with -server.
+1 to writing your own exp() implementation. That is, if this is really a bottle-neck in your application. If you can deal with a little inaccuracy, there are a number of extremely efficient exponent estimation algorithms out there, some of them dating back centuries. As I understand it, Java's exp() implementation is fairly slow, even for algorithms which must return "exact" results.
Oh, and don't be afraid to write that exp() implementation in pure-Java. JNI has a lot of overhead, and the JVM is able to optimize bytecode at runtime sometimes even beyond what C/C++ is able to achieve.
Use Java's.
Also, cache results of the exp and then you can look up the answer faster than calculating them again.
You'd want to wrap whatever loop's calling Math.exp() in C as well. Otherwise, the overhead of marshalling between Java and C will overwhelm any performance advantage.
You might be able to get it to run faster if you do them in batches. Making a JNI call adds overhead, so you don't want to do it for each exp() you need to calculate. I'd try passing an array of 100 values and getting the results to see if it helps performance.
The real question is, has this become a bottle neck for you? Have you profiled your application and found this to be a major cause of slow down? If not, I would recommend using Java's version. Try not to pre-optimize as this will just cause development slow down. You may spend an extended amount of time on a problem that may not be a problem.
That being said, I think your test gave you your answer. If jni + C is slower, use java's version.
Commons Math3 ships with an optimized version: FastMath.exp(double x). It did speed up my code significantly.
Fabien ran some tests and found out that it was almost twice as fast as Math.exp():
0.75s for Math.exp sum=1.7182816693332244E7
0.40s for FastMath.exp sum=1.7182816693332244E7
Here is the javadoc:
Computes exp(x), function result is nearly rounded. It will be correctly rounded to the theoretical value for 99.9% of input values, otherwise it will have a 1 UPL error.
Method:
Lookup intVal = exp(int(x))
Lookup fracVal = exp(int(x-int(x) / 1024.0) * 1024.0 );
Compute z as the exponential of the remaining bits by a polynomial minus one
exp(x) = intVal * fracVal * (1 + z)
Accuracy: Calculation is done with 63 bits of precision, so result should be correctly rounded for 99.9% of input values, with less than 1 ULP error otherwise.
Since the Java code will get compiled to native code with the just-in-time (JIT) compiler, there's really no reason to use JNI to call native code.
Also, you shouldn't cache the results of a method where the input parameters are floating-point real numbers. The gains obtained in time will be very much lost in amount of space used.
The problem with using JNI is the overhead involved in making the call to JNI. The Java virtual machine is pretty optimized these days, and calls to the built-in Math.exp() are automatically optimized to call straight through to the C exp() function, and they might even be optimized into straight x87 floating-point assembly instructions.
There's simply an overhead associated with using the JNI, see also:
http://java.sun.com/docs/books/performance/1st_edition/html/JPNativeCode.fm.html
So as others have suggested try to collate operations that would involve using the JNI.
Write your own, tailored to your needs.
For instance, if all your exponents are of the power of two, you can use bit-shifting. If you work with a limited range or set of values, you can use look-up tables. If you don't need pin-point precision, you use an imprecise, but faster, algorithm.
There is a cost associated with calling across the JNI boundary.
If you could move the loop that calls exp() into the native code as well, so that there is just one native call, then you might get better results, but I doubt it will be significantly faster than the pure Java solution.
I don't know the details of your application, but if you have a fairly limited set of possible arguments for the call, you could use a pre-computed look-up table to make your Java code faster.
There are faster algorithms for exp depending on what your'e trying to accomplish. Is the problem space restricted to a certain range, do you only need a certain resolution, precision, or accuracy, etc.
If you define your problem very well, you may find that you can use a table with interpolation, for instance, which will blow nearly any other algorithm out of the water.
What constraints can you apply to exp to gain that performance trade-off?
-Adam
I run a fitting algorithm and the minimum error of the fitting result is way larger
than the precision of the Math.exp().
Transcendental functions are always much more slower than addition or multiplication and a well-known bottleneck. If you know that your values are in a narrow range, you can simply build a lookup-table (Two sorted array ; one input, one output). Use Arrays.binarySearch to find the correct index and interpolate value with the elements at [index] and [index+1].
Another method is to split the number. Lets take e.g. 3.81 and split that in 3+0.81.
Now you multiply e = 2.718 three times and get 20.08.
Now to 0.81. All values between 0 and 1 converge fast with the well-known exponential series
1+x+x^2/2+x^3/6+x^4/24.... etc.
Take as much terms as you need for precision; unfortunately it's slower if x approaches 1. Lets say you go to x^4, then you get 2.2445 instead of the correct 2.2448
Then multiply the result 2.781^3 = 20.08 with 2.781^0.81 = 2.2445 and you have the result
45.07 with an error of one part of two thousand (correct: 45.15).
It might not be relevant any more, but just so you know, in the newest releases of the OpenJDK (see here), Math.exp should be made an intrinsic (if you don't know what that is, check here).
This will make performance unbeatable on most architectures, because it means the Hotspot VM will replace the call to Math.exp by a processor-specific implementation of exp at runtime. You can never beat these calls, as they are optimized for the architecture...