I have been researching the log-sum-exp problem. I have a list of numbers stored as logarithms which I would like to sum and store in a logarithm.
the naive algorithm is
def naive(listOfLogs):
return math.log10(sum(10**x for x in listOfLogs))
many websites including:
logsumexp implementation in C?
and
http://machineintelligence.tumblr.com/post/4998477107/
recommend using
def recommend(listOfLogs):
maxLog = max(listOfLogs)
return maxLog + math.log10(sum(10**(x-maxLog) for x in listOfLogs))
aka
def recommend(listOfLogs):
maxLog = max(listOfLogs)
return maxLog + naive((x-maxLog) for x in listOfLogs)
what I don't understand is if recommended algorithm is better why should we call it recursively?
would that provide even more benefit?
def recursive(listOfLogs):
maxLog = max(listOfLogs)
return maxLog + recursive((x-maxLog) for x in listOfLogs)
while I'm asking are there other tricks to make this calculation more numerically stable?
Some background for others: when you're computing an expression of the following type directly
ln( exp(x_1) + exp(x_2) + ... )
you can run into two kinds of problems:
exp(x_i) can overflow (x_i is too big), resulting in numbers that you can't add together
exp(x_i) can underflow (x_i is too small), resulting in a bunch of zeroes
If all the values are big, or all are small, we can divide by some exp(const) and add const to the outside of the ln to get the same value. Thus if we can pick the right const, we can shift the values into some range to prevent overflow/underflow.
The OP's question is, why do we pick max(x_i) for this const instead of any other value? Why don't we recursively do this calculation, picking the max out of each subset and computing the logarithm repeatedly?
The answer: because it doesn't matter.
The reason? Let's say x_1 = 10 is big, and x_2 = -10 is small. (These numbers aren't even very large in magnitude, right?) The expression
ln( exp(10) + exp(-10) )
will give you a value very close to 10. If you don't believe me, go try it. In fact, in general, ln( exp(x_1) + exp(x_2) + ... ) will give be very close to max(x_i) if some particular x_i is much bigger than all the others. (As an aside, this functional form, asymptotically, actually lets you mathematically pick the maximum from a set of numbers.)
Hence, the reason we pick the max instead of any other value is because the smaller values will hardly affect the result. If they underflow, they would have been too small to affect the sum anyway, because it would be dominated by the largest number and anything close to it. In computing terms, the contribution of the small numbers will be less than an ulp after computing the ln. So there's no reason to waste time computing the expression for the smaller values recursively if they will be lost in your final result anyway.
If you wanted to be really persnickety about implementing this, you'd divide by exp(max(x_i) - some_constant) or so to 'center' the resulting values around 1 to avoid both overflow and underflow, and that might give you a few extra digits of precision in the result. But avoiding overflow is much more important about avoiding underflow, because the former determines the result and the latter doesn't, so it's much simpler just to do it this way.
Not really any better to do it recursively. The problem's just that you want to make sure your finite-precision arithmetic doesn't swamp the answer in noise. By dealing with the max on its own, you ensure that any junk is kept small in the final answer because the most significant component of it is guaranteed to get through.
Apologies for the waffly explanation. Try it with some numbers yourself (a sensible list to start with might be [1E-5,1E25,1E-5]) and see what happens to get a feel for it.
As you have defined it, your recursive function will never terminate. That's because ((x-maxlog) for x in listOfLogs) still has the same number of elements as listOfLogs.
I don't think that this is easily fixable either, without significantly impacting either the performance or the precision (compared to the non-recursive version).
Related
This is a continuation of the two questions posted here,
Declaring a functional recursive sequence in Matlab
Nesting a specific recursion in Pari-GP
To make a long story short, I've constructed a family of functions which solve the tetration functional equation. I've proven these things are holomorphic. And now it's time to make the graphs, or at least, somewhat passable code to evaluate these things. I've managed to get to about 13 significant digits in my precision, but if I try to get more, I encounter a specific error. That error is really nothing more than an overflow error. But it's a peculiar overflow error; Pari-GP doesn't seem to like nesting the logarithm.
My particular mathematical function is approximated by taking something large (think of the order e^e^e^e^e^e^e) to produce something small (of the order e^(-n)). The math inherently requires samples of large values to produce these small values. And strangely, as we get closer to numerically approximating (at about 13 significant digits or so), we also get closer to overflowing because we need such large values to get those 13 significant digits. I am a god awful programmer; and I'm wondering if there could be some work around I'm not seeing.
/*
This function constructs the approximate Abel function
The variable z is the main variable we care about; values of z where real(z)>3 almost surely produces overflow errors
The variable l is the multiplier of the approximate Abel function
The variable n is the depth of iteration required
n can be set to 100, but produces enough accuracy for about 15
The functional equation this satisfies is exp(beta_function(z,l,n))/(1+exp(-l*z)) = beta_function(z+1,l,n); and this program approaches the solution for n to infinity
*/
beta_function(z,l,n) =
{
my(out = 0);
for(i=0,n-1,
out = exp(out)/(exp(l*(n-i-z)) +1));
out;
}
/*
This function is the error term between the approximate Abel function and the actual Abel function
The variable z is the main variable we care about
The variable l is the multiplier
The variable n is the depth of iteration inherited from beta_function
The variable k is the new depth of iteration for this function
n can be set about 100, still; but 15 or 20 is more optimal.
Setting the variable k above 10 will usually produce overflow errors unless the complex arguments of l and z are large.
Precision of about 10 digits is acquired at k = 5 or 6 for real z, for complex z less precision is acquired. k should be set to large values for complex z and l with large imaginary arguments.
*/
tau_K(z,l,n,k)={
if(k == 1,
-log(1+exp(-l*z)),
log(1 + tau_K(z+1,l,n,k-1)/beta_function(z+1,l,n)) - log(1+exp(-l*z))
)
}
/*
This is the actual Abel function
The variable z is the main variable we care about
The variable l is the multiplier
The variable n is the depth of iteration inherited from beta_function
The variable k is the depth of iteration inherited from tau_K
The functional equation this satisfies is exp(Abl_L(z,l,n,k)) = Abl_L(z+1,l,n,k); and this function approaches that solution for n,k to infinity
*/
Abl_L(z,l,n,k) ={
beta_function(z,l,n) + tau_K(z,l,n,k);
}
This is the code for approximating the functions I've proven are holomorphic; but sadly, my code is just horrible. Here, is attached some expected output, where you can see the functional equation being satisfied for about 10 - 13 significant digits.
Abl_L(1,log(2),100,5)
%52 = 0.1520155156321416705967746811
exp(Abl_L(0,log(2),100,5))
%53 = 0.1520155156321485241351294757
Abl_L(1+I,0.3 + 0.3*I,100,14)
%59 = 0.3353395055605129001249035662 + 1.113155080425616717814647305*I
exp(Abl_L(0+I,0.3 + 0.3*I,100,14))
%61 = 0.3353395055605136611147422467 + 1.113155080425614418399986325*I
Abl_L(0.5+5*I, 0.2+3*I,100,60)
%68 = -0.2622549204469267170737985296 + 1.453935357725113433325798650*I
exp(Abl_L(-0.5+5*I, 0.2+3*I,100,60))
%69 = -0.2622549205108654273925182635 + 1.453935357685525635276573253*I
Now, you'll notice I have to change the k value for different values. When the arguments z,l are further away from the real axis, we can make k very large (and we have to to get good accuracy), but it'll still overflow eventually; typically once we've achieved about 13-15 significant digits, is when the functions will start to blow up. You'll note, that setting k =60, means we're taking 60 logarithms. This already sounds like a bad idea, lol. Mathematically though, the value Abl_L(z,l,infinity,infinity) is precisely the function I want. I know that must be odd; nested infinite for-loops sounds like nonsense, lol.
I'm wondering if anyone can think of a way to avoid these overflow errors and obtaining a higher degree of accuracy. In a perfect world, this object most definitely converges, and this code is flawless (albeit, it may be a little slow); but we'd probably need to increase the stacksize indefinitely. In theory this is perfectly fine; but in reality, it's more than impractical. Is there anyway, as a programmer, one can work around this?
The only other option I have at this point is to try and create a bruteforce algorithm to discover the Taylor series of this function; but I'm having less than no luck at doing this. The process is very unique, and trying to solve this problem using Taylor series kind of takes us back to square one. Unless, someone here can think of a fancy way of recovering Taylor series from this expression.
I'm open to all suggestions, any comments, honestly. I'm at my wits end; and I'm wondering if this is just one of those things where the only solution is to increase the stacksize indefinitely (which will absolutely work). It's not just that I'm dealing with large numbers. It's that I need larger and larger values to compute a small value. For that reason, I wonder if there's some kind of quick work around I'm not seeing. The error Pari-GP spits out is always with tau_K, so I'm wondering if this has been coded suboptimally; and that I should add something to it to reduce stacksize as it iterates. Or, if that's even possible. Again, I'm a horrible programmer. I need someone to explain this to me like I'm in kindergarten.
Any help, comments, questions for clarification, are more than welcome. I'm like a dog chasing his tail at this point; wondering why he can't take 1000 logarithms, lol.
Regards.
EDIT:
I thought I'd add in that I can produce arbitrary precision but we have to keep the argument of z way off in the left half plane. If the variables n,k = -real(z) then we can produce arbitrary accuracy by making n as large as we want. Here's some output to explain this, where I've used \p 200 and we pretty much have equality at this level (minus some digits).
Abl_L(-1000,1+I,1000,1000)
%16 = -0.29532276871494189936534470547577975723321944770194434340228137221059739121428422475938130544369331383702421911689967920679087535009910425871326862226131457477211238400580694414163545689138863426335946 + 1.5986481048938885384507658431034702033660039263036525275298731995537068062017849201570422126715147679264813047746465919488794895784667843154275008585688490133825421586142532469402244721785671947462053*I
exp(Abl_L(-1001,1+I,1000,1000))
%17 = -0.29532276871494189936534470547577975723321944770194434340228137221059739121428422475938130544369331383702421911689967920679087535009910425871326862226131457477211238400580694414163545689138863426335945 + 1.5986481048938885384507658431034702033660039263036525275298731995537068062017849201570422126715147679264813047746465919488794895784667843154275008585688490133825421586142532469402244721785671947462053*I
Abl_L(-900 + 2*I, log(2) + 3*I,900,900)
%18 = 0.20353875452777667678084511743583613390002687634123569448354843781494362200997943624836883436552749978073278597542986537166527005507457802227019178454911106220050245899257485038491446550396897420145640 - 5.0331931122239257925629364016676903584393129868620886431850253696250415005420068629776255235599535892051199267683839967636562292529054669236477082528566454129529102224074017515566663538666679347982267*I
exp(Abl_L(-901+2*I,log(2) + 3*I,900,900))
%19 = 0.20353875452777667678084511743583613390002687634123569448354843781494362200997943624836883436552749978073278597542986537166527005507457802227019178454911106220050245980468697844651953381258310669530583 - 5.0331931122239257925629364016676903584393129868620886431850253696250415005420068629776255235599535892051199267683839967636562292529054669236477082528566454129529102221938340371793896394856865112060084*I
Abl_L(-967 -200*I,12 + 5*I,600,600)
%20 = -0.27654907399026253909314469851908124578844308887705076177457491260312326399816915518145788812138543930757803667195961206089367474489771076618495231437711085298551748942104123736438439579713006923910623 - 1.6112686617153127854042520499848670075221756090591592745779176831161238110695974282839335636124974589920150876805977093815716044137123254329208112200116893459086654166069454464903158662028146092983832*I
exp(Abl_L(-968 -200*I,12 + 5*I,600,600))
%21 = -0.27654907399026253909314469851908124578844308887705076177457491260312326399816915518145788812138543930757803667195961206089367474489771076618495231437711085298551748942104123731995533634133194224880928 - 1.6112686617153127854042520499848670075221756090591592745779176831161238110695974282839335636124974589920150876805977093815716044137123254329208112200116893459086654166069454464833417170799085356582884*I
The trouble is, we can't just apply exp over and over to go forward and expect to keep the same precision. The trouble is with exp, which displays so much chaotic behaviour as you iterate it in the complex plane, that this is doomed to work.
Well, I answered my own question. #user207421 posted a comment, and I'm not sure if it meant what I thought it meant, but I think it got me to where I want. I sort of assumed that exp wouldn't inherit the precision of its argument, but apparently that's true. So all I needed was to define,
Abl_L(z,l,n,k) ={
if(real(z) <= -max(n,k),
beta_function(z,l,n) + tau_K(z,l,n,k),
exp(Abl_L(z-1,l,n,k)));
}
Everything works perfectly fine from here; of course, for what I need it for. So, I answered my own question, and it was pretty simple. I just needed an if statement.
Thanks anyway, to anyone who read this.
In summary, I am trying to start at a given x and find the nearest point in the positive direction where f(x) = 0. For simplicity, solutions are only needed in the interval [initial_x, maximum_x] (the maximum is given), but any better reach is desirable. Additionally, a specific precision is not mandatory; I am looking to maximize it, but not at the cost of performance.
While this seems simple, there are a few caveats that make the solution more difficult.
Performance is the first priority, even over some precision. The zero needs to be found in the fewest possible calls to f(x), as this code will be run many times per second.
There are not guaranteed to be any specific number of zeros on this line. There may be zero, one, or many places that the function intersects the x-axis. (This is why a direct binary search will not work.)
The function f(x) cannot be manipulated algebraically, only supporting numerical evaluation at a discrete point. (This is why the solution cannot be found analytically.)
My current strategy is to define a step size that is within an acceptable loss of precision and then test in increments until an interval is found on which there is guaranteed to be at least one zero (in [a,b], a and b are on opposite sides of 0). From there, I use a binary search to narrow down the (more) exact point.
// assuming y != 0
initial_y = f(x);
while (x < maximum_x) {
y = f(x);
// test to see if y has crossed 0
if (initial_y > 0) {
if (y < 0) {
return binary_search(x - step_size, x);
}
} else {
if (y > 0) {
return binary_search(x - step_size, x);
}
}
x += step_size;
}
This has several disadvantages, mainly the fact that there is a significant trade-off between resolution and performance (the smaller step_size is, the better it works but the longer it takes). Is there a more efficient formula or strategy I can take? I thought of using the value of y to scale the step size, but I cannot figure out how to preserve precision while doing that.
The solution can be in any language because I am looking more for a strategy to find the zeros, than a specific program.
(edit:)
The function above is assumed to be continuous.
To clarify the question, I understand that this problem may be impossible to solve exactly. I am just asking for ways to improve the speed or precision of the algorithm. The one I am currently using is working quite well, even though it fails during many edge cases.
For example, a solution that requires fewer steps with similar precision or another algorithm that increases the precision or reliability with some performance impact would both be extremely helpful.
Your problem is essentially impossible to solve in the general case. For example, no algorithm can find the "first" root of sin(1/x), starting from x=0.
A tentative answer is by exponential search, i.e. starting from a small step and increase it following a geometric progression rather than an arithmetic one, until you find a change of sign. But this will fail if the first root is closer than the initial step, or if the first root is followed by a close one.
Without any information on the behavior of f, I would not even try anything (but a "standard" root finder), this is too hopeless ! (But I am sure you do have some information.)
First off, apologies if there is a better way to format math equations, I could not find anything, but alas, the expressions are pretty short.
As part of an assigned problem I have to produce some code in C that will evaluate x^n/n! for an arbitrary x, and n = { 1-10 , 50, 100}
I can always brute force it with a large number library, but I am wondering if someone with better math skills then mine can suggest a better algorithm than something with a O(n!)...
I understand that I can split the numerator to x^(n/2)x^(n/2) for even values of n, and xx^(n-1/2)*x^(n-1/2) for odd values of n. And that I can further change that into a logarithm base x of n/2.
But I am stuck for multiple reasons:
1 - I do not think that computationally any of these changes actually make a lot of difference since they are not really helping me reduce the large number multiplications I have to perform, or their overall number.
2 - Even as I think of n! as 1*2*3*...*(n-1)*n, I still cannot rationalize a good way to simplify the overall equation.
3 - I have looked at Karatsuba's algorithm for multiplications, and although it is a possibility, it seems a bit complex for an intro to programming problem.
So I am wondering if you guys can think of any middle ground. I prefer explanations to straight answers if you have the time :)
Cheers,
My advice is to compute all the terms of the summation (put them in an array), and then sum them up in reverse order (i.e., smallest to largest) -- that reduces rounding error a little bit.
Note that you can compute the k-th term from the preceding one by multiplying by x/k -- you do not need to ever compute x^n or n! directly (this is important).
(I'm not sure whether I should post this problem on this site or on the math site. Please feel free to migrate this post if necessary.)
My problem at hand is that given a value of k I'd like to numerically compute a rational function of nonlinear polynomials in k which looks like the following: (sorry I don't know how to typeset equations here...)
where {a_0, ..., a_N; b_0, ..., b_N} are complex constants, {u_0, ..., u_N, v_0, ..., v_N} are real constants and i is the imaginary number. I learned from Numerical Recipes that there are whole bunch of ways to compute polynomials quickly, in the meanwhile keeping the rounding error small enough, if all coefficients were constant. But I do not think those ideas are useful in my case since the exponential prefactors also depend on k.
Currently I calculate it in a brute force way in C with complex.h (this is just a pseudo code):
double complex function(double k)
{
return (a_0+a_1*cexp(I*u_1*k)*k+a_2*cexp(I*u_2*k)*k*k+...)/(b_0+b_1*cexp(I*v_1*k)*k+v_2*cexp(I*v_2*k)*k*k+...);
}
However when the number of calls of function increases (because this is just a part of my real calculation), it is very slow and inaccurate (only 6 valid digits). I appreciate any comments and/or suggestions.
I trust that this isn't a homework assignment!
Normally the trick is to use a loop add the next coefficient to the running sum, and multiply by k. However, in your case, I think the "e" term in the coefficient is going to overwhelm any savings by factoring out k. You can still do it, but the savings will probably be small.
Is u_i a constant? Depending on how many times you need to run this formula, maybe you could premultiply u_i * k (unless k changes each run). It's been so many decades since I took a Numerical Analysis course that I have only vague recollections of the tricks of the trade. Let's see... is e^(i*u_i*k) the same as (e^(i*u_i))^k? I don't remember the rules on imaginary numbers, or whether you'll save anything since you've got a real^real (assuming k is real) anyway (internally done using e^power).
If you're getting only 6 digits, that suggests that your math, and maybe your library, is working in single precision (32 bit) reals. Check your library and check your declarations that you are using at least double precision (64 bit) reals everywhere.
As a programmer I think it is my job to be good at math but I am having trouble getting my head round imaginary numbers. I have tried google and wikipedia with no luck so I am hoping a programmer can explain in to me, give me an example of a number squared that is <= 0, some example usage etc...
I guess this blog entry is one good explanation:
The key word is rotation (as opposed to direction for negative numbers, which are as stranger as imaginary number when you think of them: less than nothing ?)
Like negative numbers modeling flipping, imaginary numbers can model anything that rotates between two dimensions “X” and “Y”. Or anything with a cyclic, circular relationship
Problem: not only am I a programmer, I am a mathematician.
Solution: plow ahead anyway.
There's nothing really magical to complex numbers. The idea behind their inception is that there's something wrong with real numbers. If you've got an equation x^2 + 4, this is never zero, whereas x^2 - 2 is zero twice. So mathematicians got really angry and wanted there to always be zeroes with polynomials of degree at least one (wanted an "algebraically closed" field), and created some arbitrary number j such that j = sqrt(-1). All the rules sort of fall into place from there (though they are more accurately reorganized differently-- specifically, you formally can't actually say "hey this number is the square root of negative one"). If there's that number j, you can get multiples of j. And you can add real numbers to j, so then you've got complex numbers. The operations with complex numbers are similar to operations with binomials (deliberately so).
The real problem with complexes isn't in all this, but in the fact that you can't define a system whereby you can get the ordinary rules for less-than and greater-than. So really, you get to where you don't define it at all. It doesn't make sense in a two-dimensional space. So in all honesty, I can't actually answer "give me an exaple of a number squared that is <= 0", though "j" makes sense if you treat its square as a real number instead of a complex number.
As for uses, well, I personally used them most when working with fractals. The idea behind the mandelbrot fractal is that it's a way of graphing z = z^2 + c and its divergence along the real-imaginary axes.
You might also ask why do negative numbers exist? They exist because you want to represent solutions to certain equations like: x + 5 = 0. The same thing applies for imaginary numbers, you want to compactly represent solutions to equations of the form: x^2 + 1 = 0.
Here's one way I've seen them being used in practice. In EE you are often dealing with functions that are sine waves, or that can be decomposed into sine waves. (See for example Fourier Series).
Therefore, you will often see solutions to equations of the form:
f(t) = A*cos(wt)
Furthermore, often you want to represent functions that are shifted by some phase from this function. A 90 degree phase shift will give you a sin function.
g(t) = B*sin(wt)
You can get any arbitrary phase shift by combining these two functions (called inphase and quadrature components).
h(t) = Acos(wt) + iB*sin(wt)
The key here is that in a linear system: if f(t) and g(t) solve an equation, h(t) will also solve the same equation. So, now we have a generic solution to the equation h(t).
The nice thing about h(t) is that it can be written compactly as
h(t) = Cexp(wt+theta)
Using the fact that exp(iw) = cos(w)+i*sin(w).
There is really nothing extraordinarily deep about any of this. It is merely exploiting a mathematical identity to compactly represent a common solution to a wide variety of equations.
Well, for the programmer:
class complex {
public:
double real;
double imaginary;
complex(double a_real) : real(a_real), imaginary(0.0) { }
complex(double a_real, double a_imaginary) : real(a_real), imaginary(a_imaginary) { }
complex operator+(const complex &other) {
return complex(
real + other.real,
imaginary + other.imaginary);
}
complex operator*(const complex &other) {
return complex(
real*other.real - imaginary*other.imaginary,
real*other.imaginary + imaginary*other.real);
}
bool operator==(const complex &other) {
return (real == other.real) && (imaginary == other.imaginary);
}
};
That's basically all there is. Complex numbers are just pairs of real numbers, for which special overloads of +, * and == get defined. And these operations really just get defined like this. Then it turns out that these pairs of numbers with these operations fit in nicely with the rest of mathematics, so they get a special name.
They are not so much numbers like in "counting", but more like in "can be manipulated with +, -, *, ... and don't cause problems when mixed with 'conventional' numbers". They are important because they fill the holes left by real numbers, like that there's no number that has a square of -1. Now you have complex(0, 1) * complex(0, 1) == -1.0 which is a helpful notation, since you don't have to treat negative numbers specially anymore in these cases. (And, as it turns out, basically all other special cases are not needed anymore, when you use complex numbers)
If the question is "Do imaginary numbers exist?" or "How do imaginary numbers exist?" then it is not a question for a programmer. It might not even be a question for a mathematician, but rather a metaphysician or philosopher of mathematics, although a mathematician may feel the need to justify their existence in the field. It's useful to begin with a discussion of how numbers exist at all (quite a few mathematicians who have approached this question are Platonists, fyi). Some insist that imaginary numbers (as the early Whitehead did) are a practical convenience. But then, if imaginary numbers are merely a practical convenience, what does that say about mathematics? You can't just explain away imaginary numbers as a mere practical tool or a pair of real numbers without having to account for both pairs and the general consequences of them being "practical". Others insist in the existence of imaginary numbers, arguing that their non-existence would undermine physical theories that make heavy use of them (QM is knee-deep in complex Hilbert spaces). The problem is beyond the scope of this website, I believe.
If your question is much more down to earth e.g. how does one express imaginary numbers in software, then the answer above (a pair of reals, along with defined operations of them) is it.
I don't want to turn this site into math overflow, but for those who are interested: Check out "An Imaginary Tale: The Story of sqrt(-1)" by Paul J. Nahin. It talks about all the history and various applications of imaginary numbers in a fun and exciting way. That book is what made me decide to pursue a degree in mathematics when I read it 7 years ago (and I was thinking art). Great read!!
The main point is that you add numbers which you define to be solutions to quadratic equations like x2= -1. Name one solution to that equation i, the computation rules for i then follow from that equation.
This is similar to defining negative numbers as the solution of equations like 2 + x = 1 when you only knew positive numbers, or fractions as solutions to equations like 2x = 1 when you only knew integers.
It might be easiest to stop trying to understand how a number can be a square root of a negative number, and just carry on with the assumption that it is.
So (using the i as the square root of -1):
(3+5i)*(2-i)
= (3+5i)*2 + (3+5i)*(-i)
= 6 + 10i -3i - 5i * i
= 6 + (10 -3)*i - 5 * (-1)
= 6 + 7i + 5
= 11 + 7i
works according to the standard rules of maths (remembering that i squared equals -1 on line four).
An imaginary number is a real number multiplied by the imaginary unit i. i is defined as:
i == sqrt(-1)
So:
i * i == -1
Using this definition you can obtain the square root of a negative number like this:
sqrt(-3)
== sqrt(3 * -1)
== sqrt(3 * i * i) // Replace '-1' with 'i squared'
== sqrt(3) * i // Square root of 'i squared' is 'i' so move it out of sqrt()
And your final answer is the real number sqrt(3) multiplied by the imaginary unit i.
A short answer: Real numbers are one-dimensional, imaginary numbers add a second dimension to the equation and some weird stuff happens if you multiply...
If you're interested in finding a simple application and if you're familiar with matrices,
it's sometimes useful to use complex numbers to transform a perfectly real matrice into a triangular one in the complex space, and it makes computation on it a bit easier.
The result is of course perfectly real.
Great answers so far (really like Devin's!)
One more point:
One of the first uses of complex numbers (although they were not called that way at the time) was as an intermediate step in solving equations of the 3rd degree.
link
Again, this is purely an instrument that is used to answer real problems with real numbers having physical meaning.
In electrical engineering, the impedance Z of an inductor is jwL, where w = 2*pi*f (frequency) and j (sqrt(-1))means it leads by 90 degrees, while for a capacitor Z = 1/jwc = -j/wc which is -90deg/wc so that it lags a simple resistor by 90 deg.