On calculating Fibonacci numbers in C(blog.noblemail.ca)
blog.noblemail.ca
On calculating Fibonacci numbers in C
http://blog.noblemail.ca/2013/01/on-calculating-fibonacci-numbers-in-c.html
12 comments
While I agree with the gist of your comment, it's important to note that none of these give an O(lg(n)) algorithm for computing fibonacci numbers (except on a hypothetical computer that can do arbitrary size integer operations in O(1) time).
The nth fibonacci number has O(n) digits, so you can't even write it out in time less than O(n). The repeated-squaring approach will involve multiplying numbers with O(n/2) = O(n) digits in its final step, so a very coarse analysis says the complexity is at least O(n lg n).
The nth fibonacci number has O(n) digits, so you can't even write it out in time less than O(n). The repeated-squaring approach will involve multiplying numbers with O(n/2) = O(n) digits in its final step, so a very coarse analysis says the complexity is at least O(n lg n).
This is completely missing the point of Big-Oh notation. If you start trying to analyze how long integer operations take, ALL your computational complexity analysis falls apart for any sufficiently large integer arithmetic for any algorithm. This is precisely why such operations are ignored, because it becomes completely impossible to do any meaningful platform-independent analysis if you try to take that into account.
That's not to say you shouldn't take it into account, but the algorithm he described is, in fact, O(log(n)) time. It just is. That's how the math works. Now, you should note that the O(log(n)) time constraint does not take into account arbitrary integer size which will cause progressively larger constant factors to influence the efficiency of the algorithm... but it is still a O(log(n)) algorithm.
That's not to say you shouldn't take it into account, but the algorithm he described is, in fact, O(log(n)) time. It just is. That's how the math works. Now, you should note that the O(log(n)) time constraint does not take into account arbitrary integer size which will cause progressively larger constant factors to influence the efficiency of the algorithm... but it is still a O(log(n)) algorithm.
My analysis holds for any implementation for which you can perform O(1) addition and multiplication of integers of some fixed-size. That's just about as platform-independent as you can get, but it still captures the real complexity of the operation in a way that handwavy "pretend you can do arbitrary arithmetic in O(1)" analyses do not.
This isn't simply my viewpoint on the subject; even the very theoretical textbook by Papadimitriou and Vazirani takes this approach to the question (see exercise 0.4 and note that everything is carried out in terms of the complexity of arbitrary-size integer multiplication M(n)).
This isn't simply my viewpoint on the subject; even the very theoretical textbook by Papadimitriou and Vazirani takes this approach to the question (see exercise 0.4 and note that everything is carried out in terms of the complexity of arbitrary-size integer multiplication M(n)).
>progressively larger constant factors
If they are getting progressively larger with larger inputs, then they are not in fact constant factors.
If they are getting progressively larger with larger inputs, then they are not in fact constant factors.
The algorithm is also relevant if you want to compute the nth Fibonacci number mod k.
You're conflating the magnitudes. You can't directly compare the number of multiplications with the number of transistor switches or CPU cycles required for a single multiplication.
Also, the number n has O(log(n)) digits.
Also, the number n has O(log(n)) digits.
The number n has O(log(n)) digits, but that has nothing to do with the subject at hand. The nth fibonacci number has O(n) digits.
And nothing in my comment relates to "number of transistor switches or CPU cycles required for a single multiplication".
And nothing in my comment relates to "number of transistor switches or CPU cycles required for a single multiplication".
In the same vein, there's also an O(log(fib(n))) recursive algorithm:
fib :: Integer -> Integer
fib 0 = 0
fib 1 = 1
fib 2 = 1
fib n = case n `quotRem` 2 of
-- if n = 2*m
(m,0) -> (fib (m+1))^2 - (fib (m-1))^2
-- if n = 2*m + 1
(m,1) -> (fib (m+1))^2 + (fib (m ))^2If you "drop" the 1/2, you can use Z(sqrt(5)) and work with only a pair of integers. Also, the two parts of the formula are "conjugates" so it's possible to get the exact result with only one half.
Define: Im5(a+b * sqrt(5)) =b
Fib(n)= Im5 ((1+1* sqrt(5))^n) / 2^(n-1)
Define: Im5(a+b * sqrt(5)) =b
Fib(n)= Im5 ((1+1* sqrt(5))^n) / 2^(n-1)
That's how I would approach this if arbitarily large integers were needed. It's particularly nice because it readily generalizes to any other finite depth linear recurrence.
The analytic solution really should win for very large terms though, but I'm not sure how to implement it properly. Can someone outline how you would decide what precision is needed to compute the Nth Fibonacci number?
The analytic solution really should win for very large terms though, but I'm not sure how to implement it properly. Can someone outline how you would decide what precision is needed to compute the Nth Fibonacci number?
The nth term of the fibbonochi sequance can be written in the form (A^n-B^n)/sqrt(5) [1], where A=(1+sqrt(5)/2) and B=(1-sqrt(5)/2)
Lets say that A` and B` are approximations of A and B, that are within E of the actual answer, that is to say |A`-A|<E and |B`-B|<E.
The total error can be found by:
|(A^n-B^n)/sqrt(5) - (A`^n-B`^n)/sqrt(5)|
If we can make this error be less than 1/2, we can simply round to the nearest integer and have the correct answer.
|(A^n-B^n)/sqrt(5) - (A`^n-B`^n)/sqrt(5)| < 1/2
|(A^n - A`^n + B`^n - B^n)| < sqrt(5)/2
Notice that: |(A^n - A`^n + B`^n - B^n)| <= |(A^n - A`^n| + |B`^n - B^n)|
If both of those absolute values are less than sqrt(5)/4, than their sum would be less than sqrt(5)/2, satisfying the inequality. Because 1<sqrt(5/4), I will make them be less than 1. Without loss of generality, I will consider only one of the absolute values.
|A`^n - A^n| < 1
Assuming the maximum error:
|(A+E)^n - A^n| < 1
Notice that the first term when you expand the binomial is A^n, so the n-1 remaining terms are our error. Every coefficient is less than 2^n. The greatest that A factor can be is A^(n-1)<A^n. And the greatest that the E factor can be is E^1. Therefore all (n-1) factors are less than (2^n)(A^n)E.
The sum of these error term is therefore less than (n-1)(2^n)(A^n)E. So:
(n-1)(2^n)(A^n)E < 1
Because A<2, A^n<2^n we can say: (n-1)2(2^n)E < 1
E < 1/(2(n-1)(2^n))
If our value for A is accurate for d+1 (binary) digits passed the decimal point, E<=2^(-d).
2^(-d) < 1/(2(n-1)(2^n))
2^d > 2(n-1)(2^n)
d>lg(2)+lg(n-1)+lg(2^n)
d>1+lg(n-1)+n
d>2n
I made a lot of simplifications in this analysis, so you can probably get away with fewer terms. It is interesting to note that Fibonacci numbers grow exponentially, so the number of digits needed simply to right the answer grows linearly, so the extra digits do not decrease the algorithmic efficiency.
[1]http://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_ex...
Lets say that A` and B` are approximations of A and B, that are within E of the actual answer, that is to say |A`-A|<E and |B`-B|<E.
The total error can be found by:
|(A^n-B^n)/sqrt(5) - (A`^n-B`^n)/sqrt(5)|
If we can make this error be less than 1/2, we can simply round to the nearest integer and have the correct answer.
|(A^n-B^n)/sqrt(5) - (A`^n-B`^n)/sqrt(5)| < 1/2
|(A^n - A`^n + B`^n - B^n)| < sqrt(5)/2
Notice that: |(A^n - A`^n + B`^n - B^n)| <= |(A^n - A`^n| + |B`^n - B^n)|
If both of those absolute values are less than sqrt(5)/4, than their sum would be less than sqrt(5)/2, satisfying the inequality. Because 1<sqrt(5/4), I will make them be less than 1. Without loss of generality, I will consider only one of the absolute values.
|A`^n - A^n| < 1
Assuming the maximum error:
|(A+E)^n - A^n| < 1
Notice that the first term when you expand the binomial is A^n, so the n-1 remaining terms are our error. Every coefficient is less than 2^n. The greatest that A factor can be is A^(n-1)<A^n. And the greatest that the E factor can be is E^1. Therefore all (n-1) factors are less than (2^n)(A^n)E.
The sum of these error term is therefore less than (n-1)(2^n)(A^n)E. So:
(n-1)(2^n)(A^n)E < 1
Because A<2, A^n<2^n we can say: (n-1)2(2^n)E < 1
E < 1/(2(n-1)(2^n))
If our value for A is accurate for d+1 (binary) digits passed the decimal point, E<=2^(-d).
2^(-d) < 1/(2(n-1)(2^n))
2^d > 2(n-1)(2^n)
d>lg(2)+lg(n-1)+lg(2^n)
d>1+lg(n-1)+n
d>2n
I made a lot of simplifications in this analysis, so you can probably get away with fewer terms. It is interesting to note that Fibonacci numbers grow exponentially, so the number of digits needed simply to right the answer grows linearly, so the extra digits do not decrease the algorithmic efficiency.
[1]http://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_ex...
This was linked to on (yet another) Fibonacci post:
http://bosker.wordpress.com/2011/07/27/computing-fibonacci-n... and gives a somewhat better bound, and some performance tests.
Only the first 90ish fibonacci numbers fit in a 4 byte int. Use a lookup table, make it O(1).
"....since the 96th Fibonacci number is the largest to fit in an unsigned long int"
Have to agree with the first person to comment on the article's site. If we are going for speed, just pre-calc the 96 numbers and do an array lookup in the function guarded by and if.
Have to agree with the first person to comment on the article's site. If we are going for speed, just pre-calc the 96 numbers and do an array lookup in the function guarded by and if.
The article really isn't about speed, it's more that engineers shouldn't necessarily take mathematical results as the end-all to their considerations when developing an algorithm. Math is important to formulating algorithms, since it can provide, in theory, a closed form solution for something typically done iteratively (the example here: fibonacci numbers). But you still need to verify when an algorithm is going to be correct. In this case, it's pretty well known how inaccurate floats and doubles can be within X significant figures, and this is why I've always said the analytical solution doesn't really work on a computer unless you have a CAS library that can approximate an exact answer to an arbitrary number of significant figures (and it still won't be accurate if shoved into a float/double afterwards). So there's a trade-off, and choosing the theoretically best solution (a closed form solution) isn't necessarily the best possible choice for the task. In this case, yes, if we only need the first 96 fibonacci numbers, ever, in our program, it makes more sense to pre-compute them. However, if we only do the computation once in the entire program, even that doesn't necessarily make sense. It entirely depends on the application, but that's the entire point: a closed form for fibonacci numbers isn't the end of considerations.
I am curious though, why is it so many people are oblivious that there's an O(logN) time algorithm to compute fibonacci numbers iteratively on integers? It's a beautiful piece of math, and very simple so everyone should understand it. Just notice that:
I am curious though, why is it so many people are oblivious that there's an O(logN) time algorithm to compute fibonacci numbers iteratively on integers? It's a beautiful piece of math, and very simple so everyone should understand it. Just notice that:
((1, 1), (1, 0))^N -> ((f_{N+1}, f_N), (f_N, f_{N-1}))
I get the point of the article, but it seems like defaulting to a compiler optimization that might not be in other compilers is a poorer solution than examining the problem for another Math solution. You yourself have shown one.
Computer precision is a problem and programmers should learn a lot about how ints and fp work, but throwing out all Mathematical solutions is really not a good idea.
Computer precision is a problem and programmers should learn a lot about how ints and fp work, but throwing out all Mathematical solutions is really not a good idea.
I'd greatly appreciate it if you left this as a comment on the original post.
Sure, but it's not a post about the bestest solution. It's a post about how using more advanced math can ("can" not "will") lead to a worse solution for your context. Fibonacci numbers in a long int is certainly a toy example; but it's a toy example that had an actual reference.
This is basically a table lookup. You could do the same for any series of numbers like the prime numbers.
You could store them in a database table even and then pull out the result by a key with the number assigned to it that represents the Nth Fibonacci number. It might even be faster than an array loaded from a text file, and if you happen to have limited memory and cannot store the entire array in memory some how later on when you use a different data type for large numbers and go past 96 numbers, the database method may even be faster and use less memory. You might even have to store the big numbers as a blob or text or something if the database cannot handle numbers that big.
You could store them in a database table even and then pull out the result by a key with the number assigned to it that represents the Nth Fibonacci number. It might even be faster than an array loaded from a text file, and if you happen to have limited memory and cannot store the entire array in memory some how later on when you use a different data type for large numbers and go past 96 numbers, the database method may even be faster and use less memory. You might even have to store the big numbers as a blob or text or something if the database cannot handle numbers that big.
If I don't have room to store 96 numbers in an array in the function, I am going to have problems elsewhere.
You have to think ahead. You have 96 numbers now, but after 96 you have to change the data type to a larger storage to be able to store larger numbers. So when you pre-calc say 10,000 numbers what happens to your array then? What if the program is run on an embedded system for some reason that has memory limits? Like say someone finds a reason to use Fibonacci numbers on an embedded system to monitor machines and control machines and other stuff?
You don't know a reason to use Fibonacci numbers, but that doesn't mean someone else might find a way to use them.
You don't know a reason to use Fibonacci numbers, but that doesn't mean someone else might find a way to use them.
Given the limits of the data type, I really am not going to think ahead until I change the data type[1]. If the new datatype can hold 10,000 numbers, then I will probably go back to an algorithm.
I am really not sure about these embedded systems that have more RAM than ROM / EPROM / Flash. 384 bytes used for constant time performance in and embedded situation sounds pretty good to me.
1) not only would that be premature optimization, it would also change a function from constant time to variable. I do wonder about how big the stack frame is.
I am really not sure about these embedded systems that have more RAM than ROM / EPROM / Flash. 384 bytes used for constant time performance in and embedded situation sounds pretty good to me.
1) not only would that be premature optimization, it would also change a function from constant time to variable. I do wonder about how big the stack frame is.
You haven't worked with embedded systems, have you? 96 numbers is 384 bytes. That's a lot of space for some applications.
actually, in the distant past, I did. I don't remember a lot of embedded systems where Fibonacci numbers were important or where floating point was a real option (been awhile, probably same length of time since 384 bytes was the killer). I was always RAM constrained more than ROM /EPROM.
and only perform any optimizations at all after you have proved its a bottleneck for your application (unlikely) :p
I just thought it was funny the article took a shortcut on speed testing because only 96 values were possible from the function. At that point, we are no longer talking about math or math error, we are talking a simple array lookup.
I get the "less math" part of the article, but it just seemed a bit confused.
I get the "less math" part of the article, but it just seemed a bit confused.
I used to use the two bucket method to generate a Fibonacci number, thanks for showing me another way to do it.
I am working on Ackermann Numbers now, but it takes a very long time to generate them and I have to either write my own classes to handle big numbers or using a different library to use large numbers. I moved from C to C++ to do that. Your unsigned long int variable would run out of room after the 96th Fibonacci number, and you might have to move to C++ and use a custom class as well to be able to handle those big numbers.
I am working on Ackermann Numbers now, but it takes a very long time to generate them and I have to either write my own classes to handle big numbers or using a different library to use large numbers. I moved from C to C++ to do that. Your unsigned long int variable would run out of room after the 96th Fibonacci number, and you might have to move to C++ and use a custom class as well to be able to handle those big numbers.
Note that the author did write "since the 96th Fibonacci number is the largest to fit in an unsigned long int".
I still don't understand how the author jumped from a cute comparison of Fibonacci algorithms to saying programmers shouldn't learn more math. Math isn't just arithmetic and numbers. He acts as if the recursive solution isn't also math.
Developing mathematical maturity is, in my opinion, an excellent way to think more clearly, precisely, and rationally about any problem, even if it doesn't necessarily involve numbers. The author could make the case that number theory in particular isn't directly applicable to many problems (I do have a lot of fun with Project Euler though), but again, math isn't just about numbers. Math teaches you how to reason about the relationships and patterns between objects (which can be numbers, vectors, spaces, groups, rings, etc. In a certain sense, you become more comfortable with higher levels abstraction, and it requires you to be precise and explicit about assumptions.
That's jut my two cents. Math education is pretty dismal in the US, but math can be very enjoyable if you find the right textbook, teacher, or even free online class nowadays. Never fear math.
EDIT: The author has pointed out that he is making a play on words between "math" and "Math" (the mathematical library). I didn't understand that when I wrote this post.
Developing mathematical maturity is, in my opinion, an excellent way to think more clearly, precisely, and rationally about any problem, even if it doesn't necessarily involve numbers. The author could make the case that number theory in particular isn't directly applicable to many problems (I do have a lot of fun with Project Euler though), but again, math isn't just about numbers. Math teaches you how to reason about the relationships and patterns between objects (which can be numbers, vectors, spaces, groups, rings, etc. In a certain sense, you become more comfortable with higher levels abstraction, and it requires you to be precise and explicit about assumptions.
That's jut my two cents. Math education is pretty dismal in the US, but math can be very enjoyable if you find the right textbook, teacher, or even free online class nowadays. Never fear math.
EDIT: The author has pointed out that he is making a play on words between "math" and "Math" (the mathematical library). I didn't understand that when I wrote this post.
I do distinguish between math and Math. Decide for yourself what that distinction means.
The shift key?
I reread the article with that in mind, and it makes much more sense, to the point that I am inclined to agree with much of what you said now. I think you should highlight that difference closer to the beginning of the article. That is a nice, subtle play on words.
how much emphasis to give to which messages is always the hardest part for me to writing these posts. I'll keep in mind that at times I can be too subtle.
The analytic solution to the Fibonacci sequence is nice and all, but I generally dislike methods that require you to very carefully check how much precision you need. I much prefer the matrix exponentiation by repeated squaring approach [1].
1: http://ronzii.wordpress.com/2011/07/09/using-matrix-exponent...
1: http://ronzii.wordpress.com/2011/07/09/using-matrix-exponent...
I came to the exact opposite conclusion as the author. Programmers should know a lot of things, including both recursion and math since tere are many ways to approach a problem.
Wait Im confused. Wouldnt the point of using the analytic approach be that i could start at any n besides 0? Because of course if im just taking the sums from 0 to n it'd be faster to do it the tail recursive way because thats just simple addition.
The post didn't go into this, but the sum has a lovely closed form as well. The sum of the (phi^n-psi^n) terms is the difference of two geometric series, so you can apply sum(r^i, i=0..n)=(r^(n+1)-1)/(r-1) to get a formula that evaluates the sum exactly.
Re-commenting on OP @ author's request:
The article really isn't about speed, it's more that engineers shouldn't necessarily take mathematical results as the end-all to their considerations when developing an algorithm. In particular, what is considered "best" in theory is not necessarily the best choice in a given program.
Math is important to formulating algorithms, since it can provide, in theory, a closed form solution for something typically done iteratively (the example here: fibonacci numbers). But you still need to verify when an algorithm is going to be correct, and more importantly, appropriate. In this case, it's pretty well known how inaccurate floats and doubles can be within X significant figures, and this is why I've always said the analytical solution doesn't really work on a computer unless you have a CAS library that can approximate an exact answer to an arbitrary number of significant figures (and it still won't be accurate if shoved into a float/double afterwards). So there's a trade-off, and choosing the theoretically best solution (a closed form solution) isn't necessarily the best possible choice for the task. In this case, yes, as others have pointed out, if we only need the first 96 fibonacci numbers, ever, it makes more sense to pre-compute them. However, if we only do the computation once in the entire program, even that doesn't necessarily make sense. It entirely depends on the application, but that's the entire point: a closed form for fibonacci numbers isn't the end of considerations.
I am curious though, why is it so many people are oblivious that there's an O(logN) time algorithm to compute fibonacci numbers iteratively on integers? It's a beautiful piece of math, and very simple so everyone should understand it. Just notice that:
The article really isn't about speed, it's more that engineers shouldn't necessarily take mathematical results as the end-all to their considerations when developing an algorithm. In particular, what is considered "best" in theory is not necessarily the best choice in a given program.
Math is important to formulating algorithms, since it can provide, in theory, a closed form solution for something typically done iteratively (the example here: fibonacci numbers). But you still need to verify when an algorithm is going to be correct, and more importantly, appropriate. In this case, it's pretty well known how inaccurate floats and doubles can be within X significant figures, and this is why I've always said the analytical solution doesn't really work on a computer unless you have a CAS library that can approximate an exact answer to an arbitrary number of significant figures (and it still won't be accurate if shoved into a float/double afterwards). So there's a trade-off, and choosing the theoretically best solution (a closed form solution) isn't necessarily the best possible choice for the task. In this case, yes, as others have pointed out, if we only need the first 96 fibonacci numbers, ever, it makes more sense to pre-compute them. However, if we only do the computation once in the entire program, even that doesn't necessarily make sense. It entirely depends on the application, but that's the entire point: a closed form for fibonacci numbers isn't the end of considerations.
I am curious though, why is it so many people are oblivious that there's an O(logN) time algorithm to compute fibonacci numbers iteratively on integers? It's a beautiful piece of math, and very simple so everyone should understand it. Just notice that:
((1, 1), (1, 0))^N -> ((f_{N+1}, f_N), (f_N, f_{N-1}))
If im not mistaken the iterative solution of using a simple for loop would definitely be insanely faster than a recursive algorithm which is exponential.. What am i missing here?
The recursive approach isn't necessarily worse, provided the compiler can do a so-called "tail call optimization" (in the original posting, note the line that says of the compiler optimization level, "We need -O2 so gcc will recognize the tail-call.").
Tail-call optimization says that if a function directly returns the result of calling another function, then you don't need to add to the call stack in order to effect the function call-- you just modify the current stack frame as needed. For a recursive function, apparently a good compiler can essentially transform this into code very similar to what you'd get with an explicit loop.
None of this means that a simple for-loop wouldn't be appropriate, or even faster. Personally I use for-loops a lot, and, like you, I would naturally incline to writing this using a loop rather than recursion.
I see yours is a new account, so maybe this is just a cultural thing that I've got used to. To generalize grossly, recursion is more fashionable on HN than iteration. Every once in a while you'll see a post here that touches on iteration-- like maybe somebody will mention the tradeoffs between counting down and counting up a loop variable. But guaranteed at least once a week you'll see something on recursion and functional programming.
My impression is that most people here understand that a simple for-loop will get the job done, but they don't see for-loops as very interesting, and don't figure it adds anything to the discussion to bring it up.
Meanwhile, there are people here who will actively advocate against using for-loops. There's a whole theory of functional languages that says that if your program doesn't directly modify variables, then it's easier to reason about. A for-loop modifies the trip variable, so it's not a functional construct.
Tail-call optimization says that if a function directly returns the result of calling another function, then you don't need to add to the call stack in order to effect the function call-- you just modify the current stack frame as needed. For a recursive function, apparently a good compiler can essentially transform this into code very similar to what you'd get with an explicit loop.
None of this means that a simple for-loop wouldn't be appropriate, or even faster. Personally I use for-loops a lot, and, like you, I would naturally incline to writing this using a loop rather than recursion.
I see yours is a new account, so maybe this is just a cultural thing that I've got used to. To generalize grossly, recursion is more fashionable on HN than iteration. Every once in a while you'll see a post here that touches on iteration-- like maybe somebody will mention the tradeoffs between counting down and counting up a loop variable. But guaranteed at least once a week you'll see something on recursion and functional programming.
My impression is that most people here understand that a simple for-loop will get the job done, but they don't see for-loops as very interesting, and don't figure it adds anything to the discussion to bring it up.
Meanwhile, there are people here who will actively advocate against using for-loops. There's a whole theory of functional languages that says that if your program doesn't directly modify variables, then it's easier to reason about. A for-loop modifies the trip variable, so it's not a functional construct.
Yes, I understand that recursion is much more appealing and leads to much cleaner and elegant code than iterative solutions. and thank you for the introduction to the community I suppose.
Anyway I forked the original code, added an iterative version based on the comment by (Nick Craig Wood) (https://github.com/AeonAxan/benchmarking_fibonacci)
Here are the benchmarks [recursive fib: 7,765 ms] | [iterative fib: 8,047 ms] | [analytic opt fib: 54,875] | [analytic fib: 103,937 ms] |
And it appears that the recursive version is faster than the naive iterative version by a few hundred milliseconds.
Anyway I forked the original code, added an iterative version based on the comment by (Nick Craig Wood) (https://github.com/AeonAxan/benchmarking_fibonacci)
Here are the benchmarks [recursive fib: 7,765 ms] | [iterative fib: 8,047 ms] | [analytic opt fib: 54,875] | [analytic fib: 103,937 ms] |
And it appears that the recursive version is faster than the naive iterative version by a few hundred milliseconds.
Just a note, In the iterative solution, by changing the i (loop variable) from an int to an long int the times by both iterative and recursive are almost equal. Here are the new results
[Recursive - 7921 ms], [Iteratiive - 8078 ms]
So the difference is negligible.
[Recursive - 7921 ms], [Iteratiive - 8078 ms]
So the difference is negligible.
Tail call optimization does nothing to mitigate the fact that it takes an exponential number of recursive calls to calculate fibonacci. Just draw the call tree of f(5) by hand to see what happens.
>My impression is that most people here understand that a simple for-loop will get the job done, but they don't see for-loops as very interesting
Well, let's make them a bit more interesting then... The simple for loop is actually an example of dynamic programming, i.e. building a table of values from the bottom up in order to avoid repeated calculations. In this case the table only needs to have size 2, as an entry won't be read ever again after the first 2 uses.
>My impression is that most people here understand that a simple for-loop will get the job done, but they don't see for-loops as very interesting
Well, let's make them a bit more interesting then... The simple for loop is actually an example of dynamic programming, i.e. building a table of values from the bottom up in order to avoid repeated calculations. In this case the table only needs to have size 2, as an entry won't be read ever again after the first 2 uses.
1. The worst case for fibonacci is not exponential, it's actually O(fib(n)).
2. The recursive function `fib_helper` in TFA is actually iterative in an TCO environment. Read it closely. There's only on recursive call at the end, not two. `back_one` and `back_two` mean fib(x-1) and fib(x-2) respectively, the same way you would solve it with a loop. With the right `CFLAGS` it should produce more or less the same bytecode.
2. The recursive function `fib_helper` in TFA is actually iterative in an TCO environment. Read it closely. There's only on recursive call at the end, not two. `back_one` and `back_two` mean fib(x-1) and fib(x-2) respectively, the same way you would solve it with a loop. With the right `CFLAGS` it should produce more or less the same bytecode.
Take a look at the recursive code being used, it's not exactly what you think it is. The naive recursive solution makes two recursive calls to the Fib function, and requires an exponential number of calls. The 'standard' recursive solution defines a helper function that takes two arguments, so each recursive step only takes one function call and the total number of calls is linear.
In general, a recursive solution need never have a bigger big-O than an iterative solution. You just need to define a helper function that has a many input parameters as your for-loop had variables that it updates.
In general, a recursive solution need never have a bigger big-O than an iterative solution. You just need to define a helper function that has a many input parameters as your for-loop had variables that it updates.
fork the code and post the benchmark results
GMP (arbitrary precision library) implements it as a lookup table for "low" n but uses a different set of identities for larger n.
http://gmplib.org/manual/Fibonacci-Numbers-Algorithm.html
http://gmplib.org/manual/Fibonacci-Numbers-Algorithm.html
The author very casually takes a potshot at issue #41, calling it a "really silly time wasting debate". Was #41 honestly such a silly time waster? Pray, what would you have done with your precious time instead? I for one took away a lot from #41. It was a very valuable learning experience. #41 was definitely way more valuable than computing fibonaccis, if I may say so.
Another way is to interpret analytical formula in Q(sqrt(5)) ring, and implement Q(sqrt(5)) arithmetic using quadruples of integers. More concretely: suppose we have two numbers of the form a+b sqrt(5), c+d sqrt(5), where a,b,c,d are rational. then their sum is (a+c) + (b+d)sqrt(5), and their product is (ac+5bd)+(ad+bc)sqrt(5). Thus, just like we usually represent complex numbers as pairs of real numbers with special multiplication formula, we can represent elements of Q(sqrt(5)) as pairs of rationals with special multiplication formula. then we just use the analytical formula for our special number system. I've implemented it a while ago in Haskell: https://github.com/xyzzyz/FibBinet