The 1 millionth fibonacci number

Terminal - The 1 millionth fibonacci number

time echo 'n=1000000;m=(n+1)/2;a=0;b=1;i=0;while(m){e[i++]=m%2;m/=2};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]){t=a;a+=b;b=t}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc

2009-09-10 09:00:44

User: Escher

Functions: echo time

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The 1 millionth fibonacci number

EDIT: Trolling crap removed ;)

takes approx 6 secs on a Core 2 Duo @ 2GHz, and 15 secs on atom based netbooks!

uses monoid (a,b).(x,y)=(ax+bx+ay,ax+by) with identity (0,1), and recursion relations:

F(2n-1)=Fn*Fn+F(n-1)*F(n-1)

F(2n)=(Fn+2*F(n-1))*Fn

then apply fast exponentiation to (1,0)^n = (Fn,F(n-1))

Note that: (1,0)^-1=(1,-1) so (a,b).(1,0) = (a+b,a) and (a,b)/(1,0)=(a,b).(1,0)^-1=(b,a-b)

So we can also use a NAF representation to do the exponentiation,http://en.wikipedia.org/wiki/Non-adjacent_form , it's also very fast (about the same, depends on n):

time echo 'n=1000000;m=(n+1)/2;a=0;b=1;i=0;while(m>0){z=0;if(m%2)z=2-(m%4);m=(m-z)/2;e[i++]=z};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]>0){t=a;a+=b;b=t};if(e[i]<0){t=a;a=b;b=t-b}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc

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Alternatives

There are 2 alternatives - vote for the best!

Terminal - Alternatives

time echo 'n=70332;m=(n+1)/2;a=0;b=1;i=0;while(m){e[i++]=m%2;m/=2};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]){t=a;a+=b;b=t}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc

2009-09-10 08:58:47

User: Escher

Functions: echo time

-135

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The absolutely fastest nth fibonacci number

Calculates nth Fibonacci number for all n>=0, (much faster than matrix power algorithm from http://everything2.com/title/Compute+Fibonacci+numbers+FAST%2521 )

n=70332 is the biggest value at http://bigprimes.net/archive/fibonacci/ (corresponds to n=70331 there), this calculates it in less than a second, even on a netbook.

UPDATE: Now even faster! Uses recurrence relation for F(2n), see http://en.wikipedia.org/wiki/Fibonacci_number#Matrix_form

n is now adjusted to match Fn at wikipedia, so bigprimes.net table is offset by 1.

UPDATE2: Probably fastest possible now ;), uses a simple monoid operation:

uses monoid (a,b).(x,y)=(ax+bx+ay,ax+by) with identity (0,1), and recursion relations:

F(2n-1)=Fn*Fn+F(n-1)*F(n-1)

F(2n)=Fn*(2*F(n-1)+Fn)

then apply fast exponentiation to (1,0)^n = (Fn,F(n-1))

Note that: (1,0)^-1=(1,-1) so (a,b).(1,0) = (a+b,a) and (a,b)/(1,0)=(a,b).(1,0)^-1=(b,a-b)

So we can also use a NAF representation to do the exponentiation,http://en.wikipedia.org/wiki/Non-adjacent_form , it's also very fast (about the same, depends on n):

time echo 'n=70332;m=(n+1)/2;a=0;b=1;i=0;while(m>0){z=0;if(m%2)z=2-(m%4);m=(m-z)/2;e[i++]=z};while(i--){c=a*a;a=c+2*a*b;b=c+b*b;if(e[i]>0){t=a;a+=b;b=t};if(e[i]<0){t=a;a=b;b=t-b}};if(n%2)a*a+b*b;if(!n%2)a*(a+2*b)' | bc

Show sample output | Comments (5) | Add to favourites | Report as malicious

Know a better way?

If you can do better, submit your command here.

What others think

My C version is faster. Stop posting duplicates.

Comment by hank 418 weeks and 6 days ago

Stop it Escher. You're just spamming now. Quit posting this and more variants. We get it already!

Comment by linuxrawkstar 418 weeks and 6 days ago

Your C version is slower,in fact ~47secs vs my ~36secson a Core 2 Duo. I have put a message for people like you in capitals in the description section. No go away little buzzing insect.

Comment by Escher 418 weeks and 6 days ago

Don't be too hard on this MrMerry/Escher degenerate... Desperate attempts to get some attention on the internet is his only way of communicating with the world outside his mama's basement...

Comment by sklm 418 weeks and 6 days ago

Please only post comments if you have required skillz to compete against me. I know it's depressing for you to witness true greatness and realise how insignificant you are, but that's life sklm, you always were and always will be an irrelevant ant.

Comment by Escher 418 weeks and 5 days ago

"Please only post comments if you have required skillz to compete against me. I know it's depressing for you to witness true greatness and realise how insignificant you are, but that's life sklm, you always were and always will be an irrelevant ant."

LMAO!!!

Comment by sklm 418 weeks and 5 days ago

Well now. Someone has a voting script...

Comment by SuperFly 418 weeks and 5 days ago

can this be made faster by using multiple CPUs ?

Comment by alperyilmaz 418 weeks and 5 days ago

Not really, though it can be made faster by using the recurrence relations from the wikipedia article more efficiently. Check for an update soon :)

For the record, mathematica uses an algorithm which is about 4 times faster.

Comment by Escher 418 weeks and 5 days ago

I don't think this was a voting script; I'd like to think this is a reward for arrogance.

Escher: a truly great person is humble. What bothers you is that the reverse is not necessarily true. And as long as that bothers you, you will be what you are, and will not grow further.

Comment by sitaram 418 weeks and 4 days ago

Don't be so arrogant, dude !

It's just a shell command... I don't care for these numbers in any shell script... useless for me, sorry.

Comment by alvinx 418 weeks and 2 days ago

2016-6-7 leilei

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What is Tweed?

Tweed is a warm rough to touch fabric and was originated in Scotland. The reason why tweed suits in London are famous is because of its loose weave which makes it flexible and comfortable during winters. The tweed fabric is referred to the weave of fabric; therefore when you see different kinds of tweed, it would tell you the pattern it has been woven. One of the most popular kinds of tweed is the herringbone tweed which a diagonally v-shaped black and white weave. The herringbone tweed gives a rich visual texture and known for its versatility.

Ways to wear Tweed in winters

Everybody needs to wear warm clothes during the winter month to protect from the chilly weather outside. However, many people think that they might have to compromise fashion during those chilly months. Not now! There are many fabrics that you can use to protect yourself from the weather as well as still enjoy great looks. Tweed is one of the fabrics that can make you awesome even in winters as well. Below are some ways of how you can incorporate tweed with other pieces in your wardrobe

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"Casually ? if you are a starter with tweed then it is recommended to start with plain variety of tweed. You can wear a tweed jacket with dark blue jeans and high boots. Incorporate a neutral t-shirt or a winter shirt under the jacket.

At work ? Tweed suit in London are quite popular as one of the formal attires. Heavy woven tweeds can make you look older, so try buying tweed suits in London that are finely weaved and pair it with your regular shirts like white or blue with smart formal shoes. Believe it or not, it can win you praise at work.

Accessorise ? Apart from wearing tweed suits in London as attire you can even use accessories made out of tweed. Tweed caps, laptop bags, scarves, gloves are a smart way to incorporate tweed in your winter collection. It is also a safe way of using tweed fabric if you are not too comfortable wearing it. There is even footwear that has tweed as their centre fabric and looks awesome.

Taking care of Tweed

Though the fabric boasts itself about being tough and rugged, it still requires a good care to use it for coming years as well. Due to its weave structure it often pick up dirt and stains that can get hard to remove. Make sure you take your tweed clothes for dry cleaning as washing them in washing machines can make them shrink. Also, keep them in cool dry place so that they do not get attracted by moths.

Comment by leilei3915 67 weeks and 1 day ago