To check my C++ code, I would like to be able to let Boost::Random and Matlab produce the same random numbers.
So for Boost I use the code:
boost::mt19937 var(static_cast<unsigned> (std::time(0)));
boost::uniform_int<> dist(1, 6);
boost::variate_generator<boost::mt19937&, boost::uniform_int<> > die(var, dist);
die.engine().seed(0);
for(int i = 0; i < 10; ++i) {
std::cout << die() << " ";
}
std::cout << std::endl;
Which produces (every run of the program):
4 4 5 6 4 6 4 6 3 4
And for matlab I use:
RandStream.setDefaultStream(RandStream('mt19937ar','seed',0));
randi(6,1,10)
Which produces (every run of the program):
5 6 1 6 4 1 2 4 6 6
Which is bizarre, since both use the same algorithm, and same seed.
What do I miss?
It seems that Python (using numpy) and Matlab seems comparable, in the random uniform numbers:
Matlab
RandStream.setDefaultStream(RandStream('mt19937ar','seed',203));rand(1,10)
0.8479 0.1889 0.4506 0.6253 0.9697 0.2078 0.5944 0.9115 0.2457 0.7743
Python:
random.seed(203);random.random(10)
array([ 0.84790006, 0.18893843, 0.45060688, 0.62534723, 0.96974765,
0.20780668, 0.59444858, 0.91145688, 0.24568615, 0.77430378])
C++Boost
0.8479 0.667228 0.188938 0.715892 0.450607 0.0790326 0.625347 0.972369 0.969748 0.858771
Which is identical to ever other Python and Matlab value...
I have to agree with the other answers, stating that these generators are not "absolute". They may produce different results according to the implementation. I think the simplest solution would be to implement your own generator. It might look daunting (Mersenne twister sure is by the way) but take a look at Xorshift, an extremely simple though powerful one. I copy the C implementation given in the Wikipedia link :
uint32_t xor128(void) {
static uint32_t x = 123456789;
static uint32_t y = 362436069;
static uint32_t z = 521288629;
static uint32_t w = 88675123;
uint32_t t;
t = x ^ (x << 11);
x = y; y = z; z = w;
return w = w ^ (w >> 19) ^ (t ^ (t >> 8));
}
To have the same seed, just put any values you want int x,y,z,w (except(0,0,0,0) I believe). You just need to be sure that Matlab and C++ use both 32 bit for these unsigned int.
Using the interface like
randi(6,1,10)
will apply some kind of transformation on the raw result of the random generator. This transformation is not trivial in general and Matlab will almost certainly do a different selection step than Boost.
Try comparing raw data streams from the RNGs - chances are they are the same
In case this helps anyone interested in the question:
In order to the get the same behavior for the Twister algorithm:
Download the file
http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/MT2002/CODES/mt19937ar.c
Try the following:
#include <stdint.h>
// mt19937ar.c content..
int main(void)
{
int i;
uint32_t seed = 100;
init_genrand(seed);
for (i = 0; i < 100; ++i)
printf("%.20f\n",genrand_res53());
return 0;
}
Make sure the same values are generated within matlab:
RandStream.setGlobalStream( RandStream.create('mt19937ar','seed',100) );
rand(100,1)
randi() seems to be simply ceil( rand()*maxval )
Thanks to Fezvez's answer I've written xor128 in matlab:
function [ w, state ] = xor128( state )
%XOR128 implementation of Xorshift
% https://en.wikipedia.org/wiki/Xorshift
% A starting state might be [123456789, 362436069, 521288629, 88675123]
x = state(1);
y = state(2);
z = state(3);
w = state(4);
% t1 = (x << 11)
t1 = bitand(bitshift(x,11),hex2dec('ffffffff'));
% t = x ^ (x << 11)
t = bitxor(x,t1);
x = y;
y = z;
z = w;
% t2 = (t ^ (t >> 8))
t2 = bitxor(t, bitshift(t,-8));
% t3 = w ^ (w >> 19)
t3 = bitxor(w, bitshift(w,-19));
% w = w ^ (w >> 19) ^ (t ^ (t >> 8))
w = bitxor(t3, t2);
state = [x y z w];
end
You need to pass state in to xor128 every time you use it. I've written a "tester" function which simply returns a vector with random numbers. I tested 1000 numbers output by this function against values output by cpp with gcc and it is perfect.
function [ v ] = txor( iterations )
%TXOR test xor128, returns vector v of length iterations with random number
% output from xor128
% output
v = zeros(iterations,1);
state = [123456789, 362436069, 521288629, 88675123];
i = 1;
while i <= iterations
disp(i);
[t,state] = xor128(state);
v(i) = t;
i = i + 1;
end
I would be very careful assuming that two different implementations of pseudo random generators (even though based on the same algorithms) produce the same result. There could be that one of the implementations use some sort of tweak, hence producing different results. If you need two equal "random" distributions I suggest you either precalculate a sequence, store and access from both C++ and Matlab or create your own generator. It should be fairly easy to implement MT19937 if you use the pseudocode on Wikipedia.
Take care ensuring that both your Matlab and C++ code runs on the same architecture (that is, both runs on either 32 or 64-bit) - using a 64 bit integer in one implementation and a 32 bit integer in the other will lead to different results.
Related
I'm using a C++ compiler but writing code in C (if that helps)
There's a series of numbers
(-1^(a-1)/2a-1)B^(2a-1)
A and X are user defined... A must be positive, but X can be anything (+,-)...
to decode this sequence... I need use exponents/powers, but was given some restrictions... I can't make another function, use recursion, or pow() (among other advanced math functions that come with cmath or math.h).
There were plenty of similar questions, but many answers have used functions and recursion which aren't directly relevant to this question.
This is the code that works perfectly with pow(), I spent a lot of time trying to modify it to replace pow() with my own code, but nothing seems to be working... mainly getting wrong results. X and J are user inputted variables
for (int i = 1; i < j; i++)
sum += (pow(-1, i - 1)) / (5 * i - 1) * (pow(x, 5 * i - 1));
}
You can use macros to get away with no function calls restriction as macros will generate inline code which is technically not a function call
however in case of more complex operations macro can not have return value so you need to use some local variable for the result (in case of more than single expression) like:
int ret;
#define my_pow_notemp(a,b) (b==0)?1:(b==1)?a:(b==2)?a*a:(b==3)?a*a*a:0
#define my_pow(a,b)\
{\
ret=1;\
if (int(b& 1)) ret*=a;\
if (int(b& 2)) ret*=a*a;\
if (int(b& 4)) ret*=a*a*a*a;\
if (int(b& 8)) ret*=a*a*a*a*a*a*a*a;\
if (int(b&16)) ret*=a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a;\
if (int(b&32)) ret*=a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a;\
}
void main()
{
int a=2,b=3,c;
c=my_pow_notemp(a,b); // c = a^b
my_pow(a,b); c = ret; // c = a^b
}
as you can see you can use my_pow_notemp directly but the code is hardcoded so only up to a^3 if you want more you have to add it to code. The my_pow is accepting exponents up to a^63 and its also an example on how to return value in case of more complex code inside macro. Here are some (normal) ways on how to compute powers in case you need non integer or negative exponents (but to convert it to unrolled code will be insanely hard without loops/recursion):
Power by squaring for negative exponents
In case you want to get away with recursion and function calls you can use templates instead of macros but that is limited to C++.
template<class T> T my_pow(T a,T b)
{
if (b==0) return 1;
if (b==1) return a;
return a*my_pow(a,b-1);
}
void main()
{
int a=2,b=3,c;
c=my_pow(a,b);
}
As you can see templates have return value so no problem even with more complex code (more than single expression).
To avoid loops you can use LUT tables
int my_pow[4][4]=
{
{1,0,0,0}, // 0^
{1,1,1,1}, // 1^
{1,2,4,8}, // 2^
{1,3,9,27}, // 3^
};
void main()
{
int a=2,b=3,c;
c=my_pow[a][b];
}
If you have access to FPU or advanced math assembly you can use that as asm instruction is not a function call. FPU usually have log,exp,pow functions natively. This however limits the code to specific instruction set !!!
Here some examples:
How to: pow(real, real) in x86
So when I consider your limitation I think the best way is:
#define my_pow(a,b) (b==0)?1:(b==1)?a:(b==2)?a*a:(b==3)?a*a*a:0
void main()
{
int a=2,b=3,c;
c=my_pow(a,b); // c = a^b
}
Which will work on int exponents b up to 3 (if you want more just add (b==4)?a*a*a*a: ... :0) and both int and float bases a. If you need much bigger exponent use the complicated version with local temp variable for returning result.
[Edit1] ultimative single expression macro with power by squaring up to a^15
#define my_pow(a,b) (1* (int(b&1))?a:1* (int(b&2))?a*a:1* (int(b&4))?a*a*a*a:1* (int(b&8))?a*a*a*a*a*a*a*a:1)
void main()
{
int a=2,b=3,c;
c=my_pow(a,b); // c = a^b
}
In case you want more than a^15 just add sub term (int(b&16))?a*a*a*a*a*a*a*a*a*a*a*a*a*a*a*a:1 and so on for each bit of exponent.
It is a series. Replace pow() based on the previous iteration. #Bathsheba
Code does not need to call pow(). It can form pow(x, 5 * i - 1) and pow(-1, i - 1), since both have an int exponent based on the iterator i, from the prior loop iteration.
Example:
Let f(x, i) = pow(x, 5 * i - 1)
Then f(x, 1) = x*x*x*x
and f(x, i > 1) = f(x, i-1) * x*x*x*x*x
double power_n1 = 1.0;
double power_x5 = x*x*x*x;
for (int i = 1; i < j + 1; i++)
// sum += (pow(-1, i - 1)) / (5 * i - 1) * (pow(x, 5 * i - 1));
sum += power_n1 / (5 * i - 1) * power_x5;
power_n1 = -power_n1;
power_x5 *= x*x*x*x*x;
}
In C++, I should write a program where the app detects which numbers are divisible by 3 from 1 till 10 and then multiply all of them and print the result. That means that I should multiply 3,6,9 and print only the result, which is 162, but I should do it by using a "While" loop, not just multiplying the 3 numbers with each other. How should I write the code of this? I attached my attempt to code the problem below. Thanks
#include <iostream>
using namespace std;
int main() {
int x, r;
int l;
x = 1;
r = 0;
while (x < 10 && x%3==0) {
r = (3 * x) + 3;
cout << r;
}
cin >> l;
}
Firstly your checking the condition x%3 == 0 brings you out of your while - loop right in the first iteration where x is 1. You need to check the condition inside the loop.
Since you wish to store your answer in variable r you must initialize it to 1 since the product of anything with 0 would give you 0.
Another important thing is you need to increment the value of x at each iteration i.e. to check if each number in the range of 1 to 10 is divisible by 3 or not .
int main()
{
int x, r;
int l;
x = 1;
r = 1;
while (x < 10)
{
if(x%3 == 0)
r = r*x ;
x = x + 1; //incrementing the value of x
}
cout<<r;
}
Lastly I have no idea why you have written the last cin>>l statement . Omit it if not required.
Ok so here are a few hints that hopefully help you solving this:
Your approach with two variables (x and r) outside the loop is a good starting point for this.
Like I wrote in the comments you should use *= instead of your formula (I still don't understand how it is related to the problem)
Don't check if x is dividable by 3 inside the while-check because it would lead to an too early breaking of the loop
You can delete your l variable because it has no affect at the moment ;)
Your output should also happen outside the loop, else it is done everytime the loop runs (in your case this would be 10 times)
I hope I can help ;)
EDIT: Forget about No.4. I didn't saw your comment about the non-closing console.
int main()
{
int result = 1; // "result" is better than "r"
for (int x=1; x < 10; ++x)
{
if (x%3 == 0)
result = result * x;
}
cout << result;
}
or the loop in short with some additional knowledge:
for (int x=3; x < 10; x += 3) // i know that 3 is dividable
result *= x;
or, as it is c++, and for learning purposes, you could do:
vector<int> values; // a container holding integers that will get the multiples of 3
for (int x=1; x < 10; ++x) // as usual
if ( ! x%3 ) // same as x%3 == 0
values.push_back(x); // put the newly found number in the container
// now use a function that multiplies all numbers of the container (1 is start value)
result = std::accumulate(values.begin(), values.end(), 1, multiplies<int>());
// so much fun, also get the sum (0 is the start value, no function needed as add is standard)
int sum = std::accumulate(values.begin(), values.end(), 0);
It's important to remember the difference between = and ==. = sets something to a value while == compares something to a value. You're on the right track with incrementing x and using x as a condition to check your range of numbers. When writing code I usually try and write a "pseudocode" in English to organize my steps and get my logic down. It's also wise to consider using variables that tell you what they are as opposed to just random letters. Imagine if you were coding a game and you just had letters as variables; it would be impossible to remember what is what. When you are first learning to code this really helps a lot. So with that in mind:
/*
- While x is less than 10
- check value to see if it's mod 3
- if it's mod 3 add it to a sum
- if not's mod 3 bump a counter
- After my condition is met
- print to screen pause screen
*/
Now if we flesh out that pseudocode a little more we'll get a skeletal structure.
int main()
{
int x=1//value we'll use as a counter
int sum=0//value we'll use as a sum to print out at the end
while(x<10)//condition we'll check against
{
if (x mod 3 is zero)
{
sum=x*1;
increment x
}
else
{
increment x
}
}
//screen output the sum the sum
//system pause or cin.get() use whatever your teacher gave you.
I've given you a lot to work with here you should be able to figure out what you need from this. Computer Science and programming is hard and will require a lot of work. It's important to develop good coding habits and form now as it will help you in the future. Coding is a skill like welding; the more you do it the better you'll get. I often refer to it as the "Blue Collar Science" because it's really a skillset and not just raw knowledge. It's not like studying history or Biology (minus Biology labs) because those require you to learn things and loosely apply them whereas programming requires you to actually build something. It's like welding or plumbing in my opinion.
Additionally when you come to sites like these try and read up how things should be posted and try and seek the "logic" behind the answer and come up with it on your own as opposed to asking for the answer. People will be more inclined to help you if they think you're working for something instead of asking for a handout (not saying you are, just some advice). Additionally take the attitude these guys give you with a grain of salt, Computer Scientists aren't known to be the worlds most personable people. =) Good luck.
I am trying to write a simple gradient descent algorithm in C++ (for 10,000 iterations). Here is my program:
#include<iostream>
#include<cmath>
using namespace std;
int main(){
double learnrate=10;
double x=10.0; //initial start value
for(int h=1; h<=10000; h++){
x=x-learnrate*(2*x + 100*cos(100*x));
}
cout<<"The minimum is at y = "<<x*x + sin(100*x)<<" and at x = "<<x;
return 0;
}
The output ends up being: y=nan and x=nan. I tried looking at the values of x and y by putting them into a file, and after a certain amount of iterations, I am getting all nans (for x and y). edit: I picked the learning rate (or step size) to be 10 as an experiment, I will use much smaller values afterwards.
There must be something wrong with your formula. Already the first 10 values of x are increasing like hell:
-752.379
15290.7
-290852
5.52555e+06
-1.04984e+08
1.9947e+09
-3.78994e+10
7.20088e+11
-1.36817e+13
2.59952e+14
No matter what starting value you choose the absolute value of the next x will be bigger.
|next_x| = | x - 20 * x - 100 * cos(100*x) |
For example consider what happens when you choose a very small starting value (|x|->0), then
|next_x| = | 0 - 20 * 0 - 100 * cos ( 0 ) | = 100
Because at h=240 the variable "x" exceeds the limits of double type (1.79769e+308). This is a diverging arithmetic progression. You need to reduce your learn rate.
A couple of more things:
1- Do not use "using namespace std;" it is bad practice.
2- You can use "std::isnan() function to identify this situation.
Here is an example:
#include <iomanip>
#include <limits>
int main()
{
double learnrate = 10.0;
double x = 10.0; //initial start value
std::cout<<"double type maximum=" << std::numeric_limits<double>::max()<<std::endl;
bool failed = false;
for (int h = 1; h <= 10000; h++)
{
x = x - learnrate*(2.0*x + 100.0 * std::cos(100.0 * x));
if (std::isnan(x))
{
failed = true;
std::cout << " Nan detected at h=" << h << std::endl;
break;
}
}
if(!failed)
std::cout << "The minimum is at y = " << x*x + std::sin(100.0*x) << " and at x = " << x;
return 0;
}
Print x before the call to the cosine function and you will see that the last number printed before NaN (at h = 240) is:
-1.7761e+307
This means that the value is going to infinity, which cannot be represented (thus Not a Number).
It overflows the double type.
If you use long double, you will succeed in 1000 iterations, but you will still overflow the type with 10000 iterations.
So the problem is that the parameter learnrate is just too big. You should do let steps, while using a data type with larger range, as I suggested above.
The "learn rate" is far too high. Change it to 1e-4, for example, and the program works, for an initial value of 10 at least. When the learnrate is 10, the iterations jump too far past the solution.
At its best, gradient descent is not a good algorithm. For serious applications you want to use something better. Much better. Search for Brent optimizer and BFGS.
I have the following code (the xorshift128+ code from Wikipedia modified to use vector types):
#include <immintrin.h>
#include <climits>
__v8si rand_si() {
static auto s0 = __v4du{4, 8, 15, 16},
s1 = __v4du{23, 34, 42, 69};
auto x = s0, y = s1;
s0 = y;
x ^= x << 23;
s1 = x ^ y ^ (x >> 17) ^ (y >> 26);
return (__v8si)(s1 + y);
}
#include <iostream>
#include <iomanip>
void foo() {
//Shuffle a bit. The result is much worse without this.
rand_si(); rand_si(); rand_si(); rand_si();
auto val = rand_si();
for (auto it = reinterpret_cast<int*>(&val);
it != reinterpret_cast<int*>(&val + 1);
++it)
std::cout << std::hex << std::setfill('0') << std::setw(8) << *it << ' ';
std::cout << '\n';
}
which outputs
09e2a657 000b8020 1504cc3b 00110040 1360ff2b 00150078 2a9998b7 00228080
Every other number is very small, and none have the leading bit set. On the other hand, using xorshift* instead:
__v8si rand_si() {
static auto x = __v4du{4, 8, 15, 16};
x ^= x >> 12;
x ^= x << 25;
x ^= x >> 27;
return x * (__v4du)_mm256_set1_epi64x(0x2545F4914F6CDD1D);
}
I get the much better output
0889632e a938b990 1e8b2f79 832e26bd 11280868 2a22d676 275ca4b8 10954ef9
But according to Wikipedia, xorshift+ is a good PRNG, and produces better pseudo-randomness than xorshift*. So, do I have a bug in my RNG code, or am I using it wrong?
I think you should not judge a random generator by looking at 8 numbers it generated. Furthermore, generators usually needs good seeding (your seeding can be considered bad - your seeds start with almost all bits zeros. Calling rand_si() just a few times is not enough for the bits to "spread").
So I recommend you to use proper seeding (for example, a simple solution is to call rand_si() even more times).
xorshift* look like behaving better because of the final multiplication, so it doesn't have easily spotted bad behavior because of inadequate seeding.
A tip: Compare the numbers your code generates with the original implementation. This way you can be sure that your implementation is correct.
geza's answer was exactly right, the seeding was the culprit. It worked much better to seed it using a standard 64-bit PRNG:
void seed(uint64_t s) {
std::mt19937_64 e(s);
s0 = __v4du{e(), e(), e(), e()};
s1 = __v4du{e(), e(), e(), e()};
}
Both guys above are mistaken. The xorshift+ generator works fine even when the initial base (seed) is 1, 2, 3, ... and other simplest values. Generator fails on a zero valued seed only. Check your 64-bit variables representation and correctly work of the binary operators.
I am making a program for nth Fibonacci number. I made the following program using recursion and memoization.
The main problem is that the value of n can go up to 10000 which means that the Fibonacci number of 10000 would be more than 2000 digit long.
With a little bit of googling, I found that i could use arrays and store every digit of the solution in an element of the array but I am still not able to figure out how to implement this approach with my program.
#include<iostream>
using namespace std;
long long int memo[101000];
long long int n;
long long int fib(long long int n)
{
if(n==1 || n==2)
return 1;
if(memo[n]!=0)
return memo[n];
return memo[n] = fib(n-1) + fib(n-2);
}
int main()
{
cin>>n;
long long int ans = fib(n);
cout<<ans;
}
How do I implement that approach or if there is another method that can be used to achieve such large values?
One thing that I think should be pointed out is there's other ways to implement fib that are much easier for something like C++ to compute
consider the following pseudo code
function fib (n) {
let a = 0, b = 1, _;
while (n > 0) {
_ = a;
a = b;
b = b + _;
n = n - 1;
}
return a;
}
This doesn't require memoisation and you don't have to be concerned about blowing up your stack with too many recursive calls. Recursion is a really powerful looping construct but it's one of those fubu things that's best left to langs like Lisp, Scheme, Kotlin, Lua (and a few others) that support it so elegantly.
That's not to say tail call elimination is impossible in C++, but unless you're doing something to optimise/compile for it explicitly, I'm doubtful that whatever compiler you're using would support it by default.
As for computing the exceptionally large numbers, you'll have to either get creative doing adding The Hard Way or rely upon an arbitrary precision arithmetic library like GMP. I'm sure there's other libs for this too.
Adding The Hard Way™
Remember how you used to add big numbers when you were a little tater tot, fresh off the aluminum foil?
5-year-old math
1259601512351095520986368
+ 50695640938240596831104
---------------------------
?
Well you gotta add each column, right to left. And when a column overflows into the double digits, remember to carry that 1 over to the next column.
... <-001
1259601512351095520986368
+ 50695640938240596831104
---------------------------
... <-472
The 10,000th fibonacci number is thousands of digits long, so there's no way that's going to fit in any integer C++ provides out of the box. So without relying upon a library, you could use a string or an array of single-digit numbers. To output the final number, you'll have to convert it to a string tho.
(woflram alpha: fibonacci 10000)
Doing it this way, you'll perform a couple million single-digit additions; it might take a while, but it should be a breeze for any modern computer to handle. Time to get to work !
Here's an example in of a Bignum module in JavaScript
const Bignum =
{ fromInt: (n = 0) =>
n < 10
? [ n ]
: [ n % 10, ...Bignum.fromInt (n / 10 >> 0) ]
, fromString: (s = "0") =>
Array.from (s, Number) .reverse ()
, toString: (b) =>
b .reverse () .join ("")
, add: (b1, b2) =>
{
const len = Math.max (b1.length, b2.length)
let answer = []
let carry = 0
for (let i = 0; i < len; i = i + 1) {
const x = b1[i] || 0
const y = b2[i] || 0
const sum = x + y + carry
answer.push (sum % 10)
carry = sum / 10 >> 0
}
if (carry > 0) answer.push (carry)
return answer
}
}
We can verify that the Wolfram Alpha answer above is correct
const { fromInt, toString, add } =
Bignum
const bigfib = (n = 0) =>
{
let a = fromInt (0)
let b = fromInt (1)
let _
while (n > 0) {
_ = a
a = b
b = add (b, _)
n = n - 1
}
return toString (a)
}
bigfib (10000)
// "336447 ... 366875"
Expand the program below to run it in your browser
const Bignum =
{ fromInt: (n = 0) =>
n < 10
? [ n ]
: [ n % 10, ...Bignum.fromInt (n / 10 >> 0) ]
, fromString: (s = "0") =>
Array.from (s) .reverse ()
, toString: (b) =>
b .reverse () .join ("")
, add: (b1, b2) =>
{
const len = Math.max (b1.length, b2.length)
let answer = []
let carry = 0
for (let i = 0; i < len; i = i + 1) {
const x = b1[i] || 0
const y = b2[i] || 0
const sum = x + y + carry
answer.push (sum % 10)
carry = sum / 10 >> 0
}
if (carry > 0) answer.push (carry)
return answer
}
}
const { fromInt, toString, add } =
Bignum
const bigfib = (n = 0) =>
{
let a = fromInt (0)
let b = fromInt (1)
let _
while (n > 0) {
_ = a
a = b
b = add (b, _)
n = n - 1
}
return toString (a)
}
console.log (bigfib (10000))
Try not to use recursion for a simple problem like fibonacci. And if you'll only use it once, don't use an array to store all results. An array of 2 elements containing the 2 previous fibonacci numbers will be enough. In each step, you then only have to sum up those 2 numbers. How can you save 2 consecutive fibonacci numbers? Well, you know that when you have 2 consecutive integers one is even and one is odd. So you can use that property to know where to get/place a fibonacci number: for fib(i), if i is even (i%2 is 0) place it in the first element of the array (index 0), else (i%2 is then 1) place it in the second element(index 1). Why can you just place it there? Well when you're calculating fib(i), the value that is on the place fib(i) should go is fib(i-2) (because (i-2)%2 is the same as i%2). But you won't need fib(i-2) any more: fib(i+1) only needs fib(i-1)(that's still in the array) and fib(i)(that just got inserted in the array).
So you could replace the recursion calls with a for loop like this:
int fibonacci(int n){
if( n <= 0){
return 0;
}
int previous[] = {0, 1}; // start with fib(0) and fib(1)
for(int i = 2; i <= n; ++i){
// modulo can be implemented with bit operations(much faster): i % 2 = i & 1
previous[i&1] += previous[(i-1)&1]; //shorter way to say: previous[i&1] = previous[i&1] + previous[(i-1)&1]
}
//Result is in previous[n&1]
return previous[n&1];
}
Recursion is actually discommanded while programming because of the time(function calls) and ressources(stack) it consumes. So each time you use recursion, try to replace it with a loop and a stack with simple pop/push operations if needed to save the "current position" (in c++ one can use a vector). In the case of the fibonacci, the stack isn't even needed but if you are iterating over a tree datastructure for example you'll need a stack (depends on the implementation though). As I was looking for my solution, I saw #naomik provided a solution with the while loop. That one is fine too, but I prefer the array with the modulo operation (a bit shorter).
Now concerning the problem of the size long long int has, it can be solved by using external libraries that implement operations for big numbers (like the GMP library or Boost.multiprecision). But you could also create your own version of a BigInteger-like class from Java and implement the basic operations like the one I have. I've only implemented the addition in my example (try to implement the others they are quite similar).
The main idea is simple, a BigInt represents a big decimal number by cutting its little endian representation into pieces (I'll explain why little endian at the end). The length of those pieces depends on the base you choose. If you want to work with decimal representations, it will only work if your base is a power of 10: if you choose 10 as base each piece will represent one digit, if you choose 100 (= 10^2) as base each piece will represent two consecutive digits starting from the end(see little endian), if you choose 1000 as base (10^3) each piece will represent three consecutive digits, ... and so on. Let's say that you have base 100, 12765 will then be [65, 27, 1], 1789 will be [89, 17], 505 will be [5, 5] (= [05,5]), ... with base 1000: 12765 would be [765, 12], 1789 would be [789, 1], 505 would be [505]. It's not the most efficient, but it is the most intuitive (I think ...)
The addition is then a bit like the addition on paper we learned at school:
begin with the lowest piece of the BigInt
add it with the corresponding piece of the other one
the lowest piece of that sum(= the sum modulus the base) becomes the corresponding piece of the final result
the "bigger" pieces of that sum will be added ("carried") to the sum of the following pieces
go to step 2 with next piece
if no piece left, add the carry and the remaining bigger pieces of the other BigInt (if it has pieces left)
For example:
9542 + 1097855 = [42, 95] + [55, 78, 09, 1]
lowest piece = 42 and 55 --> 42 + 55 = 97 = [97]
---> lowest piece of result = 97 (no carry, carry = 0)
2nd piece = 95 and 78 --> (95+78) + 0 = 173 = [73, 1]
---> 2nd piece of final result = 73
---> remaining: [1] = 1 = carry (will be added to sum of following pieces)
no piece left in first `BigInt`!
--> add carry ( [1] ) and remaining pieces from second `BigInt`( [9, 1] ) to final result
--> first additional piece: 9 + 1 = 10 = [10] (no carry)
--> second additional piece: 1 + 0 = 1 = [1] (no carry)
==> 9542 + 1 097 855 = [42, 95] + [55, 78, 09, 1] = [97, 73, 10, 1] = 1 107 397
Here is a demo where I used the class above to calculate the fibonacci of 10000 (result is too big to copy here)
Good luck!
PS: Why little endian? For the ease of the implementation: it allows to use push_back when adding digits and iteration while implementing the operations will start from the first piece instead of the last piece in the array.