Related
To clarify first:
2^3 = 8. That's equivalent to 2*2*2. Easy.
2^4 = 16. That's equivalent to 2*2*2*2. Also easy.
2^3.5 = 11.313708... Er, that's not so easy to grok.
Want I want is a simple algorithm which most clearly shows how 2^3.5 = 11.313708. It should preferably not use any functions apart from the basic addition, subtract, multiply, or divide operators.
The code certainly doesn't have to be fast, nor does it necessarily need to be short (though that would help). Don't worry, it can be approximate to a given user-specified accuracy (which should also be part of the algorithm). I'm hoping there will be a binary chop/search type thing going on, as that's pretty simple to grok.
So far I've found this, but the top answer is far from simple to understand on a conceptual level.
The more answers the merrier, so I can try to understand different ways of attacking the problem.
My language preference for the answer would be C#/C/C++/Java, or pseudocode for all I care.
Ok, let's implement pow(x, y) using only binary searches, addition and multiplication.
Driving y below 1
First, take this out of the way:
pow(x, y) == pow(x*x, y/2)
pow(x, y) == 1/pow(x, -y)
This is important to handle negative exponents and drive y below 1, where things start getting interesting. This reduces the problem to finding pow(x, y) where 0<y<1.
Implementing sqrt
In this answer I assume you know how to perform sqrt. I know sqrt(x) = x^(1/2), but it is easy to implement it just using a binary search to find y = sqrt(x) using y*y=x search function, e.g.:
#define EPS 1e-8
double sqrt2(double x) {
double a = 0, b = x>1 ? x : 1;
while(abs(a-b) > EPS) {
double y = (a+b)/2;
if (y*y > x) b = y; else a = y;
}
return a;
}
Finding the answer
The rationale is that every number below 1 can be approximated as a sum of fractions 1/2^x:
0.875 = 1/2 + 1/4 + 1/8
0.333333... = 1/4 + 1/16 + 1/64 + 1/256 + ...
If you find those fractions, you actually find that:
x^0.875 = x^(1/2+1/4+1/8) = x^(1/2) * x^(1/4) * x^(1/8)
That ultimately leads to
sqrt(x) * sqrt(sqrt(x)) * sqrt(sqrt(sqrt(x)))
So, implementation (in C++)
#define EPS 1e-8
double pow2(double x, double y){
if (x < 0 and abs(round(y)-y) < EPS) {
return pow2(-x, y) * ((int)round(y)%2==1 ? -1 : 1);
} else if (y < 0) {
return 1/pow2(x, -y);
} else if(y > 1) {
return pow2(x * x, y / 2);
} else {
double fraction = 1;
double result = 1;
while(y > EPS) {
if (y >= fraction) {
y -= fraction;
result *= x;
}
fraction /= 2;
x = sqrt2(x);
}
return result;
}
}
Deriving ideas from the other excellent posts, I came up with my own implementation. The answer is based on the idea that base^(exponent*accuracy) = answer^accuracy. Given that we know the base, exponent and accuracy variables beforehand, we can perform a search (binary chop or whatever) so that the equation can be balanced by finding answer. We want the exponent in both sides of the equation to be an integer (otherwise we're back to square one), so we can make accuracy any size we like, and then round it to the nearest integer afterwards.
I've given two ways of doing it. The first is very slow, and will often produce extremely high numbers which won't work with most languages. On the other hand, it doesn't use log, and is simpler conceptually.
public double powSimple(double a, double b)
{
int accuracy = 10;
bool negExponent = b < 0;
b = Math.Abs(b);
bool ansMoreThanA = (a>1 && b>1) || (a<1 && b<1); // Example 0.5^2=0.25 so answer is lower than A.
double accuracy2 = 1.0 + 1.0 / accuracy;
double total = a;
for (int i = 1; i < accuracy* b; i++) total = total*a;
double t = a;
while (true) {
double t2 = t;
for(int i = 1; i < accuracy; i++) t2 = t2 * t; // Not even a binary search. We just hunt forwards by a certain increment
if((ansMoreThanA && t2 > total) || (!ansMoreThanA && t2 < total)) break;
if (ansMoreThanA) t *= accuracy2; else t /= accuracy2;
}
if (negExponent) t = 1 / t;
return t;
}
This one below is a little more involved as it uses log(). But it is much quicker and doesn't suffer from the super-high number problems as above.
public double powSimple2(double a, double b)
{
int accuracy = 1000000;
bool negExponent= b<0;
b = Math.Abs(b);
double accuracy2 = 1.0 + 1.0 / accuracy;
bool ansMoreThanA = (a>1 && b>1) || (a<1 && b<1); // Example 0.5^2=0.25 so answer is lower than A.
double total = Math.Log(a) * accuracy * b;
double t = a;
while (true) {
double t2 = Math.Log(t) * accuracy;
if ((ansMoreThanA && t2 > total) || (!ansMoreThanA && t2 < total)) break;
if (ansMoreThanA) t *= accuracy2; else t /= accuracy2;
}
if (negExponent) t = 1 / t;
return t;
}
You can verify that 2^3.5 = 11.313708 very easily: check that 11.313708^2 = (2^3.5)^2 = 2^7 = 128
I think the easiest way to understand the computation you would actually do for this would be to refresh your understanding of logarithms - one starting point would be http://en.wikipedia.org/wiki/Logarithm#Exponentiation.
If you really want to compute non-integer powers with minimal technology one way to do that would be to express them as fractions with denominator a power of two and then take lots of square roots. E.g. x^3.75 = x^3 * x^(1/2) * x^(1/4) then x^(1/2) = sqrt(x), x^(1/4) = sqrt(sqrt(x)) and so on.
Here is another approach, based on the idea of verifying a guess. Given y, you want to find x such that x^(a/b) = y, where a and b are integers. This equation implies that x^a = y^b. You can calculate y^b, since you know both numbers. You know a, so you can - as you originally suspected - use binary chop or perhaps some numerically more efficient algorithm to solve x^a = y^b for x by simply guessing x, computing x^a for this guess, comparing it with y^b, and then iteratively improving the guess.
Example: suppose we wish to find 2^0.878 by this method. Then set a = 439, b = 500, so we wish to find 2^(439/500). If we set x=2^(439/500) we have x^500 = 2^439, so compute 2^439 and (by binary chop or otherwise) find x such that x^500 = 2^439.
Most of it comes down to being able to invert the power operation.
In other words, the basic idea is that (for example) N2 should be basically the "opposite" of N1/2 so that if you do something like:
M = N2
L = M1/2
Then the result you get in L should be the same as the original value in N (ignoring any rounding and such).
Mathematically, that means that N1/2 is the same as sqrt(N), N1/3 is the cube root of N, and so on.
The next step after that would be something like N3/2. This is pretty much the same idea: the denominator is a root, and the numerator is a power, so N3/2 is the square root of the cube of N (or the cube of the square root of N--works out the same).
With decimals, we're just expressing a fraction in a slightly different form, so something like N3.14 can be viewed as N314/100--the hundredth root of N raised to the power 314.
As far as how you compute these: there are quite a few different ways, depending heavily on the compromise you prefer between complexity (chip area, if you're implementing it in hardware) and speed. The obvious way is to use a logarithm: AB = Log-1(Log(A)*B).
For a more restricted set of inputs, such as just finding the square root of N, you can often do better than that extremely general method though. For example, the binary reducing method is quite fast--implemented in software, it's still about the same speed as Intel's FSQRT instruction.
As stated in the comments, its not clear if you want a mathematical description of how fractional powers work, or an algorithm to calculate fractional powers.
I will assume the latter.
For almost all functions (like y = 2^x) there is a means of approximating the function using a thing called the Taylor Series http://en.wikipedia.org/wiki/Taylor_series. This approximates any reasonably behaved function as a polynomial, and polynomials can be calculated using only multiplication, division, addition and subtraction (all of which the CPU can do directly). If you calculate the Taylor series for y = 2^x and plug in x = 3.5 you will get 11.313...
This almost certainly not how exponentiation is actually done on your computer. There are many algorithms which run faster for different inputs. For example, if you calculate 2^3.5 using the Taylor series, then you would have to look at many terms to calculate it with any accuracy. However, the Taylor series will converge much faster for x = 0.5 than for x = 3.5. So one obvious improvement is to calculate 2^3.5 as 2^3 * 2^0.5, as 2^3 is easy to calculate directly. Modern exponentiation algorithms will use many, many tricks to speed up processing - but the principle is still much the same, approximate the exponentiation function as some infinite sum, and calculate as many terms as you need to get the accuracy that is required.
I am writing a program for Catalan number. So here is the formula for that:
I decided to use the middle part of the formula, because the other parts are too abstract for my knowledge (maybe I slept too much in math classes).
Actually my program works fine for n = 0;,n = 5;, n = 10; But if I enter n = 15; - here comes the boom - the output is 2 when it should be 9694845.
So here is my child:
using System;
namespace _8_Numbers_of_Catalan
{
class CatalanNumbers
{
static void Main()
{
Console.Write("n: ");
int n = int.Parse(Console.ReadLine());
Console.WriteLine("Catalan({0})", n);
//calculating the Catan number from the formula
// Catan(n) = [(2*n)!]/[(n+1)! * n!]
Console.WriteLine((factorial(2 * n)) / (factorial(n + 1) * factorial(n)));
}//finding the factorial
private static ulong factorial(int n)
{
ulong fact = 1;
for (int i = 1; i <= n; i++)
{
fact *= (ulong)i;
}
return fact;
}
}
}
Thank you in advance for understanding me if there is something obviously wrong. I am new in programming.
That is because you are performing calculation of these using integer variables that can contain at most 64 bits.
Your call to factorial(15 * 2) is 30! which would result in a value of
265,252,859,812,191,058,636,308,480,000,000
Much more than fits in a 64 bit integer variable:
18,446,744,073,709,551,615 (0xFFFFFFFFFFFFFFFF).
The options you have are to use a System.Numerics.BigInteger type (slow) or a double (up to a maximum value of 1.7976931348623157E+308). Which means you will loose some precision, that may or may not be relevant.
Another option you have is to use an algorithm to approximate the value of large factorials using an asymptotic approximation such as the Schönhage–Strassen algorithm used by Mathematica.
You may also want to check out some existing online resources for calculation of big factorials in .NET
As a last but not least option (and I have not thoroughly checked) it seems likely to me that specific algorithms exists that allow you to calculate (or approximate to a sufficient accuracy and precision) a Catalan number.
you should use a System.Numerics.BigInteger for this. (add System.Numerics as reference in your project).
private static BigInteger factorial(int n)
{
BigInteger fact = 1;
for (int i = 1; i <= n; i++)
{
fact *= i;
}
return fact;
}
// output: 9694845
I'm messing around with writing a class similar to mpz (C) or BigInteger (Java). This is just for fun, so please don't go on about how I shouldn't be writing my own.
I have a class similar to:
public class HugeInt
{
public List<Integer> digits;
public HugeInt(String value)
{
// convert string value into its seperate digits.
// store them in instance variable above
}
}
Now, doing the add() and subtract() method of this class are pretty simple. Here is an example:
private List<Integer> add(List<Integer> a, List<Integer> b)
{
List<Integer> smallerDigits = (compareDigits(a,b) < 0) ? a : b;
List<Integer> largerDigits = (compareDigits(a,b) >= 0) ? a : b;
List<Integer> result = new ArrayList<Integer>();
int carry = 0;
for(int i = 0; i < largerDigits.size(); i++)
{
int num1 = largerDigits.get(i);
int num2 = (i < smallerDigits.size()) ? smallerDigits.get(i) : 0;
result.add((num1 + num2 + carry) % 10);
carry = ((num1 + num2 + carry) / 10);
}
if (carry != 0) result.add(carry);
return result;
}
Similarly, doing the multiply wasn't that hard either.
I see on wikipedia there is a page on Division Algorithms, but I'm not sure which one is appropriate for what I'm trying to do.
Because these positive integers (represented as digits) can be arbitrarily long, I want to make sure I don't attempt to do any operations on anything other than digit-by-digit basis.
However, can anyone point me in the right direction for doing a division of two numbers that are represented as List<Integer>'s? Also, I can ignore the remainder as this is integer division.
You could just do long division, but this certainly isn't the optimal way to do it (edit: although it seems that something like this is a good way to do it). You could look at other implementations of big integer libraries, and a bit of Googling turns up a fair bit of useful information.
This may be a slight overkill, but if this is the kind of things you do for fun, you'll enjoy reading this:
http://www.fizyka.umk.pl/nrbook/c20-6.pdf
(that's "Arithmetic at Arbitrary Precision" from "Numerical recipes in C").
Pretty fascinating, as is most of this book, with good explanations and lots of code.
Since I assume you're just dealing with integer division it's not very hard. Multiplication is repeated addition, division is the opposite - repeated subtraction. So what you'll do is check how many times you can subtract the divisor from the dividend. For example, 3 can be subtracted from 10 3 times without going <0, so the integer division quotient is 3.
This article A Larger Integer does not show how to implement digit by digit operations for "larger integers", but it does show how to implement a (apparently fully functional) 128 bit integer in terms of two Int64 types. I would imagine that it would not be too hard to extend the approach to use an array of Int64 types to yield an arbitary length integer. I just spent a few minutes looking back over the article and the implementation of multiply looks like it could get pretty involved for arbitrary length.
The article shows how to implement division (quotient and remainder) using binary division.
I need to be able to calculate (a^b) % c for very large values of a and b (which individually are pushing limit and which cause overflow errors when you try to calculate a^b). For small enough numbers, using the identity (a^b)%c = (a%c)^b%c works, but if c is too large this doesn't really help. I wrote a loop to do the mod operation manually, one a at a time:
private static long no_Overflow_Mod(ulong num_base, ulong num_exponent, ulong mod)
{
long answer = 1;
for (int x = 0; x < num_exponent; x++)
{
answer = (answer * num_base) % mod;
}
return answer;
}
but this takes a very long time. Is there any simple and fast way to do this operation without actually having to take a to the power of b AND without using time-consuming loops? If all else fails, I can make a bool array to represent a huge data type and figure out how to do this with bitwise operators, but there has to be a better way.
I guess you are looking for : http://en.wikipedia.org/wiki/Montgomery_reduction
or the simpler way based on Modular Exponentiation (from wikipedia)
Bignum modpow(Bignum base, Bignum exponent, Bignum modulus) {
Bignum result = 1;
while (exponent > 0) {
if ((exponent & 1) == 1) {
// multiply in this bit's contribution while using modulus to keep result small
result = (result * base) % modulus;
}
// move to the next bit of the exponent, square (and mod) the base accordingly
exponent >>= 1;
base = (base * base) % modulus;
}
return result;
}
Fast Modular Exponentiation (I think that's what it's called) might work.
Given a, b, c and a^b (mod c):
1. Write b as a sum of powers of 2. (If b=72, this is 2^6 + 2^3 )
2. Do:
(1) a^2 (mod c) = a*
(2) (a*)^2 (mod c) = a*
(3) (a*)^2 (mod c) = a*
...
(n) (a*)^2 (mod c) = a*
3. Using the a* from above, multiply the a* for the powers of 2 you identified. For example:
b = 72, use a* at 3 and a* at 6.
a*(3) x a*(6) (mod c)
4. Do the previous step one multiplication at a time and at the end, you'll have a^b % c.
Now, how you're going to do that with data types, I don't know. As long as your datatype can support c^2, i think you'll be fine.
If using strings, just create string versions of add, subtract, and multiply (not too hard). This method should be quick enough doing that. (and you can start step 1 by a mod c so that a is never greater than c).
EDIT: Oh look, a wiki page on Modular Exponentiation.
Here's an example of Fast Modular Exponentiation (suggested in one of the earlier answers) in java. Shouldn't be too hard to convert that to C#
http://www.math.umn.edu/~garrett/crypto/a01/FastPow.html
and the source...
http://www.math.umn.edu/~garrett/crypto/a01/FastPow.java
Python has pow(a,b,c) which returns (a**b)%c (only faster), so there must be some clever way to do this. Maybe they just do the identity you mentioned.
I'd recommend checking over the Decimal documentation and seeing if it meets your requirements since it is a built in type and can use the mod operator. If not then you're going to need an arbitrary precision library like java's Bignum.
You can try factoring 'a' into sufficiently small numbers.
If the factors of 'a' are 'x', 'y', and 'z', then
a^b = (x^b)(y^b)(z^b).
Then you can use your identity: (a^b)%c = (a%c)^b%c
It seems to me like there's some kind of relation between power and mod. Power is just repeated multiplication and mod is related to division. We know that multiplication and division are inverses, so through that connection I would assume there's a correlation between power and mod.
For example, take powers of 5:
5 % 4 = 1
25 % 4 = 1
125 % 4 = 1
625 % 4 = 1
...
The pattern is clear that 5 ^ b % 4 = 1 for all values of b.
It's less clear in this situation:
5 % 3 = 2
25 % 3 = 1
125 % 3 = 2
625 % 3 = 1
3125 % 3 = 2
15625 % 3 = 1
78125 % 3 = 2
...
But there's still a pattern.
If you could work out the math behind the patterns, I wouldn't be surprised if you could figure out the value of the mod without doing the actual power.
You could try this:
C#: Doing a modulus (mod) operation on a very large number (> Int64.MaxValue)
http://www.del337ed.com/blog/index.php/2009/02/04/c-doing-a-modulus-mod-operation-on-a-very-large-number-int64maxvalue/
Short of writing your own fast modular exponentiation, the simplest idea I can come up with, is to use the F# BigInt type: Microsoft.FSharp.Math.Types.BigInt which supports operations with arbitrarily large scale - including exponentiation and modular arithmetic.
It's a built-in type that will be part of the full .NET framework with the next release. You don't need to use F# to use BitInt - you can make use of it directly in C#.
Can you factor a, b, or c? Does C have a known range?
These are 32 bit integers! Go check this site
For instance, here is how you get the mod of n%d where d 1>>s (1,2,4,8,...)
int n = 137; // numerator
int d = 32; // denom d will be one of: 1, 2, 4, 8, 16, 32, ...
int m; // m will be n % d
m = n & (d - 1);
There is code for n%d where d is 1>>s - 1 (1, 3, 7, 15, 31, ...)
This is only going to really help if c is small though, like you said.
Looks like homework in cryptography.
Hint: check out Fermat's little theorem.
I've been wrestling with Project Euler Problem #16 in C# 2.0. The crux of the question is that you have to calculate and then iterate through each digit in a number that is 604 digits long (or there-abouts). You then add up these digits to produce the answer.
This presents a problem: C# 2.0 doesn't have a built-in datatype that can handle this sort of calculation precision. I could use a 3rd party library, but that would defeat the purpose of attempting to solve it programmatically without external libraries. I can solve it in Perl; but I'm trying to solve it in C# 2.0 (I'll attempt to use C# 3.0 in my next run-through of the Project Euler questions).
Question
What suggestions (not answers!) do you have for solving project Euler #16 in C# 2.0? What methods would work?
NB: If you decide to post an answer, please prefix your attempt with a blockquote that has ###Spoiler written before it.
A number of a series of digits. A 32 bit unsigned int is 32 binary digits. The string "12345" is a series of 5 digits. Digits can be stored in many ways: as bits, characters, array elements and so on. The largest "native" datatype in C# with complete precision is probably the decimal type (128 bits, 28-29 digits). Just choose your own method of storing digits that allows you to store much bigger numbers.
As for the rest, this will give you a clue:
21 = 2
22 = 21 + 21
23 = 22 + 22
Example:
The sum of digits of 2^100000 is 135178
Ran in 4875 ms
The sum of digits of 2^10000 is 13561
Ran in 51 ms
The sum of digits of 2^1000 is 1366
Ran in 2 ms
SPOILER ALERT: Algorithm and solution in C# follows.
Basically, as alluded to a number is nothing more than an array of digits. This can be represented easily in two ways:
As a string;
As an array of characters or digits.
As others have mentioned, storing the digits in reverse order is actually advisable. It makes the calculations much easier. I tried both of the above methods. I found strings and the character arithmetic irritating (it's easier in C/C++; the syntax is just plain annoying in C#).
The first thing to note is that you can do this with one array. You don't need to allocate more storage at each iteration. As mentioned you can find a power of 2 by doubling the previous power of 2. So you can find 21000 by doubling 1 one thousand times. The doubling can be done in place with the general algorithm:
carry = 0
foreach digit in array
sum = digit + digit + carry
if sum > 10 then
carry = 1
sum -= 10
else
carry = 0
end if
digit = sum
end foreach
This algorithm is basically the same for using a string or an array. At the end you just add up the digits. A naive implementation might add the results into a new array or string with each iteration. Bad idea. Really slows it down. As mentioned, it can be done in place.
But how large should the array be? Well that's easy too. Mathematically you can convert 2^a to 10^f(a) where f(a) is a simple logarithmic conversion and the number of digits you need is the next higher integer from that power of 10. For simplicity, you can just use:
digits required = ceil(power of 2 / 3)
which is a close approximation and sufficient.
Where you can really optimise this is by using larger digits. A 32 bit signed int can store a number between +/- 2 billion (approximately. Well 9 digits equals a billion so you can use a 32 bit int (signed or unsigned) as basically a base one billion "digit". You can work out how many ints you need, create that array and that's all the storage you need to run the entire algorithm (being 130ish bytes) with everything being done in place.
Solution follows (in fairly rough C#):
static void problem16a()
{
const int limit = 1000;
int ints = limit / 29;
int[] number = new int[ints + 1];
number[0] = 2;
for (int i = 2; i <= limit; i++)
{
doubleNumber(number);
}
String text = NumberToString(number);
Console.WriteLine(text);
Console.WriteLine("The sum of digits of 2^" + limit + " is " + sumDigits(text));
}
static void doubleNumber(int[] n)
{
int carry = 0;
for (int i = 0; i < n.Length; i++)
{
n[i] <<= 1;
n[i] += carry;
if (n[i] >= 1000000000)
{
carry = 1;
n[i] -= 1000000000;
}
else
{
carry = 0;
}
}
}
static String NumberToString(int[] n)
{
int i = n.Length;
while (i > 0 && n[--i] == 0)
;
String ret = "" + n[i--];
while (i >= 0)
{
ret += String.Format("{0:000000000}", n[i--]);
}
return ret;
}
I solved this one using C# also, much to my dismay when I discovered that Python can do this in one simple operation.
Your goal is to create an adding machine using arrays of int values.
Spoiler follows
I ended up using an array of int
values to simulate an adding machine,
but I represented the number backwards
- which you can do because the problem only asks for the sum of the digits,
this means order is irrelevant.
What you're essentially doing is
doubling the value 1000 times, so you
can double the value 1 stored in the
1st element of the array, and then
continue looping until your value is
over 10. This is where you will have
to keep track of a carry value. The
first power of 2 that is over 10 is
16, so the elements in the array after
the 5th iteration are 6 and 1.
Now when you loop through the array
starting at the 1st value (6), it
becomes 12 (so you keep the last
digit, and set a carry bit on the next
index of the array) - which when
that value is doubled you get 2 ... plus the 1 for the carry bit which
equals 3. Now you have 2 and 3 in your
array which represents 32.
Continues this process 1000 times and
you'll have an array with roughly 600
elements that you can easily add up.
I have solved this one before, and now I re-solved it using C# 3.0. :)
I just wrote a Multiply extension method that takes an IEnumerable<int> and a multiplier and returns an IEnumerable<int>. (Each int represents a digit, and the first one it the least significant digit.) Then I just created a list with the item { 1 } and multiplied it by 2 a 1000 times. Adding the items in the list is simple with the Sum extension method.
19 lines of code, which runs in 13 ms. on my laptop. :)
Pretend you are very young, with square paper. To me, that is like a list of numbers. Then to double it you double each number, then handle any "carries", by subtracting the 10s and adding 1 to the next index. So if the answer is 1366... something like (completely unoptimized, rot13):
hfvat Flfgrz;
hfvat Flfgrz.Pbyyrpgvbaf.Trarevp;
pynff Cebtenz {
fgngvp ibvq Pneel(Yvfg<vag> yvfg, vag vaqrk) {
juvyr (yvfg[vaqrk] > 9) {
yvfg[vaqrk] -= 10;
vs (vaqrk == yvfg.Pbhag - 1) yvfg.Nqq(1);
ryfr yvfg[vaqrk + 1]++;
}
}
fgngvp ibvq Znva() {
ine qvtvgf = arj Yvfg<vag> { 1 }; // 2^0
sbe (vag cbjre = 1; cbjre <= 1000; cbjre++) {
sbe (vag qvtvg = 0; qvtvg < qvtvgf.Pbhag; qvtvg++) {
qvtvgf[qvtvg] *= 2;
}
sbe (vag qvtvg = 0; qvtvg < qvtvgf.Pbhag; qvtvg++) {
Pneel(qvtvgf, qvtvg);
}
}
qvtvgf.Erirefr();
sbernpu (vag v va qvtvgf) {
Pbafbyr.Jevgr(v);
}
Pbafbyr.JevgrYvar();
vag fhz = 0;
sbernpu (vag v va qvtvgf) fhz += v;
Pbafbyr.Jevgr("fhz: ");
Pbafbyr.JevgrYvar(fhz);
}
}
If you wish to do the primary calculation in C#, you will need some sort of big integer implementation (Much like gmp for C/C++). Programming is about using the right tool for the right job. If you cannot find a good big integer library for C#, it's not against the rules to calculate the number in a language like Python which already has the ability to calculate large numbers. You could then put this number into your C# program via your method of choice, and iterate over each character in the number (you will have to store it as a string). For each character, convert it to an integer and add it to your total until you reach the end of the number. If you would like the big integer, I calculated it with python below. The answer is further down.
Partial Spoiler
10715086071862673209484250490600018105614048117055336074437503883703510511249361
22493198378815695858127594672917553146825187145285692314043598457757469857480393
45677748242309854210746050623711418779541821530464749835819412673987675591655439
46077062914571196477686542167660429831652624386837205668069376
Spoiler Below!
>>> val = str(2**1000)
>>> total = 0
>>> for i in range(0,len(val)): total += int(val[i])
>>> print total
1366
If you've got ruby, you can easily calculate "2**1000" and get it as a string. Should be an easy cut/paste into a string in C#.
Spoiler
In Ruby: (2**1000).to_s.split(//).inject(0){|x,y| x+y.to_i}
spoiler
If you want to see a solution check
out my other answer. This is in Java but it's very easy to port to C#
Here's a clue:
Represent each number with a list. That way you can do basic sums like:
[1,2,3,4,5,6]
+ [4,5]
_____________
[1,2,3,5,0,1]
One alternative to representing the digits as a sequence of integers is to represent the number base 2^32 as a list of 32 bit integers, which is what many big integer libraries do. You then have to convert the number to base 10 for output. This doesn't gain you very much for this particular problem - you can write 2^1000 straight away, then have to divide by 10 many times instead of multiplying 2 by itself 1000 times ( or, as 1000 is 0b1111101000. calculating the product of 2^8,32,64,128,256,512 using repeated squaring 2^8 = (((2^2)^2)^2))) which requires more space and a multiplication method, but is far fewer operations ) - is closer to normal big integer use, so you may find it more useful in later problems ( if you try to calculate the last ten digits of 28433×2^(7830457)+1 using the digit-per int method and repeated addition, it may take some time (though in that case you could use modulo arthimetic, rather than adding strings of millions of digits) ).
Working solution that I have posted it here as well: http://www.mycoding.net/2012/01/solution-to-project-euler-problem-16/
The code:
import java.math.BigInteger;
public class Euler16 {
public static void main(String[] args) {
int power = 1;
BigInteger expo = new BigInteger("2");
BigInteger num = new BigInteger("2");
while(power < 1000){
expo = expo.multiply(num);
power++;
}
System.out.println(expo); //Printing the value of 2^1000
int sum = 0;
char[] expoarr = expo.toString().toCharArray();
int max_count = expoarr.length;
int count = 0;
while(count<max_count){ //While loop to calculate the sum of digits
sum = sum + (expoarr[count]-48);
count++;
}
System.out.println(sum);
}
}
Euler problem #16 has been discussed many times here, but I could not find an answer that gives a good overview of possible solution approaches, the lay of the land as it were. Here's my attempt at rectifying that.
This overview is intended for people who have already found a solution and want to get a more complete picture. It is basically language-agnostic even though the sample code is C#. There are some usages of features that are not available in C# 2.0 but they are not essential - their purpose is only to get boring stuff out of the way with a minimum of fuss.
Apart from using a ready-made BigInteger library (which doesn't count), straightforward solutions for Euler #16 fall into two fundamental categories: performing calculations natively - i.e. in a base that is a power of two - and converting to decimal in order to get at the digits, or performing the computations directly in a decimal base so that the digits are available without any conversion.
For the latter there are two reasonably simple options:
repeated doubling
powering by repeated squaring
Native Computation + Radix Conversion
This approach is the simplest and its performance exceeds that of naive solutions using .Net's builtin BigInteger type.
The actual computation is trivially achieved: just perform the moral equivalent of 1 << 1000, by storing 1000 binary zeroes and appending a single lone binary 1.
The conversion is also quite simple and can be done by coding the pencil-and-paper division method, with a suitably large choice of 'digit' for efficiency. Variables for intermediate results need to be able to hold two 'digits'; dividing the number of decimal digits that fit in a long by 2 gives 9 decimal digits for the maximum meta-digit (or 'limb', as it is usually called in bignum lore).
class E16_RadixConversion
{
const int BITS_PER_WORD = sizeof(uint) * 8;
const uint RADIX = 1000000000; // == 10^9
public static int digit_sum_for_power_of_2 (int exponent)
{
var dec = new List<int>();
var bin = new uint[(exponent + BITS_PER_WORD) / BITS_PER_WORD];
int top = bin.Length - 1;
bin[top] = 1u << (exponent % BITS_PER_WORD);
while (top >= 0)
{
ulong rest = 0;
for (int i = top; i >= 0; --i)
{
ulong temp = (rest << BITS_PER_WORD) | bin[i];
ulong quot = temp / RADIX; // x64 uses MUL (sometimes), x86 calls a helper function
rest = temp - quot * RADIX;
bin[i] = (uint)quot;
}
dec.Add((int)rest);
if (bin[top] == 0)
--top;
}
return E16_Common.digit_sum(dec);
}
}
I wrote (rest << BITS_PER_WORD) | big[i] instead of using operator + because that is precisely what is needed here; no 64-bit addition with carry propagation needs to take place. This means that the two operands could be written directly to their separate registers in a register pair, or to fields in an equivalent struct like LARGE_INTEGER.
On 32-bit systems the 64-bit division cannot be inlined as a few CPU instructions, because the compiler cannot know that the algorithm guarantees quotient and remainder to fit into 32-bit registers. Hence the compiler calls a helper function that can handle all eventualities.
These systems may profit from using a smaller limb, i.e. RADIX = 10000 and uint instead of ulong for holding intermediate (double-limb) results. An alternative for languages like C/C++ would be to call a suitable compiler intrinsic that wraps the raw 32-bit by 32-bit to 64-bit multiply (assuming that division by the constant radix is to be implemented by multiplication with the inverse). Conversely, on 64-bit systems the limb size can be increased to 19 digits if the compiler offers a suitable 64-by-64-to-128 bit multiply primitive or allows inline assembler.
Decimal Doubling
Repeated doubling seems to be everyone's favourite, so let's do that next. Variables for intermediate results need to hold one 'digit' plus one carry bit, which gives 18 digits per limb for long. Going to ulong cannot improve things (there's 0.04 bit missing to 19 digits plus carry), and so we might as well stick with long.
On a binary computer, decimal limbs do not coincide with computer word boundaries. That makes it necessary to perform a modulo operation on the limbs during each step of the calculation. Here, this modulo op can be reduced to a subtraction of the modulus in the event of carry, which is faster than performing a division. The branching in the inner loop can be eliminated by bit twiddling but that would be needlessly obscure for a demonstration of the basic algorithm.
class E16_DecimalDoubling
{
const int DIGITS_PER_LIMB = 18; // == floor(log10(2) * (63 - 1)), b/o carry
const long LIMB_MODULUS = 1000000000000000000L; // == 10^18
public static int digit_sum_for_power_of_2 (int power_of_2)
{
Trace.Assert(power_of_2 > 0);
int total_digits = (int)Math.Ceiling(Math.Log10(2) * power_of_2);
int total_limbs = (total_digits + DIGITS_PER_LIMB - 1) / DIGITS_PER_LIMB;
var a = new long[total_limbs];
int limbs = 1;
a[0] = 2;
for (int i = 1; i < power_of_2; ++i)
{
int carry = 0;
for (int j = 0; j < limbs; ++j)
{
long new_limb = (a[j] << 1) | carry;
carry = 0;
if (new_limb >= LIMB_MODULUS)
{
new_limb -= LIMB_MODULUS;
carry = 1;
}
a[j] = new_limb;
}
if (carry != 0)
{
a[limbs++] = carry;
}
}
return E16_Common.digit_sum(a);
}
}
This is just as simple as radix conversion, but except for very small exponents it does not perform anywhere near as well (despite its huge meta-digits of 18 decimal places). The reason is that the code must perform (exponent - 1) doublings, and the work done in each pass corresponds to about half the total number of digits (limbs).
Repeated Squaring
The idea behind powering by repeated squaring is to replace a large number of doublings with a small number of multiplications.
1000 = 2^3 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9
x^1000 = x^(2^3 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9)
x^1000 = x^2^3 * x^2^5 * x^2^6 * x^2^7 * x^2*8 * x^2^9
x^2^3 can be obtained by squaring x three times, x^2^5 by squaring five times, and so on. On a binary computer the decomposition of the exponent into powers of two is readily available because it is the bit pattern representing that number. However, even non-binary computers should be able to test whether a number is odd or even, or to divide a number by two.
The multiplication can be done by coding the pencil-and-paper method; here I'm using a helper function that computes one row of a product and adds it into the result at a suitably shifted position, so that the rows of partial products do not need to be stored for a separate addition step later. Intermediate values during computation can be up to two 'digits' in size, so that the limbs can be only half as wide as for repeated doubling (where only one extra bit had to fit in addition to a 'digit').
Note: the radix of the computations is not a power of 2, and so the squarings of 2 cannot be computed by simple shifting here. On the positive side, the code can be used for computing powers of bases other than 2.
class E16_DecimalSquaring
{
const int DIGITS_PER_LIMB = 9; // language limit 18, half needed for holding the carry
const int LIMB_MODULUS = 1000000000;
public static int digit_sum_for_power_of_2 (int e)
{
Trace.Assert(e > 0);
int total_digits = (int)Math.Ceiling(Math.Log10(2) * e);
int total_limbs = (total_digits + DIGITS_PER_LIMB - 1) / DIGITS_PER_LIMB;
var squared_power = new List<int>(total_limbs) { 2 };
var result = new List<int>(total_limbs);
result.Add((e & 1) == 0 ? 1 : 2);
while ((e >>= 1) != 0)
{
squared_power = multiply(squared_power, squared_power);
if ((e & 1) == 1)
result = multiply(result, squared_power);
}
return E16_Common.digit_sum(result);
}
static List<int> multiply (List<int> lhs, List<int> rhs)
{
var result = new List<int>(lhs.Count + rhs.Count);
resize_to_capacity(result);
for (int i = 0; i < rhs.Count; ++i)
addmul_1(result, i, lhs, rhs[i]);
trim_leading_zero_limbs(result);
return result;
}
static void addmul_1 (List<int> result, int offset, List<int> multiplicand, int multiplier)
{
// it is assumed that the caller has sized `result` appropriately before calling this primitive
Trace.Assert(result.Count >= offset + multiplicand.Count + 1);
long carry = 0;
foreach (long limb in multiplicand)
{
long temp = result[offset] + limb * multiplier + carry;
carry = temp / LIMB_MODULUS;
result[offset++] = (int)(temp - carry * LIMB_MODULUS);
}
while (carry != 0)
{
long final_temp = result[offset] + carry;
carry = final_temp / LIMB_MODULUS;
result[offset++] = (int)(final_temp - carry * LIMB_MODULUS);
}
}
static void resize_to_capacity (List<int> operand)
{
operand.AddRange(Enumerable.Repeat(0, operand.Capacity - operand.Count));
}
static void trim_leading_zero_limbs (List<int> operand)
{
int i = operand.Count;
while (i > 1 && operand[i - 1] == 0)
--i;
operand.RemoveRange(i, operand.Count - i);
}
}
The efficiency of this approach is roughly on par with radix conversion but there are specific improvements that apply here. Efficiency of the squaring can be doubled by writing a special squaring routine that utilises the fact that ai*bj == aj*bi if a == b, which cuts the number of multiplications in half.
Also, there are methods for computing addition chains that involve fewer operations overall than using the exponent bits for determining the squaring/multiplication schedule.
Helper Code and Benchmarks
The helper code for summing decimal digits in the meta-digits (decimal limbs) produced by the sample code is trivial, but I'm posting it here anyway for your convenience:
internal class E16_Common
{
internal static int digit_sum (int limb)
{
int sum = 0;
for ( ; limb > 0; limb /= 10)
sum += limb % 10;
return sum;
}
internal static int digit_sum (long limb)
{
const int M1E9 = 1000000000;
return digit_sum((int)(limb / M1E9)) + digit_sum((int)(limb % M1E9));
}
internal static int digit_sum (IEnumerable<int> limbs)
{
return limbs.Aggregate(0, (sum, limb) => sum + digit_sum(limb));
}
internal static int digit_sum (IEnumerable<long> limbs)
{
return limbs.Select((limb) => digit_sum(limb)).Sum();
}
}
This can be made more efficient in various ways but overall it is not critical.
All three solutions take O(n^2) time where n is the exponent. In other words, they will take a hundred times as long when the exponent grows by a factor of ten. Radix conversion and repeated squaring can both be improved to roughly O(n log n) by employing divide-and-conquer strategies; I doubt whether the doubling scheme can be improved in a similar fastion but then it was never competitive to begin with.
All three solutions presented here can be used to print the actual results, by stringifying the meta-digits with suitable padding and concatenating them. I've coded the functions as returning the digit sum instead of the arrays/lists with decimal limbs only in order to keep the sample code simple and to ensure that all functions have the same signature, for benchmarking.
In these benchmarks, the .Net BigInteger type was wrapped like this:
static int digit_sum_via_BigInteger (int power_of_2)
{
return System.Numerics.BigInteger.Pow(2, power_of_2)
.ToString()
.ToCharArray()
.Select((c) => (int)c - '0')
.Sum();
}
Finally, the benchmarks for the C# code:
# testing decimal doubling ...
1000: 1366 in 0,052 ms
10000: 13561 in 3,485 ms
100000: 135178 in 339,530 ms
1000000: 1351546 in 33.505,348 ms
# testing decimal squaring ...
1000: 1366 in 0,023 ms
10000: 13561 in 0,299 ms
100000: 135178 in 24,610 ms
1000000: 1351546 in 2.612,480 ms
# testing radix conversion ...
1000: 1366 in 0,018 ms
10000: 13561 in 0,619 ms
100000: 135178 in 60,618 ms
1000000: 1351546 in 5.944,242 ms
# testing BigInteger + LINQ ...
1000: 1366 in 0,021 ms
10000: 13561 in 0,737 ms
100000: 135178 in 69,331 ms
1000000: 1351546 in 6.723,880 ms
As you can see, the radix conversion is almost as slow as the solution using the builtin BigInteger class. The reason is that the runtime is of the newer type that does performs certain standard optimisations only for signed integer types but not for unsigned ones (here: implementing division by a constant as multiplication with the inverse).
I haven't found an easy means of inspecting the native code for existing .Net assemblies, so I decided on a different path of investigation: I coded a variant of E16_RadixConversion for comparison where ulong and uint were replaced by long and int respectively, and BITS_PER_WORD decreased by 1 accordingly. Here are the timings:
# testing radix conv Int63 ...
1000: 1366 in 0,004 ms
10000: 13561 in 0,202 ms
100000: 135178 in 18,414 ms
1000000: 1351546 in 1.834,305 ms
More than three times as fast as the version that uses unsigned types! Clear evidence of numbskullery in the compiler...
In order to showcase the effect of different limb sizes I templated the solutions in C++ on the unsigned integer types used as limbs. The timings are prefixed with the byte size of a limb and the number of decimal digits in a limb, separated by a colon. There is no timing for the often-seen case of manipulating digit characters in strings, but it is safe to say that such code will take at least twice as long as the code that uses double digits in byte-sized limbs.
# E16_DecimalDoubling
[1:02] e = 1000 -> 1366 0.308 ms
[2:04] e = 1000 -> 1366 0.152 ms
[4:09] e = 1000 -> 1366 0.070 ms
[8:18] e = 1000 -> 1366 0.071 ms
[1:02] e = 10000 -> 13561 30.533 ms
[2:04] e = 10000 -> 13561 13.791 ms
[4:09] e = 10000 -> 13561 6.436 ms
[8:18] e = 10000 -> 13561 2.996 ms
[1:02] e = 100000 -> 135178 2719.600 ms
[2:04] e = 100000 -> 135178 1340.050 ms
[4:09] e = 100000 -> 135178 588.878 ms
[8:18] e = 100000 -> 135178 290.721 ms
[8:18] e = 1000000 -> 1351546 28823.330 ms
For the exponent of 10^6 there is only the timing with 64-bit limbs, since I didn't have the patience to wait many minutes for full results. The picture is similar for radix conversion, except that there is no row for 64-bit limbs because my compiler does not have a native 128-bit integral type.
# E16_RadixConversion
[1:02] e = 1000 -> 1366 0.080 ms
[2:04] e = 1000 -> 1366 0.026 ms
[4:09] e = 1000 -> 1366 0.048 ms
[1:02] e = 10000 -> 13561 4.537 ms
[2:04] e = 10000 -> 13561 0.746 ms
[4:09] e = 10000 -> 13561 0.243 ms
[1:02] e = 100000 -> 135178 445.092 ms
[2:04] e = 100000 -> 135178 68.600 ms
[4:09] e = 100000 -> 135178 19.344 ms
[4:09] e = 1000000 -> 1351546 1925.564 ms
The interesting thing is that simply compiling the code as C++ doesn't make it any faster - i.e., the optimiser couldn't find any low-hanging fruit that the C# jitter missed, apart from not toeing the line with regard to penalising unsigned integers. That's the reason why I like prototyping in C# - performance in the same ballpark as (unoptimised) C++ and none of the hassle.
Here's the meat of the C++ version (sans reams of boring stuff like helper templates and so on) so that you can see that I didn't cheat to make C# look better:
template<typename W>
struct E16_RadixConversion
{
typedef W limb_t;
typedef typename detail::E16_traits<W>::long_t long_t;
static unsigned const BITS_PER_WORD = sizeof(limb_t) * CHAR_BIT;
static unsigned const RADIX_DIGITS = std::numeric_limits<limb_t>::digits10;
static limb_t const RADIX = detail::pow10_t<limb_t, RADIX_DIGITS>::RESULT;
static unsigned digit_sum_for_power_of_2 (unsigned e)
{
std::vector<limb_t> digits;
compute_digits_for_power_of_2(e, digits);
return digit_sum(digits);
}
static void compute_digits_for_power_of_2 (unsigned e, std::vector<limb_t> &result)
{
assert(e > 0);
unsigned total_digits = unsigned(std::ceil(std::log10(2) * e));
unsigned total_limbs = (total_digits + RADIX_DIGITS - 1) / RADIX_DIGITS;
result.resize(0);
result.reserve(total_limbs);
std::vector<limb_t> bin((e + BITS_PER_WORD) / BITS_PER_WORD);
bin.back() = limb_t(limb_t(1) << (e % BITS_PER_WORD));
while (!bin.empty())
{
long_t rest = 0;
for (std::size_t i = bin.size(); i-- > 0; )
{
long_t temp = (rest << BITS_PER_WORD) | bin[i];
long_t quot = temp / RADIX;
rest = temp - quot * RADIX;
bin[i] = limb_t(quot);
}
result.push_back(limb_t(rest));
if (bin.back() == 0)
bin.pop_back();
}
}
};
Conclusion
These benchmarks also show that this Euler task - like many others - seems designed to be solved on a ZX81 or an Apple ][, not on our modern toys that are a million times as powerful. There's no challenge involved here unless the limits are increased drastically (an exponent of 10^5 or 10^6 would be much more adequate).
A good overview of the practical state of the art can be got from GMP's overview of algorithms. Another excellent overview of the algorithms is chapter 1 of "Modern Computer Arithmetic" by Richard Brent and Paul Zimmermann. It contains exactly what one needs to know for coding challenges and competitions, but unfortunately the depth is not equal to that of Donald Knuth's treatment in "The Art of Computer Programming".
The radix conversion solution adds a useful technique to one's code challenge toolchest, since the given code can be trivially extended for converting any old big integer instead of only the bit pattern 1 << exponent. The repeated squaring solutiono can be similarly useful since changing the sample code to power something other than 2 is again trivial.
The approach of performing computations directly in powers of 10 can be useful for challenges where decimal results are required, because performance is in the same ballpark as native computation but there is no need for a separate conversion step (which can require similar amounts of time as the actual computation).