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Java Compute C(n,k) and factorial with biginteger

I want to compute the answer of C(n,k),such as C(10,2)=10*9/2*1 = 45
If I test my code by small numbers like 10, the code works.
However, when I try to compute C(1000,900), it compiles
Exception in thread "main" java.lang.ArithmeticException: / by zero
I've seen someone says it should use BigInteger,But after I tried, it still has errors.
For example: I change int factorial into BigInteger factorial,
while the for loop in cSelect, I can not change int i into BigInteger type,
As result, the answer up/factorial(y) has errors.
Please help me to fix this problem. Thanks!!
public class Test {
// Write a factorial function
static int factorial(int m) {
int result =1;
for (int i=2; i<=m; i++) {
result = result*i;
}
return result;
}
// Caculate C(x,y)
static int cSelect(int x, int y) {
int up=1;
for(int i=x; i>=(x-y+1); i--) {
up = up*i;
}
return up/factorial(y);
}
public static void main(String[] args) {
System.out.println(cSelect(1000,900));
}
}

Your code is fairly easy to translate in factorial. Start with ONE, multiply by the BigInteger.valueOf(long) for each i in your loop. Like,
// Write a factorial function
static BigInteger factorial(int m) {
BigInteger result = BigInteger.ONE;
for (int i = 2; i <= m; i++) {
result = result.multiply(BigInteger.valueOf(i));
}
return result;
}
Your other function does exactly the same, plus a division by the result of factorial(y). Like,
// Caculate C(x,y)
static BigInteger cSelect(int x, int y) {
BigInteger up = BigInteger.ONE;
for (int i = x; i >= (x - y + 1); i--) {
up = up.multiply(BigInteger.valueOf(i));
}
return up.divide(factorial(y));
}
With no other changes I get
63850511926305130236698511142022274281262900693853331776286816221524376994750901948920974351797699894319420811933446197797592213357065053890
Which I assume is correct.

First, return value must be BigInteger, because result of C(1000,900) far exceeds the range on an int.
Second, you don't need separate factorial() method. Doing the division as you iterate will improve memory footprint by not creating excessively large intermediate values (at cost of doing multiple divisions, but even so it might actually be faster).
Like this:
static BigInteger cSelect(int x, int y) {
BigInteger v = BigInteger.ONE;
for (int i = x, j = 1; j <= y; i--, j++)
v = v.multiply(BigInteger.valueOf(i)).divide(BigInteger.valueOf(j));
return v;
}
By counting i down and j up, there will never be a fraction from the division.
Test
System.out.println(cSelect(10, 2));
System.out.println(cSelect(1000, 900));
Output
45
63850511926305130236698511142022274281262900693853331776286816221524376994750901948920974351797699894319420811933446197797592213357065053890

You have to use BigInteger to do the calculation.
The value you are trying to compute is approximately 6.385051192630516e+139 and it is not representable as a Java primitive integer value.
Even if the result was representable, the reason you are getting divide by zero errors is that the divisor expression 900! ∗ 100! is overflowing to zero. You then divide by that zero.
The reason that it overflows to zero is that it is divisible by 2^32 and 2^64. That can be proven by using some simple algebra to compute the number of factors of 2 there are in 900! and 100!

how to create an Exp(-x^2) function?

I am using the "think java" book and I am stuck on exercise 7.6. The goal here is to write a function that can find . It gives you a couple hints:
One way to evaluate is
to use the infinite series expansion:
In other words, we need to add up a series of terms where the ith term
is equal to
Here is the code I came up with, but it is horribly wrong (when compared to Math.exp) for anything other than a power of 1. I don't understand why, as far as I can tell the code is correct with the formula from the book. I'm not sure if this is more of a math question or something related to how big of a number double and int can hold, but I am just trying to understand why this doesn't work.
public static void main(String[] args) {
System.out.println("Find exp(-x^2)");
double x = inDouble("Enter x: ");
System.out.println("myexp(" + -x*x + ") = " + gauss(x, 20));
System.out.println("Math.exp(" + -x*x + ") = " + Math.exp(-x*x));
}
public static double gauss(double x, int n) {
x = -x*x;
System.out.println(x);
double exp = 1;
double prevnum = 1;
int prevdenom = 1;
int i = 1;
while (i < n) {
exp = exp + (prevnum*x)/(prevdenom*i);
prevnum = prevnum*x;
prevdenom = prevdenom*i;
i++;
}
return exp;
} // I can't figure out why this is so inacurate, as far as I can tell the math is accurate to what the book says the formula is
public static double inDouble(String string) {
Scanner in = new Scanner (System.in);
System.out.print(string);
return in.nextDouble();
}

I am about to add to the comment on your question. I do this because I feel I have a slightly better implementation.
Your approach
Your approach is to have the function accept two arguments, where the second argument is the number of iterations. This isn't bad, but as #JamesKPolk pointed out, you might have to do some manual searching for an int (or long) that won't overflow
My approach
My approach would use something called the machine epsilon for a data type. The machine epsilon is the smallest number of that type (in your case, double) that is representable as that number. There exists algorithm for determining what that machine epsilon is, if you are not "allowed" to access machine epsilon in the Double class.
There is math behind this:
The series representation for your function is
Since it is alternating series, the error term is the absolute value of the first term you choose not to include (I leave the proof to you).
What this means is that we can have an error-based implementation that doesn't use iterations! The best part is that you could implement it for floats, and data types that are "more" than doubles! I present thus:
public static double gauss(double x)
{
x = -x*x;
double exp = 0, error = 1, numerator = 1, denominator = 1;
double machineEpsilon = 1.0;
// calculate machineEpsilon
while ((1.0 + 0.5 * machineEpsilon) != 1.0)
machineEpsilon = 0.5 * machineEpsilon;
int n = 0; //
// while the error is large enough to be representable in terms of the current data type
while ((error >= machineEpsilon) || (-error <= -machineEpsilon))
{
exp += error;
// calculate the numerator (it is 1 if we just start, but -x times its past value otherwise)
numerator = ((n == 0) ? 1 : -numerator * x);
// calculate the denominator (denominator gets multiplied by n)
denominator *= (n++);
// calculate error
error = numerator/denominator;
}
return exp;
}
Let me know how this works!

Java: double rounding algorithm

I got curious about a rounding algorithm, because in CS we had to emulate an HP35 without using the Math library. We didn't include a rounding algorithm in our final build, but I wanted to do it anyway.
public class Round {
public static void main(String[] args) {
/*
* Rounds by using modulus subtraction
*/
double a = 1.123599;
// Should you port this to another method, you can take this as a parameter
int b = 5;
double accuracy = Math.pow(10, -b);
double remainder = a % accuracy;
if (remainder >= 5 * accuracy / 10) // Divide by ten is important because remainder is smaller than accuracy
a += accuracy;
a -= remainder;
/*
* Removes round off error done by modulus
*/
String string = Double.toString(a);
int index = string.indexOf('.') + b;
string = string.substring(0, index);
a = Double.parseDouble(string);
System.out.println(a);
}
}
Is this a good algorithm, or are there any better ones? I don't care about the ones defined in the Java API, I just wanted to know how it was done.
[EDIT]
Here's the code I came up with after looking over EJP's answer
public class Round {
public static void main(String[] args) {
double a = -1.1234599;
int b = 5;
boolean negative = a < 0;
if (negative) a = -a;
String string = Double.toString(a);
char array[] = string.toCharArray();
int index = string.indexOf('.') + b;
int i = index;
int value;
if (Character.getNumericValue(array[index +1]) >= 5) {
for (; i > 0; i--) {
value = Character.getNumericValue(array[i]);
if (value != -1) {
++value;
String temp = Integer.toString(value)
array[i] = temp.charAt(temp.length()-1);
if (value <= 9) break;
}
}
}
string = "";
for (int j=0; j < index + 1 ; j++) {
string += array[j];
}
a = Double.parseDouble(string);
if (negative) a =-a;
System.out.println(a);
}
}

Floating-point numbers don't have decimal places. They have binary places, and the two are not commensurable. Any attempt to modify a floating-point variable to have a specific number of decimal places is doomed to failure.
You have to do the rounding to a specified number of decimal places after conversion to a decimal radix.

There are a different ways to round numbers. The RoundingMode documentation for Java (introduced in 1.5) should give you a brief introduction to the different methods people use.
I know you said you don't have access to the Math functions, but the simplest rounding you can do is:
public static double round(double d)
{
return Math.floor(d + 0.5);
}
If you don't want to use any Math functions, you could try something like this:
public static double round(double d)
{
return (long)(d + 0.5);
}
Those two probably behave differently in some situations (negative numbers?).

Need of typecasting before ceil

I am going through the following code in Java:
public float a1 = 0.10f;
int Increments;
if(TotalTime > 0)
{
Increments = (int) ceil(TotalTime / Increment1);
amount = round(Increments * a1, 4);
}
Where,
TotalTime and Increment1 are integers
My questions is, Why do we need to typecast to integer just before ceil as done in the following line of code above?
Increments = (int) ceil(TotalTime / Increment1);

Because Math.ceil (I suppose it's a static import there) returns a double. As you make a restriction of the converted value (by converting to a less precise type), you need to do the conversion explicitly.
Also, please note that the argument you pass to ceil is the ratio of two integers, which is also an integer. So you're in danger of losing precision. Moreover, applying ceil to an integer is redundant, as it will return that integer itself. And the last remark: ceil expects a double as its argument. In this case, you pass an int, but the conversion is done implicitly, because it is done from a less precise to a more precise type.

The reason is that Math.ceil returns a double. You cannot implicitly cast from a double to an int. Trying to compile this:
double d = 2.1;
int i = d;
Gives you:
dur.java:4: possible loss of precision
found : double
required: int
int i = d;
^
1 error
As the error message states, the reason is that you might lose precision, so the compiler wants to be sure that you really want to allow this loss. You do this by making the cast explicit:
int i = (int)d;
Note that even so, your code won't do what you expect.
int i = 10;
int j = 3;
int k = (int)Math.ceil(i/j);
System.out.println(k); //outputs '3'
The reason is that i / j performs integer division if both i and j are integers, which rounds down. i / j is already 3, and all Math.ceil does is return 3.0, which then again gets casted to 3. You want to do floating point division instead, which is done when one or both of i and j are floating point types. You can achieve this by casting one of the integers to a double before doing the division:
int i = 10;
int j = 3;
int k = (int)Math.ceil((double)i/j);
System.out.println(k); //outputs '4'

Are there any functions for truncating a double in java?

Is there a Java Library function which can be used to truncate a number to an arbitrary number of decimal places?
For Example.
SomeLibrary.truncate(1.575, 2) = 1.57
Thanks

Try setScale of BigDecimal like so:
public static double round(double d, int decimalPlace) {
BigDecimal bd = new BigDecimal(d);
bd = bd.setScale(decimalPlace, BigDecimal.ROUND_HALF_UP);
return bd.doubleValue();
}

Incredible no one brought this up yet, Java API has had DecimalFormat for ages now for this exact purpose.

For most numbers, you won't be able to get an exact representation of xxx.yyyy unless you use a decimal class with guaranteed accuracy, such as BigDecimal.

There's one in commons-math. Check out http://commons.apache.org/math/apidocs/org/apache/commons/math/util/MathUtils.html:
public static double round(double x,
int scale)
It's implemented using BigDecimal, and is overloaded to allow specifying a rounding method, so you can use it to truncate, like this:
org.apache.commons.math.util.MathUtils.round(1.575, 2,
java.math.BigDecimal.ROUND_DOWN);
Update:
In the last version (Math3), this method is in the class Precision.
org.apache.commons.math3.util.Precision.round(double x, int scale, int roundingMethod)

Simply remove the fractional portion
public double trunk(double value){
return value - value % 1;
}

Use this simple function
double truncateDouble(double number, int numDigits) {
double result = number;
String arg = "" + number;
int idx = arg.indexOf('.');
if (idx!=-1) {
if (arg.length() > idx+numDigits) {
arg = arg.substring(0,idx+numDigits+1);
result = Double.parseDouble(arg);
}
}
return result ;
}

I just want to add to ubuntudroid's solution.
I tried it and it wouldn't round down, so I had to add
df.setRoundingMode(RoundingMode.FLOOR);
for it to work.

here is a short implementation which is many times faster than using BigDecimal or Math.pow
private static long TENS[] = new long[19];
static {
TENS[0] = 1;
for (int i = 1; i < TENS.length; i++) TENS[i] = 10 * TENS[i - 1];
}
public static double round(double v, int precision) {
assert precision >= 0 && precision < TENS.length;
double unscaled = v * TENS[precision];
if(unscaled < Long.MIN_VALUE || unscaled > Long.MAX_VALUE)
return v;
long unscaledLong = (long) (unscaled + (v < 0 ? -0.5 : 0.5));
return (double) unscaledLong / TENS[precision];
}
Delete the assert'ions to taste. ;)

Actually, this sort of thing is easy to write:
public static double truncate(double value, int places) {
double multiplier = Math.pow(10, places);
return Math.floor(multiplier * value) / multiplier;
}
Note that it's Math.floor, because Math.round wouldn't be truncating.
Oh, and this returns a double, because that's what most functions in the Math class return (like Math.pow and Math.floor).
Caveat: Doubles suck for accuracy. One of the BigDecimal solutions should be considered first.

To do it 100% reliably, you'd have to pass the argument as string, not as floating-point number. When given as string, the code is easy to write. The reason for this is that
double x = 1.1;
does not mean that x will actually evaluate to 1.1, only to the closest exactly representable number.

created a method to do it.
public double roundDouble(double d, int places) {
return Math.round(d * Math.pow(10, (double) places)) / Math.pow(10, (double)places);
}